Star products, deformation quantization and Toeplitz operators

Transcription

Cours de M2: Star products, deformation
quantization and Toeplitz operators
L. Boutet de Monvel
(état provisoire - ne pas diffuser)
Dans ce cours nous décrivons la théorie des star-produits, les méthodes
qu’elle utilise, ainsi que les exemples les plus utiles. Les star produits ont été
inventés (cf. en particulier dans [6]) pour décrire comment une algèbre commutative, par exemple l’algèbre des fonctions différentiables sur une variété, ou un
algèbre “d’observables” de la physique classique, se déforme en une algèbre non
commutative. Ils servent à décrire comment la mécanique classique hamiltonienne est limite de la mécanique quantique (analyse semi-classique). Le calcul des
opérateurs pseudo-différentiels a été développé à partir de 1965 par de nombreux
auteurs : il donne lieu à un calcul asymptotique “algébrique” essentiellement
identique à celui des star-produits (opérateurs pseudo-différentiels, opérateurs
de Toeplitz, analyse microlocale). Ce calcul fournit des solutions asymptotiques,
par exemple des développements asymptotiques aux hautes fréquences de solutions d’équations aux dérivées partielles; il explique bien par exemple comment
l’optique géométrique est limite de l’optique ondulatoire. Dans ce calcul le rôle
du petit paramètre de déformation est joué par la taille d’une petite longueur
d’onde (inverse d’une haute fréquence); la principale différence est qu’il n’y a
plus de “petit paramètre” de déformation qui commute avec toutes les autres
opérations, comme c’est le cas pour des star-produits.
A une star-algèbre est toujours associé un crochet de Poisson, qui décrit
la limite de la loi des commutateurs (dans le cas d’une déformation: {f, g} =
d
dt (f ∗t g − g ∗t f )|t=0 ). Un des problèmes classiques de cette thèorie est de
classifier, isomorphisme près, les star-produits. Ce problème a été résolu par
M. De Wilde M. et P. Lecomte [38] dans le cas de la déformation d’un crochet
de Poisson symplectique (par V. Guillemin et moi-même [27] dans le cadre
“Toeplitz” indiqué ci-dessus), et par M. Kontsevich [91] dans le cas général.
Dans le cas sympectique B.V. Fedosov [60] a donné une solution très élégante,
qui est celle que nous suivrons ici.
Dans la deuxième partie du cours, nous illustrerons cette théorie et la théorie
des opérateurs de Toeplitz.
1
Keywords: star-products, deformation quantization, symplectic geometry, contact manifolds, CR geometry, Toeplitz operators, residual trace.
Mathematics Subject Classification (2000): 16S32, 16S80, 32A25, 32V05, 35S05,
53C05, 53D10, 53D55, 58J40.
Contents
1 Introduction
6
2 Star Algebras
2.1 Cones . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Star products and star algebras . . . . . . . . . .
2.3 Poisson bracket . . . . . . . . . . . . . . . . . . .
2.3.1 Reminder of differential calculus notations
2.3.2 Poisson brackets . . . . . . . . . . . . . .
2.3.3 Symplectic cones and contact manifolds .
2.3.4 Commutators . . . . . . . . . . . . . . . .
2.4 Functional calculus . . . . . . . . . . . . . . . . .
2.5 Comments . . . . . . . . . . . . . . . . . . . . . .
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3 Models and examples
3.1 Moyal star product . . . . . . . . . . . . . .
3.2 Star-product defined by a formal group law
3.3 Formal pseudo-differential operators . . . .
3.4 Pseudo-differential operators . . . . . . . .
3.5 Semi-classical operators . . . . . . . . . . .
3.6 Toeplitz operators . . . . . . . . . . . . . .
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4 Homomorphisms, automorphisms
4.1 Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Subprincipal Symbol . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Automorphisms of symplectic deformation algebras . . . . . . . .
4.6 Automorphisms of symplectic algebras . . . . . . . . . . . . . . .
4.7 Automorphisms preserving a subprincipal symbol or an involution
4.8 Fourier integral operators . . . . . . . . . . . . . . . . . . . . . .
4.8.1 As functional operators. . . . . . . . . . . . . . . . . . .
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35
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5 Classification
5.1 Hochschild cohomology . . . . . . . . . . . . . .
5.2 Non commutative cohomology . . . . . . . . . . .
5.3 Symplectic algebras are locally isomorphic . . . .
5.4 Classification . . . . . . . . . . . . . . . . . . . .
5.5 Classification of symplectic algebras . . . . . . .
5.6 Classification of symplectic deformation algebras
5.7 Algebras of pseudo-differential type . . . . . . . .
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37
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6 Fedosov Connections
c.
6.1 Valuations and relative tangent algebra W
c
6.2 Automorphisms and Derivations of W . . .
6.3 Embeddings . . . . . . . . . . . . . . . . . .
c . . . .
6.4 Vector Fields with Coefficients in W
6.5 Fedosov Connections . . . . . . . . . . . . .
6.6 Fedosov curvature . . . . . . . . . . . . . .
6.7 Base-point . . . . . . . . . . . . . . . . . . .
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48
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7 Traces
7.1 Residual integral. . . . . . . . . . . . . .
7.2 Trace for Moyal products . . . . . . . .
7.3 Canonical trace on symplectic algebras.
7.4 Trace for deformation algebras . . . . .
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56
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8 Vanishing of the Logarithmic Trace.
8.1 Notations . . . . . . . . . . . . . . . . . . . . . .
8.2 Adapted Fourier Integral Operators . . . . . . .
8.3 Model Example . . . . . . . . . . . . . . . . . . .
8.4 Generalized Szegö projectors . . . . . . . . . . .
8.5 Residual trace and logarithmic trace . . . . . . .
8.6 Trace on a Toeplitz algebra A and on End A (M )
8.7 Embedding . . . . . . . . . . . . . . . . . . . . .
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9 Asymptotic equivariant index
of Toeplitz operators.
9.1 Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 Microlocal model . . . . . . . . . . . . . . . . . . . .
9.1.2 Generalized Szegö projectors . . . . . . . . . . . . .
9.1.3 Holomorphic case . . . . . . . . . . . . . . . . . . . .
9.2 Equivariant trace and index . . . . . . . . . . . . . . . . . .
9.2.1 Equivariant Toeplitz algebra . . . . . . . . . . . . .
9.2.2 Equivariant trace . . . . . . . . . . . . . . . . . . . .
9.2.3 Equivariant index . . . . . . . . . . . . . . . . . . .
9.2.4 Asymptotic index . . . . . . . . . . . . . . . . . . . .
9.3 K-theory and embedding . . . . . . . . . . . . . . . . . . . .
9.3.1 A short digression on Toeplitz algebras and modules
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9.3.2
Embedding . . . . . . . . . . . . . . . . . . . . . . . . . .
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10 Asymptotic equivariant index :
Atiyah-Weinstein index formula.
10.1 Equivariant trace and index . . . . . . . . . . . . . . . . . . .
10.1.1 Equivariant Toeplitz Operators. . . . . . . . . . . . . .
10.1.2 G-trace . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1.3 G index . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 K-theory and embedding . . . . . . . . . . . . . . . . . . . . .
10.2.1 A short digression on Toeplitz algebras . . . . . . . .
10.2.2 Asymptotic trace and index . . . . . . . . . . . . . . .
10.2.3 E-modules . . . . . . . . . . . . . . . . . . . . . . . . .
10.2.4 Embedding . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Relative index . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Holomorphic setting . . . . . . . . . . . . . . . . . . .
10.3.2 Collar isomorphisms . . . . . . . . . . . . . . . . . . .
10.3.3 Embedding . . . . . . . . . . . . . . . . . . . . . . . .
10.3.4 Index . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.1 Contact isomorphisms and base symplectomorphisms .
10.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4.3 Final remarks . . . . . . . . . . . . . . . . . . . . . . .
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11 Complex Star Algebras
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Star Algebras . . . . . . . . . . . . . . . . . . . . . . . .
11.2.1 Cones . . . . . . . . . . . . . . . . . . . . . . . .
11.2.2 Star Products on a Real or Complex Cone. . . .
11.2.3 Associated Poisson bracket . . . . . . . . . . . .
11.3 Pseudo-differential Algebras . . . . . . . . . . . . . . . .
11.3.1 E-algebras . . . . . . . . . . . . . . . . . . . . . .
11.3.2 Differential Operators and D-algebras . . . . . .
11.3.3 Automorphisms and Symbols of Automorphisms
11.3.4 Non Commutative Cohomology Classes . . . . .
11.3.5 Symbols . . . . . . . . . . . . . . . . . . . . . . .
11.3.6 Filtrations . . . . . . . . . . . . . . . . . . . . .
11.4 E-Algebras on T ∗ X, dim X ≥ 2 . . . . . . . . . . . . . .
11.4.1 General Results. . . . . . . . . . . . . . . . . . .
11.4.2 The case dim X ≥ 3 . . . . . . . . . . . . . . . .
11.4.3 The case dim X = 2 . . . . . . . . . . . . . . . .
11.5 E-Algebras over Curves (dim X = 1) . . . . . . . . . . .
11.5.1 Open curves . . . . . . . . . . . . . . . . . . . . .
11.5.2 Curves of genus g ≥ 2 . . . . . . . . . . . . . . .
11.5.3 Curves of genus 1 . . . . . . . . . . . . . . . . . .
11.5.4 The projective line . . . . . . . . . . . . . . . . .
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12 Related symplectic star algebras
12.1 Geometric quantization . . . . . . . . .
12.2 Homomorphisms between Star Algebras
12.3 Action of a Compact Group . . . . . . .
12.4 Circle Action . . . . . . . . . . . . . . .
12.5 Elliptic Circle Action . . . . . . . . . . .
12.6 End of description . . . . . . . . . . . .
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13 Toeplitz operators and asymptotic
equivariant index
13.1 Szegö projectors, Toeplitz operators
13.1.1 Example 1: Microlocal model
13.2 Example 2 : holomorphic model . . .
13.3 Main properties . . . . . . . . . . . .
13.4 Miscellaneous . . . . . . . . . . . . .
13.5 Equivariant Toeplitz algebra . . . . .
13.6 Equivariant trace . . . . . . . . . . .
13.7 Equivariant index . . . . . . . . . . .
13.8 Asymptotic index . . . . . . . . . . .
13.9 K-theory and embedding . . . . . . .
13.10Embedding and transfer . . . . . . .
13.11Relative index . . . . . . . . . . . . .
13.12Enlargement . . . . . . . . . . . . .
13.13Collar isomorphism . . . . . . . . . .
13.14Embedding . . . . . . . . . . . . . .
13.15Index . . . . . . . . . . . . . . . . .
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1
Introduction
Deformation algebras and star products were introduced in [6], so as the main
problems they pose; a closely related formulation was given by F.A. Berezin
[10]. Typically a deformation algebra is given by the following data :
- an initial algebra A, usually A = C ∞ (X), the algebra of smooth functions
on a manifold X; more generally A could be the algebra of holomorphic
(resp. algebraic) functions on a holomorphic (resp. algebraic) space X.
P n
- A formal multiplication law B =
~ Bn where for each integer n, Bn :
(f, g) 7→ Bn (f, g) is a C-bilinear map : A×A → A (later on we will always
suppose that the Bn are bidifferential operators)
- a formal family A~ of algebras depending
a formal parameter ~ : its
P on
elements are formal power series f =
fn ~n ∈ A[[~]]; the multiplication
P n
law (f, g) 7→ f ∗ g = B(f, g) is a C[[~]]-bilinear operator B =
~ Bn :
A[[~]]
×
A[[~]]
→
A[[~]]
(the
B
are
C-bilinear
:
A
×
A
→
A,
if f =
n
P p
P q
P
~ fp , g =
~ gq , B(f, g) = ~n+p+q Bn (fp , gq ).
The product B must satisfy the following conditions :
i) for ~ = 0 we get the initial law : B0 (f, g) = f g
ii) the law is associative : (f ∗ g) ∗ h = f ∗ (g ∗ h), i.e. for all n,
X
Bp (Bq (f, g), h) − Bp (f, Bq (g, h)) = 0.
p+q=n
iii) the multiplicative unit is 1 : 1 ∗ f = f ∗ 1 = f i.e. Bn (1, f ) = Bn (f, 1) = O
if n > 0 (the bidifferential operator Bn has no term of order 0 for n > 0).
(The third condition could be omitted; in fact the first two conditions imply
that there is a unit, i.e. a formal P
series a = 1 + O(~) such that a = a ∗ a, and a
∞
suitable linear bijection f 7→ f + 1 will give an equivalent product for which
the unit is 1).
Here we will only deform algebras of functions on manifolds, and we will
always suppose that the Bn are bidifferential
(local) operators, i.e. in local
P
coordinates of the form Bn (f, g) = an,α,β ∂ α f ∂ β g.
In physics ~ usually denotes the Planck constant (an action). We have kept
this notation although here (and in all the sequel) ~ will usually denote a small
formal parameter. In W KB asymptotic expansions from quantum physics, the
small parameter is the ratio of the Planck constant to actions at the scale or
the phenomenon described, which is very small for phenomenons well above
the atomic scale. In geometrical optics, one encounters asymptotic expansions,
where ~ measures the size of a small wavelength, the inverse of a large frequency:
this is very small for optic waves, which explains that geometric optics is relevant
at our scale; much less for radio waves and not at all for acoustic waves.
6
The calculus of pseudo-differential operators enters essentially in the same
description, except there is no longer a small central parameter ~. In order to
include it we will slightly broaden the description above, replacing formal series
in the “small parameter” ~ by more general asymptotic expansions with respect
to a large quantity - e.g. the size of a “large frequency”, playing the role of ~−1 .
Remark 1 It is natural to ask if a star algebra corresponds
to a true (not
P
formal) family of algebras, i.e. the formal symbols
fk ~k (or at least some
of them) represent the asymptotics for ~ → 0 of functions (or distributions)
f (x, h) with a product law. This is often the case, although one usually does not
expect convergent series, and it is completely unusual that a symbol determines
uniquely the corresponding distribution (e.g. the trivial deformation product
corresponds to the usual commutative product of smooth functions f (x, h), but
the Taylor series of f only determines f up to functions vanishing of infinite
order for ~ = 0)
The last sections are extracted from recent articles of the author.
7
2
Star Algebras.
2.1
Cones
Definition 1 A real cone is a C ∞ principal bundle Σ with group R×
+ . The basis
is X = XΣ = Σ/R×
+.
×
The trivial cone with basis X is X ×R×
+ with the action of R+ (homotheties)
given by λ.(x, r) = (x, λr).
Any point of X has a neighborhood U such that Σ|U = p−1 (U ) is isomorphic
to the trivial cone U × R×
+ (p denotes the projection map Σ → U ). If X is
paracompact, any cone Σ with basis X is trivial, i.e. isomorphic to the trivial
cone (it has a section), but the product structure is not part of the data, only the
fibration. In the sequel we will only consider paracompact cones and manifolds
if Dω is the long real line, T ∗ Dω −(its zero section) is a non trivial real cone.
In some instances it will be convenient to use the complex line bundle Σc
extending Σ :
Σc = Σ ×R× C×
(1)
+
b
Homogeneous functions of integral degree f ∈ O(n) or formal series f ∈ O
obviously extend to Σc .
For complex or algebraic geometry, one need a slightly broader definition : the
basis X is a complex manifold, resp. an algebraic variety over a field k; a cone Σ is a
line bundle over X, not necessarily trivial, deprived of its zero section. Since we will be
using differential operators, X will usually be supposed smooth (without singularities)
and k of characteristic zero.
On a cone Σ the group of homotheties has an infinitesimal generator : this
is the radial vector field ρ corresponding to the derivation Lρ ) such that
Lρ f (x) =
∂
f (λx)|λ=1
∂λ
(ρ = r
∂
∂r
in any trivialisation Σ = X × R+ ). (2)
Definition 2 (i)
L We denote O(m) the sheaf of homogeneous functions of degree
m on Σ, O =
O(m) (f ∈ O(m) ⇐⇒ ρf = mf ).
b the sheaf on XΣ of formal symbols :
(ii) We denote O
X
b if f =
f ∈O
fm with fm ∈ O(m), m an integer, m → −∞ (3)
m≤m0
b are asymptotic expansions for ξ → ∞ on Σ. We will usually
Elements f ∈ O
b is of degree m, its symbol (or
refer to them as “total symbols”. If f ∈ O
principal symbol to avoid confusions) is its leading term σm (f ) = fm ; we will
usually write σ(f ) if there is no ambiguity.
8
O(m) is a vector space, and O is a graded commutative algebra where the
product is defined by the natural multiplication maps O(m)⊗O(n) → O(m+n).
b is equipped with a canonical decreasing filtration O
bm (elements of degree
O
L b
L
b
b
≤ m); the graded associated sheaf is gr O =
Om /Om−1 =
O(m)
b are sheaves (of vector spaces or algebras) on the basis X.
O(m), O and O
We recall that “F is a sheaf on X” means that F(U ) is well defined for
any open subset U of X (↔ open sub-cone of Σ) (it is a set, a group, a vector
space, an algebra, or or an object of some category); the restriction maps ρU V :
O(V ) → P(U ) are defined
S for U ⊂ V , with ρU W = ρU V ρV W if U ⊂ V ⊂QW , and
for any covering U = Uj , the product of restriction map
Q O(U ) → Q O(Uj )
identifies O(U ) with the equalizer (kernel) of the two maps O(Uj ) ⇒ O(Ui ∩
Uj ) : in other words two sections of O which are equal in some neighborhood
of each point of U are equal, and a family fi ∈ O(Ui ) which agree in each
intersection Ui ∩ Uj can be patched together to produce a global section f ∈
O(U ). If p : E → X is a continuous map, the sheaf of sections is F(U ) =
Γ(X, U ) =the continuous sections f → E (functions such that pf = Id U is a
sheaf. Any sheaf of sets is isomorphic to some sheaf of sections, with p etale
(locally an isomorphism) (cf. [70]). Elements of F(U ) are usually called sections
of F (over U ), and F(U ) is often denoted Γ(U, F).
b
Definition 3 (iii) For any integer k ≥ 1 we denote
P Dk the sheaf (on XΣ ) of
formal k-differential operators : P (f1 , . . . , fk ) =
m≤m0 Pm (f1 , . . . , fk ) with
Pm a k-linear differential operator homogeneous of degree m with respect to
b
homotheties, m an integer, m → −∞. For k = 1 we just write D.
Locally we may choose homogeneous local coordinates xj on Σ : Pm (f1 , . . . , fk )
is a sum of homogeneous monomials
aα (x) ∂1α1 . . . ∂nαn
P
i.e. aα is homogeneous of degree m + αk deg (xk ) (in the notation above α is
a multi-index : α = (α1 , . . . , αn ) ∈ Nn , ∂ α = ∂ α1 . . . ∂ αn )
Two cases will be useful :
1) x1 , . . . , xn−1 are homogeneous of degree 0 and are local coordinates on
the basis XΣ , xn is homogeneous of degree 1 or −1;
2) the xj are all of degree 21 . There is no restriction on the order of Pm .
Below “degree” will always refer to the degree w.r. to homotheties; thus
bk each term Pm of degree m is of finite order, although the resulting
if P ∈ D
infinite sum P may be of infinite order.
We will denote
b× ⊂ D
b
D
(4)
b × is invertthe sheaf of invertible formal differential operators : P =
Pk ∈ D
ible iff its leading term σ(P ) = Pm0 is invertible, i.e. Pm0 is of order 0, the
multiplication by a non-vanishing function homogeneous of degree m0 .
P
9
b × the sub-sheaf of those invertible P such
Definition 4 (iv) We denote by D
0
that P (1) = 1, i.e. P is of degree 0, P0 = 1, Pm (1) = 0 if m < 0.
2.2
Star products and star algebras
Definition 5 Let Σ be a cone. A star product on Σ (or on the basis X) is
P
b2 (i.e. for each integer n, Bn is
a formal bilinear product B =
Bn ∈ D
a bidifferential operator, homogeneous of degree −n), defining a product law :
b×O
b→O
b - (f, g) 7→ f ∗ g = B(f, g), such that
O
i) B0 (f, g) = f g (or B0 = 1, B is a deformation of the usual product)
ii) the law is associative
: (f ∗ g) ∗ h = f ∗ (f ∗ h), i.e. B(B ⊗ 1) = B(1 ⊗ B),
P
or for all n, p+q=n Bp (Bq ⊗ 1 − 1 ⊗ Bq ) = 0.
iii) the multiplicative unit is 1 : 1 ∗ f = f ∗ 1 = f i.e. Bn (1, f ) = Bn (f, 1) = 0
if n > 0 (the bidifferential operator Bn has no term of order 0)
(Here again the first two conditions imply that there is a unit u with leading
term u0 = 1, and we can fix this unit equal to 1, e.g. replacing B by B 0 (f, g) =
u−1 B(uf, ug)).
Example 1 A typical example of star product is the Leibniz rule for the composition of differential operators on an open set P
U ⊂ E, E a vector space (e.g.
E = Rn ) : a differential operator P = P (x, ∂) = aα (x)∂ α , with aα ∈ C ∞ (U )
is characterized by its total symbol
X
p(x, ξ) = e−x.ξ P (ex.ξ ) =
aα (x)ξ α
which is a function, polynomial in ξ, on the cotangent bundle T ∗ U = U ×E ∗ ; the
cone is Σ = T • U = T ∗ U −the zero section{ξ = 0}, with x, resp. ξ homogeneous
of degree 0, resp. 1.
If P, Q are two operators, the total symbol r of R = P ◦Q is given by Leibniz’
rule :
X 1
∂ α p ∂xα q
r(x, ξ) =
α! ξ
b
This formula still makes sense as a formal series if p, q ∈ O(Σ)
are symbols
(x, ξ of degree 0 resp. 1) : this is the composition law for pseudo-differential
operators (see below §3.4.
Definition 6 A star-algebra is a sheaf of unitary associative algebras on the
b equipped with a star-product, where the strucbasis XΣ , locally isomorphic to O
b× .
tural patching sheaf of groups is D
0
If A is a star algebra, we can pick local frames (i.e. an isomorphism A(U ) →
b ) over a small open set U ); in such a frame the product is given by a
O(U
b2 (B = P Bk (f, g) with Bk
bidifferential operator : f ∗ g = B(f, g), with B ∈ D
10
a bidifferential operator homogeneous of degree −k → −∞, B0 = 1). Transition
b× .
maps from one frame to another are given by invertible operators P ∈ D
0
b × respects the natural filtration of O, a star-algebra A is
Since any P ∈ D
0
equipped with a natural filtration. We will denote Am the set of elements of
b × is 1, the
degree ≤ m (they form a sheaf). Since the leading term of any P ∈ D
0
leading term of any a ∈ A is well defined; if a ∈ Am we denote it σm (a) ∈ O(m)
(or σ(a) if there is no confusion) and call it the principal symbol of a. The
L
b = L O(m) is an isomorphism of
symbol map : gr A =
Am /Am−1 → gr O
graded algebras : σm+n (ab) = σm (a)σn (b)).
b of degree
A total symbol on A is a formal differential operator σtot : A → O
0 such that σtot (1) = 1. Total symbols exist locally by definition, hence also
globally ; if Xi isPan open covering of the basis X, and σi a total symbol on
Xi , then σtot =
φi σi is well defined and is a total symbol, for φi a smooth
partition of 1 subordinate to the covering. This does not work for algebraic
or holomorphic star algebras. In fact the algebra of differential (or pseudodifferential) operators on the complex projective line P1 (C) does not have a
total symbol.
Example 2 differential operators or pseudo-differential operators on a manifold
form a star algebra, for which there is no preferred total symbol (see below §3.4).
The distinction between star-products and star-algebras is not really essenb equipped
tial on real manifolds, where a star-algebra is always isomorphic to O
with a star-product because a “total symbol” i.e. a global isomorphism P :
b with P ∈ D
b locally always exists (locally, by definition, and on a real
A→
˜ O
manifolds these can be patched together using a partition of unity); however, as
mentioned in [23] this is no longer true over holomorphic manifolds : the most
usual star-algebras such as the algebra of pseudo-differential or semi-classical
pseudo differential operators on a manifold are not equipped with a canonical
total symbol. For functorial manipulations it is more convenient to deal with
star-algebras rather than star-products.
2.3
2.3.1
Poisson bracket
Reminder of differential calculus notations
If X is a manifold, we denote T X the tangent bundle, T ∗ X the cotangent
bundle. We will often use the cotangent bundle deprived of its zero section,
which we denote T • X.
V
L Vk
The sections of the P
exterior algebra T ∗ X =
T X form the sheaf of
differential forms Ω =
Ωk . This is a graded anti-commutative algebra, on
which several canonical operations are defined :
If ξ is a vector field, it defines a derivation Lξ of the algebra OX of functions
on X. More generally the Lie derivative Lξ u is defined for any differential
11
object u which is functorially defined, e.g. differential forms, tensors, differential
d −tξ
e∗ u|t = 0 where etξ denotes the germ of group with
operators etc. : Lξ u = dt
infinitesimal generator ξ (i.e. t 7→ etξ x is the solution of the differential equation
d
dt x(t) = ξ, x(0) = x), and ∗ denotes the push-forward.
The exterior derivation d is the unique anti-derivation of degree 1 such that
d2 = 0 and for any function f ∈ Ω0 , hdf, ξi = Lξ (f ) (anti-derivation means :
¯
d(ab) = da b + (−1)dā a db, where x̄ denotes the degree of x).
If ξ is a vector field, Iξ denotes the anti-derivation of degree −1 such that
Iξ (ω) = hξ, ωi for ω ∈ Ω1 (also noted ξyω).
On Ω, Lξ is the unique derivation (of degree 0) such that [d, Lξ ] = 0 and
Lξ (f ) = hdf, ξi for f a function. and we have
Lξ = [d, Iξ ]
(in a graded anti-commutative algebra [..] denotes the super bracket : [a, b] =
ab − (−1)da db ba with dx the degree of x - here [d, Iξ ] = dIξ + Iξ d).
The exterior derivative d is (locally) exact, i.e. if dω = 0 then any point
has a neighborhood in which ω is a derivative (ω = dµ). On a cone we will
need homogeneous primitives: if ω is closed, homogeneous of degree k 6= 0, it
is globally exact: ω = k1 Lρ ω = k1 dIρ ω. Let us choose r > 0 homogeneous of
degree 1 (⇔ a trivialization of Σ): ρ = r∂r . Any form ω ∈ Ωk , homogeneous of
degree 0 is ω = dr
r µ + ν with µ, ν pull-backs of forms on the basis X (of degree
k − 1 resp. k). Ω is closed, resp. exact iff µ and ν are. In particular for k = 1,
ω is closed iff ν is closed and µ is a locally constant function; it is locally exact
iff µ = 0.
The bracket [ξ, η] of two vector fields is defined by L[ξ,η] = [Lξ , Lη ] (i.e.
the corresponding first order operator is the commutator of Lξ and
V Lη). It
has a natural extension to the algebra of multivectors (sections of T X): the
Schouten-Nijenhuis (super)bracket ([106, 109]). The extension extension is defined
as follows : first I : ξ → Iξ as extend as a homorphism of algebras
V
T → L(Ω) (Iξ1 ..ξk = Iξ1 ◦ . . . Iξk , so hξ1 . . . ξk , ωi = Iξk . . . Iξ1 (ω) if ω ∈ Ωk ):
the Schouten-Nijenhuis bracket is characterized by
I[ξ,η] = [Iξ [d, Iη ]]
(one the right hand side [.[.]] denotes again the super bracket). For instance if
c, c0 are two bivectors (identified with antisymmetric bidifferential operators),
the tridifferential operator defined by [c, c0 ] is
[c, c0 ](f, g, h) = c(c0 (f, g), h) + c(c0 (g, h), f ) + c(c0 (h, f ), g) +
+ c0 (c(f, g), h) + c0 (c(g, h), f ) + c0 (c(h, f ), g)
2.3.2
Poisson brackets
Definition 7 A Poisson bracket on a manifold X is a bilinear map (f, g) 7→
c(f, g) on C ∞ (X) (often denoted c(f, g) = {f, g}) such that
12
(i) c(f, f ) = 0 (c is skew symmetric, {f, g} = −{g, f })
(ii) c is a derivation w.r. to g (or f ) : {f, gh} = {f, g}h + g{f, h}
(iii) c satisfies the Jacobi identity : {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0
P
An equivalent definition is that c is a bivector (locally =
cij ∂i ∂j such
[c, c] = 0 where [c, c] is the Nijenhuis-Schouten bracket.
If f is a smooth function, the derivation = vector field Hf (g) = {f, g} is
called the hamitonian vector field of f . Condition (iii) is equivalent to
[Hf , Hg ] = H{f,g}
equivalently, that {..} is invariant by Hf : Hf ({g, h} = {Hf g, h} + {g, Hf h}.
A Poisson bracketP
on a manifold X is called symplectic if its matrix is invertible, i.e. locally c =
cij ∂i ∧ ∂j with (cij ) an invertible antisymmetric matrix
with smooth coefficients. Equivalently c defines an antisymmetric isomorphism
T X → T ∗ X. The inverse then corresponds to a symplectic 2-form according to
the following definition:
We will call symplectic star algebra a star algebra on a cone Σ with symplectic Poisson bracket. σ is then even dimensional. Typical example aree the
pseudo-differential algebras, or the Toeplitz algebras mod. smoothing operators
(see below).
If A is a deformation algebra on a manifold X, its Poisson bracket on the
cone Σ = X × R+ is of the form {..}Σ = ~{..}X with {..} a Poisson bracket
on X. We will say that it is symplectic is {..}X is symplectic. X is then even
dimensional (and dim Σ is odd). A typical example is the algebra of semiclassical
pseudo-differential operatos (see below).
Definition 8 A symplectic manifold is a manifold X equipped with a 2-form
ω ∈ Ω2 (X) such that
P
(i) ω is invertible, i.e. locally ω =
aij dxi dxj where the matrix (aij ) is
skew-symmeric, invertible (this implies that dim X is even).
(ii) dω = 0
One readily checks that the condition dω = 0 is equivalent to condition (iii)
above (Jacobi identity). Thus a symplectic manifold is the same thing as a
manifold equipped with an invertible Poisson bracket. In mechanics Poisson
brackets are not always symplectic, especially if they depend on parameters
A Poison cone is a cone equipped with a Poisson bracket homogeneous of
degree −1. A symplectic cone is a Poisson Σ cone whose Poisson bracket is
invertible, Σ is equipped with a symplectic form ω homogeneous of degree 1.
Such a form is always exact: if ρ is the radial vector (infinitesimal generator
of homotheties), we have ω = Lρ ω = dIρ ω. The Liouville form λ = Iρ ω is the
unique primitive of ω which is orthogonal to ρ.
13
Example 3 Let X be a manifold. The Liouville form λ on T ∗ X is the tautological form: for any smooth function f , df = f¯∗ λ where f¯ is the section X → T ∗ X
∗
defined
denotes the pull-back; in any system of local coordinates
P by df ,
P
λ=
ξj dxj . The canonical symplectic form of T ∗ X is ω = dλ =
dξj dxj .
The corresponding Poisson bracket is the canonical Poisson bracket of T ∗ X :
λ=
X
ξj dxj ,
ω=
X
dξj dxj ,
{f, g} =
X ∂f ∂g
∂f ∂g
−
∂ξj ∂xj
∂xj ∂ξj
(5)
Unless otherwise stated, we will always suppose that the Poisson bracket
or symplectic form are real. In PDE theory they are pure imaginary, but this
makes only insignificant differences for the calculus. However many of the statements and constructions below work with minor modifications, replacing Σ by
its complexification Σc .
V
Exercise A multivector p (section of T X) can be viewed as a superfunction p(x, ξ on T ∗ X (its symbol), super meaning that the local cotangent
coordinates ξj anti-commute (this is tautological. On the algebra of these functions we dispose locally of the derivations ∂xj (of degree 0), ∂ξj , which is an
anti-derivation of degree −1. The symbol of the Schouten-Nijenhuis bracket is
X
[a, b] =
∂ξj a ∂xj b − (−1)(ā−1)(b̄−1) ∂ξj b ∂xj a
A Poisson bracket c is an antisymmetric map T X V
→ T ∗ X and
V defines a
homomorphism of graded antisymmetric algebras C : T X → T ∗ X, and
Ω → multivectors. Then we have C(dω) = [c, Cω] (both are C derivations,
which coincide on functions, and 1-forms).
Note that C −1 is homogeneous of degree 1: it takes multivectors homogeneous of degree k to forma of degree k + 1. We may translate the local exactness
of the exterior differential d:
Lemma 9 if X is a multivector homogeneous of degree k =1 such that [c, X] =
0, there exists locally Y of degree k + 1 such that X = [c, Y ]
The same holds on a symplectic manifold, without degree conditions.
A theorem of Liouville states :
Theorem 10 Two symplectic manifolds, resp. symplectic cones of the same
dimension are locally isomorphic
We briefly recall the proof : first recall that two invertible skew symmetric
invertible bilinear forms on a vector space are always isomophic (in particular
they are isomorphic to the canonical form c((x, y), (x0 , y 0 ) = x.y 0 −x0 .y on V ∗ ×V
- this is true for any ground field).
Let X be a symplectic of dimension n, and (cjk ) an invertible antisymmetric
matrix. We construct functions xj in a neighborhood of a given point (origin)
as follows: x1 is chosen arbitrarily with dx1 6= 0; Then we construct inductively
14
xk so that Hxj (xk ) = cjk for j < k, and the Hamiltonian fields Hxj are linearly
independent. This is possible because the induction hypothesis implies that the
Hxj commute ([Hxj , Hxk = Hcjk = 0) ahd the Frobenius integrability condition
is satisfied. The independence condition is ensured by fixing appropriate initial
conditions, e.g. fixing the values of the Hamiltonian fields (i.e. the dxj at the
origin).
This works in the same manner on a symplectic cone; in the inductive construction, one also asks that Hxj , j < n be independent of the radial vector ρ to
ensure that there is a homogeneous solution. Thus a symplectic cone is always
locally isomorphic to the cotangent bundle T • Rn ; so as symplectic manifold
(without the homogeneity condition)..
2.3.3
Symplectic cones and contact manifolds
A contact form on a manifold X of dimension 2n − 1 is a 1-form λ such that
λ(dλ)n−1 is a volume form. Liouville’s theorem also states that two contact
forms of the same dimension are locally isomorphic (it is proved in the same
manner). A 1-form λ on X is a contact form iff the cone Σ = R+ λ ⊂ T • X (the
positive multiples of λ- is symplectic.
Let us call call contact morphism u : X → X 0 a smooth map such that λ is
a positive multiple of u∗ λ0 ; likewise a symplectic morphism Σ → Σ0 is a smooth
homogeneous map such that ωΣ = u∗ (ωΣ0 . (Such a map is not necessarily an
isomorphism but it is necessarily an immersion: its derivative is injective at each
point of X (in particular if such a map exists we have dim X ≤ dim X 0 (resp.
dim Σ ≤ dim Σ0 ). If X, X 0 is the basis of Σ, Σ0 , a contact map X → X 0 has a
unique symplectic lifting Σ → Σ0 , and conversely. In other words the functor
which to a symplectic cone (resp. map) assigns its basis (resp. the base map) is
an equivalence. When dealing with symplectic cones, it will often be practical
to replace it by its contact basis, and conversely.
2.3.4
Commutators
Let A be a star algebra on Σ. As mentioned above A has a canonical filtration
b by homogeneity degrees, and there is a canonical
coming from the filtration of O
isomorphism :
M
b=
gr A ' gr O
O(m)
b × induces the identity on gr O).
b
because the patching sheaf of groups D
−
The leading term of the commutator law :
{f, g} = B1 (f, g) − B1 (g, f )
is a Poisson bracket on Σ, homogeneous of degree −1, and gr A is a Poisson
algebra (i.e. equipped with a Lie bracket which is a bidfferentiation w.r. to the
multiplication).
15
Conditions (ii) and (iii) immediately follow from the fact that on any algebra
the bracket [a, b] = ab − ba is a Lie bracket: ad a : b 7→ [a, b] is a derivation
([a, bc] = [a, b]c + b[a, c]), and the Jacobi identity holds: [a, [b, c]] = [[a, b], c] +
[b, [a, c]] (equivalent to [ad a, ad b] = ad [a, b]).
2.4
Functional calculus
Let A be a star algebra, a ∈ AfPa smooth function. P
We want to define f (a).
k
E.g. If f is a poynomial
f
(T
)
=
f
T
,
then
f
(a)
=
fk ak is defined for any
k
P∞
P
k
k
a ∈ A. If f = 0 fk T a formal series, f (a) =
fk a is also well defined if
deg a < 0.
The requirements are the following:
• f 7→ f (a) must be a homomorphism of algebras, i.e. (f + g)(a) = f (a) +
g(a), f g(a) = f (a) ∗ g(a);
P
P
• if f (T ) =
fk T k is a polynomial, then f (a) =
fk ak Plus some continuity condtions which will be made precise later.
If f is holomorphic, we define f (a) in a neighborhood of a point x ∈ X as
Z
1
γf (t)(t − a)−1 dt,
(6)
2iπ
where γ is any small path/loop of index 1 around the value σa (x). This does not
depend on the choice of the loop, so these definitions patch together to define
f (a); f should be holomorphic in a neighborhood of each value of σa .
For any y close to x, we can rewrite this integral as
Z
X
X 1
1
f (a) =
f (t)
(t − a0 (y)−k−1 (a − a0 (y))k dt =
f (k) (y)(a − a(y))k
2iπ
k!
k
(a0 (y) is a number so t − a0 (y) and a − a0 (y) commute).
We denote Iy A0 the ideal of elements a ∈ A0 whose principal symbol vanish
at y. We will limit to the case where
P A is defined by a star product B of
pseudo-differential type, i.e. B =
Bn where the n-th component Bn is of
order ≤ (n, P
n) (see below - where we will also see that this is always possible).
m
Then a =
k≤0 ∈ Iy if for each k, ak vanishes of order ≥ m − 2k (for a
symplectic algebra,
converse is true).
Pthe
1 (k)
Then the sum
f
(y)(a − a(y))ky=x the k-th term (a − a(y))ky=x is i fact
k!
k
of degree ≤ − 2 .
So f (a) is again well defined if a0 is real, and f is a smooth function of a
real variable (more generally if f is smooth on the range of a0 and holomorphic
inside). We have f (a) − f (b) ∈ IYm if a − b ∈ IYm ; so the “Taylor series” of f (a)
at a point y only depends on that of a (and depends continuously on it).
16
2.5
Comments
Let X be a Poisson manifold, i.e. a manifold X equipped with a Poisson bracket
{f, g} (see below). A
quantization of X is a unitary associative
Pdeformation
P n
product law f ∗ g =
hn Bn (f, g) on the space of formal series f =
h fn ∈
C ∞ (X)[[h]], where each Bn is a bilinear differential operator on X, B0 (f, g) =
f g, the unit is 1 (Bn has no “term of order 0” if n > 0), and the leading term in
the commutator law is given by the Poisson bracket of X : B1 (f, g) − B1 (g, f ) =
{f, g}.
The corresponding cone is X × R×
+ equipped with the Poisson bracket hcX ,
1
cX the Poisson bracket of X; the canonical coordinate of R×
+ is h , homogeneous
of degree 1 (in asymptotic analysis it corresponds to a large frequency). We
will also refer to these as deformation, or “semi-classical” star-products or staralgebras.
In a deformation quantizations (or semi-classical algebra), the deformation
parameter h (the “Planck constant”, or its inverse h−1 ) is central and part of
the data; for these we only allow automorphisms which preserve h.
Existence of star algebras with a given Poisson bracket, and classification of
these, was an important problem of the theory.
Existence of a global star-algebra on a real symplectic cone Σ was proved by
V. Guillemin and myself in [27] (see also [19]).
Existence of deformations of symplectic Poisson brackets (symplectic semiclassical star algebras) were proved by M. De Wilde and P. Lecomte ([38, 40]),
who also described their classification. For these a very elegant method was
developped by B.V. Fedosov [60].
In [91] M. Kontsevitch proved that any Poisson bracket comes from a starproduct in the real semiclassical case. More precisely he proves that there is
a one to one correspondence between isomorphic classes of star-products and
isomorphic classes of formal families of Poisson brackets depending on the “small
parameter” h. His result extends, without changing a word, to star-products
on a real cone with the definition above ; families of Poisson brackets should be
b on Σ :
replaced by formal Poisson brackets (with coefficients in O)
X
c=
cm
(with cm homogeneous of degree m).
(7)
k≤−1
The main point of Kontsevitch’s construction is a formula giving a star
product from a Poisson bracket on an affine space; this also works on the complex
affine spaces, and adapts easily to the conic setting. Going from a local existence
or classification result to a global one requires some gluing argument. Below we
will follow the method of B.G. Fedosov (loc. cit.), which adapts easily to the real
conic framework. The gluing argument uses real constructions, such partitions
of unity, global connections, or tubular neighborhoods of subvarieties, which do
17
not exist in the holomorphic or algebraic category: for these it is not known
if a global star product with given Poisson bracket in general exists, even in
the symplectic case, nor what the classification of such algebras looks like; see
however [88], where it is shown that even if E may not exist, the category of
sheaves of E-modules is can be defined as a stack.
18
3
Models and examples.
3.1
Moyal star product
A graded vector
L space is a vector space equipped with a direct sum decomposition E =
En (n ∈ R) (for most purposes R could be replaced by anything).
Equivalently E is equipped with a degree operator d, a diagonalisable linear operator: En = ker (d−n), the n-eigenspace; equivalently E is equipped
with
P
P a oneparameter group of linear transformations (action of R+ ): L
λ( xn ) = λn xn .
In algebraic cases the grading will be integral (e.g.
O(m) above, and
homotheties define an action of the multiplicative group (z.x = z d x for z ∈ C×
resp. z ∈ k × , k the ground field).
If E is a graded vector space, Σ = E − E0 is a cone, with the action
of R+ above (one must exclude E0 to get a free action of R+ ). The spaces
b
O(m)(Σ), O(Σ)
are well defined as above.
L
(Note that the grading of O =
O(m) is integral, by definition. The grading of
E is not necessarily so; occasionally it will be convenient to choose “ocal coordinates”
homogeneous of non integral degree - in particular of degree 21 )
The Moyal star product is a typical
example of star product s : let E be
P
a graded vector space. Let b =
bij ∂i ∂j ∈ V ⊗ V a 2-tensor of degree −1
(or of integral degree ≤ −1), defining a bidifferential operator with constant
coefficients
X
b(f, g) =
bij ∂i f ∂j g.
The Moyal product defined by b lives on Σ = E − E0 :
f ∗b g = eb(∂ξ ,∂η ) f (ξ)g(η)
|η=ξ
=
X 1
bn (∂ξ , ∂η )f (ξ)g(η)
n!
|η=ξ
(8)
the formal sum is well defined since b is of negative degree; it is a star product
on V : associativity follows from the bilinearity of b: we have
b(u + v, w) + b(u, v) = b(u, v + w) + b(v, w),
which implies
1
(f ∗ g) ∗ h = eb(∂ξ ,∂η ) e 2 b(∂ξ ,∂η,∂ζ ) f (ξ)g(η)h(ζ)
|η=ζ=ξ
= · · · = f ∗ (g ∗ h)
The corresponding Poisson bracket is the top-order part (of degree −1) of
the antisymmetrization of b :
{f, g}(ξ) = (b(∂ξ , ∂η ) − b(∂η , ∂ξ )) f (ξ)g(η)
|η=ξ
= b(f, g) − b(g, f )
(9)
For example the normal law (Leibniz rule, example 1) for differential operators on an open set U ⊂ Rn is a typical example of Moyal star product :
f ∗g =
X 1 (α)
∂ f ∂x(α) g = f ∗bn g
α! ξ
19
P
with bn (f, g) =
∂ξk f ∂xk g − ∂xk f ∂ξk g. Its Poisson bracket is the canonical
Poisson bracket of T ∗ U : {f, g} = fξ .gx − fx .gξ (cf. 5).
Proposition 11 Two bidifferential operators b, b0 with the same antisymmetric
part yield isomorphic Moyal products.
Indeed if p = b0 −b is symmetric, we have (b0 −b)(u, v) = 21 (p(u+v)−p(u)−p(v))
i.e.
1
1
1
p(u + v) + b(u, v) = b0 (u, v) + p(u) + p(v),
2
2
2
The second order operator P = p(∂ξ , ∂ξ ) is homogeneous of degree −1, as b and
b0 , so the formal sum
X 2−k
1
e2P f =
P kf
k!
is well defined. We have
0
1
1
1
e 2 P (∂ξ +∂η ) eb(∂ξ ,∂η ) = eb (∂ξ ,∂η e 2 P (∂ξ ) e 2 P (∂η ) ,
from which immediately follows the equality
1
1
1
e 2 P (f ∗b g) = (e 2 P f ) ∗b0 (e 2 P g)
It is often convenient to use the equivalent but more symmetric Weyl calculus: this is the Moyal product corresponding to the antisymmetrisation of
cn :
1 X
bw (f, g) = (
∂ξk f ∂xk g − ∂xk f ∂ξk g)
2
1
f ∗w g = e 2 (∂ξ .∂y −∂η ∂x ) f (x, ξ)g(y, η)|(y,η)=(x,ξ)
On a graded vector space where all coordinates are of degree 12 this allows more
obvious symmetries, in particular the product is invariant by the full symplectic
group.
We will see later on that all star products with a real (or pure imaginary)
symplectic Poisson bracket are locally isomorphic, in particular they are locally
isomorphic to the pseudo-differential star product (normal law), or to the Weyl
star product.
For deformation star products the canonical Moyal model is the following :
let X be a vector space (no grading
P - coordinates xk all of degree 0) equipped
with bidifferential operator b =
bjk ∂j ⊗ ∂k with constant coefficients. The
star-product, with central formal deformation variable ~ is :
f ∗ g(ξ) = e~c(∂ξ ,∂η ) f (ξ)g(η)|η=ξ
(10)
Here again the corresponding Poisson bracket is ~(b(f, g)−b(g, f )); and two bidifferential operators with the same antisymmetric part yield isomorphic Moyal
products.
20
3.2
Star-product defined by a formal group law
Let G be a Lie group. Then the left invariant pseudo-differential operators on G
(see below) obviously form a sub-algebra of the algebra of all pseudo-differential
operators. For these the symbol (principal or total) is completely determined
by its restriction to the cotangent fiber at the origin, and produce a star algebra
on the dual of the Lie algebra G of G.
Here we will produce here an “explicit” formula for this star-product
Let V be a finite dimensional vector space equipped with a formal group
law :
x ◦ y = x + y + c(x, y)
(11)
where c is a formal power series in the variables x = (x1 , ..xn ) y = (y1 , ..yn ) in V .
Associativity (x◦y)◦z = x◦(y◦z) implies that the error term c(x, y) = x◦y−x−y
is O(|x||y|). The group law defines a Lie algebra structure on V .
Let V ∗ be the dual of V . If F is a distribution on V with support the origin,
its Fourier-Laplace transform is the polynomial
f (ξ) = hF, ex.ξ i (so that F = f (−∂x ) δ)
(12)
If F and G are two such distribution, the convolution product F ∗G is defined
by
hF ∗ G, ui = hF (y) G(z), u(y ◦ z)i = f (∂y ) g(∂x ) u(y ◦ z)
|y=z=0
(13)
This is defined for u a smooth function on V , but only depends on the Taylor
series of u at the origin and makes sense for u ∈ C[[V ∗ ]], an arbitrary formal
series.
In particular the Fourier transform of F ∗ G, which we will still denote f ∗ g
(with f, g the Fourier transforms of F, G) is
f ∗ g(ξ) = hF (y) G(z), eξ.(y◦z) i = f (∂y ) g(∂x ) exp ξ.(y ◦ z)
|y=z=0
= exp ξ. (∂η ◦ ∂ζ ) f (η) g(ζ)
|η=ζ=0
(14)
Let us rewrite ∂η ◦ ∂ζ = ∂η + ∂ζ + c(∂η , ∂ζ ). Taking into account the fact that
all these differential operators commute and that we have eξ. ∂u f (u) |u=0 = f (ξ)
(Taylor formula), we get
f ∗ g (ξ) = eξ.c(∂η ,∂ζ ) f (η) g(ζ)
|η=ζ=ξ
(15)
b
Now c is of order
P ≥ 2 , so this
P final expression still makes sense for f, g ∈ O
symbols , ie. f = fm−k , g = gm−k formal series of homogeneous functions
of degree → −∞ (ξ. c(∂η , ∂ζ ) is of degree ≤ −1). This is the star product
associated to our group law. The cone is Σ = V ∗ − {0}, the Poisson bracket is
the standard Poisson bracket of V ∗ (if f, g are smooth functions on V ∗ , ξ ∈ V ∗ ,
21
the derivatives df (ξ), dg(ξ) belong to the Lie algebra V : we have {f, g}(ξ) =
h[df (ξ), dg(ξ)], ξi).
Isomorphic group laws give rise to equivalent star-products, i.e. up to isomorphism the star product above only depends on the Lie algebra V , or equivalently
on the Poisson bracket of V ∗ . A canonical construction consists in choosing the
group law given by the Campbell-Hausdorff formula.
One can also replace the group law by the one deduced by homothety :
−1
eλ ξ.c(λ ∂η , λ ∂ζ )..|η=ζ=ξ .
For the standard Fourier transform of real analysis λ = ±i)
Exercise - one recovers the law for semiclassical pseudo-differential operators,
using the (formal) Heisenberg group or more generally Moyal laws :
u ◦ v = u + v + q(u, v) e
(16)
with q a bilinear form on V, e ∈ V a central vector (the Heisenberg group
corresponds to the case where q is non-degenerate mod. e).
3.3
Formal pseudo-differential operators
Let U be an open set of Rn . We mentioned above that the composition of
differential operators (Leibniz rule, example 1)
p ∗ q(x, ξ) =
X 1
∂ α p ∂xα q = e∂ξ .∂y f (x, ξ)g(y, η)|y=x
η=ξ
α! ξ
b
is a star product, which extends to symbols p, q ∈ O(Σ)
(Σ = T • U = U × Rn
minus its zero section U × {0}, with x, ξ of degree 0 resp. 1). We will denote EU
this algebra. If X is a manifold, Xj an atlas of X i.e. a covering of X by open
sets isomorphic to open subsets of some numeric space Rn , the EXj glue together
canonically (see below) and produce the pseudo-differential algebra EX ; for this
again the cone is T • X, and the Poisson bracket is the canonical one.
An oscillatory asymptotic expansion with phase φ is an expression of the
form
P m−k
A = eλφ(x) a(λ, x)
with a =
λ
am−k (x)
b +×U
where λ is a formal parameter; a(λ, ξ) is a symbol in O(R
Differential operators act on such expansions: if P = p(x, d) is a differential
operator, we have P (eφ a) = eφ Pφ (a) with
X
Pφ = e−λφ P eλφ = p(x, d + dφ) =
pα (x) (d + dφ(x))α
(caution: d and dφ do not commute).
22
b
Proposition
P m−k 12 We have Pφ ∈ D(U × R+ ), i.e. Pφ has formal expansion
Pφ =
λ
Pk where the Pk are differential operators (of order ≤ 2k) on U
(m is the order of P ).
Let φ2 (x, y) be defined by φ(x) = φ(y) + (x − y).φ0 (y) + φ2 (x, y): for any fixed
y we have Pφ = eλ−φ2 P (x, d + λφ0 (y))eλφ2 , hence
X 1 γ
∂ p(x, λφ0 (y)) ∂xγ (eλφ2 a)|y=x
(17)
Pφ (a) =
γ! ξ
Now φ2 vanishes of order ≥ 2 for y = x so ∂ γ (eλφ2 a)|y=x is a polynomial of
γ
0
γ λφ2
degree ≤ |γ|
a)|y=x is a symbol of degree ≤ m − |γ|
2 , and ∂ξ p(x, λφ (y))∂x (e
2 .
If dφ 6= 0 the sum above is still defined and converges as a formal series when
p is symbol, thus defining the action of formal pseudo-differential operators on
asymptotic expansions.
If y = χ(x) is a local change of coordinates (i.e. the derivative ψ 0 is invertible,
the total symbol of a differential operator P = p(x, d) in the new coordinates y
is p0 (y, η) = e−y.η P (ey.η ): this is still well defined if P is a formal differential
operator, so the sheaf of formal differential operators on an arbitrary manifold X
is well defined as announced; they can be viewed as operators on all asymptotic
expansions as above, with dφ 6= 0.
The oscillating asymptotic expansions corresponding to φ form a sheaf of
EX -modules, with support the cone Λφ = R+ {dφ} ⊂ T • X (if φ is complex
valued this is still well defined, but M φ rather lives on the complex cotangent
bundle T ∗ X ⊗ C)
3.4
Pseudo-differential operators
The theory of pseudo-differential operators, which gives main example of symplectic star algebra, was developed since the 1960’s in particular by J.J. Kohn,
L. Nirenberg, L. Hörmander, and widely used by many others. We describe
now how it works and gives rise to the star algebra above. Results are stated
without proofs, and we send back to the literature for further details.
Pseudo-differential makes use of exponentials eix.ξ with pure imaginary exponent, because they are the only bounded ones; they are those which are enter
in the
√ Fourier transformation. Below we have followed custom and put a sign
i = −1 in the exponents; an even better solution is to say that calculus lives
on the set of pure imaginary covectors. This makes no significant difference for
the formal calculus (so we omit the sign i for formal calculus), but of course in
real analysis it is essential.
Let U be an open set of Rn . A symbol of degree m on T ∗ U = U × Rn is a
smooth function on admitting for ξ → ∞ an asymptotic expansion
p(x, ξ) ∼
∞
X
k=0
23
pm−k (x, ξ)
with pm−k smooth, homogeneous
of degree m − k w.r. to ξ.
P
(∼ means that p− m−k>−N pk has the same size as a homogeneous function
of degree −N , and likewise for any derivative
X
∂xα ∂ξβ (p −
pk ) = O(|ξ|−N −β ) for ξ → ∞).
k≥−N
Above k is an integer; m should be an integer for compatibility with the definition above, but occasionally it may be any complex number. There are many
other useful and more general classes of symbols which are “micro-local”; the
main point is that high order derivatives of p should be integrable w.r. to ξ.
Here we have described the minimal class for which symbolic calculus works and
which are obviously linked with star algebras.
The pseudo-differential P = p(x, D) is defined by the formula
Z
p(x, D)f = (2π)−n eix.ξ p(x, ξ)fˆ(ξ)dξ
where fˆ is the Fourier transform of f - on Rn :
Z
Z
fˆ(ξ) = e−ix.ξ f (x)dx
f (x) = (2π)−n eix.ξ fˆ(ξ)dξ
P
α
For instance if p(x,
P ξ) = α aα (x)ξ is 1a polynomial in ξ, p(x, D) is the
differential operator
aα (x)D , with D = i ∂x .
A pseudo-differential operator P acts continuously from the space C0∞ of
smooth functions to C ∞ : it extends to distributions, and diminishes singular
supports : if f is a distribution, P f is C ∞ in any open set where f is.
The Schwartz-kernel of P is the distribution P̃ such that
Z
P f (x) = P̃ (x, y)f (y)dy
R
It is the inverse Fourier transform of p : P̃ = (2π)−n ei(x−y),ξ p(x, ξ)dξ. (the
integral only converges in distribution sense). It is always a smooth function
outside of the diagonal {x = y}.
We will write P ∼ Q if P − Q has a smooth Schwartz-kernel (equivalently;
P − Q extends as a continuous linear operator from the space of distributions
to the space of smooth functions). P is always equivalent to a “proper” pseudodifferential operator (i.e. the two projections from the support of P̃ to U are
proper maps; if P is proper, it acts on the space functions or distributions with
compact support and extends to all distributions (for any compact set K ⊂ U
there exists another one L such that supp f ⊂ K ⇒ supp P f ⊂ L and f = 0
out of L ⇒ g = 0 out of K).
The total symbol of P = p(x, D) is the asymptotic formal sum σ(P ) =
b
pm−k ; it belongs to O(Σ),
with Σ = T • U = U × Rn (in the sense of definition
2), where the variables xj , resp. ξj are homogeneous of degree 0 resp. 1.
P
24
σ(P ) completely defines P mod. smoothing operators (P ∼ 0 ⇔ σ(P ) = 0,
i.e. p(x, ξ) if of rapid decrease for ξ → ∞ so as all its derivatives - we will again
write p ∼ 0). In fact we have the asymptotic formula (exact for differential
operators):
p(x, ξ) ∼ e−ix.ξ P (eix.ξ )
Proper pseudo-differential operators form an algebra. The total symbol of a
product is again given asymptotically by the Leibniz rule (exact for differential
operators), same as above (except for the i sign): if P = p(x, D), Q = q(x, D)
then P ◦ Q = R(x, D) with
σ(P ◦ Q) =
X i −α
∂ξα σ(P )∂xα σ(Q)
α!
The set of pseudo differential operators is invariant by diffeomorphisms, so
pseudo-differential operators on a manifold X are well defined. Modulo smoothing operators, they form a star algebra on the cotangent cone T • X = T ∗ X
deprived of its zero section, isomorphic to the forma star algebra above: the
principal symbol (leading term) is well defined as a homogeneous function on
T • X, although there is no canonical or preferred total symbol.
There is an analogue of the oscillatory asymptotic expansion result: if P is a
pseudo-differential operator and a(x, λ) a symbol, we have P (eλiφ a) ∼ eλiφ Pφ (a)
i.e. the difference between the left and right hand sides is of rapid decrease for
λ → ∞. This requires that φ be real. It still works if Im φ is ≥ 0 (eλiφ bounded)
with a suitable modification of Pφ (which is at first only defined where dφ is
real).
3.5
Semi-classical operators
Semi-classical analysis produces an important example of deformation algebra,
which is closely linked to the preceding one. Let X be a smooth real manifold at first an open set of Rn . A semiclassical differential operator is a differential
operator depending on a formal parameter ~:
X
P = P (x, ~D) =
ak,α (x)~k (~D)α
k≥m
~ is a formal small parameter; in real analysis D = 1i ∂x . For a differential
operator there are only finitely many terms; the degree of P is m (we allow
m < 0).
The total symbol is the symbol on T ∗ X × R+ :
X
i
i
p = p(~, x, ξ) =
akα ~k ξ α = e− ~ x.ξ P (e ~ x.ξ )
The total symbol of P ◦Q is again given by the Leibniz rule (or Moyal formula) :
p∗q =
X i−γ
γ!
∂ξ p∂xγ q = exp(−i∂ξ ∂y )p(~, x, ξ)q(~, y, η)|y=x
η=ξ
25
for any smooth f or formal series
P (f ) =
P
~k fk we have
X i −|α|
~|α| ∂ξα ∂xα f
α!
and more generally an oscillatory asymptotic expansion:
X
i
i
P (e ~ φ a) = e ~ φ Pφ a (a =
~k ak a formal series)
This is exact is P is a differential operator, and still well defined if p is a symbol,
φ real.
i
i
As above Pφ = e− ~ φ P e ~ φ is a formal operator on X × R+ , which only
depends on the Taylor expansion of p along the section dφ.
For real analysis one needs a link between these formal expansions and true
functions : this goes as follows: a symbol on X × R is a smooth function
a(x, ~) (the corresponding formal symbol is just the Taylor expansion along
~ = 0; one could allow a pole of finite order for ~ = 0). The corresponding
i
oscillatory object is e ~ φ a, and the Taylor expansion of a gives good bounds if φ
is real (or Im φ ≥ 0 so that the exponential is bounded). One can define semiclassical pseudo-differential operators (but for many problems in PDE theory,
consideration of the formal symbols is enough).
Some problems that are hard or impossible to solve exactly may become
much easier to solve for asymptotic solutions. For example : with P = 1 − ~2 ∆
i
i
the problem P (e ~ φ a = e ~ φ b i.e. Pφ a = b is very easy, because the leading term
(principal symbol) of Pφ is 1 + |dφ|2 6= 0, so Pφ is invertible and we dispose
of explicit (if not short) formulas to write the inverse and solve the problem
(ARRANGER)
As above semi-classical operators allow changes of coordinates and are still
well defined on manifolds.
In fact the can be identified with a sub-algebra of the preceding section as
follows: let Y = X × R, and denote (abusively) (x, t) the variables of Y , Ξ, τ the
dual cotangent variables. Then As c is isomorphic to the open half-space τ > 0
of T ∗ Y .
3.6
Toeplitz operators
Toeplitz algebras give an example of a symplectic star algebra on a symplectic
cone which is not a cotangent bundle. We describe here the standard example,
which comes from the theory of several complex variables. We will see later
that a canonical Toeplitz algebra (i.e. well defined, up to isomorphism) can be
defined on any symplectic cone (cf.[27]).
Let U ⊂ Cn be a complex domain with smooth boundary X (more generally
U could be a Stein space with isolated singularities away fro its boundary). U is
26
strictly pseudo-convex if it can be defined (locally or globally) by an inequality
¯ 0 (i.e. the
u < 0 with u a smooth function such that du 6= 0 on X, and ∂ ∂u
matrix (∂zp ∂z̄q is hermitian 0), e.g. U is strictly convex.
The boundary value fX of holomorphic function on U is well defined if f is
continuous on the closure Ū = U ∪ X, they are stil defined as a distribution if f
is of moderate growth (f = O(|u|−N for some N - as for any harmonic function
1
.
If U is strictly pseudo-convex, the boundary values of holomorphic functions
are (locally or globally) exactly those that satisfy the tangent Cauchy-Riemann
¯ = P ∂z dz̄j ; ∂¯b
equations ∂¯b g = 0. ∂¯ denotes the antiholomorphic part of d: ∂f
j
¯ mod. u, ∂u.
¯ Let us denote
is the induced system on X it defines: ∂¯b fX = ∂f
Os (X) the space of holomorphic boundary values which lie in the Sobolev space
H s (X). It is known (cf. [80]) that the micro-singularities of boundary values
of holomorphic functions (f ∈ ker ∂b i.e.f ∈ Os (X) for some s) lie on the halfline bundle Σ ⊂ T • X of positive multiples of λ = −i∂u|X ; this is real since
¯ |X = 0, strict pseudo-convexity implies that λ = −i∂u|X is a contact
(∂u + ∂u)
form, and Σ is symplectic.
The Szegö projector S is the orthogonal projector L2 (X) → O0 (X) (the
definition requires choosing a smooth density on X to define the L2 norm). It
was proved in [30] that it is a Fourier integral operator with complex canonical
relation, well behaved with respect to pseudo-differential operators, in particular
it is continuous H s (X) → Os (X) for all s.
The canonical relation of S is a complex (formal) Lagrangian manifold C ⊂
T • X × T • X 0 ; it is the outflow of ∂¯b × ∂b out of Id Σ .
Toeplitz operators are the operators of the form f 7→ TP (f ) = SP S(f )
with P a pseudo-differential operator on X. Equivalently they are the Fourier
integral operators A with canonical relation C such that A = SAS.
We will say that A is of degree ≤ m if if is of degree ≤ m as a Fourier integral
operator (⇔ A = TP with P of degree ≤ m); then A is continuous Os → Os−m .
Toeplitz operators form an algebra. Mod. smoothing operators (operators
of degree −∞) they are localized, and they define a symplectic star algebra over
Σ, locally isomorphic to the pseudo-differential albebra in n real variables. The
principal symbbol of A = TP is σ(P )Σ and we have
Σ(AB) = σ(A)σ(B),
σ[A, B] = −i[σ(A), σ(B)}Σ
1 without the growth condition, the boundary value would be defined as a hyperfunction if
X is real analytic
27
4
4.1
Homomorphisms, automorphisms.
Morphisms
Let A, A0 be two star algebras, Σ, Σ0 the corresponding cones, with base X, X 0 ,
and Poisson brackets c, c0 . A homomorphism U : A → A0 is a linear map which
preserves the products, and the filtrations (U (ab) = U (aU (b), deg U f ≤ deg f )
The symbol map gr U : gr A → gr B, is then a Poisson algebra homomorphism, i.e. it preserves products and Poisson brackets. Its restriction to O(0)
corresponds to a smooth map uX : X 0 → X (because X is the maximal spectrum
of O(0) ∼ C ∞ (X) : gr U f = f ◦ uX if f is of degree 0). U is a homomorphism
of sheaves of algebras A → A0 over the smooth map uX : X 0 → X.
0
gr U itself comes from
L a smooth homogeneous map u : Σc → Σc , because
the spectrum of O =
O(m) is the complexified cone Σc (gr U f = f ◦ u). u is
compatible with the Poisson brackets : c0 (f ◦ u, g ◦ u) = c(f, g) ◦ u; equivalently
the Poisson bracket c is projectable and u∗ c = c0 , or the graph of u in (Σ×Σop )c
is isotropic. Σ0op denotes the opposite cone, i.e. Σ0 equipped with −c0 . (Σc is
the line complexification of Σ). 2
Remark 2 u is a map Σ0 → Σ iff gr U is real positive, i.e. σ(U f ) ≥ 0 if
σ(f ) ≥ 0. This is what happens for Fourier integral transformations. If gr U
ie real, it is either positive or negative (i.e. σ(U f ) ≤ 0 is f is of degree 1 and
σ(f ) ≥ 0; then u(Σ ⊂ −Σ or equivalently u defines a map Σ0 → Σop , as for
involutions below.
Note that the fact that gr U is real is not directly related to the fact that
the Poisson bracket is real; the Poisson bracket of pseudo-differential theory is
pure imaginary.
Example A typical example of homomorphism is the formula of change of coordinates for the total symbol of pseudo-differential operators or semi-classical PDO that
can be derived from (17)
If B, B 0 are two star algebras on cones Σ, Σ0 , the star algebra B ⊗ B 0 on
Σ × Σ0 is well defined (locally the product law is the exterior product of the two
bidifferential operators, B ⊗ B 0 (f, g) = Bx Bx0 0 (f (x, x0 )g(x, x0 ); which obviously
glue together.
There are canonical injective homomorphisms U resp. U 0 : A resp. A0 →
A ⊗ A0 whose range commute, and whose geometric support are the projections
from Σ × Σ0 to Σ resp. Σ0 (f 7→ f ⊗ 1 resp. 1 ⊗ f ; if we are dealing with star
products, the pull-back by either projection). The exterior product A ⊗ A0 is
well defined (up to unique isomorphism) by this data. Its Poisson bracket is
c + c0 (exterior sum of two bidifferential operators).
Note that the basis of Σ × Σ0 is not X × X 0 ).
2 In fact the definition above should be completed by the condition that locally U is described locally by formal differential operators: the total symbol is given by (U )f (x0 ) = (P f )◦u
with P a suitable formal differential operator on X; this is automatic for the symplectic cases
we usually consider
28
If U is a homomorphism A → A0 it defines a monogenous (A0 , Aop)-bimodule
Mu with one generator e and relations ea = U a e. The support of Mu is the
graph of U . It is free, of rank 1 on A0 . It will sometimes be convenient to view
homomorphisms via such bimodules.
4.2
Automorphisms
An automorphism of a star product is by definition a formal differential operb which preserves the star-product, the filtration and the principal
ator U ∈ D
b × , gr U = Id , and the corresponding Poisson map is u = Id Σ .
symbols, so U ∈ D
0
Automorphisms are local operators : the germ of U f at a point only depends
on the germ of f at this point 3 so the definition extends immediately :
Definition 13 Let A be a star-algebra : an automorphism U of A is a linear sheaf automorphism which preserves the star-product and the filtration (the
second condition is in fact automatic)
U is a homomorphism A → A and the associated Poisson map is Id Σ .
Since U = 1 mod. operators of negative degree, Log U is well defined:
Log U =
∞
k−1
X
(−1)
1
k
(U − 1)k
it is a derivation, i.e. d(f ∗ g) = df ∗ g + f ∗ dg. U 7→ D = Log U is a bijection
from the set of automorphisms
P 1 n to the set of derivations of degree ≤ −1; the
reciprocal is D 7→ eD =
n! D .
Any other series of (U − 1) also makes sense, in particular the fractional
powers U s , s ∈ C :
X s
s
U = exp(sLog U ) =
(U − 1)k
k
which form a one parameter group of automorphisms.
P
Remark 3 If U is an automorphism of a star product, a = ak , the homogeneous components of U s a are polynomials of s, as the binomial coefficients.
We have tr (Ad A)s P = tr (P ) for all s ∈ Z, hence also for all s ∈ C.
Inner derivations are those of the form ad a : f 7→ [a, f ] = a ∗ f − f ∗ a;
inner automorphisms are those of the form Ad a : f 7→ a ∗ f ∗ a−1 . We have
Adea = exp ad a, LogAd a = ad log a.
3 there may also exist global linear operators preserving products which are not local, but
we will never consider these
29
Let A be a deformation algebra. We denote A× the sheaf of its invertible
elements. Inner automorphisms do not form a sheaf, but they generate a sheaf
of groups, isomorphic to the quotient sheaf A× /ZA where ZA is the center of
A× .
The automorphisms of a semi-classical algebra B preserve ~, by definition.
We will denote Aut ~ (B) the group of such automorphisms.
Since ~ is central we have Ad ~m a = Ad a for all m, P
and any inner au∞ m
tomorphism is of the form Ad a with a of degree 0 (a =
0 ~ am with a0
invertible).
Inner automorphisms Ad a with a of integral degree 6= 0 also exist, but for
these Ad as is not, even locally, an inner automorphism. If U = Ad a is an
inner automorphism, D = Log U is locally an inner derivation D = ad b with
b = Log a, a of degree 0.
4.3
Involutions
If A is a star algebra, the opposite algebra Aop is the same sheaf, with the
symmetric product law (for a star product; B op (f, g) = B(g, f ). It is obviously
also a star algebra, with the opposite Poisson bracket [f, g]op = [g, f ] = −[f, g].
The corresponding cone is Σop , i.e. Σ with the opposite Poisson bracket.
Definition 14 An anti-involution of a star product is a formal differential opb × , such such that
erator J ∈ D
0
(i) J(a ∗ b) = J(b) ∗ J(a)
(ii)gr Ja(ξ) = a(−ξ) (i.e. σm (Ja) = (−1)m σm (a) if a is of degree m).
This definition extends to star algebras: an anti-involution J of A is a sheaf
isomorphism (over the basis X) A → Aop which is locally an anti-involution of
star products.
The corresponding base map is jX = Id X , i.e. J preserves principal symbols
of degree 0. The corresponding cone map is the antipodal map j : Σc → Σop
c
(j(ξ) = −ξ).
X
X
gr J(
fk ) =
(−1)n fk
if fk is of degree k
j reverses the Poisson bracket c. In fact condition (i) implies J[a, b] = [Jb, Ja] =
−[Ja, Jb] so jc = −c. Since c is homogeneous of degree −1 this automatically
implies that j is the antipodal map if c does not vanish. We have made it part
of the definition in all cases; with this definition the identity map is never an
anti-involution.
For an anti-involution on a deformation algebra we require J(~) = −~.
30
Examples
1. On the Moyal model (§3.1) defined by an antisymmetric bidifferential
operator b(∂ξ , ∂η ) (half the PoissonP
bracket), there is a canonical antiinvolution: f (ξ) 7→ Jf = f (−ξ) = (−1)k fk . Indeed the star product
defined by b is f ∗b g = eb(∂ξ ,∂η ) f (ξ)g(η)η=ξ and since b is is homogeneous
of degree −1, its image by the antipodal map ξ 7→ −ξ is −b = b(∂η, ∂ξ ),
which implies J(f ∗b g) = Jg ∗ bJf
2. Likewise for the Moyal model deformation star product (10) with antisymmetric b, there is a canonical involution : a(x, ~) 7→ a(x, −~).
n
3. On the algebra of
operators
there is a canonical involuPdifferential
P on R
α
t
α α
tion P (x, ∂x ) = aα (x)∂ 7→ P = (−1) ∂ aα (x).
Proposition 15 (i) Any two anti-involutions on A are conjugate (globally).
(ii) If A has an anti-automorphism, it also has an anti-involution.
Proof : (i) if J, J 0 are two anti-involutions, U = J 0 J is an automorphism and
J 0 = U J = JU −1 , so J 0 = U 1/2 JU −1/2 .
(ii) If A is an anti-automorphism, A2 is an automorphism which commutes
1
1
with A, so as the automorphism (A2 )− 2 , and J = (A2 )− 2 A is an anti-involution.
4.4
Subprincipal Symbol
n
The subprincipal symbol
of a pseudo-differential operator
P
P P on R with total
1
symbol p(x, ξ) =
pm−k (x, ξ) is sub P = pm−1 − 2i
∂ξj ∂xj pm . Together
with the principal symbol, it determines P up to order m − 2.
b
A symbol map of order 2 on a star algebra is a total symbol P : A → O
such that for a ∈ Ap , b ∈ Aq we have
1
(i) P (a ∗ b) = P (a)P (b) + {P (a), P (b)} mod. terms of degree ≤ p + q − 2
2
(ii) P ([a, b]) = {P (a), P (b)} mod. terms of degree ≤ p + q − 3
This only concerns P mod. operators of degree ≤ −2, which we can forget in
what follows.
If a ∈ Am the leading term of P (a) is the principal symbol σm (a) ∈ O(m).
The next term is the subprincipal symbol sub m (a) ∈ O(m−1) (associated to P ).
Together with the principal, it determines a mod. Am−2 . Subpricipal symbols
are characterized by the fact that for all m they define maps Am → O(m − 1)
(component of a symbol map as above) such that sub m (a) = 0 if a ∈ Am−2 ,
sub m (a) = σm−1 (a) if a ∈ Am−1 . Conditions (i), (ii) above can be rewritten :
(i)bis if a, b are of degree m, m0 , we have
1
sub m+m0 (a∗b) = sub (a)σ(b)+σ(a)sub (b)+ {σ(a), σ(b)} ∈ O(m+m0 −1)
2
31
(ii)bis if a, b are of degree m, m0 , we have
sub m+m0 −2 [a, b] = sub (a), σ(b)} + {σ(a), sub (b) ∈ O(m + m0 − 2)
The subprincipal symbol of pseudo-differential operators mentioned above
satisfies these conditions (in fact it is invariant by diffeomorpfism for differential
operators acting on half desnsities - for which the trnsformation law by χ is
χ∗, 21 P = (Ad
1
chi0 |) 2 χ∗ P )
The Moyal star productPf ∗b g with b with antisymmetric b has a canonical
subprincipal symbol : f =
fm−k ∈ Am 7→ sub f = fm−1 . Indeed the product
P
−k
law is B =
Bk with Bk (f, g) = 2k! (b(∂ξ , ∂η )k (f (ξ)g(η))η=ξ =; the bracket
law is [f, g] = 2b(f, g) + B2 (f, g) − B2(g, f ) + . . . . If b is antisymmetric, the
Poisson bracket is {..} = b and B2 is symmetric, so for f ∈ Ap , g ∈ Aq we have
f ∗ g = f g + 21 {f, g} mod. Ap+q−2 , [f, g] = {f, g} mod. Ap+q−3 .
Proposition 16 If P, Q are two symbol maps of order 2, P − Q is of degree
≤ 0 and its principal part δ = σ−1 (P − Q) is a Poisson vector field, i.e.
δ(f g) = δ(f )g + f δ(g),
δ({f, g} = {δ(f ), g} + {f, δ(g)}
the first equality follows immediately from (i), and the second from (ii).
Remark 4 A map satisfying (i) exists on any star algebra (cf. below ref).
Equivalently any star product is equivalent to a star product of the form f ∗ g =
f g + 21 {f, g} + . . . ). But this is often not sufficient for global computations
and does not deserve the name “subprincipal symbol”. We will see that on
symplectic algebras two subprincipal symbols are always conjugate; for this
condition (i) alone is not enough.
Subprincipal symbols, satisfying (ii), do not exist on all star algebras (even
symplectic); in particular they do not always exist on the canonical Toeplitz
algebra constructed in [27].
4.5
Automorphisms of symplectic deformation algebras
Theorem 17 Derivations and automorphisms of symplectic deformation algebras are locally inner derivations resp. automorphisms.
Since Adea = exp ad a and LogAd a = ad log a, the two statements are equivalent.
If D is a derivation of degree m, its symbol is of the form ~−m d with δ
Poisson vector field on the basis X, i.e. a vector field (δ(f g) = δf g + f δg)
such that δ{f, g} = {δf, g} + {f, δg}. If X is symplectic, this means that the
1-form ω corresponding to δ is closed (dω = 0), so locally there exists a function
φ such that dφ : ω, Hφ = δ; then for any a ∈ A with symbol ~)mφ, D − ad a
32
is of lower degree. By induction we get a (as a convergent formal series) such
that D = ad a.
Thus the sheaf Aut A coincides with the sheaf Int A of automorhisms wich
are locally inner inner automorphism of the form Ad a with a ∈ A×
0 , the sheaf
of invertible elements of degree 0.
×
Lemma
18 The center of A×
(its elements are the formal series
0 is C[[~]]
P∞
k
a
~
with
a
locally
constant,
a
=
6
0)
k
k
0
0
of A is the algebra of formal series ofP
~ alone (of the form
P The center
k
k
a
~
with
a
locally
constant).
Indeed
if
f
=
k
k≥k0 k
k≥k0 ~ fk is central,
and fk is constant for k < m, we have {fm , g} = 0 for all g, which implies that
fm is (locally) constant since the Poisson bracket is symplectic.
Thus we have an exact sequence of sheaves
0 → C[[~]]× → A×
0 → Aut A → 0
(18)
It will be convenient to modify this, using the sheaf Ae×
0 whose sections are
ϕ
pairs ϕ, f with f ∈ A×
0 invertible, ϕ ∈ O(0); e = σ(f ) (logarithms of sections
f ∈ A×
0 ) : we have an exact sequence of sheaves :
0 → C[[~]] → Ae×
0 → Aut A → 0
(19)
Although Ae×
0 is not commutative, it is a soft sheaf (it has“partitions of
unity”, and its cohomology is essentially trivial (cf. below)
4.6
Automorphisms of symplectic algebras
∗
Let Σ be a cone, X its basis. We denote Hhom
(Σ) the De Rham cohomology of
k
homogeneous forms: Hhom
(Σ, C) is the space of closed k-forms with coefficients
b mod. exact forms.
in O
k
Lemma 19 Hhom
(Σ, C) is canonically isomorphic to H k (X, C) ⊕ H k−1 (X, C).
P
P
Proof : if ω = k≤k0 ωk with ω homogenous of degree k, we have dω = dωk ,
so ω is closed resp. exact iff each ωk is so.
P
If ρ is the radial vector, we have Lρ ω =
kωk . So if ω is
P(dIρ 1+ Iρ d)ω =
closed, it is cohomologous to ω0 = ω − dIρ k6=0 k ωk
Let ω ∈ Ωk be homogeneous of degree 0: ω = µ + dr
r ν where µ, ν are pull
backs of forms of degree k rep. k − 1 on X (r denotes a “vertical” coordinate,
i.e. a smooth positive function homogeneous of degree 1). Then dω = dµ− dr
r dν
so ω is closed, resp. exact iff both µ and ν are, hence the lemma.
We now turn to automorphisms and derivations of symplectic algebras. Let
A be a symplectic star algebra on Σ, with symplectic Poisson bracket. We
33
denote A×
0 the sheaf of invertible elements of degree 0 (i.e. sections a of degree
×
0 such that σ0 (a) never vanishes). We denote A×
− ⊂ A0 the sheaf of elements
with symbol 1 (deg (a − 1) < 0).
If D is a derivation of degree m of A, its symbol (leading term) δ = σm (D)
is a Poisson derivation δ of degree m, i.e. a vector field such that [c, δ] = 0.
These form a sheaf (on the basis X) isomorphic to the sheaf of closed 1-forms
homogeneous of degree m + 1.
If U is an automorphism, we define its symbol as the symbol of Log U :
σ(U ) ∈ ω0
Proposition 20 Let A be a symplectic star algebra. (i) A derivation of degree
< −1 is an inner derivation. A automorhism U such that U − 1 is of degree
< −1 is an inner automorphism : there exists a unique a ∈ A of degree ≤ −1
such that U = ad a.
(ii) Any automorphism is locally of the form U = (Ad a)s Ad P with P invertible of degree 0, a a fixed elliptic element of degree 1, s a constant, Any
automorphism U such that U − Id is of degree < −1 is an inner automorphism.
(iii) We have an exact sequence of sheaves
0 → A×
− → Aut A → ω0
(iv) Any section of ωm+1 is the symbol of a global derivation. Any section
of ω0 is the symbol of an automorphism.
Let δ be a Poisson derivation homogeneous of degree m. Then the corresponding 1-form ω is closed, homogeneous of degree 6= 0 hence exact: ω = dφ
with φ homogeneous of degree m + 1, and δ = Hφ . Then δ is the leading term
of ad a if σ(a) = φ.
Thus if D is a derivation of degree m < −1 we can construct by successive
approximations a ∈ A such that D = ad a, and if U − 1 is of degree < −1, then
so is D = Log U . hence (i). Note that if A is symplectic, its center is reduced
to the constants C, so the element a ∈ A−1 such that U = Ad a is unique.
If m = −1, the closed 1-form ω of degree 0 corresponding to δ is of the form
dµ + s dr
r with s a constant, µ closed of degree 0. Then locally µ has a primitive
φ and δ is the symbol of Ad a(Ad b)s if σ(a) = eφ , σ(b) = r (b is of degree 1,
(Ad b)s is well defined). Hence (ii) and (iii).
The last assertion follows from the fact that the sheaf A×
− is soft. Here
is a softened proof: the closed form µ above has a multi-valued primitive φ
(a function on the simply connected cover X̃ of X) whose branches differ by
constants. We may suppose that A defined by a star product. Let a be the
section with total symbol φ: although a is only defined on X̃, ad a is well
defined, and δ is the leading term of ad a + sLog Ad b.
If U is an automorphism, its symbol is the closed 1-form ω homogeneous of
degree 0 corresponding to δ = σ−1 (Log U ). The exponent of U is the coefficient
s = ρyω ∈ C of the vertical part s dr
r (if X is not connected, it is a locally
constant function)number s in 20
34
4.7
Automorphisms preserving a subprincipal symbol or
an involution
Proposition 21 Let A be a symplectic sub-algebra
1) An automorphism (resp derivation) preserving a subprincipal symbol is an
inner automorphism (resp. inner derivation).
2) An automorphism U which preserves an anti-involution J is of the form Ad P
−1
with P ∈ A×
; in particular its symbol is 0. A derivation preserving
− , JP = P
J is an inner deerivation ad a with Ja = −a.
3) Two subprincipal symbols are conjugate.
Proof 1) If U preserves a subprincipal symbol, then obviously U − Id is of
degree < −1 so by proposition 20 U is an inner automorphism, of the form
Ad (1 + P ) with deg P ≤ −1.
2) Let J. If U is an automorphism preserving J, it is an inner automorphism
of the form Ad a with σ(a) = 1. The condition U = JU J i.e. Ad a = Ad Ja−1
implies Ja = ca−1 with c a constant, so c = 1 since σ(a) = 1.
3) If sub 0 = sub + δ are two subprincipal symbols, δ is a Poisson vector field
(of degree −1) so it is the leading term of of a derivation D, and sub 0 = sub ◦eD .
Note that 1) and 2) are incorrect for deformation algebras. 3) holds for
deformation algebras, because the Poisson derivation, difference between two
subprincipal symbols, is still the leadig term of a star derivation os degree −1
(not necessarily inner); but is less useful because as we will see later (ref) two
symplectic algebras possessing an involution or a subprincipal symbol are isomorphic, but this is not the case for deformation algebras. We will see later
that the base point algebra from Fedosov’s construction in involutive.
4.8
4.8.1
Fourier integral operators
As functional operators.
1) Fourier integral operators, and more generally Fourier integral distributions,
were precisely introduced and described by L. Hörmander [82, 46], M.Sato,
T.Kawai, M. Kashiwara [108, 86]. A first, somewhat less precise definition,
was proposed by V.I. Maslov [99], and J. Egorov [48] showed how they forcibly
introduce symplectic geometry in the picture.
In the main case, a Fourier F integral operator is associated to a homogeneous symplectic map φ : T ∗ X → T ∗ Y (where X, Y are manifolds - the zero
sections should be removed). They act on distributions and preserve microsupports (wavefront sets): if f is a distribution (or a microfunction) on X, F f is a
distribution on Y , and SS(F f ) ⊂ φ(SS(f )).
There is a notion of ”elliptic” Fourier integral operator (the ”principal symbol” must be invertible, as for pseudodifferential operators). If F is elliptic
is has a parametrix, i.e. a Fourier integral operator G, associated to φ−1 , inverse to F mod smoothing operators (i.e. F G − Id and GF − Id have smooth
Schwartz-kernels).
35
When this is the case, if P is a pseudodifferential operator, its pushforward
Q = F P G is also a pseudodifferential operator, with principal symbol the pushforward σ(Q) = σ(P ) ◦ φ−1 (Egorov’s theorem).
2) Because pseudodifferential operators and Fourier integral operators preserve microsupports, mod smoothing operators they are local on the cotangent
bundles (deprived of zero sections). This makes it possible to localize PDE
problems on the cotangent bundle.
Analogues or generalizations of Fourier integral transformations mod smoothing operators (or transformations defined by (EX − EY )-bimodules) are used
to describe isomorphisms between symplectic algebras.
36
5
Classification.
5.1
Hochschild cohomology
bk we
We will use repeatedly some elements of Hochschild cohomology.If P ∈ D
b
define δP ∈ Dk+1 :
δP (f0 , . . . , fk ) = f0 P (f1 , . . . , fk − P (f0 f1 , f2 , . . . fk ) . . .
+ (−1)k−1 P (f0 , . . . ; fk−1 fk ) + (−1)k P (f0 , .., fk )fk+1
(20)
e.g. if B is a star product and P ∈ D1 is of negative degree, the top order term
P
−P
of eP
f, e−P g) − B(f, g) = P (f g) − −P (f )gf P (g)+
∗ B − B is −δP , i.e. e B(e
terms of degree < −N
R ∈ D2 (k = 1) is of negative degree, δR = 0 is the leading (linear) term in
the condition ensuring that B + R is associative).
Lb
Theorem 22 1) δ is a differential on
Dk i.e. δ 2 = 0. We have δP = 0 if
P is of order 1, in particular if it is the antisymmetric k-differential operator
defined by a multi-vector.
V
Pb
2) The canonical injection T X →
Dk , taking a k-multi-vector to the
antisymmetric k-differential operator it defines, is a quasi-isomorphism, i.e. it
induces isomorphisms on the cohomology ker δ/Im δ.
proof: for the general case we refer to the literature [40]. We will only need
the cases k = 1, 2, for which we give a short, although not very illuminating
proof
If k = 1 δP = 0 i.e. P (f, g) = f P (g) − P (f g) + P (f )g = 0 for all f, g means
tat P is a derivation (vector field).
Let k = 2. If P is of total order N , and PN (ξ, η) is its symbol, a homogeneous
polynomial of degree N in ξ, η with smooth coefficients, δP = 0 implies
PN (η, ζ) − PN (ξ + η, ζ) − +PN (ξ, η + ζ) − −PN (ξ, η) = 0
(21)
Taking three successive derivatives with respect to ξ, ζ, η gives, for η = 0
(∂ξi − ∂ζi )∂ξj ∂ζk = 0
(22)
We may dismiss the case N = 1 (P = C = δC, C a constant), and N = 1
(δP = 0 ⇒ P = 0).
If N = 2, PN is of the form PN (ξ, η) = Aξ.ξ + Bξ.η + Cη.η with A, C
symmetric matrices. (22) gives −Aξ.(ξ +2η)+Cζ.(ζ +2η) = 0 so A = C = 0: P2
is bilinear. If P2 is bilinear symmetric, we have P2 = 21 δQ with Q(ξ) = P (ξ, ξ).
Note that δQ is always symmetric, so a P2 bilinear antisymmetric is not a
coboundary.
P (22) implies that the pij = ∂ξ ∂etaj p are polynomials of ξ + η; the 1-forms
i Pij dξj are closed, so the primitives are of the form ∂ξj p = pj (ξ + η) − qj (η),
i.e. dη p = ω(ξ + η) − ω 0 (η).
37
Then we have dη ω(ξ + η) = dη ω(η) which implies that both ω and ω 0 have
constant coefficients; if N > 2 they vanish. Integrating one last time with
respect to η shows that PN is of the form a(η) − b(ξ + η) + c(ξ), and (21)
requires a = b = c i.e. PN = δa.
By induction on N we get theorem 22 (for k=2).
5.2
Non commutative cohomology
In this section we recall the elementary resuls of noncommutative cohomology
that we will use (for more information see [69]). Let X a manifold (or a paracompact topological space) and G a sheaf of groups on X (e.g. the sheaf of
continuous functions with values in a topological group G or the sheaf of sections of a group bundle on X). We denote H 0 (X, G) = G(X, G) the set of global
sections of G over X : this is a group.
We denote H 1 (X, G) the set of equivalence classes of cocycles
uij ∈ G(Xi ∩ Xj , G) such that uij ujk = uik
S
associated to open coverings X = Xi ; two cocycles are equivalent if, after
a suitable refinement of the covering, we have uij = ui u0ij u−1
for some family
j
ui ∈ G(Xi , G).
H 1 (X, G) classifies the set of isomorphy classes of G principal homogeneous
right G torsors, i.e. sheaves P on X, equipped with a right action of G, locally
isomorphic to G considered as a right G-sheaf
S : if P is a torsor, it is locally trivial,
i.e. there exists an open covering X = Xj and isomorphisms Uj : P → G
over Xj . The transition map Uij is an isomorphism of G → G over Xi ∩ Xj ,
commuting with right multiplications: Uij f = uij .f where uij ∈ G(Xi ∩ Xj )
is the section Uij (1). The family (uij is a cocycle (with coefficients in G) , i.e.
uik = uij ujk over Xi ∩ Xj ∩ Xk . Changing the local trivialisations (Vi f =
vi Ui f with vi ∈ G(Xi ) leads to the new equivalent cocycle vij = Vi uij vj−1 , and
H 1 (X, G) is the inductive limit of the sets of equivalent classes of cocycles for
finer and finer coverings.
A torsor is also used to twist left G sheaves : if E is such a sheaf, i.e. there
is a sheaf-group action G × E → E (for each open set U a group action G(U ) ×
E(U ) → E(U ) compatible with the sheaf restriction maps), the twisted sheaf is
EP = P ×G E. In particular the sheaf of automorphisms of P is isomorphic to
the twisted group GP = P ×Ad G G which acts on the left on EP .
Proposition 23 Let
u
v
0 → A→
B→
C→0
be an exact sequence of sheaves of groups on X, with A normal in B. Then
there is a exact cohomology sequence ;
0 → H 0 (X, A) → H 0 (X, B) → H 0 (X, C) → H 1 (X, A) →
→ H 1 (X, B) → H 1 (X, C)
38
(23)
Th second and fith arrows are defined bt u, the third and sixth by v, the middle
arrow (coboundary) is defined below. The sequence is exact in the following
sense :
i) it is exact at the first three places (the H 0 are groups); the H 1 are pointed
sets. and (obviously) the range of one map maps to the next basepoint.
ii) The group H 0 (X, C) acts on the set H 1 (X, A), and its orbits are the fibers
of the map H 1 (X, A) → H 1 (X, B). The action is given by c · (aij ) = (bi aij b−1
j )
if c is a global section of C, and bi ∈ B(Xi ) a lifting of c to B over a fine enough
covering Xi (we have bi aij bj−1 ∈ A since bi = bj mod. A and A is normal in B).
Two cocycle (aij , (a0ij ) have the same image in H 1 (X, B) iff a0ij = bi aij b−1
for
j
some family bj ∈ B(Xj ) (if the covering (Xj ) is fine enough); then the cj = v(bj )
patch together into a section c ∈ C(X).
iii) If β ∈ H 1 (X, B) maps to the base point, it can be defined by a cocycle
bij which maps to the trivial cocycle cij = 1, i.e. bij ∈ A
More generally any element β ∈ H 1 (X, B) defines a torsor Pβ and a twisted
sheaf of groups Bβ = Isom B (Pβ , Pβ = Pβ ×Ad B B, a normal subsheaf of groups
Aβ = Pβ ×Ad B A ⊂ Bβ , and a quotient Cβ = Bβ /Aβ = Pβ ×Ad B C. One sees
immedialely that two elements β, β 0 ∈ H 1 (X, C) iff the Bβ -torsor Isom (Pβ 0 , Pβ
maps to the base-point of H 2 (X, Cβ (the trivial torsor). The fiber of the map
H 1 (X, B) → H 1 (X, C) is the image of H 1 (X, Aβ ) in H 1 (X, C).
A sheaf of groups G on X is soft if any germ of section over a closed set
F ⊂ X is the germ of a global section, i.e. for any section s ∈ G(U ), with U
an open neighborhood of F , there exists a global section s0 ∈ G(X) such that
s = s0 in some neighborhood V , F ⊂ V ⊂ U ). G is flabby if any section on
any open set extends to the whole of X. Soft is a weaker condition than being
flabby, but occurs often in real analysis - e.g. the sheaf of continuous or smooth
sections of a vector bundle is soft, because there are partitions of 1, but it is not
flabby unless X is discrete.
Proposition 24 If G is soft, then H 1 (X, G) = 1
In other words if G is soft, any G-torsor P has a global section, all 2-cocycles
are equivalent to the trivial cocycle uij = 1. In fact a locally soft sheaf as P
is soft : indeed let (Yi )i∈I be a locally finite closed coverig on X such that for
each i, P is trivial, hence soft, in some open neighborhood of Yi . Then is s is a
germ of section along
S a closed set F , it extends as a germ to F ∪ Yi ; since (Yi )
is locally finite, F ∪ j∈J Yj is closed for any subset J ∈ I, so by Zorn’s lemma
s extends to the whole of X.
In this paper the noncommutative cohomology sequence stops there, and
we will not use higher cohomology objects H j , j ≥ 2 whose definition is more
elaborate. Exact sequences concerning non commutative cohomology as above
were introduced by J. Frenkel [65, 66]. See also [69].
39
However, the sheaves we will work with, such G = Aut A, A, Ae×
0 have a
completed descending filtration for which gr G is commutative, and possibly
soft. This will make it possible to push the exact sequence (23) a little further.
If A, B, C are commutative, the higher cohomology groups H k , k ≥ 0 are
well defined commutative groups: e.g. H k (X, G) is the inductive limit for
finer and finer coverings of the set of cocycle (k-cochains f = (fi0 ,...,ik ∈
G(Xi0 ,...,ik ) such that the (k + 1)-cochain df vanishes, with df i0 , . . . , ik+1 =
P
(−1)j f i0 , . . . , ibj , . . . , ik+1 , mod. coboundaries dg). For more details we refer
to the literature f, e.g. [11, 12, 69, 70].
We will use the long cohomology exact sequence in that case, up to order 2:
if 0 → A → B → C → 0 is an exact sequence of commutative group sheaves,
there is a long exact sequence of homology groups:
0 → H 0 (X, A) → H 0 (X, B) → H 0 (X, C) → H 1 (X, A) → H 1 (X, B) → (24)
→H 1 (X, C) → H 2 (X, A) → H 2 (X, B) → H 2 (X, C) → H 3 (X, A) → . . .
(exact means that the range of on arrow is the kernel of the next).
5.3
Symplectic algebras are locally isomorphic
Theorem 25 Let Σ be a symplectic cone with Poisson bracket c. Two star
algebras over Σ (with this same Poisson bracket c are locally isomorphic. The
same holds for deformation algebras over a symplectic manifold.
We will use the following result (which holds for any star algebra):
Lemma 26 Any star product is equivalent to a star product of the form f ∗ g =
f g + 21 {f, g} + . . . 4
Proof: let Bf, g) = f g + B1 (f, g) + . . . be a star product. Associativity writes
B(B ⊗ 1) − B(1 ⊗ B); there is no term of degree 0, an the term of degree −1 is
B0 (B1 (f, g),h) + B1 (B0 (f, g), h) − B0 (f, B1 (g, h)) − B1 (f, B0 (g, h)) =
B1 (f, g)h + B1 (f g, h) − f B1 (g, h) − B1 (f, gh) = 0
Thus δB1 = 0: B1 is a Hochschild cocycle, so there exists a differential operator
P homogeneous of degree −1 such that B1 −δP = C is a bivector (antisymmetric
P
bidifferential operator of order (1, 1)). Recall that eP
∗ B is defined by e∗ B(f, g) =
P p
p
P
0
0
e (e −P f, e −P g): we get e∗ B = B0 + B 1 + . . . with B 1 = B1 − δP = C.
Since the Poisson bracket has not changed we have necessarily C(f, g) = 12 {f, g}.
Proof of the theorem : Suppose we have two star products B, B 0 with the
same symplectic Poisson bracket c. By Lemma26 we may suppose that both
4 Much of the literature requires B = 1 {..} in the definition star products. We did not
1
2
do it here because it is not true for very natural laws such as the normal composition law of
differential operators.
40
begin by f ∗ g = f g + 12 {f, g}. . . . , so B 0 = B + R with R of degree N ≤ −2
(R ∈ D2 ). We will show that on any disk-neighborhood V of a point the two
products are equivalent. (the exact condition is H 2 (V, C) = 0 for deformation
algebras, H(2 V, C) + H 1 (V, C) = 0 for symplectic algebras).
We will use repeatedly (inductivly) the following result;
Lemma 27 Let B be a star product, and P a formal differential operator of
degree −N . Then
(i) The leading term of ep∗ B − B (of degree −N ) is −δPN .
(ii) If P is a vector field, the leadig term for the error in the bracket law,
(eP − 1)(B −tB), of degree −N − 1, is −[C, P ] (the Nijenhuis-Schouten
bracket - C denotes the Poisson bracket of B, of deree −1).
Proof: we have eP (f, g) = B(f, g)+P B(f, g)−B(P f, g)−B(f, P g)+ lower order
terms. The leading term in the error, of degree −N , is P (f g) − P (f )g − f P (g)
as announced.
This vanishes if P is a vector field. In this case eP behaves as a diffeomorphism; we have for the leading term of commutators we get eP ([e−P f, e−P g]) −
[f, g] = PN ({[f, g}) − {PN f, g} − {f, PN g} i.e. the error is [PN , C] = −[C, PN ].
We will kill the leading part RN of the error R in two steps.
We first kill the antisymmetric part. We put β(f, g) = B(f, g) − B(g, f )),
ρ(f, g) = R(f, g) − R(g, f ). ρ is an antisymmetric cocycle, so is is a bi-vector
P
(bidifferential operator of order
P 1).The Jacobi identity for brackets is 3 (β +
ρ)(f ; (β + ρ)(g, h) = 0 where 3 is the sum over cyclic permutations of f, g, h.
The leading part for the error is then
X
{f, ρN (g, h)} + ρN (f, {g, h} = 0
3
i.e. [C, ρN ] = 0.
Now if the Poisson bracket C is symplectic, it is isomorphic to the exterior
differential d. So since ρN is closed ([C, ρn ] = 0) there exists a vector field PN −1
homogeneous of degree 1 − N < 0 such that ρN = [C, PN −1 ].
P
If we set B” = e∗ N −1 B 0 −B the leading part R”N of B”−B is now symmetric.
By theorem22 it is of the form δP for some formal differential operator of degree
−N , Q
and eP
∗ B” − B is of degree ≤ −N − 1. By induction, we get an operator
0
P = Pk with deg Pk → −∞ such that eP
∗ B = B.
Remark If the Poisson bracket is real (or pure imaginary) there exist canonical local homogeneous coordinates, i.e. homogeneous functions xj - of degree 0
or 1, or all of degree 21 , and cij = {xi , xj } is constant. Then it is easy to construct
by successive approximations elements Xj such that σ(Xj ) = xj , [Xi , Xj ] = cij
(or ~ci j in the deformatioon case). One recovers theorem5.3 again by sucessive
approximation, or using the functional calculus of section2.4.
41
5.4
Classification
Theorem 5.3 helps for the classifiction of symplectic algebras (or deformation
algebras): let Σ be a symplectic cone with basis X (resp. X a symplectic
manifold). We denote Alg Σ or Alg X the set of equivalence classes of star
algebras (resp. deformation algebras) with the given Poisson bracket.
If A, B ∈ Alg X they are A, B locally isomorphic, so If Xi is a sufficiently fine
covering of X there exist isomorphisms Ui : B → A over Xi defining a 2-cocycle
Uij = Ui Uj−1 with coefficients in Aut A. We will denote δ(B, A) the element
of H 1 (X, Aut A) it defines.(the corresponding torsor is the sheaf Isom (B, A) of
local isomorphisms from B to A; conversely the algebra defined by a torsor P is
the sheaf of A-automorphisms of P ). Clearly two algebras B, B 0 are isomorphic
iff δ(B 0 , B 0 ) = 0, and this is equivalent to δ(B 0 , A) = δ(B, A).
Theorem 28 If A is a symplectic algebra, the map B 7→ δ(B, A) is a bijection
Alg X → H 1 (X, Aut A)
Note that this map depends on A. We will see below that H 1 (X, Aut A) is
a commutative group, which acts freely on Alg X , and the element δ(B 0 , B) is
really the difference of the classes of B and A.
The theorem does not ensure the existence of an algebra A; this will be
proved later, together with Fedosov’s very elegant description of the classification.
Let us push our analysis a little further. We make two separate cases.
5.5
Classification of symplectic algebras
Proposition 29 Let A be a star algebra on a symplectic cone Σ: then there is
2
a canonical bijection ch A : Alg Σ = H 1 (X, Aut A) → Hhom
(Σ).
Proof : we have seen (proposition20 that there is an exat sequence
0 → A×
− → Aut A → ω0 → 0,
where A×
− is the sheaf of invertible elements with principal symbol 1, ω0 the sheaf
2
of closed forms homogeneous of degree 0. We have H 1 (X, ω0 ) = Hhom
(Σ) (by
2
1
definition) and this in turn is isomorphic to ∼ H (X, C) ⊕ H (X, C) (proposion
Hhom). The map H 1 (X, Aut A) → H 1 (X, ω0 ) comes from the cohomology
exact sequence (prop. 23; it is a bijection because A×
− is a soft sheaf.
Here is a more detailed proof.
The map is injective (prop. 23); in fact if two algebras B, B 0 have the cocycles
the same image in H 1 (ω0 ) the image of δ(B 0 , B) vanishes, i.e. B 0 can be defined
by a cocyle with coefficients in B0× . Now B0× is soft, so B and B 0 are equivalent.
42
The map is onto: let αij be a 2- cocycle with coefficients in ω0 . This can be
lifted to a family Uij of automorphisms. Set Uijk = Uij Ujk Uki : Uijk − 1 is of
degree < −1 since αij (its symbol is αij + αjk + αki = 0). Suppose it is of degree
−N − 1 < −1; then the symbols αijk of Uijk are closed 1-forms homogeneous
−1
−1
of degree −N , and form a 2 cocycle (we have Uijk Uikl Uijl
Uij = Ujkl Uij
=1
so αijk − αijl + αikl − αjkl = 0). Now we have seen that homogeneous forms of
degree 6= 0 have no cohomology, so αijk is a coboundary: αijk = βij + βjk − βki .
If uij is a family of automorphisms with symbol βij (uij = Ad (1 + aij ) where
0
= uij Uij patch togetherto
aij is of degree −N and dσ−N (aij ) = βij , the Uij
0
0 −1
0
order −N − 2 (i.e. Uij Ujk Uik − 1 is of degree ≤ −N − 2). By succesive
approximations we finally get a cocycle Uij with symbol αij .
Complement
Proposition 30 Two syplectic star algebras possessing a subprincipal symbol
(resp. an anti-involution) are isomorphic.
If A, B are two symplectic star algebras possessing subprincipal symbols, it
follows from proposition21 that they are locally isomorphic, and that the sheaf
Isom sub (A, B) of local isomorphisms preserving sub is a torsor under A×
− . This
is soft, because A×
is
soft,
so
it
has
a
global
section,
i.e.there
exists
a
global
−
isomorphism A → B preserving sub .
This applies also to anti-involutive algebras : such algebras possess a sub
so they are isomorphic, and two anti-involutions are always globally conjugate
(proposition15). (note that the sheaf Aut J A of local autoisomorphisms preserving an antiinvolution J is soft : for such an automorphism U there is a unique
a ∈ A−1 such that U = Ad ea , Ja = −a, and these obviously for a soft sheaf).
5.6
Classification of symplectic deformation algebras
Proposition 31 Let A be a deformation algebra on a symplectic manifold X.
There is a canonical bijection ch A : Alg X → H 2 (X, C[[~]]).
Proof : we have seen (theorem17) that any automorphism of A is locally inner
(Aut A = Int A), and there is an exact sequence (19)
0 → C[[~]] → Ae×
0 → Aut A → 0
×
where Ae×
0 is the sheaf whose sections are pairs ϕ, f with f ∈ A0 invertible,
ϕ
ϕ ∈ O(0); e = σ(f ) (“logarithms” of invertible elements of A0 ).
Because C[[~]] is commutative, the “cohomology exact sequence” (23) extends one step further, and the resulting map ch : Alg X → H 2 (X, C[[~]]) is a
bijection because Ae×
0 is a soft sheaf.
Since we did not define the homology objects of order ≥ 2, here is a more
detailed proof.
43
First if αij = Ad aij is a cocyle with coefficients in Aut A, the aij ∈ Ae×
0 patch
together mod. “constants”: aij ajk = ecijk aik . The family (cijk ) is a cocycle with
coefficients in C[[~]]. This defines the map ch : H 1 (X, Aut A) → H 2 (X, C[[~]]).
The map ch is injective : If two cocycles Ad (α = (aij ), Ad (α0 = (a0ij ))
have equivalent images cijk , c0ijk , the difference c0ijk − cijk is a coboundary :
c0ijk − cijk = cij + cjk − cik for some family cij (for a sufficiently fine covering). Replacing ; then, with αij = (cij , ecij ), we have (αij aij ) is a cocycle with
coefficients in Ae×
0.
Remark 5 Let A be a symplectic deformation algebra. Aut ~ A is the sheaf of
automorpisms. It is useful to compare it to the sheaf Aut A af all star algebra
automorphisms. If U is such an automorphism (not necessarily preserving ~),
it still preserves the center C[[~]], and its restriction to C[[~]] is tangent to the
identity : we have U (~) = ~ + O(~2 ), because U preserves symbols. We denote
Aut 0 (C[[~]]) the group of such automorphisms. Any u ∈ Aut 0 C[[~]] is of the
form eδ where δ = Log u is a derivation of the form δ = φ(~)~∂~ .
In fact there is an exact sequence of sheaves
0 → Aut ~ B → Aut A → Aut 0 C[[~]] → 0
(25)
i.e. any automorphism u ∈ Aut 0 C[[~]] (such that u(~) = ~+O(~2 ) can be lifted,
at least locally. Indeed u∗ A is another symplectic deformation algebra and it
is locally isomorphic to A (we will see below (section6) that if the symplectic
form of X is not exact, the push-forward u∗ A is never globally isomorphic to A
if u 6= Id ).
Similarily we have there is an exact sequence for derivations of degree 0 :
0 → Der ~ A → Der A → Der C[[~]] .
5.7
Algebras of pseudo-differential type
1. Let X be a cone. A pseudodifferential operator
on X (or operator of pseudoP
differential type)is a formal operator P =
Pm−k where the componant Pm−k
homogeneous of degree m − k is of order ≤ k.
In a local P
coordinate system x1 , . . . xn homogeneous of degree 1, P is of ψD
type iff P =
aα (x)∂xα where the aα are all symbols of degree ≤ m (aα ∂ α is
b
then of degree ≤ m − |α| and the sum converges in D).
The total symbol of P in this coordinate system is
X
p(x, ξ) =
aα (x)ξ α = e−x.ξ P (ex.ξ )
b The principal symbol
This is a formal power series in ξ with coefficients in O.
is
X
σm P = pm (x, ξ) =
σm (aα )ξα
44
bX (a subPseudo-differential operators obviously form a subalgebra EX ⊂ D
sheaf).The total symbol of a product is given by Leibniz’ rule; in particular
σm+m0 (P ◦ Q) = σm (P )σm (Q).
The total symbol of P depends on the choice of a coordinate system (not
the fact it is of pseudo-differentiel type); the principal symbol does not, it is a
smooth function on T ∗ X, or rather a jet of infinite order along the zero section
ξ = 0, homogeneous of degree m if we define ξ as homogeneous of degree 0.
EX is a star algebra (the jet of infinite order of -) on the cone Σ = T ∗ X where
homotheties are given by λ(x, ξ) = (λx, ξ)(= λhλ∗ (x, ξ)) (in the last notation
hλ is the homothety x 7→ λx, hλ∗ its extension (pushforwrd) to covectors)
2. Pseudo-differential star products
b is of
Let Σ be a cone. We will say that a formal differential operator P ∈ D
pseudo-differential
type (or that it is a pseudo-differential operator) of degree
P
m if P =
Pm−k with Pm−k homogeneous
degree m − k and of order ≤ k.
P of
The deformations translation is P =
~k−m Pk with Pk of order ≤ k.
These operators obviously form an algebra EΣ (a sheaf).
If Σ = Rn − {0} with all x-variables homogeneous of degree 1, this means
that
X
bm , a symbol of degree ≤ m
P =
aα (x)∂ α
with aα ∈ O
P
The total symbol of P is
aα (x)ξ α , which is a jet of infinite order (formal
Taylor series) along ξ = 0 in T ∗ Σ; the principal symbol is the leading part,
homogeneous of degree m (counting that xi is of degree 1, ξi of degree 0).
The product is again given by the Leibniz rule, hence the name. (This is
locally isomorphic to the algebra of jets of standard pseudo-differential operators
on Rn along the x = 0 fiber, except we have exchanged the x and ξ variables.)
Definition 32 We will say that a star product B is of pseudo-differential type
if for any symbol a, (of degree m) the left mutliplication operator La : b 7→ a ∗ b
is of pseudo-differential type (of degree m).
Then the map a 7→ La is an embedding A → E. Recall that such an embedding
has a (formal) geometric support u; this is a (homogeneous) Poisson map T ∗ Σ →
Σ (or rather Tc∗ Σ → Σc ).
Theorem 33 1) Any star product is equivalent to a pseudo-differential star
product. Equivalently any star algebra on Σ can be embedded in EΣ
2) Two such embeddings are conjugate by an isomorphism. More precisely
two embeddings with the same geometric support are conjugate trough a P ∈ E,
and this is unique if we prescribe P (1) = 1.
Before we prove the theorem, we need some geometric preparation.
Lemma? If X is a symplectic manifold, F a foliation, we denote OF the sheaf
of smooth functions constant on the leaves of F. F is a Poisson foliation if OF is
a Poisson algebra (i.e. locally {f, g} is constant on te leaves if f and g are. Then
there is an orthogonal foliation F ⊥ : its tangent distribution is T F ⊥ , spanned
45
by the Hamiltonian fields Hf , f ∈ F (this satisfies the Frobenius integrability
condition since [Hh , Hg ] = H{f,g} and {f, g} ∈ F. OF ⊥ is the commutator of
OF , i.e. the sheaf of functions φ such that (locally) {f, φ} = 0 for all f ∈ OF .
The level sets of a map u as above define a Poisson foliation (they are
smooth because u is a fibration). The orthogonal foliation is in fact a fibration
(the tangent spaces of fibers of u are linear complements of T X ⊂ T (T ∗ X) so
their orthogonals are linear complements of T (T ∗ X)⊥ = T (T ∗ X). It defines a
Poisson map u⊥ : T ∗ X → X
Lemma 34 Let X be a Poisson manifold. Then there exists a (formal) Poisson
map u : T ∗ X → X along the zero section (T ∗ X is equipped with its usual
symplectic structure).
If u1 , u2 are two such maps, there is a unique isomorphism u of T ∗ X such
that u2 = u1 ◦ u, u|X = Id
Proof 1) by successive approximations
or: outflow of the Hamiltonian of a suitable “phase function”
2) there is a unique map u such that u|X = Id which takes the Hamiltonian
of f ◦ u1 (and its flow) to that of f ◦ u2 for f a smooth function on X (the flow
of the Hamiltonians lie in the orthogonal foliations); this map u is symplectic.
Note
Proof of theorem33 Let A be a star algebra with Poisson product C. We will
say that a “differential operator” U : A → E (locally of the form f 7→ P (f ◦ u)
is a homomorphism of order N if R(f, g) = U f U g − U (f ∗ g) is of degree
≤ deg f + deg g − N (R = B(U ⊗ U ) − U B of degree ≤ −N ).
R can be viewed as the restriction of a formal differential on E to OF (the
functions constant along the fibers of u).
For N = 1 this just means U f = f ◦ u+ terms of lower degree. Lemma 34
shows that we can choose u so that it preserves Poisson brackets, so an there
exists an embedding of order N = 2, because both the product law of A and
that of E are equivalent to a law if the form f ∗ g = f g + 21 {f, g}).
Associativity of both products gives
U f U gU h − U (f ∗ g ∗ h) = U f R(g, h) + R(f, g ∗ h) = R(f, g)U h + R(f ∗ g, h
for the leading term (homogeneous of degree −N this gives
f RN (g, h) − RN (f g, h) + RN (f, gh) − RN (f, g)h = 0
(26)
i.e. RN is a Hochschild cocycle.
The commutator error is Γ(f, g) = R(f, g) − R(g, f ). For its leading term
we get
X
ΓN (f, {g, h} + {f, ΓN (g, h)} = 0
(27)
3
(
P
3
denotes the sum over cyclic permutations of f, g, h).
46
Suppose U is an embedding of order N ≥ 2. We construct an embedding U 0
of order N + 1, such that U 0 − U is of degree ≤ −N + 1 in two steps.
We first get rid of the antisymmetric part of R : (26) shows that ΓN is an
antisymmetric Hochschild cocyle, i.e. a bivector, and (27) means that we have
[c, RN ] = 0, with c the Poisson bracket of T ∗ Σ). Since the formal leaves of
u are contractible, there exists a unique vector field p homogeneous of degree
−N + 1 (vanishing on X) such that ΓN = [c, p]. Then U ” = (1 + p)U is another
embedding of order N , and for this the error RN ” is symmetric.
Suppose finally that RN is symmetric : then by 22 there exists a differential
operator P homogeneous of degree −N (not necessarily of pseudo-differential
type) P such that ΓN = δP i.e. ΓN (f, g) = f P (g − −P (f g) + P (f )g. Then
U 0 = (1 + P )U ” is an embedding of order N + 1 (R0 is of degree ≤ −N − 1)/
Thus by succesive approximations we get an embedding U : A → E.
Uniqueness. Let U1 , U2 be two embeddings, with geometrical supports u1 , u2 .
By theorem 33 there exists a (unique) (formal) symplectic map u taking u1 to
u2 . Now there exists (4.8) a invertible “Fourier integral operator” F with canonical relation the graph if u, so F U2 F −1 has the same geometric support as U1
(Note: formally Fourier integral operators as above are of the form eP where
P ∈ E is of degree 1, and the symbol σ1 (P ) vanishes of order 2 on Σ (i.e. for
ξ = 0, so that the flow of HP fixes Σ)).
Suppose now that U1 and U2 have the same geometric support u, and that
U1 − U2 is of degree −N (N ≥ 1). Then the error term r =. Ite leading term
rN os a Poisson vector field, and there exists a unique φ homogeneous of degree
−N − 1, vanishing for ξ = 9, such that rN = Hφ . Then p ∈ E, σ−N +1 (p) = φ,
ep U1 e−p − U2 is of degree −N − 1. By successive approximations we get p (of
degree ≤ 0 such that U2 = ep U1 e−p .
47
6
Fedosov Connections.
In [60], Fedosov gave a very elegant and geometric description for the existence
and classification of semi-classical algebras : a deformation algebra B over a
real symplectic manifold X can be embedded as the subalgebra of flat sections
c over
of a suitable “Fedosov connection” ad ∇ in a “universal” star algebra W
the tangent bundle T X (the definitions are recalled below); the curvature of ∇
is a closed central 2-form with leading term h−1 ωX , whose cohomology class
determines B up to isomorphism; in particular this construction singles out a
base-point (the algebra whose curvature is exactly h−1 ωX ). This description
can easily be adapted to symplectic algebras (more generally to Poisson cones
of constant rank [20]). We will use this adapted description, and recall here how
it works. In fact we give here a direct proof of the embedding theorem.
6.1
c
Valuations and relative tangent algebra W
Let Σ be a Poisson cone. On the tangent bundle T Σ each fiber Tx Σ inherits
of a Poisson bracket cx with constant coefficients; there collection is a Poisson
bracket cT which is homogeneous of degree −1 with respect to homotheties.
We will denote W the sheaf generated by homogeneous functions on on T Σ
that are polynomial in the fibres: if we choose homogeneous local coordinates
x = (x1 , . . . , xn ) on Σ and ξ = (ξ1 , . . . , ξn ) denote the corresponding tangent
coordinates, a section of W is locally a finite sum of homogeneous monomials
aα (x)ξ α . The degree of such a monomial is
X
deg aα ξ α = deg aα +
αk deg ξk
(28)
The degree valuation is
p(f ) = inf(−deg fα )
if f is sum of homogeneous monomials fα
(29)
We define the weight (or order) of a homogeneous monomial aα (x)ξ α as
w(aα ξ α ) = − deg (aα ξ α ) +
|α|
2
(30)
The order valuation is
w(f ) = inf w(fα )
if f is sum of homogeneous monomials fα
(31)
Thus W is equipped with two valuations p ≤ w.
The Poisson brackets cx define fiberwise a Weyl product on W:
1
f ∗ g (x, ξ) = exp cx (∂ξ , ∂η ) f (x, ξ) g(x, η) |η=ξ
(32)
2
Since we will be dealing simultaneously with several product laws, we will adopt
the following notations, to avoid confusions if need be:
∗
or ∗W , ∗A
for the star-product,
48
× (or no sign) for the usual product
(33)
Both p and w are valuations for this product, i.e.
p(f ∗ g) ≥ p(f ) + p(g), w(f ∗ g) ≥ w(f ) + w(g)
(34)
L
The graded algebra gr p W = k (Wp≥k /Wp>k ) is commutative (its product
is the usual product since p([f, g]) ≥ p(f ) + p(g) + 1).
The graded algebra gr w W is not commutative; in fact the star-product ∗ is
homogeneous for w, i.e. if f, g are sums of monomials of pure weight k resp. k 0 ,
then f ∗ g and [f, g] are sums of monomials of pure weight k + k 0 .
cΣ (or W
c if there is no confusion) the completion
Definition 35 We denote W
of W for the valuation w.
c is a set of formal functions on T Σ, equipped with a star-product.
Note that W
c
W is obviously functorial, i.e. if u : Σ → Σ0 is a morphism of Poisson cones
(a smooth homogeneous map Σ → Σ0 compatible with the Poisson brackets, i.e.
u∗ cΣ = cΣ0 or {f ◦ u, g ◦ u} = {f, g} ◦ u for any smooth functions f, g on Σ0 )
cΣ0 → W
cΣ (f 7→ f ◦ T u) is a homomorphism af algebras, both for
then u∗ : W
the star-product and the ordinary product.
For deformation algebras on X, we will rather use, as Fedosov, the complech = W
ch (X) for the valuation w of the sheaf on X of algebras, whose
tion W
P
sections are polynomials f =
fk,α (x)hk ξ α ; (the valuations p, w are defined as
ch is
above, (p(h) = 1, functions on X are homogeneous of degree 0 (p = 0)). W
−1
5
c
canonically a quotient of WΓ : a 7→ h , α 7→ 0.
6.2
c
Automorphisms and Derivations of W
Let Σ be a symplectic cone (resp. X a symplectic manifold).
cΣ (resp.
In this section we examine derivations and automorphisms of W
c
Wh ). We only consider automorphisms U such that w(U − Id ) > 0.
c (i.e. O
bΣ , resp.
Any derivation or automorphism preserves the center of W
b
OX ((h))). We will say that D resp. U is an O-derivation (resp. automorphism)
if it fixes the center (i.e. D or U − Id vanishes on the center).
Proposition 36 All O-derivations, resp. automorphisms are inner derivations
(resp. inner automorphisms). More precisely, if D is an O-derivation, resp. U
c such that
an O-automorphism, there exists a unique section d resp. u ∈ W
d = 0, resp. u = 1 for ξ = 0, and D = ad d resp. U = Ad u (i.e. Df = [d, f ],
U f = u ∗ f ∗ u−1 .
the deformation case (X a Poisson manifold), the relevant cone is Γ = X ×R×
+ equipped
with the Poisson bracket hcX with h = a−1 , a the canonical coordinate of R×
+ . We have
cΓ is the commutative algebra, completion of C[a, α] for
T Γ = T X × T R× ; the center of W
5 In
+
−1 , α the
the order valuation w, where (a, α) are the canonical coordinates of T R×
+ (a = h
1
corresponding tangent coordinate; w(a) = −1, w(α) = − 2 ).
49
Proof : this is obvious on the local model (Σ or X a graded vector space),
b
where all O-derivations
(resp. automorphisms) are inner, because ad d = ad (d−
U
d(x, 0)), Ad U = Ad U (x,0)
. The uniqueness property just reflects the fact that
bS resp. OX ((h)). The global statement follows (uniqueness
the center is O
ensures that objects constructed on a covering patch together).
b
Note that if U is an O-automorphism
(w(U − Id ) > 0), D = Log U is well
defined, w(D) > 0, so D = ad d with some d such that w(d) > 0 and U = Ad u
with u = ed ; however d(x, 0) = 0 is not equivalent to ed (x, 0) = 1 (the set of
sections which vanish for ξ = 0 is not an ideal for ∗).
An immediate consequence of this proposition is that the sheaf (on Y = BΣ
b
b
or X) of O-derivations
or O-automorphisms
is soft .
6.3
Embeddings
Let A be a symplectic algebra over Σ (resp. B a deformation algebra over X).
c (resp. B → W
ch ) is a
We will say that a homomorphism of algebras A → W
good embedding if it respects the weight w (w(u) ≥ 0 i.e. w(uf ) ≥ p(f )) and
locally, in any set of homogeneous coordinates xj we have u(xj ) = xj + ξj + rj
with w(rj ) ≥ p(xj ) + 1 (in the deformation case we require u(h) = h). The
second condition does not depend on the choice of homogeneous coordinates.
Theorem 37 1) For any symplectic algebra A (resp. deformation algebra B)
cΣ (resp. B → W
ch ).
there exists a good embedding A → W
b
2) If u1 , u2 are two good embeddings there exists a unique O-automorphism
U
such that u2 = U u1 (two good embeddings are conjugate).
Proof : locally the theorem is immediate: if (xj ) is a set of local homogeneous
symplectic coordinates ({xi , xj } = cij = constant), a model good embedding
is f (x) 7→ uf = f (x + ξ). As mentioned it is often convenient to choose the
xj , ξj homogeneous of degree 21 ; we may then use them only as intermediate
tools since the monomials of W H must be homogeneous of integral degree.
If v is another good embedding, v(xj ) = xj + ηj , we have [ηi , ηj ] = [ξi , ξj ] =
c so by the universal property of the Weyl
cij because the xj are central in W,
algebra there exists a unique algebra homomorphism U such that U ξj = ηj
(U f = f if f is central). Since w(ηj − ξj ) > w(ξj ), U respects the valuation w
c is bijective.
and its unique continuous extension to W
This unique conjugacy statement is obviously also global.
In the general case there exists an open covering Σi of Σ (resp. Xi of X)
c over Σi (resp. ...). For each pair
and for each i a good embedding ui : A → W
b
(i, j) there is a unique O-automorphism
Uij over Σi ∩ Σj such that ui = Uij uj ,
and this is a cocycle (Uij Ujk = Uik because Uij Ujk uk = Uik uk = ui ). Since
b
the sheaf of O-automorphisms
(or derivations) is soft, (Uij ) is a coboundary:
Ui Uij = Uj for a suitable family (Ui ) so that the Ui ui patch together to produce
a global good embedding.
50
Remark 6 Another way to state this result is the following: the completion of
c the valuation used to define the completion is the
O ⊗ A is isomorphic to W;
unique valuation such that w(f ⊗ g) = p(f ) + p(g), w(1 ⊗ f − f ⊗ 1) = p(f ) + 12
if 1 ⊗ f − f ⊗ 1 6= 0 i.e. f is not a constant.
Remark 7 In the above construction we did not require that the embedding u
preserve the homogeneity valuation p. It is easy to show that there also exists
a good embedding u which preserves p (see §4.5 below). When this is the case
we have σ(uf ) = σ(f ) ◦ u where u is a smooth homogeneous map T Σ → Σ
compatible with the Poisson brackets, and tangent to the map (x, ξ) 7→ x + ξ
along the zero-section of T Σ. It is still true that two such embeddings u1 , u2
are conjugate: u2 = U u1 with U an O-automorphism (w(U − Id ) > 0), but U
cannot preserve the homogeneity valuation p if the geometric maps u1 , u2 are
not equal.
6.4
c
Vector Fields with Coefficients in W
To describe the other derivations or automorphisms, it is convenient to choose
a homogeneous symplectic connection ∇s on Σ (resp. X). We can choose it
homogeneous of degree 0 (with respect to homotheties) and torsionless, but any
b0 ⊗ sp (degree ≤ 0) will do. ∇s
symplectic connection with coefficients in O
c and if V is a vector field on Σ (homogeneous or with
extends canonically to W,
b ∇s is a derivation, both for the star-product and the usual
coefficients in O),
V
product, (it respects both valuations p, w if V is homogeneous of degree 0).
We identify the Lie algebra of sections of sp(T Σ) with the sub-Lie algebra W2
cΣ of homogeneous polynomials of degree 2 with respect to ξ: γ = P γjk ξj ξk
of W
(acting by ad : ad γf = [γ, f ]). Thus locally, if we choose homogeneous symplectic coordinates xj ({xi , xj } = cij = cst), we have
∇s f (x, ξ) =
with γ =
P
X
dxj (
X
∂f
+
[γijk ξj ξk , f ] )
∂xj
(35)
dxi γijk (x)ξj ξk homogeneous of degree 1 (ad γ of degree 0).
cΣ , its restriction to the center is a derivation of O
bΣ
If D is a derivation of W
b so we have D = ∇s + ad d for
i.e. a vector field V on Σ with coefficients in O,
V
c
some sections d ∈ W.
c
It is useful to introduce, as Fedosov, “derivations with coefficients in W”.
These are well defined because the symplectic Lie algebra sp is identified with the
c They form a Lie algebra VectW which is an extension of
subalgebra W2 ⊂ W.
s
c
the Lie algebra Der W
c of derivations of W. Its sections can be written ∇V + d,
s
where the rule for changes of coordinates for ∇V is the rule for vectors with
c The bracket is
coefficients in the Lie algebra sp(T Σ) = W2 ⊂ W.
[∇sV1 + d1 , ∇sV2 + d2 ] = [∇sV1 , ∇sV2 ] + ∇sV1 (d2 ) − ∇sV2 (d1 ) + [d1 , d2 ]
51
(36)
where the first bracket is given by the rule for connections with coefficients in
sp = W2 :
[∇sV1 , ∇sV2 ] − ∇s[V1 ,V2 ] = Rs (V1 ∧ V2 ) ∈ W2
(37)
c we denote ad V the derivation it
If V is a derivation with coefficients in W
b
defines. We have ad V = 0 if and only if V is central (of the form ∇s0 +f, f ∈ O),
s
c has a unique representative D = ad (∇ + f )
and any derivation D ∈ Der W
V
with f (x, 0) = 0.
6.5
Fedosov Connections
b Σ (resp Ω
b X ) denote the exterior algebra of differential forms on Σ (resp.
Let Ω
b (resp O((h))).
X) with coefficients in O
c such a conAs for derivations we define connections with coefficients in W:
c It
nection is of the form ∇ = ∇s + γ with γ a 1-form with coefficients in W.
c by
acts on forms with coefficients in W
ad ∇f = ∇s (f ) + [γ, f ]
c We
The curvature is R = ∇2 , identified with a 2-form with coefficients in W.
2
have (ad ∇) = ad R so ad ∇ is flat (integrable) if and only if the curvature R
b (independent of ξ). We will then
is central, i.e. a 2-form with coefficients in O
say, as Fedosov, that ∇ is abelian.
Definition 38 1) Let (xj ) be local coordinates on Σ, (ξj ) the corresponding
tangent coordinates. We denote ξj∨ the dual linear fiber coordinates on TΣ , such
that [ξi∨ , ξj ] = δij .
c dual to the canonical form
2) We denote τ the 1-form with coefficients in W,
of Σ with coefficients in T Σ (resp. T X):
τ=
X
dxj ξj∨
(resp.
τ=
1X
dxj ξj∨ )
h
(38)
τ is the unique 1-form with linear coefficients (in ξ) such that for any differential
c
form f with coefficients in W:
[τ, f ] =
X
dxj
∂f
∂ξj
(39)
We have p(τ ) = −1, w(τ ) = − 21 (τ is homogeneous of degree 1 and its coefficients
vanish of order 1 on the zero section). Also τ 2 = 12 [τ, τ ] is the symplectic form:
τ 2 = ωΣ
(resp. h−1 ωX )
(40)
c
Definition 39 A Fedosov connection is a connection ∇ with coefficients in W
of the form
∇ = ∇s − τ + γ with w(γ) ≥ 0.
52
Theorem 40 Let A be a symplectic algebra on Σ (resp. B a deformation algebra on X). Then
c (resp. ...) there exists a unique abelian
1) For any good embedding u : A → W
Fedosov connection ∇ such that ∇u = 0. This is unique up to a central form.
c is a symplectic
2) Conversely if ∇ is an abelian Fedosov connection, ker ∇ ⊂ W
(resp. deformation) algebra.
1) Locally, let us choose symplectic coordinates (e.g. homogeneous of degree
1
2 ). The standard local embedding u0 : f (x) 7→ u0 f = f (x + ξ) is killed by
∇0 = dx − τ , and obviously this is the only Fedosov connection which kills
it, with coefficients vanishing on the zero-section {ξ = 0}. If u is another
good embedding, it is conjugate to u0 : u = Ad U −1 u0 so it is killed by the
Fedosov connection ∇ = ∇0 + γ with γ = U −1 ∇0 (U ); we have w(γ) > 0 since
w(U − Id ) > 0. Note that γ may not vanish for ξ = 0, but we can replace it by
γ − γ(x, 0).
2) The converse is immediate: if ∇ is an abelian Fedosov connection, the map
˜
b there exists a unique
f ∈ ker ad ∇ 7→ f˜(x, 0) is clearly one to one (for each f ∈ O
˜
˜
˜
c
f ∈ W such ∇f = 0, f (x, 0) = f (x) because ∇ is “transversal” to the zerosection; further since ∇ is a Fedosov connection, we have locally w(f˜−f (x+ξ) ≥
w(f + 1)).
(The proofs for the deformation case are the same).
6.6
Fedosov curvature
Fedosov connections provide a one to one correspondence between symplectic
or deformation algebras and 2-forms on Σ (resp. X):
Theorem 41 (Fedosov) 1) Any closed 2-form R = ωΣ +r on Σ with coefficients
in O (resp. ωX + r)..) is the curvature of an abelian Fedosov connection.
2) Two abelian connections have the same curvature if and only if they are
conjugate. They define isomorphic algebras if and only if their curvatures differ
by an exact form.
Proof : 1) For the sake of completeness, we repeat the proof of Fedosov’s, by
successive approximations, improving the weight w of the error: if ∇ is a Fedosov
connection, the leading term (term of lowest weight w) of ad ∇ is
ad τ : f 7→
X
dxj
∂f
∂ξj
1
(of weight − )
2
If ∇2 = R + ρN with w(ρN ) = N2 ≥ 0, one looks for a 1-form γN +1 such that
w(∇ + γN +1 )2 − R) ≥ N2+1 , i.e.
[τ , γN +1 ] = ρN + (w ≥ N )
53
(41)
Let Iξ denote the interior product: Iξ f = ξj ∂xj yf (the usual product, not the
c Then if f = P dxi ...dxik fα (x) ξ α we have
star-product of W).
1
X
Iξ ad τ + ad τ Iξ f = Lξ f =
(|α| + k)dxi1 ...dxik fα ξ α
Since we have w([τ, ρN ]) > N 2−1 , equation 41 has a solution (in fact it has a
unique solution of pure weight N2+1 killed by Iξ ). By successive approximations
we get an exact solution - in fact there is a unique and canonical solution γ such
that Iξ γ = 0 (which implies [Iξ , ∇] = 0).
Remark 8 The denomination “canonical” is abusive, because it depends in
fact on the choice of a torsionless homogeneous symplectic connection ∇s on
Σ to begin with. In ambiguous cases we will refer to “the canonical Fedosov
connection associated with ∇s ”.
2) Obviously if ∇1 = U ∇U −1 and R = ∇2 is central, we have ∇21 = U RU −1 =
R. The converse (i.e. the existence of U such that ∇1 = U ∇U −1 if ∇ and
∇1 have the same central curvature) is proved by successive approximation as
above.
Note that ker ad ∇ = ker ad ∇1 means that ∇1 = ∇ + γ with γ central,
w(γ) ≥ 0; this implies R1 = R + dγ, hence the last assertion of 2).
Remark 9 The canonical solution is in fact of weight p(ad ∇) ≥ 0 (p(∇) ≥
−1), so that the corresponding embedding u is associated to a geometrical map
u : T Σ → Σ, preserving Poisson brackets and tangent to the map (x, ξ) 7→ x + ξ.
Remark 10 If A is a symplectic or deformation algebra, an embedding u deb (locally
fines a total symbol σU (f ) = uf (x, 0) which is an isomorphism A → O
×
b
in D0 ), and a star-product Bu (σu (f ∗A g) = Bu (σu f, σu g)). If we replace u by
c σu is replaced by PU,u σu for some well defined
U u, U an automorphism of W,
b × (PU V,u = PU,V u PV,u ), and Bu is replaced by
asymptotic operator PU,u ∈ D
0
PU,u Bu ((P B)(f, g) = P B(P −1 f, P −1 g)).
b × ; for example we
The set of transition operators PU,u is a P
strict subset of D
0
α
have PU,u = 1 if U is even, i.e. U = 1 + uα (x)ξ , α > 0 even). Likewise
b × σu of all possible
the set of all total symbols σu is a strict subset of the set D
0
total symbols equivalent to σu , and the set of all corresponding star-products
Bu is a strict subset of the set of all star-products equivalent to a given one;
for instance one always has Bu (f, g) = f g + 21 {g, g} + · · · . I will not describe
further here this special class of star-products. In fact I do not know if it has
been distinguished before.
6.7
Base-point
Fedosov’s construction gives a canonical base-point in the set of star-algebras,
viz. the algebra corresponding to a connection with “trivial curvature” R = ωΣ
(resp. R = h−1 ωX ).
54
c has a canonical involution (and a sub-principal
Note that the Weyl algebra W
c if ∇ resymbol, fiberwise). The algebra A∇ is a sub-involutive algebra of W
spects the involution. If this is the case the curvature
P R also respects the involution, i.e. it is odd (an odd power series h−1 ωX + h2k+1 ωk in the deformation
case, R ∼ 0 in the symplectic case). Conversely if R is odd, Fedosov’s construction obviously yields a connection ∇ which respects the involution. Thus the
base-point is involutive; in the symplectic case, it is the unique involutive algebra; in the deformation case we have seen that there are many other nontrivial
involutive algebras.
For further use let us also note the following result: let Gh denote the group
of symbol preserving automorphisms of C((h)) (i.e. automorphisms U of C((h))
with its standard commutative algebra structure, such that w(U (h) − h) > 0).
Gh obviously acts on the set of deformation algebras on X (replacing h by
U (h)).
Proposition 42 If X is compact, the action of Gh on h-Alg (X) is free
On any symplectic manifold X, a symbol-preserving isomorphism Ũ preserves the center, hence induces an automorphism U ∈ Gh . Recall that the
local model for deformation algebras is the algebra B0 of (jets along {ξ = 0}
∂
of) pseudodifferential operators on Rn × R which commute with h−1 = ∂t
∂
(X = T ∗ Rn , ξ = h ∂x
). Clearly any U ∈ Gh lifts to Aut B0 (e.g. Ũ such that
∂
∂
∂
, Ũ (t) such that [U ( ∂t
), Ũ (t)] = 1).
Ũ (x) = x, Ũ ( ∂x ) = ∂x
The proposition states that if X is compact, an isomorphism which preserves
symbols, but not necessarily h, in fact fixes h.
ch . If R is the Fedosov curvature of a deformation
Proof : if U ∈ Gh , it acts on W
algebra B, the curvature of U (B) is U (R). Now the leading term of R is h−1 ωX .
∂
If U 6= 0 the leading term of U − 1 is c hk ∂h
for some integer k ≥ 2 and constant
c 6= 0, so that the leading term of U (R) − R is −c hk−2 ωX ; this is 6= 0 if X is
compact (ωX is not a coboundary).
Therefore if U ∈ Gh and U B is isomorphic to B, then U (h) = h i.e. U = Id .
Equivalently if Ũ is a symbol-preserving isomorphism of B, it fixes the center
(U (h) = h).
55
7
Traces.
If A is an algebra, a trace on A is a linear form T : A → C such that T (ab) =
T (ba) for any a, b ∈ A. If IA is a two sided ideal, a trace (relative to I) is
a linear form I → C such that T (ab) = T (ba) for any a ∈ I, b ∈ A. For
deformation algebras we will use an obvious extension: an ~-trace is an ~-linear
map A → C((~)) such that T (ab) = T (ba) for any a, b ∈ A (there is also a
notion of trace relative to an ideal).
7.1
Residual integral.
P
Let Σ be a oriented cone of dimension n, and µ =
µk an n-form (density)
b and compact conic support.
with coefficients in O
R res
Definition 43 The residual integral
µ is defined as
Z res
Z
µ=
ρyµ0
X
where µ0 is the component homogeneous of degree 0 of µ, ρ denotes the radial
vector field; the contraction ρyµ0 is the pull-back of a (n − 1)-form on X
PR
1
µ (γ = X × S1 ⊂ Σc ).
It is also the contour integral 2iπ
γ k
It behaves like an integral :
b µ is exact, i.e. µ = dν with ν a
- if µ is an n-form with coefficients in O,
R res
(n − 1)-form with compact conic support iff
dµ = 0 (indeed if µ is of degree
k 6= 0 it is exact: we have µ = k1 dIρ µ;R if µ is Rof degree 0, we have µ = dr
r ν with
res
ν = ρyµ. then µ is exact iff ν is, i.e. X ν = Σ µ = 0).
R res
b
- in particular
LvRµ = 0 for any vector
R res fiels with coefficients in O, so we
res
may integrate by parts:
µLv f = −
Lv µ.f .
7.2
Trace for Moyal products
Let E be a graded vector space, Σ = E −E0 . We equip
P E with a volume element
dv = c dx1 . . . dxn (this is homogeneous of degree
deg xi ). Let b be bilinear
form with constant coefficients, homogeneous of degree −1 on E, and A be the
star algebra associated to the the Moyal star product ∗b on Σ.
Pres
Proposition 44 The residual integral T (a) =
f dv is a trace on the ideal
of elements with compact conic support.
P 1 k
Proof: we have f ∗ g =
k! b (∂ξ , ∂η )f (ξ)g(η)|η=ξ , so
[f, g] =
X 1
(bk (∂ξ , ∂η ) − bk (∂η , ∂ξ ))f (ξ)g(η)|η=ξ
k!
56
Integrating by parts we get
Z
Z res
[bk (∂ξ , ∂η )f (ξ)g(η)]|η=ξ =
res
[bk (∂ξ , −∂ξ )f (ξ)]g(η)|η=ξ
Now we have b(−u, u) = b(u, −u) so the two integrals cancel.
Note that if b is antisymmetric
we even have bk (∂ξ , −∂ξ) = 0 so all terms
R res
R res
vanish for k > 0 and
f ∗g =
f g: then we have T (a ∗ b) = T (ab).
7.3
Canonical trace on symplectic algebras.
Proposition 45 Suppose that c = b −t b is a symplectic Poisson bracket with
constant coefficient, homogeneous of degree −1 (dim E = 2n). Then the trace
T (a) above is, up to a constant factor, the unique race on A.
Proof: we show that the set of sums of brackets in A is the kernel of T , a subvector space of codimension 1. Let ξk be a hommogeneous basis of coordinates,
ξ˜k the dual basisR i.e. {ξ˜j , ξk } = δjk . If a has compact conic support, a dv is
res
exact iff T (a) =
adv = 0. Then there exist aj with compact conic support
P
c ..dξn i.e. a = P L∂ aj = P[ξ˜j , aj ].
such that a dv = d (−1)j−1 aj dξ1 ..dξ
j
j
Definition 46 With notations as above, the canonical residual trace on A is :
Z
ωn
tr res a = (2π)−n
a
res n!
where ω is the symplectic form inverse of the Poisson bracket c (ω is of degree
n
1, dv of degre n, the density is dv = ωn! ).
Example: let E have canonicalP
homogeneous coordinates ξk , xk ({ξp , ξq } =
{xp , xq } = {ξp , xq } − δpq = 0, ω =
dξj dxj ).
P
∂ξj ∧∂xj (the coordinates
The Weyl product corresponds to the case b = 21
ξj , xj are all of degree 21 ). We have trres a ∗w b = trresP
ab
The normal product corresponds to the case b =
∂ξj ⊗ ∂xj (ξ of degree
1, x of degree 0). We have trres [a, b] = 0 (but not trres an b = trres ab).
Theorem 47 tr res is invariant by all isomorphisms of star algebras
Proof: let U, V be open subcones of E − E0 and U : A|U → A|V an isomorphism. Since trres is unique except for a constant factor, we have U∗ trres =
ctrres . Now the geometric support u of U is a homogeneous symplectic map
n
U → V ; it treserves ω and ωn! , so also the leading term of trres (trres a =
R res
n
σ−n (a) ωn! if a is of degree −n), hence c = 1.
It follows that any symplectic star algebra A possesses a canonical (residual)
trace (Σ is a union of open subcones Σk isomorphic to open subcones of symplectic graded vector spaces as above; then the traces on A|Σk are well defined
and glue together).
We complete remark3 as follows:
57
P
Remark 11 If a =
ak ∈ A and U is an automorphism of A, tr U s P is a
polynomial of s (of degree ≤ n + deg a).
7.4
Trace for deformation algebras
Since deformation algebras are C((~))-algebras, it is natural to study traces with
values in C((~)). Most of what was said above can be repeated:
if X is a vector
with a bidifferential operator b with constant
Pspace equipped
coefficients, a =
ak (x)hk , dv a volume element (with constant coefficients),
the integral
Z
X Z
T (a) =
adv =
~k
ak dv ∈ C((~))
X
X
is a trace on the deformation algebra A corresponding to the Moyal product
∗~b, i.e. T [a, b] = 0 (T (a ∗ b) = T (ab) if b is antisummetric).
If the Poisson bracket c = b −t b is symplectic, the trace is unique up to
a “constant” factor, i.e. an invertible element of C((~)), and we define the
canonical trace as
Z
ωn
tr(a) = (2π~)−n a
n!
where ω is the symplectic form inverse of c. (We will sometimes use the residual
trace, with values in C: this is the coefficient of ~n in tr a; it agrees with
the residual trace for Toeplitz operators when A can be nicely embedded in a
Tpeplitz algebra).
The analogue of proposition 45 still holds, but the proof above no longer
works because a trace is not determined by its leading term (no more than an
invertible element of C((~))). We repeat Fedosov’s proof ([61]: first note that the
theorem is local on X, so as the integral defining the trace. tr is also invariant
by inner automorphisms (tr (aba−1 − b) = tr [ab, a−1 ] = 0], so it is invariant by
all automorphisms (which are locally inner). The Weyl deformation product is
obviously invariant by affine symplectic isomorphisms, so as the trace.
We now turn to general isomorphisms: if X, Y are open subsets of a symplectic vector space E and U : AX → AY an isomorphisms of the two deformation
Weyl algebras, the geometric support u : Y → X is is a symplectic isomorphism.
Let y0 be a point of Y , x0 = u(y0 ). The affine map v tangent to u preseves the
star product and tr . Replacing U by v −1 ◦ U we are reduced to the case where
u is tangent ti the identity map at ξ0
Then one can choose a smooth family ut of symplectic diffeomorphisms defined in a fixed neighborhood of ξ0 so that u0 = Id , u1 = u. This in turn can
du
be lifted in a smooth family Ut of isomorphisms : u−1
t dt is a smooth family of
symplectic vector fields, which can be lifted to a smooth family of derivations
of degree 0 : Dt = ad ~−1 at (a0 = 0); this in turn can be integrated to a smooth
t
family of isomorphisms Ut = Ad eat above ut , with U0 = Id ( dU
dt = Dt Ut ).
d
Then dt tr Ut a = Dt Ut a = 0 so tr Ut a = tr a. Finally tr U a = tr U1 a since
U −1 U1 is an automorphism (its geometric support is the identity map).
58
8
Vanishing of the Logarithmic Trace.
In this section we show that the logarithmic trace of Szegö projectors introduced
by K. Hirachi [78] for CR structures and extended in [25] to contact structures
vanishes identically.6
In [78] K. Hirachi showed that the logarithmic trace of the Szegö projector
is an invariant of the CR structure. In [25] I showed that it is also defined
for generalized Szegö projectors associated to a contact structure (definitions
recalled below, sect.8.4), that it is a contact invariant, and that it vanishes if the
base manifold is a 3-sphere, with arbitrary contact structure (not necessarily the
canonical one). Here we show that it always vanishes. For this use the fact that
this logarithmic trace is the residual trace of the identity (definitions recalled
below, sect.8.5), and show that it always vanishes, because the Toeplitz algebra
associated to a contact structure can be embedded in the Toeplitz algebra of a
sphere, where the identity maps of all ‘good’ Toeplitz modules have zero residual
trace.
8.1
Notations
We first recall the notions that we will use. Most of the material below in §1-5
is not new; we have just recalled briefly the definitions and useful properties,
and send back to the literature for further details (cf. [82, 83, 102, 86]).
If X is a smooth manifold we denote T • X ⊂ T ∗ X the set of non-zero
covectors. A complex subspace Z corresponds to an ideal IZ ⊂ C ∞ (T • X, C)). Z
is conic (homogeneous) if it is generated by homogeneous functions. It is smooth
if IZ is locally generated by k = codim Z functions with linearly independent
derivatives. If Z is smooth, it isPinvolutive if IZ is stable by the Poisson bracket
∂f ∂g
∂f ∂g
(in local coordinates {f, g} =
∂ξj ∂xj − ∂xj ∂ξj ); it is 0 if locally IZ has
generators ui , vj (1 ≤ i ≤ p, 1 ≤ j ≤ q, p + q =codim Z) such that the vj are
real, ui complex, and the matrix 1i ({uk , ūl ) is hermitian 0. The real part
ZR is then a smooth real submanifold of T • X, whose ideal is generated by the
Re ui , Im ui , vj . If I is 0, it is exactly determined by its formal germ (Taylor
expansion) along the set of real points of ZR .
A Fourier integral operator (FIO) from Y to X, is a linear operator from
functions or distributions on Y to same on X defined as a locally finite sum of
oscillating integrals
Z
f 7→ F f (x) = eiφ(x,y,θ) a(x, y, θ)f (y) dθdy ,
where φ is a phase function (homogeneous w.r. to θ), a a symbol function.
Here we will only consider regular symbols, i.e. which are asymptotic sums
6 arXiv:math.AP/0604166 v1; Proceedings of the Conference in honor of T. Kawai, Springer
Verlag, 2006
59
P
a ∼ k≥0 am−k where am−k is homogeneous of degree m − k, k an integer. m
could be any complex number. There is also a notion of vector FIO, acting on
sections of vector bundles, which we will not use here.
The canonical relation of F is the image of the critical set of φ (dθ φ = 0) by
the differential map (x, y, θ) 7→ (dx φ, −dy φ) - this is always assumed to be an
immersion from the critical set onto a Lagrangian sub-manifold of T • X × T • Y 0
(the sign 0 means that we have reversed the sign of the canonical symplectic
form; likewise if A is a ring, A0 denotes the opposite ring). We will also use FIO
with complex positive phase function: then the canonical relation is defined by
its ideal (the set of complex functions u(x, ξ, y, η which lie in the ideal generated
by the coefficients of dθ φ, ξ − dx φ, η + dy φ which do not depend on θ); it should
not be confused with its set of real points.
Following Hörmander [82, 83], the degree of F is defined as
1
deg F = deg (adθ) − (nx + ny + 2nθ ),
4
with nx , ny , nθ the dimensions of
Pthe x, y, θ-spaces, deg (adθ) the degree of the
differential form adθ = deg a + νk (νk = deg θk , usually νk = 1 but could be
any real number - not all 0); this only depends on F and not on its representation
by oscillating integrals). In what follows we will always require that the degree
be an integer (which implies m ∈ Z/4).
8.2
Adapted Fourier Integral Operators
The Toeplitz operators and Toeplitz algebras used here, associated to a CR or
a contact structure, were introduced and studied in [17, 27], using the analysis
of the singularity of the Szegö kernel (cf. [15, 87]), or in a weaker form, the
“Hermite calculus” of [14, 71]. The terminology “adapted” is taken from [27]:
lacking anything better I have kept it.
For k = 1, 2, let Σk ⊂ T • Xk be smooth symplectic sub-cones of T • Xk and
u : Σ1 → Σ2 an isomorphism.
Definition 48 A Fourier integral operator A is adapted to u if its canonical
relation C is complex 0, with real part the graph of u. It is elliptic if its
principal symbol does not vanish (on Re C).
As above, a conic complex Lagrangian sub-manifold Λ of T • X is 0 if
its defining ideal IΛ is locally generated by n = dim X homogeneous functions u1 , . . . , un (n = dim X) with independent derivatives, u1 , . . . , uk complex,
uk+1 , . . . , un real for some k (1 ≤ k ≤ n), and the matrix ( 1i {up , ūq })1≤p,q≤k is
hermitian 0; equivalently; the intersection C ∩ C¯ is clean, and on the tangent
bundle the hermitian form 1i ω(U, V̄ ) is positive, with kernel the complexification
of the tangent space of Re C.
Pseudo-differential operators are a special case of adapted FIO (X1 = X2 =
X, u = IdT • X ); so are Toeplitz operators on a contact manifold (see below).
Adapted FIO always exist (cf [27]), more precisely
60
Proposition 49 For any symplectic isomorphism u as above, there exists an
elliptic FIO adapted to u.
In fact if Λ is a complex 0 Lagrangian sub-manifold of T • X, in particular
if it is real, it can always be defined by a global phase function with positive
imaginary part (Im φ % dist (., ΛR )2 ) living on T • X: it is easy to see that such
phase functions exist locally, and the positivity condition makes it possible to
glue things together using a homogeneous partition of the unity. Once one has
chosen a global phase function, it is obviously always possible to choose an
elliptic symbol - of any prescribed degree (cf. also [27])7 . Note that elliptic
only means that the top symbol is invertible on the real part of the canonical
relation, not that the operator is invertible mod. smoothing operators (for this
the canonical relation must be real: X1 , X2 have the same dimension, Σk =
T • Xk and C is the graph of an isomorphism).
8.3
Model Example
Here is a generic example of adapted FIO: let X1 , X2 , Z be three vector spaces
Σk = TXk Xk × T • Z ⊂ T • (Xk × Z)
(k = 1, 2, TXk Xk the zero section),
u the identity map IdT • Z : Σ1 → Σ2 .
(42)
If C 0 is a complex canonical relation with real part the graph of u, the
complex formal germ along Σ of the restriction to C of the projection map
(x, ξ, z, ζ, z 0 , ζ 0 , y, η) 7→ (x, z, ζ 0 , y) is an isomorphism (the dimensions are right,
and it is an immersion: if v = (0, ξ, 0, ζ, z 0 , 0, 0, η) is a complex vector with zero
projection, it is orthogonal to v̄ (because these vectors form a real Lagrangian
space), so if it is tangent to C 0, it is tangent to the real part, i.e. the diagonal
of T • Z, and this obviously implies v = 0).
So we can choose the phase function as
φ =< z − z 0 , ζ 0 > +iq(x, z, ζ 0 , y)
where q is smooth complex function of x, z, ζ 0 , y alone, homogeneous of degree
1 w.r. to ζ 0 , vanishing of order 2 for x = y = 0, and Re q % (x2 + y2 ) |ζ 0 | (it is
easy to check that conversely any such phase function corresponds to a positive
adapted canonical relation as above). The operator is
Z
0 0
0
F f (x, z) = ei<z−z ,ζ >−q(x,z,ζ ,y) a(x, z, ζ 0 , y)f (z 0 , y)dζ 0 dz 0 dy ,
(43)
with a a symbol as above.
7 the
intrinsic differential-geometric description of the symbol is elaborate: it is a section of a
line bundle whose definition incorporates half densities and the Maslov index or an elaboration
of this in the case of complex canonical relations. However on real manifolds this line bundle
is always topologically trivial
61
Since any symplectic sub-cone of a cotangent manifold is always locally
equivalent to T • Z ⊂ T • (X × Z), the model above is universal i.e. any adapted
FIO is micro-locally equivalent to F1 ◦ A ◦ F2 where F1 , F2 are elliptic invertible
FIO with real canonical relations, graphs of local symplectic isomorphisms, and
F is as the model above.
For adapted FIO the Hörmander degree coincides with the degree in the scale
of Sobolev spaces, i.e. if F is of degree s it is continuous H m (Y ) → H m−Re s (X);
this is easily seen on the model example above (F is L2 continuous if its degree
is 0 i.e. a is of degree − 41 (nx + ny )). (This not true for general FIOs - in fact
for a FIO with a real canonical relation C, this is only true if C is locally the
graph of a symplectic isomorphism.)
The following result also immediately follows from the positivity condition:
Proposition 50 Let X1 , X2 , X3 be three manifolds, Σk ⊂ T • Xk symplectic
sub-cones, u resp. v a homogeneous symplectic isomorphism X1 → X2 resp.
X2 → X3 , F, G FIO (with compact support) adapted to u, v. Then G ◦ F is
adapted to v ◦ u; its canonical relation is transversally defined and positive. It
is elliptic if F and G are elliptic.
This is mentioned in [27]; the crux of Rthe matter is that if Q(y) is a quadratic
form with 0 real part, the integral e−|ξ|Q(y) dy does not vanish: it is an
1
− 21
elliptic symbol of degree − 21 ny , equal to disc( Q
|ξ|− 2 ny .
π)
8.4
Generalized Szegö projectors
These were called “Toeplitz projectors” in [24, 25]. C. Epstein suggested the
present name, which is better. References: [30, 17, 27, 21].
Definition 51 Let X be a manifold, Σ ⊂ T • X a symplectic sub-cone. A generalized Szegö projector (associated to Σ) is an elliptic FIO S adapted to IdΣ
which is a projector (S 2 = S)
(Note that “elliptic” (or “of degree 0”) is part of the definition; otherwise
there exist many non-elliptic projectors, of degree > 0 as FIOs). The case we
are most interested in is the case where Σ is the half line bundle corresponding
to a contact structure on X (i.e. the set of positive multiples of the contact
form). But everything works as well in the slightly more general setting above.
We will not require here that S be an orthogonal projector; this makes sense
anyway only once one has chosen a smooth density to define L2 -norms.
If S is a generalized Szegö projector, its canonical relation C ⊂ T • X × T • X
is idempotent, positive, and can be described as follows: the first projection is a
complex positive involutive manifold Z+ with real part Σ; the second projection
is a complex negative manifold Z− with real part Σ (Z− = Z¯+ if S is selfadjoint). The characteristic foliations define fibrations Z± → Σ (the fibers are the
characteristic leaves; they have each only one real point so are “contractible”
62
(they vanish immediately in imaginary domain), and there is no topological
problem for them to build a fibration). Finally we have C = Z+ ×Σ Z−
Generalized Szegö projectors always exist, so as orthogonal ones (cf. [27,
21]). As mentioned in [25], generalized Szegö projectors mod. smoothing operators form a soft sheaf on Σ, i.e. any such projector defined near a closed conic
subset of T • X or Σ is the restriction of a globally defined such projector.
8.5
Residual trace and logarithmic trace
The residual trace was introduced by M. Wodzicki [116]. It was extended to
Toeplitz operators and suitable Fourier integral operators by V. Guillemin [73]
(cf. also [74, 117]). It is related to the first example of ‘exotic’ trace given by J.
Dixmier [43].
Let C be a canonical relation in T • X × T • X. A family As (s ∈ C) of FIOs
s
of degree s belonging to C is holomorphic if s 7→ (∆)− 2 As is a holomorphic
map from C to FIO of fixed degree (in the obvious sense). If As has compact
support, the trace tr AS is then well defined and depends holomorphically on s
if Re s is small enough (As is then of trace class). Often, e.g. if the canonical
relation is real analytic, this will extend as a meromorphic function of s on the
whole complex plane, but this is not very easy to use because the poles are hard
to locate and usually not simple poles.
Proposition 52 If C is adapted to the identity IdΣ , with Σ ⊂ T • X a symplectic
sub-cone, and As a holomorphic family, as above, then tr As has at most simple
poles are the points s = −n − deg A0 + k, k ≥ 0 an integer, n = 12 dim Σ (the
degree deg A0 is defined as above)
Proof: this is obviously true is As is of degree −∞ (there is no pole at all).
In general we can write As as a sum of FIO with small micro-support (mod
smoothing operators), and a canonical transformation reduces us to the model
case, where result is immediate.
Definition 53 If A is a FIO adapted to C, the residual trace trres A is the
residue at s = 0 of any holomorphic family As as above, with A = A0 .
This does not depend on the choice of a family As : indeed if A0 = 0, the
family As is divisible by s i.e. As = sBs where Bs is another holomorphic
family, and since tr Bs has only simple poles, tr As has no pole at all at s = 0.
Proposition 54 The residual trace is a trace, i.e. if A and B are adapted
Fourier integral operators, we have trres AB = trres BA.
Indeed with the notations above tr ABs and tr Bs A are well defined and
equal for Re s small, so their meromorphic extensions and poles coincide.
Logarithmic trace (contact case)
63
Let Σ ⊂ T • X be a symplectic half-line bundle, defining a contact structure
on X. A complex canonical relation C 0 adapted to IdΣ is always the conormal bundle of a complex hypersurface Y of X × X, with real part the diagonal
(rather the positive half)8 , so if A is a FIO adapted to IdΣ , its Schwartz kernel
can be defined by a one dimensional Fourier integral:
Z ∞
Ã(x, y) =
e−T φ(x,y) a(x, y, T ) dT ,
0
with φ = 0 an equation of the hypersurface
P Y , φ = 0 on the diagonal, Re φ ≥
cst dist(., diag)2 , and a is a symbol: a ∼ k≤N ak (x, y)T k−1 (N = deg A).
Its singularity has a typically holonomic form:
f (x, y)(φ + 0)−N + g(x, y) Log (
1
),
φ+0
(44)
with f, g smooth functions on X × X, and in particular g(x, x) = a0 (x, x).
Proposition 55 With notations as above, the residual trace of A is the trace
of the logarithmic coefficient:
Z
trres A =
g(x, x) .
(45)
X
An obvious holomorphic family extending A (mod. a smoothing operator) is
the family As with Schwartz kernel
Z ∞
Ãs (x, y) =
e−T φ(x,y) a(x, y, T ) T s dT .
1
Since as (x, x, T ) ∼
P
k≤N
T s+k−1 ak (x, x) and φ(x, x) = 0, we get
Ãs (x, x) ∼
X ak (x, x)
s+k
,
with an obvious notation: the meromorphic extension
R of the trace just has just
simple poles at each integer j ≥ −N , with residue X a−j (x, x). In particular
the residue for s = 0 is the logarithmic trace.
The residual trace is also equal to the logarithmic trace in the case of pseudodifferential operators, or in the model case. In general the residual trace is well
defined, but I do not know if the logarithmic coefficient can be reasonably defined
e.g. if the projection Σ → X is not of constant rank. For the equality with the
residual trace, and for theorem 58 below, the sign is important: the logarithmic
trace is the integral of the coefficient of Log φ1 , not the opposite.
8 indeed a complex vertical vector v is as before orthogonal to v̄; if it is tangent to C at a
point of CR = diag Σ, it is tangent to the real part CR = diag Σ since C 0, but this implies
that v is the radial vector, because the radial vector is the only vertical vector tangent to Σ.
So the projection C → X × X is of maximal rank 2n − 1 and the image is a hypersurface.
64
8.6
Trace on a Toeplitz algebra A and on End A (M )
If S is a generalized Szegö projector associated to Σ ⊂ T • X. The corresponding
Toeplitz operators are the Fourier integral operators of the form TP = SP S, P
a pseudo-differential operator (equivalently, the set of Fourier integral operators
A with the same canonical relation, such that A = SAS). They form an algebra
A on which the residual trace is a trace: trres AB = trres BA. Mod. smoothing
operators, this can be localized, and the Toeplitz algebra AΣ is this quotient; it
is a sheaf on Σ (or rather on the basis).
Proposition 56 The Toeplitz algebra, so as the residual trace of S only depend
on Σ and not on the embedding Σk ⊂ T • X
Indeed if Σ → T • X 0 is another embedding, S 0 a corresponding Szegö projector,
it follows from prop. 49 that there exist elliptic adapted FIO F, F 0 from X
to X 0 resp. X 0 to X such that F = S 0 F S, F 0 = SF 0 S 0 , F F 0 ∼ S, F 0 F ∼ S
so S, S 0 have the same residual trace, and A 7→ F AF 0 is an isomorphism of
the two Toeplitz algebras. In the lemma wa could as well embed Σ in another
symplectic cone endowed with a Toeplitz structure.
The definition of the residual trace extends in an obvious way to End A (M )
when M is a free A-module (End A (M
P) is isomorphic to a matrix algebra with
coefficients in A0 , where trres (aij ) = trres aii is obviously a trace, independent
of the choice of a basis of M ). It extends further to the case where M is locally
free (a direct summand of a free module), and to the case where M admits
(locally or globally) a finite locally free resolution: if
d
d
0 → LN −
→ . . . L1 −
→ L0 → 0
is such a resolution, i.e. a complex of are locally free A-modules Lj , exact in
degree 6= 0, with a given isomorphism : L0 /dL1 → M : Then End A (M ) is
isomorphic to Rhom0 (L, L), i.e. any a ∈ End A (M ) extends as morphism ã of
complexes of L (ã = (aj ), aj ∈ End A (Lj )) (if a has compact support, ã can
be chosen with compact support). Any two such extensions ã, ã0 differ by a
super-commutator [d, s] = ds + sd (s, of degree 1, can be chosen with compact
support if ã − ã0 has compact support). The super-trace
X
supertrres (ã) =
(−1)j trres (aj )
is then well defined; it only depends on a, because the super-trace of a supercommutator [s, d] vanishes: this defines the trace in End (M ). Below we will
use “good” modules, i.e. which have a global finite locally free resolution for
which this is already seen on the principal symbols (i.e. M and the Lj are
equipped with good filtrations for which gr d is a locally free resolution of gr M
(in the analytic setting this always exists if M is coherent and has a global good
filtration, and the base manifold is projective).
65
Alternative description of the residual trace
Let S be a generalized Szegö projector associated to a symplectic cone Σ ⊂
T • X. Then the left annihilator of S is a 0 ideal I in the pseudo-differential
algebra of X; its characteristic set is the first projection Σ+ of the complex
canonical relation of S; as mentioned earlier it is involutive 0 with real part
Σ. (there is a symmetric statement for the right annihilator).
Proposition 57 Let M be the EX -module M = E/I. Then the Toeplitz algebra
is canonically isomorphic to End E (M ).
Proof: let eM be the image of 1 ∈ E (it is a generator of M ). It is elementary
that End E (M ) is identified with opposite algebra ([E : I]/I)0 where [E : I]
denotes the set of P ∈ E such that IP ⊂ I (to P corresponds the endomorphism
aP such that aP (e) = P e). It is also immediate that the map u which to aP
assigns the Toeplitz operator TP = SP S = P S is an isomorphism (both algebras
have a complete filtration by degrees, and the associated graded algebra in both
cases is the algebra of symbols on Σ); clearly u is a homomorphism of algebras,
of degree ≤ 0, and gr u = Id.
Now M is certainly “good” in the sense above: it is locally defined by transverse equations and has, locally, a resolution whose symbol is a Koszul complex.
So the residual trace is well defined on End E (M ). Since the trace on an algebra
of Toeplitz type is unique up to a constant factor, there exists a constant C
such that
trres aP = C trres TP .
(46)
Below we only only need C 6= 0; however with the conventions above we have:
Theorem 58 The constant C above is equal to one (C = 1).
Proof: to the resolution of M above corresponds a complex of pseudo-differential
operators
D
D
0 → C ∞ (x) −
→ C ∞ (X, E1 ) → . . . −
→ C ∞ (X, EN ) → 0 ,
(47)
exact in degree > 0 and whose homology in degree 0 is the range of S (mod.
smoothing operators), i.e. there exists a micro-local operator E on (Ek ) such
that DE + ED ∼ 1 − S (cf. [14]; E is a pseudo-differential operator of type 21 ,
not a “classical” pseudo-differential operator, but it preserves micro-supports).
It is elementary that one can modify D, E, and if need be S, by smoothing
operators so that (47) is exact (on global sections) in degree 6= 0, and ker D0
is the range of S. Then if as = (ak , s) is a holomorphic family of pseudodifferential homomorphisms P
of degree s, and Tas is the Toeplitz operator Tas =
a0,s |ker S , we have tr Tas = (−1)k tr ak,s for Re s 0 hence also equality for
the meromorphic extensions and residues.
Here is an alternative proof (slightly more in the spirit of the paper because
it really uses operators mod. C ∞ rather than true operators). Notice first that
theorem 58 is (micro) local, and since locally all bundles are trivial, we can
66
reason by induction on codim Σ. Thus it is enough to check the formula for one
example, where codim Σ = 2. Note also that above we had embedded in the
algebra of pseudo-differential operators on a manifold, but we could just as well
embed in another Toeplitz algebra.
We choose Σ corresponding to the standard contact (CR) sphere S2n−1 of
Cn , embedded as the diameter z1 = 0 in the sphere of Cn+1
The Toeplitz space Hn+1 is the space of holomorphic functions in the unit
ball of Cn+1 (more correctly: their restrictions to the sphere); we choose Hn
the subspace of functions independent of z1 . There is an obvious resolution:
0 → Hn+1 → Hn+1 → 0 (Hn P
= ker ∂1 ). We choose on Hn the operator
a, restriction of ρ− n, with ρ =
zj ∂zj (this is the simplest operator with a
nonzero residual - our convention is that ρσ kills constant functions for all σ).
Lemma 59 On the sphere S2n−1 the residual trace of the Toeplitz operator ρ−n
1
.
is trres ρ−n = (n−1)!
Proof: the standard Szegö kernel is S(z, w) =
have the obvious identity
ρ(ρ + 1) . . . (ρ + n − 1) Log
1
vol S2n−1 (1
− z.w̄)−n . Now we
1
= (n − 1)!((1 − z w̄)−n − 1) ,
1 − z.w̄
so that the leading coefficient of the logarithmic part of ρ−n is
1
whose integral over the sphere is (n−1)!
.
1
vol S2n−1 (n−1)! ,
On the sphere S2n+1 we have ∂1 ρ = (ρ + 1)∂1 so we choose for (ã) the pair
(a0 = ρ−n , a1 = (ρ + 1)−n ). Since terms of degree < −n − 1 do not contribute
on S2n+1 , the super-trace is
supertrres (ã) = trres (ρ−n − (ρ + 1)−n ) = trres (nρ−n−1 ) =
8.7
n
= trres a .
n!
Embedding
If Σ1 , Σ2 are two symplectic cones, with contact basis X1 , X2 , symplectic embeddings Σ1 → Σ2 exactly correspond to contact embeddings X1 → X2 , i.e.
an embedding u : X1 → X2 such that the inverse image u∗ (λ2 ) is a positive
multiple of the contact form λ1 (the corresponding symplectic map take the
section u∗ λ2 of T • X1 to the section λ2 of T • X2 . With this in mind we have
Lemma 60 If X is a compact oriented contact manifold, it can be embedded in
the standard contact sphere.
Proof: the standard
(2N − 1)-sphere of radiusPR has coordinates
P 2 contact
xj , yj (1 ≤ j ≤ N ,
xj + yj2 = R2 ) and contact form λ = xj dyj − yj dxj (or
a positive multiple of this).
PmIf X is a compact contact manifold, its contact form
can always be written 2P 2 xj dyj for some suitable choice of smooth functions
m
xj , yj , or just as well
1 xj dyj − yj dxj , setting for instance x1 = 1, y1 =
67
Pm
xj yj (m may be larger than the dimension). Adding suitably many other
P
1
pairs (xj , yj ) with yj = 0, xN = (R2 − x2j + yj2 ) 2 , for R large enough, we get
an embedding in a contact sphere of radius R.
2
Theorem 61 For any generalized Szegö projector Σ associated to a symplectic
cone with compact basis, we have trres S = 0. In particular if Σ corresponds to
a contact structure on a compact manifold, the logarithmic trace vanishes.
By lemma 60 we can suppose that Σ is embedded in the symplectic cone of
a standard odd contact sphere. Let B be the canonical Toeplitz algebra on the
sphere: then by prop. 57, the Toeplitz algebra A of S is isomorphic to EndB (M )
where M is a suitable good B module. Now on the sphere any good locally free
B-module is free (any complex vector bundle on an odd sphere is trivial), and
the Szegö projector has no logarithmic term, so and trres 1M = 0, for any free
hence also for any good B-module M .
This result is rather negative since it means that the logarithmic trace cannot
define new invariants distinguishing CR or contact manifolds. Note however that
that it is not completely trivial: it holds for the Toeplitz algebras associated
to a CR or contact structure, as constructed in [27], but a contact manifold
carries many other star algebras which are locally isomorphic to the Toeplitz
algebra (I showed in [24] how Fedosov’s classification of star products [60] can
be adapted to classify these algebras). Any such algebra A carries a canonical
trace, because the residual trace is invariant by all isomorphisms, so that local
traces glue together. If the contact basis is compact, the trace trres 1A is well
defined, but there are easy examples showing that it is not always zero.
68
9
Asymptotic equivariant index
of Toeplitz operators.
In this section we describe the asymptotic equivariant trace and index of Toeplitz
operators invariant under the action of a compact group G. This theory is an
avatar of M.F. Atiyah’s index theory for relatively elliptic pseudodifferential operators [5] on a G-manifold. Atiyah’s theory does not apply directly to Toeplitz
operators on a contact manifold, because the function space on which they act
(Toeplitz space) is only defined up to a space of finite dimension from symbolic calculus, so the absolute index or trace do not make much sense. The
G-asymptotic trace and index are weaker forms (Atiyah’s trace or index is a
distribution on G, the asymptotic trace or index is its singularity). The advantage of the asymptotic index is that it is well defined for Toeplitz operators,
whereas the “absolute” index is not, and it still contains useful information. We
have recently used it with E. Leichtnam, X. Tang, A. Weinstein [28], to give a
new “simple” proof of the Atiyah-Weinstein conjecture. We refer to loc. cit.
for further details about this formula, for which a proof was recently given by
C. Epstein [54], using “Heisenberg pseudodifferential calculus”.
9.1
Toeplitz operators
In this section we recall the mechanism of generalized Szegö projectors and
Toeplitz operators. We refer to [27, 21, 26] for more details.
As in [27, 21, 26], we call symplectic cone a smooth (paracompact) manifold
which is a principal R×
+ bundle, equipped with a symplectic form ω homogeneous
of degree 1. The Liouville form is its horizontal primitive λ = ρy ω (ω =
dλ), where ρ denotes the radial (Euler) vector field, infinitesimal generator of
homotheties. The basis X = Σ/R×
+ is an oriented contact manifold; its contact
form λX (any pull-back of λ by a smooth section) is defined up to a smooth
positive factor, and Σ is canonically identified with the set of positive multiples
of λX in T ∗ X.
9.1.1
Microlocal model
We first describe the microlocal model for generalized Szegö projectors given
in [14]. Let (x, y) = (x1 , . . . , xp , y1 , . . . , yq ) denote the variable in Rp+q . We
consider the system of pseudodifferential operators D = (Dj ) with
Dj = ∂yj + |Dx |yj (j = 1, . . . , q)
The Dj commute; the complex involutive variety char D is defined by the complex equations ηj − i|ξ|yj = 0; it is 0, in the sense of [102, 101]. Its real part
is the symplectic manifold Σ : {ηj = yj = 0}.
The kernel of D in L2 is the range of the Hermite operator H (in the sense
of [14]) defined by its partial Fourier transform:
q
2
1
f ∈ L2 (Rp ) 7→ Hf with Fx Hf (ξ, y) = (π −1 |ξ|) 4 e− 2 |ξ|y fˆ(ξ)
69
The orthogonal projector on ker D is S = HH ∗ :
Z
q
0
2
02
1
f 7→ (2π)−p
ei(<x−x ,ξ>+i 2 (y +y )) (π −1 |ξ|) 2 f (x0 , y 0 )dx0 dy 0 dξ
R2p+q
It is a Fourier integral operator, so as H; its complex canonical relation is 0,
with real part the graph of Id Σ (Fourier integral operators are described in
[82], Fourier integral operators with complex canonical relation are described in
[102, 101]).
9.1.2
Generalized Szegö projectors
Let M be a compact manifold, and Σ ⊂ T • M a symplectic subcone (T • M
denotes T ∗ M deprived of its zero section). A generalized Szegö projector associated to Σ (or Σ-Szegö projector) is a self adjoint9 elliptic Fourier integral
projector S of degree 0 (S = S ∗ = S 2 ), whose complex canonical relation C
is 0, with real part the diagonal diag Σ (elliptic means that the principal
symbol of S does not vanish on Σ).
From [27, 21, 26], we recall:
1) A Σ-Szegö projector S always exists. It is microlocally isomorphic (mod.
some elliptic FIO transformation) to the model above.
We will denote H ⊂ C −∞ (M ) its range. Modulo C ∞ , it defines a sheaf
µH on Σ - a subsheaf supported by Σ of the sheaf of microfunctions on
T •M .
2) Toeplitz operators defined by S are the operators on H of the form u ∈
H 7→ TP (u) = SP S(u) with P a pseudodifferential operator on M . More
generally, if P is any FIO whose canonical relation is complex positive,
with real part containing diag Σ, then SP S is a Toeplitz operator.
Modulo operators of degree −∞ (smoothing operators), Toeplitz operators
form a sheaf AΣ of algebras on Σ, acting on µH; (AΣ , µH) is locally
isomorphic to the sheaf of pseudodifferential operators in p real variables
(2p = dim Σ), acting on the sheaf of microfunctions. The principal symbol
(principal part) of TP is σ(P )|Σ .
3) If S, S 0 are two Σ-Szegö projectors with range H, H0 , S 0 induces a quasi
isomorphism H → H0 (the restriction of SS 0 to H is a positive (≥ 0) elliptic
Toeplitz operator).
More generally, if Σ ⊂ T • M, Σ0 ⊂ T • M 0 are two symplectic cones and
f : Σ → Σ0 a homogeneous symplectic isomorphism, there always exists a
Fourier integral operator F from M to M 0 , inducing an “elliptic” Fredholm
map H → H0 , e.g. there exists a complex canonical relation C 0 with
real part the graph of f , and we may take F = S 0 ◦ F 0 where F 0 is any
elliptic FIO with canonical relation C (such elliptic FIO exist, they were
called “adapted” in [27, 21]).
9 the
requirement that S be self adjoint is convenient but not essential
70
Thus the pair (AΣ , µH) consisting of the sheaf of micro Toeplitz operators
(i.e. mod smoothing operators), acting on µH is well defined, up to (non unique)
isomorphism: it only depends on the symplectic cone Σ, not on the embedding.
9.1.3
Holomorphic case
A first example of Toeplitz structure is Σ = T • M (M a compact manifold),
S = Id : the Toeplitz algebra is the algebra of pseudodifferential operators
acting on the sheaf of microfunctions on M .
In general, as noted above, the basis X = Σ/R×
+ of Σ is a contact manifold,
and Σ can be canonically embedded in T • X as the set of positive multiples of
the contact form. An important particular case is the holomorphic case: X
is the smooth, strictly pseudoconvex boundary of a Stein complex manifold;
the contact form of X is the form induced by Im ∂φ where φ is any defining
function (φ = 0, dφ 6= 0 on X, φ < 0 inside - e.g. if X is the unit sphere
bounding the unit ball of Cn , with defining function z̄ · z − 1, the contact form
is Im z̄ · dz|X ). Then the Szegö projector S is the orthogonal projector on the
space of boundary values of holomorphic functions in L2 (X) (the fact that it is
Fourier integral operator as above was proved in [15]).
The pseudodifferential algebra is a special case of holomorphic Toeplitz algebra: if M is a manifold, it has a real analytic compact manifold; if M c is a
complexification of M , small tubular neighborhoods of M in M c (for some hermitian metric) are Stein manifold with strictly complex boundary X ∼ S ∗ M ,
and the pseudodifferential algebra of M acting on microfunctions is isomorphic
to the Toeplitz algebra of X acting on H. In fact there exists an adapted Fourier
integral operator from M to X which defines an isomorphism from C −∞ (M )
to H(X)10 and interchanges pseudodifferential operators on M and Toeplitz
operators on X.
Note: the Atiyah-Weinstein problem can be described as follows: If X is a
compact contact manifold, and S, S 0 two Szegö projectors defined by two embeddable CR structures giving the same contact structure, then the restriction
of S 0 to H is a Fredholm operator H → H0 (SS 0 induces an elliptic Toeplitz operator on H). The Atiyah-Weinstein conjecture computes the index in terms of
topological data of the situation (topology of the holomorphic fillings of which
X is the boundary).
9.2
9.2.1
Equivariant trace and index
Equivariant Toeplitz algebra
R
Let G be a compact Lie group, dg its Haar measure ( dg = 1), g its Lie algebra.
Let Σ be a G-symplectic cone (with compact basis), ω its (invariant) symplectic form, λ the Liouville form (ω = dλ). As mentioned above, the basis
10 e.g.
eiA with A =
√
−∆ for some real analytic Riemannian metric on M , cf [16].
71
X = Σ/R×
+ is a G-compact oriented contact manifold; replacing it by its Gmean, we may choose an invariant form λX defining the contact structure, and
Σ is canonically identified with the set of positive multiples of λX in T ∗ X.
As was shown in [27, 21], the statements of §1 allow a compact group action: if M is a compact G-manifold and Σ is embedded as an invariant symplectic subcone of T • M , there exists a G-invariant generalized Szegö projector associated to Σ 11 ; if S 0 is another one, it induces an equivariant Fredholm map H → H0 , and more generally if u is an equivariant isomorphism
Σ ⊂ T • M → Σ0 ⊂ T • M 0 , there exists an equivariant adapted FIO F inducing
an equivariant elliptic Toeplitz FIO H → H0 .
If S is an equivariant generalized Szegö projector, G acts on H and on the
Toeplitz algebra, so as on their microlocalization µH, AΣ . The infinitesimal
generators of G (vector fields image of elements ξ ∈ g) define Toeplitz operators
Tξ of degree 1 on H. The elements of G act as unitary Fourier integral operators
- or “Toeplitz-FIO’s”.
The Toeplitz space H (and its Sobolev counterparts) splits according to
L
the irreducible representations of G: H = c Hα (the same will hold for the
equivariant “Toeplitz bundles” below).
9.2.2
Equivariant trace
The G-trace and G-index (relative index in [5]) were introduced by M.F. Atiyah
in [5] for equivariant pseudo-differential operators on a G-manifold. The G-trace
of P is a distribution on G, describing tr (g ◦ P ). Here we adapt this to Toeplitz
operators.
Below we will use the following extension: an equivariant Toeplitz bundle
is the range of an equivariant Toeplitz projector P of degree 0 on some HN .
The symbol of E is the range of the principal symbol of P ; it is an equivariant
vector bundle on X; any equivariant vector bundle on X is the symbol of an
equivariant Toeplitz bundle. We will denote by E(s) its space of Sobolev H s
sections.
If E, F are two equivariant Toeplitz bundles, there is an obvious notion of
Toeplitz (matrix) operator P : E → F, and of its principal symbol σd (P ) if it
is of degree d, which is a homogeneous vector-bundle homomorphism E → F
on Σ. P is elliptic if its symbol is invertible; then it is a Fredholm operator
E(s) → F(s−d) and has an index which does not depend on s.
Definition 62 We denote char g (characteristic set of g) the closed subcone of
Σ where all symbols of infinitesimal operators Tξ , ξ ∈ g vanish.
char g contains the fixed point set ΣG , whose basis is the fixed point set X G
(because G is compact). The base Z of char g is the set of points of X where all
11 e.g.
the Szegö projector of an invariant embeddable CR structure is invariant.
72
Lie generators Lξ , ξ ∈ g are orthogonal to λX . Note that ΣG is always a smooth
symplectic cone and its base X G a smooth contact manifold; char g and Z may
be singular.
Let E be an equivariant Toeplitz bundle. If P : E → E is a Toeplitz operator
of trace class (deg P < −n), the trace function TrG
P (g) = tr (g◦P ) is well defined;
it is a continuous function on G. It is smooth if P is of degree −∞ (P ∼ 0). If
P is equivariant, its Fourier coefficient for the representation α is d1 α tr P|Hα (dα
the dimension of α).
The following result is an immediate adaptation of the similar result of [5]
for pseudo-differential operators.
Proposition 63 Let P : E → E be a Toeplitz operator, with P ∼ 0 near char g.
Then TrG
P (g) = tr g◦P is well defined as a distribution on G. If P is equivariant,
tr P |Hα is well defined (finite), and we have, in distribution sense:
TrG
P =
X 1
tr P|Hα χα
dα
(48)
where α runs over the set of irreducible representation of G, with dimension dα
and character χα .
We have seen above that this is true if P is of trace class. Let DG be a
bi-invariant elliptic operator of order m > 0 on G, e.g. the Casimir of a faithful
representation (with m = 2); its image DX on X defines an invariant Toeplitz
operator E → F, with characteristic set char g.
If P ∼ 0 near Σ, we can divide it repeatedly by DX (mod. smoothing
operators) and get for any N :
N
P = DX
Q+R
with R ∼ 0
The degree of Q is deg P − mN , so it is of trace class if N is large enough. We
G
G
N
set TrG
P = DG TrQ + TrR : this is well defined as a distribution; the fact that it
does not depend on the choice of DG , N, Q, R is immediate.
Formula 48 for equivariant operators, obviously follows. Note that the series
converges in distribution sense, i.e. the coefficients have at most polynomial
growth (with respect to the eigenvalues of DG ).
More generally assume that we have an equivariant Toeplitz complex of finite
length:
d
(E, d) : · · · → Ej −
→ Ej+1 → . . .
i.e. E is a finite sequence Ek of equivariant Toeplitz bundles, d = (dk : Ek →
Ek+1 ) a sequence of Toeplitz operators such that d2 = 0. Then for a Toeplitz
operator
P : E → E, P ∼ 0 near char g, its equivariant supertrace TrG
P =
P
k
(−1) TrG
Pk is well defined; it vanishes if P is a supercommutator.
73
9.2.3
Equivariant index
Let E0 , E1 be two equivariant Toeplitz bundles. We will say that an equivariant
Toeplitz operator P : E0 → E1 is G-elliptic (relatively elliptic in [5]) if it is
elliptic on char g, i.e. the principal symbol σ(P ), which is a homogeneous equivariant vector bundle homomorphism E0 → E1 , is invertible on char g. Then
there exists an equivariant Q : E0 → E1 such that QP ∼ 1E0 , P Q ∼ 1E1 near
G
char g. The G-index IndIPG is then defined as the distribution TrG
1−QP −Tr1−P Q .
More generally, an equivariant complex (E, d) as above is G-elliptic if the
principal symbol σ(d) is exact on char g. Then there exists an equivariant
Toeplitz operator s = (sk : Ek → Ek−1 ) such that 1 − [d, s] ∼ 0 near char g
G
([d, s] = ds + sd). The index (Euler characteristic) is the super trace I(E,d)
=
P
G
j
supertr (1 − [d, s]) = (−1) Tr(1−[d,s])j .
If P is G-elliptic, for any irreducible representation α, the restriction Pα :
E0,α → E1,α is a Fredholm operator: its kernel, cokernel and index Iα are
finite dimensional (resp. more generally the cohomology Hα∗ of d|Eα is finite
dimensional), and we have
Ind IPG =
9.2.4
X Iα
χα
dα
G
(resp. Ind I(E,d)
=
X
dim Hαj
(−1)j
χα )
dα
(49)
Asymptotic index
The G-index Ind G
P is obviously invariant under compact perturbation and deformation, so for fixed Ej it only depends on the homotopy class of the symbol
σ(P ). However it does depend on the choice of Szegö projectors: as mentioned,
the Toeplitz bundles Ej are known in practice only through their symbols Ej ,
and are only determined up to a space of finite dimension, so as the Toeplitz
spaces H. However if E, E0 are two equivariant Toeplitz bundles with the same
symbol, there exists an equivariant elliptic Toeplitz operator U : E → E0 with
quasi-inverse V (i.e. V U ∼ 1E , U V ∼ 10E ). This may be used to transport
equivariant Toeplitz operators from E to E0 : P 7→ Q = U P V . Then if P ∼ 0
on X0 , Q = U P V and V U P have the same G-trace, and since P ∼ V U P , we
have TPG − TQG ∈ C ∞ (G). Thus the equivariant G-trace or index are ultimately
well defined up to a smooth function on G.
Definition 64 We define the asymptotic G-trace AsTrG
P as the singularity of
G
∞
the distribution TrG
(i.e.
Tr
mod.
C
(G)).
P
P
If P ∼ 0, we have TrG
P ∼ 0, i.e. the sequence of Fourier coefficients is of
rapid decrease, O(cα )−m for all m, where cα is the eigenvalue of DG in the
representation α (where DG is as above a bi-invariant elliptic operator on G).
Definition 65 If P is elliptic on char g, the asymptotic G-index AsIndG
P is
defined as the singularity of IndG
.
P
74
It only depends on the homotopy class of the principal symbol σ(P ), and since
it is obviously additive we get :
G
Theorem 66 The asymptotic index defines an additive map from KX−Z
(X)
−∞
∞
to Sing(G) = C
/C (G)(Z ⊂ X denotes the basis of char g).
G
KX−Z
(X) denotes the equivariant K-theory of X with compact support in X −
Z, i.e. the group of stable classes of triples (E, F, u) where E, F are equivariant
G-bundles on X, u an equivariant isomorphism E → F defined near Z, with
the usual equivalence relations ((E, F, a) ∼ 0 if a is stably homotopic near Z to
an isomorphism on the whole of X). The asymptotic index is also defined for
equivariant Toeplitz complexes, exact near Z.
Note the sequence of Fourier coefficients d1α tr Pα is at most of polynomial
growth with respect to the eigenvalues of DG ; if P ∼ 0 it is of rapid decrease.
The Fourier coefficients of the asymptotic index are integers, so they are completely determined, except for a finite number of them, by the asymptotic index:
G
AsIndG
P = 0 means that the Fourier series of IndP has finite support.
Example : let Σ be a symplectic cone, with free positive elliptic action of
U (1), i.e. the Toeplitz generator A = 1i ∂θ is elliptic with positive symbol (this is
the situation studied in [27]). Then the algebra of invariant Toeplitz operators
(mod. C ∞ ) is a deformation star algebra, setting as deformation “parameter”
~ = A−1 . char g is empty and the asymptotic trace or indexP
is always defined.
∞
kiθ
The asymptotic trace of any element a is the series
, ak =
−∞ ak e
tr a|Hk , mod. smooth functions of θ, i.e. the sequence (ak ) is known mod.
rapidly decreasing sequences. It is standard knowledge that the sequence (ak )
has an asymptotic expansion:
X
ak ∼
αj k −j .
(50)
k≤k0
In this case the asymptotic trace is just as well defined by this asymptotic
expansion, which encodes essentially the same thing as the residual trace.
Remark. For a general the circle group action, with generator A = eiθ ,
all simple representations are powers of the identity representation, denoted T ,
and all representations occurring as indices can be written as sums.
X
nk T k (mod. finite sums)
(51)
k∈Z
In fact, using the sphere embedding below, it can be seen that the positive and
negative parts of the series have a weak periodicity property: they are of the
form
P± (T, T −1 )
(1 − T ±k )k
for a suitable polynomial P± and some integer k; in other words they represent
rational functions whose poles are roots of 1, and whose Taylor series have
integral coefficients.
75
9.3
K-theory and embedding
It will be convenient (even though not technically indispensable) to reformulate
some constructions above in terms of sheaves of Toeplitz algebras and modules,
in particular to follow the index in an embedding (§9.3.2).
9.3.1
A short digression on Toeplitz algebras and modules
As above we use the following notation: for distributions, f ∼ g means that
f − g is C ∞ ; for operators, A ∼ B (or A = B mod. C ∞ ) means that A − B
is of degree −∞, i.e. has a smooth Schwartz kernel. If M is a manifold, T • M
denotes the cotangent bundle deprived of its zero section; it is a symplectic cone
with base the cotangent sphere S ∗ M = T • M/R+ .
As pointed out above, if Σ is a G-symplectic cone, the micro sheaf AΣ of
Toeplitz operators acting on µH are well defined with the action of G, up to
(non unique) isomorphism, independently of any embedding Σ → T • M . The
G
asymptotic trace AsTrG
P resp. index AsIndP are well defined for a section P of
AΣ vanishing (resp. invertible) near char g.
If M is a G-manifold and X = S ∗ M (Σ = T • M ), AΣ identifies with the
sheaf of pseudodifferential operators acting on the sheaf µH of microfunctions
on X (note that even in that case the exact index problem does not make sense:
a Toeplitz bundle E on X corresponds to a vector bundle on the cotangent E
on X, not necessarily the pull-back of a vector bundle on M , so E is in general
at best defined up to a space of finite dimension).
It will be convenient to use the language of E-modules. In the C ∞ category
E is not coherent and general E-module theory is not practical. We will just
stick to two useful examples.12
If M is an A-module, resp. a complex of A modules, it corresponds to a system of pseudodifferential (resp. Toeplitz) operators, whose sheaf of solutions is
Hom A (M, µH). E.g. a locally free complex of (E, d)-modules defines a Toeplitz
complex (E, D) = Hom (L, H).
More generally we will say that a E-module
M is “good” if it isTfinitely
S
generated, equipped with a filtration M = Mk (i.e. Ep Mq = Mp+q , Mk =
0) such that the symbol gr M has a finite locally free resolution. We denote
σ(M) = M0 /M−1 , which is a sheaf of C ∞ modules on the basis X; since there
exist global elliptic sections of E, gr M is completely determined by the symbol,
so as the resolution.
It is elementary that a resolution of σ(M) lifts to a “good resolution” of M,
i.e. a good finite locally free resolution of M13 . It is also standard that two
resolutions of σ(M) are homotopic, and if σ(M) has locally finite locally free
12 In proof of the Atiyah-Weinstein conjecture we need to patch together two smooth embedded manifolds near their boundaries: this cannot be done in the real analytic category,
where things work slightly better
13 the converse is not true: if d is a locally free resolution of M its symbol is not necessarily
a resolution of the symbol of M - if only because filtrations must be defined to define the
symbol and can be modified rather arbitrarily.
76
resolutions it also has a global one (because we are working in the C ∞ category
on a compact manifold or cone with compact support, and dispose of partitions
of unity); this lifts to a global good resolution of M.
If M is “good”, it defines a K-theoretical element [M] ∈ KY (X) (Y =
supp σ(M)), viz. the K-theoretical element defined by the symbol of any good
resolution (this does not depend on the resolution of σ(M ) since any two such
are homotopic).
This works just as well in presence of a G-action (one must choose invariant
filtrations etc.).
The asymptotic trace and index extend in an obvious manner to endomorphisms of good complexes or modules:
• if M = AN is free, End A (M) identifies with the ring of N × N matrices
with coefficients in the opposite ring Aop , and if A = (Aij ) vanishes near
P
char g we set AsTrG (A) = AsTrG (Ajj ).
• If M is isomorphic to the range P N of a projector P in a free module N
(this does not depend on the choice of N ), or if A ∈ End A (M) we set
AsTrG (A) = AsTrG (P A).
• If (L, d) is a locally free complex and A isPa A = (Ak ) endomorphism,
vanishing near char g, we set AsTrG (A) = (−1)k AsTrG (Ak ) (the Euler
characteristic or super trace; if A, B are endomorphisms of opposite de2
grees m, −m, we have AsTrG [A, B] = 0, where [A, B] = AB − (−1)m BA
is the superbracket).
• If M is a good A-module, (L, d) a good locally free resolution of M,
e where A
e is any extension
A ∈ End A (M), we set AsTrG (A) = AsTrG (A),
of A to (L, d) (such an extension exists, and is unique up to homotopy i.e.
up to a supercommutator).
• Finally if M is a locally free complex with symbol exact on char g, or a
good A-module with support outside of char g, it defines a K-theoretical
element [M] ∈ KZG (X), and its asymptotic index (the supertrace of the
identity), is the image of [M] by the index map of Theorem 66.
Remark. The equivariant trace or index are defined just as well for modules
admitting a projective resolution (projective meaning direct summand of some
AN , with a projector not necessarily of degree 0). What does not work for these
more general objects is the relation to topological K-theory.
9.3.2
Embedding
Let Σ be a G-symplectic cone, embedded equivariantly in T • M with M a compact G-manifold, and S an equivariant Szegö projector. As recalled in §1, the
77
range µH of S is the sheaf of solutions of an ideal I ⊂ EM . The corresponding
EM -module M = EM /I is good as one can see on the microlocal model.
We have End E (M) = [I : I], the set of ψDO a such that Ia ⊂ I, acting
G
on the right. The map a 7→ TrG
a (Tra f (1) = f a(1)) is an isomorphism from
End E (M) to the algebra of Toeplitz operators mod. C ∞ . M is a E, E 0 bimodule.
If P is a (good) E 0 -module, the transfered module is M ⊗E 0 P, which
has the same solution sheaf (Hom (M ⊗ P, H) = Hom (P, Hom (M, H)) and
Hom (M, H) = H0 ). Thus the transfer preserves traces and indices.
This extends obviously to the case where Σ is embedded equivariantly in
another symplectic cone Σ ⊂ Σ0 : the small Toeplitz sheaf µH is realized as
Hom AΣ (M, µH0 ), with M = E/I and I ⊂ E is the annihilator of the Szegö
projector S of Σ.
Theorem 67 Let X 0 , X be two compact contact G-manifolds and f : X →
X 0 be an equivariant embedding. Then the K-theoretical push-forward (Bott
G
G
0
homomorphism) KX−Z
(X) → KX
0 −Z 0 (X ) commutes with the asymptotic G
index.
Let F : AΣ → A0Σ be an equivariant embedding of the corresponding Toeplitz
algebras (above f ), and let M be the A0Σ -module associated with the Szegö
projector SΣ . We have seen that transfer P 7→ M ⊗ P preserves the asymptotic
index.
Lemma 68 The K-theoretical element (with support in Σ) [M] ∈ KΣG (T • M )
is precisely the Bott element used to define the Bott isomorphism K G (X) →
G
(X 0 ).14
KX
Proof: We have already noticed that M is good; it has, locally (and globally), a good resolution. Its symbol is a locally free resolution of σ(M) =
C ∞ (X)/σ(I). Let us identify a small equivariant tubular neighborhood of Σ
with the normal tangent bundle N of Σ in Σ0 ; N is a symplectic bundle; the
ideal I endows it with a compatible positive complex structure N c , i.e. the first
order jet of elements of σ(I) are holomorphic in the fibers of N c ; if a, b are such
symbols we have {a, b}N = 0; 1i {a, ā}N 0. In such a neighborhood a good
symbol resolution is homotopic toVthe Koszul complex : the Koszul complex
−p
is the complex (E, d) with Ep =
(N c∗ ) (0 if p > 0), the differential d at
a point with complex coordinates z of N is the interior product (contraction)
0
dω = zyω. The K-theoretical element [(E, d)] ∈ GG
Σ (Σ ) is precisely the Bott
element.
14 if f : X → Y is a map between manifolds (or suitable spaces), the K-theoretical pushforward is the topological translation of the Grothendieck direct image in K-theory (for algebraic or holomorphic spaces). Its definition requires a spinc structure on the virtual normal
of f (cf [29], §1.3) and this always exists (canonically) if X, Y are almost symplectic or almost
complex, or as here if f is an immersion whose normal tangent bundle is equipped with a
symplectic or complex structure.
78
E.g. if Σ0 = CN − {0}, with Liouville form Im z̄.dz 15 , with basis the unit
sphere X = S 2N −1 , H the space of holomorphic functions on the sphere X 0 =
S 2N −1 , X ⊂ X the diameter z1 = · · · = zk = 0, Σ0 , H0 = the functions
independent of z1 , . . . , zk , I is the ideal spanned by the Toeplitz operators T∂k .
Pk
The transfer module M is A/I with I = 0 zj A, its resolution is the standard
Koszul complex.
Remark. It is always possible to embed a compact contact manifold in a
canonical contact sphere with linear G-action:
Proposition 69 Let Σ be a G cone (with compact base), λ a G-invariant horizontal 1-form homogeneous of degree 1, i.e. Lρ λ = λ, ρyλ = 0, where ρ is
the radial vector field, generating homotheties. Then there exists an equivariant
homogeneous embedding x 7→ Z(x) of Σ in a complex unitary representation V c
of G such that λ = Im Z̄.dZ
In this construction, Z must be homogeneous of degree 12 as above. This applies
of course if Σ is a symplectic cone, λ its Liouville form (the symplectic form is
ω = dλ and λ = ρyω). We first choose a smooth equivariant function Y = (Yj ),
homogeneous of degree 12 , realizing an equivariant embedding of Σ in V − {0},
where V is a real unitary G-vector space (this always exists if the basis is
compact). Then there exists a smooth function X = (Xj ) homogeneous of
degree 21 Rsuch that λ = 2X.dY . We can suppose X equivariant, replacing it by
its mean g.X(g −1 x) dg if need be. Since Y is of degree 12 we have 2ρydY = Y
hence X.Y = ρy(2X.dY ) = 0. Finally we get
λ = Im Z̄.dZ
with Z = X + iY
(the coordinates zj on V are homogeneous of degree
form Im Z̄.dZ is of degree 1)
15 the
coordinates zj are homogeneous of degree
79
1
.
2
1
2
so that the canonical
10
Asymptotic equivariant index :
Atiyah-Weinstein index formula.
This section introduces and illustrates the ”equivariant asymptotic index of
Toeplitz operators”.16
Let Ω, Ω0 be two bounded Stein domains (or manifolds) with smooth strictly
pseudoconvex boundaries X0 , X00 (these are compact contact manifolds), and f0
a contact isomorphism X0 → X00 . If H0 , H00 denote the spaces of CR functions
(or distributions) on X0 , X00 (boundary values of holomorphic functions), S, S 0
the Szegö projectors,17 the map E0 : u 7→ S 0 (u ◦ f0−1 ) : H0 → H00 is Fredholm
(it is an elliptic Toeplitz FIO). The index of E0 was introduced by C. Epstein
[52, 53, 54, 55], who called it the relative index of the two CR structures. A
formula for the index was proposed in [114]. A special case was established
in [56], and a proof of this index formula in the general case was given by C.
Epstein in [54], based on an analysis of the situation using the “Heisenbergpseudodifferential calculus”. (Another proof based on deformation quantization
should also be possible, using the ideas in [23] and [24].) In this paper we
propose a simpler proof based on equivariant Toeplitz-operator calculus, which
gives a straightforward view. Our formula is described in section 10.3.4. It is is
essentially equivalent to the formula proposed in [114], which was stimulated by
a problem in the theory of Fourier Integral Operators, a subject in which Hans
Duistermaat was a pioneer [46].
It is awkward to keep track of the index in the setting of Toeplitz operators
on X0 and X00 alone, because we are dealing with several Szegö projectors, and
Toeplitz operator calculus controls the range H of a generalized Szegö projector
at best up to a vector space of finite rank18 .
e ⊂ C × Ω defined by tt̄ < φ where
To make up for this, we use the ball Ω
t is the coordinate on C, φ is a smooth defining function (φ = 0, dφ 6= 0 on
X0 = ∂Ω, φ > 0 inside - note that this is the opposite sign from the usual
one) chosen so that Log φ1 is strictly plurisubharmonic, so that the boundary
e is strictly pseudoconvex; such a defining function always exists, e.g.
X = ∂Ω
we can choose φ strictly pseudoconvex. X is then a compact contact manifold
with (positive) action of the circle group U (1). We will identify X0 with the
submanifold {0} × X0 of X.
We perform the same construction for Ω0 : we will see that there exists
an equivariant germ near X0 of equivariant contact isomorphism f : X →
X 0 extending f0 such that t0 ◦ f is a positive multiple of t, and an elliptic
equivariant Toeplitz FIO E extending E0 , associated 19 to the contact map f ;
16 to appear in ’Geometric Aspects of Analysis and Mechanics’, a conference in honor of
Hans Duistermaat, Progress in Math, Birkhaser 2010.
17 The definition of S requires choosing a smooth positive density on X ; nothing of what
0
follows depends on this choice.
18 There is no index formula for a vector bundle elliptic Toeplitz operator, although there
is one for matrix Toeplitz operators - a straightforward generalisation of the Atiyah-Singer
formula, cf. [17].
19 f is associated to E in the same manner as a canonical map is associated to a FIO.
80
the holomorphic spaces H, H0 split in Fourier components Hk , H0k on which the
index is repeated infinitely many times. This construction has the advantage of
taking into account the geometry of the two fillings Ω, Ω0 , which obviously must
come into the picture.
The final result can be then expressed in terms of an asymptotic version of
the relative index (G-index) of E, derived from the equivariant theory of M.F.
Atiyah and I.M. Singer [5]: the asymptotic index, described in §10.3.4, ignores
finite dimensional spaces and is well defined for Toeplitz operators or Toeplitz
systems; it is also preserved by suitable contact embeddings.
The asymptotic equivariant trace and index are described in §10.1,10.2. The
relative index formula is described and proved in §10.3 (Theorem 126 and 84).
10.1
Equivariant trace and index
10.1.1
Equivariant Toeplitz Operators.
R
Let G be a compact Lie group with Haar measure dg ( dg = 1), g its Lie
algebra, and X a smooth compact co-oriented contact manifold with an action
of G: this means that X is equipped with a contact form λ (two forms define
the same oriented contact structure if they are positive multiples of each other);
G acts smoothly on X and preserves the contact structure, i.e.
R for any g the
image g∗ λ is a positive multiple of λ; replacing λ by the mean g∗ λdg, we may
suppose that it is invariant. The associated symplectic cone Σ is the set of
positive multiples of λ in T ∗ X, a principal R+ bundle over X, a half-line bundle
over X.
We also choose an invariant measure dx with smooth positive density on X,
so L2 norms are well defined. The results below will not depend on this choice.
It was shown in [27] that there always exists an invariant generalized Szegö
projector S which is a self adjoint Fourier-integral projector whose microsupport
is Σ, mimicking the classical Szegö projector. S extends or restricts to all
Sobolev spaces; for s ∈ R we will denote by H(s) the range of S in the Sobolev
space H s (X), and by H the union.
A Toeplitz operator of degree m on H is an operator of the form f 7→
TQ f = SQf , where Q is a pseudodifferential operator of degree m. Here we use
pseudodifferential operators in a strict sense, i.e. in any P
local set of coordinates
the total symbol has an asymptotic expansion q(x, ξ) ∼ k≥0 qm−k (x, ξ) where
qm−k is homogeneous of degree m − k with respect to ξ, and the degree m
and k ≥ 0 are integers 20 . A Toeplitz operator of degree m is continuous
H(s) → H(s−m) for all s. Recall that Toeplitz operators give rise to a symbolic
calculus, microlocally isomorphic to the pseudodifferential calculus, that lives
on Σ (cf. [27]).
In particular, the infinitesimal generators of G (vector fields determined by
elements ξ ∈ g) define Toeplitz operators Tξ of degree 1 on H. An element P of
20 We will occasionally use as multipliers operators of degree m =
number), with k still an integer in the expansion.
81
1
2
(or any other complex
the universal enveloping algebra U (g) acts as a higher order Toeplitz operator
PX (equivariant if P is invariant), and the elements of G act as unitary Fourier
integral operators - or “Toeplitz-FIO”.
H (with its Sobolev counterparts) splits according to the irreducible repreL
sentations of G: H = c Hα .
Below we will use the following extended notions: an equivariant Toeplitz
bundle E is the range of an equivariant Toeplitz projector P of degree 0 on a
direct sum HN . The symbol of E is the range of the principal symbol of P ; it is
an equivariant vector bundle on X. Any equivariant vector bundle on X is the
symbol of an equivariant Toeplitz bundle (this also follows from [27]).
10.1.2
G-trace
The G-trace and G-index (relative index in [5]) were introduced by M.F. Atiyah
in his joint work with I.M. Singer [5] for equivariant pseudodifferential operators
on G-manifolds. The G-trace of such an operator A is a distribution on G,
describing tr (g ◦ A). Here we adapt this to Toeplitz operators. Because the
Toeplitz spaces H or E are really only defined up to a finite dimensional space,
their G-trace or index are ultimately only defined up to a smooth function,
i.e. they are distribution singularities on G (distributions mod C ∞ ); they are
described below, and renamed “asymptotic G-trace or index”.
If E, F are equivariant Toeplitz bundles, there is an obvious notion of Toeplitz
(matrix) operator P : E → F, and of its principal symbol σd (P ) (if it is of degree
d), a homogeneous vector-bundle homomorphism E → F over Σ. P is elliptic
if its symbol is invertible; it is then a Fredholm operator Es → Fs−d and has an
index which does not depend on s.
If E is an equivariant Toeplitz bundle and P : E → E is a Toeplitz operator
of trace class21 (deg P < −n), the trace function22 TrG
P (g) = tr (g ◦ P ) is well
defined; it is a continuous function on G. It is smooth if P is of degree −∞
(P ∼ 0). If P is equivariant, its Fourier coefficient for the representation α is
1
dα tr P |Eα (with dα the dimension of α, Eα the α-isotypic component of E).
Definition 70 We denote by char g ⊂ Σ the characteristic set of the G-action,
i.e. the closed subcone where all symbols of infinitesimal operators Tξ , ξ ∈ g
vanish (this contains the fixed point set ΣG ). The base of char g is the set of
points of X where all Lie generators Lξ , ξ ∈ g, are orthogonal to the contact
form λ; in the sequel we will usually denote it by Z ⊂ X.
21 dim X = 2n − 1. The Toeplitz algebra is microlocally isomorphic to the algebra of
pseudodifferential operators in n real variables, and operators of degree < −n are of trace
class.
22 We still denote by g the action of a group element g through a given representation; for
example if we are dealing with the standard representation on functions, gf = f ◦ g −1 , also
denoted by g∗ f , g ∗−1 f , or g −1∗ f .
82
The fixed point set X G is the base of ΣG because G is compact (there is an
invariant section). Z contains the fixed point set X G . Note that ΣG is always
a smooth symplectic cone and its base X G is a smooth contact manifold; char g
and Z may be singular.
The following result is an immediate adaptation of the similar result for
pseudodifferential operators in [5].
Proposition 71 Let P : E → E be a Toeplitz operator, with P ∼ 0 near char g
(i.e. its total symbol vanishes near char g). Then TrG
P = tr (g ◦P ) is well defined
as a distribution on G. If P is equivariant, we have, in distribution sense:
TrG
P =
X 1
(tr P |Eα ) χα
dα
(52)
where α runs over the set of irreducible representations, dα is the dimension
and χα the character.
We have seen above that this is true if P is of trace class. For the general case,
let DG be a bi-invariant elliptic operator of order m > 0 on G (e.g. the Casimir
of a faithful representation, with m = 2). Since DG is in the center of U (g),
the Toeplitz operator DX : E → E it defines is invariant, with characteristic set
char g.
If P ∼ 0 near char g, we can divide it repeatedly by DX (modulo smoothing
operators) and get for any N :
N
P = DX
Q+R
with R ∼ 0.
The degree of Q is deg P −N deg (DG ), so it is of trace class if N is large enough.
G
G
N
We set TrG
P = DG TrQ + TrR : this is well defined as a distribution; the fact that
this does not depend on the choice of DG , N, Q, R is immediate.
Formula (52) for equivariant operators is obvious for trace class operators,
N
N
and the general case follows by application of DX
and DG
. Note that the series
in the formula converges in the distribution sense, i.e. the coefficients have at
most polynomial growth.
Slightly more generally, let (E, d)
d
· · · → Ej −
→ Ei+1 → · · ·
be an equivariant Toeplitz complex of finite length, i.e. E is a finite sequence Ek
of equivariant Toeplitz bundles, d = (dk : Ek → Ek+1 ) a sequence of Toeplitz
operators such that d2 = 0. If the (degree zero) endomorphism
P = {Pk } of the
P
G
k
complex E is ∼ 0 near char g, its supertrace TrG
=
(−1)
Tr
P
Pk is well defined;
it vanishes if P = [P1 , P2 ] is a supercommutator with one factor ∼ 0 on char g.
83
10.1.3
G index
Let E0 , E1 be two equivariant Toeplitz bundles. An equivariant Toeplitz operator P : E0 → E1 is G- elliptic (relatively elliptic in [5]) if it is elliptic on char g,
i.e. the principal symbol σ(P ), which is a homogeneous equivariant bundle homomorphism E0 → E1 , is invertible on char g. Then there exists an equivariant
Q : E1 → E0 such that QP ∼ 1E0 , P Q ∼ 1E1 near char g. The G-index IPG is
G
defined as the distribution TrG
1−QP − Tr1−P Q .
More generally,23 an equivariant complex E as above is G-elliptic if the
G
([d, s] = ds + sd). The index (Euler characteristic) is the super trace I(E,d)
=
P
G
str (1 − [d, s]) = (−1)j Tr(1−[d,s])j .
For any irreducible representation α, the restriction Pα : E0,α → E1,α is a
Fredholm operator with index Iα , (resp. the cohomology Hα∗ of d |Eα is finite
dimensional), and we have
IPG =
X 1
Iα χα
dα
G
(resp. I(E,D)
=
X (−1) j
j,α
dα
dim Hαj χα ).
G
IA
is obviously invariant under compact perturbation and deforThe G-index
mation, so it only depends on the homotopy class of σ(P ) once Ej has been
chosen; it does depend on a choice of Ej (on the projector that defines it, or on
the Szegö projector), because Ej is determined by its symbol bundle only up to
a finite dimensional space; this inconvenience is removed with the asymptotic
index below.
It is sometimes convenientPto note an index as an infinite representation
(mod finite representations)
nα χα . For the circle group U (1), all simple
representations are powers of the tautological representation, denoted J, and
all representations occurring as indices have a generating series
X
nk J k (mod finite sums)
(53)
k∈Z
In fact the positive and negative partsQ
of the series have a weak periodicity property: they are of the form P± (J ±1 )/ i (1 − (J ±1 )ki ) for a suitable polynomial
P± and positive integers ki . 24
HereP
in our relative index problem, only very simple representations of the
∞
form m 0 J k = m(1 − J)−1 (for some integer m) will occur.
23 This reduces to the case of a single operator where the complex is concentrated in degrees
0 and 1.
24 This notation denotes the series expansion in positive powers of J ±1 ; it is obviously
abusive but suggestive - especially if one thinks of the extension to a multidimensional torus;
it also represents a rational function whose poles are roots of 1, and whose Taylor series has
integral coefficients; of which the corresponding distribution on G is the boundary value from
one or the other side of the circle in the complex plane. Something similar occurs for any
compact group, cf. [5].
84
10.2
A crucial point in the proof of the Atiyah-Singer index theorem [3] consists in
showing how one can embed an elliptic system A in a simpler manifold where the
index theorem is easy to prove, preserving the index and keeping track of the Ktheoretic element [A]. The new embedded system F+ A is analogous to a derived
direct image (as in algebraic geometry), and the K-theoretic element [F+ A]
is the image of [A] by the Bott homomorphism constructed out of R. Bott’s
periodicity theorem (cf.[3]). Here we will do the same for Toeplitz operators.
The direct image F+ A is even somewhat more natural, as is its relation to the
Bott homomorphism (§10.2.4). The direct image for elliptic systems does not
preserve the exact index, since this is not defined (because the Toeplitz space
H is at best only defined mod a space of finite rank); but it does preserve the
asymptotic equivariant index.
10.2.1
A short digression on Toeplitz algebras
We use the following notation: for distributions, f ∼ g means that f − g is
C ∞ ; for operators, A ∼ B (or A = B mod C ∞ ) means that A − B is of degree
−∞, i.e. has a smooth Schwartz-kernel. If M is a manifold, T • M denotes the
cotangent bundle deprived of its zero section; it is a symplectic cone with base
S ∗ M = T • M/R+ , the cotangent sphere bundle.
As mentioned above, a compact contact G-manifold always possesses an
invariant generalized Szegö projector. More generally, if M is a G manifold,
Σ ⊂ T • M an invariant symplectic cone, there exists an associated equivariant
Szegö projector (cf [27]). If Σ ⊂ T • M, Σ0 ⊂ T • M 0 and f : Σ → Σ0 is an
isomorphism of symplectic cones, there always exists an “adapted FIO” F which
defines a Fredholm map u 7→ F̃ u = S 0 (F u) : H → H0 and an isomorphism of
the corresponding Toeplitz algebras (A 7→ F̃ AF̃ −1 , mod C ∞ ).
One can choose F equivariant if f is. Indeed any adapted FIO can be defined
using a global phase function φ on T • (M × M 0op ) such that25
1) φ vanishes on the graph of f , and dφ coincides with the Liouville form
ξ · dx − η · dy there;
2) Im φ 0, i.e. Im φ > 0 outside of the graph of f , and the transversal
hessian is R 0; replacing φ by its mean gives an invariant phase; we may set
F f (x) = eiφ af (y)dy dηdξ where the density a(x, ξ, y, η)dy dηdξ is a symbol,
invariant and positive elliptic (F is of Sobolev degree deg (a dy dη dξ)− 34 (nx +ny )
(cf. L. Hörmander [82]), so a is possibly of non integral degree if we want F of
degree 0). The transfer map from H to H0 is S 0 F S.
If M is a manifold and X = S ∗ M , the cotangent sphere, X carries a canonical Toeplitz algebra, viz. the sheaf ES ∗ M of pseudo-differential operators acting
on the sheaf µ of microfunctions. In general, if X is a contact manifold, we will
denote by EX (or just E) the algebra of Toeplitz operators on X. It is a sheaf
25
op in M 0op refers to the change of sign in the symplectic form on T ∗ M 0 .
85
of algebras on X acting on µH = H mod C ∞ , which is a sheaf of vector spaces
on X; the pair (EX , µH) is locally isomorphic to the pair of sheaves of pseudodifferential operators acting on microfunctions. If X is a G- contact manifold,
we can choose the Szegö projector invariant, so G acts on EX and µX .
For a general contact manifold, EX is well defined up to isomorphism, independently of any embedding - but no better than that. The corresponding
Szegö projector (not mod C ∞ ) is defined only up to a compact operator (a little
better than that - see below).
10.2.2
Asymptotic trace and index
The symbol bundles Ej of the Toeplitz bundles Ej only determines these up to
a space of finite dimension (because, as mentioned above, both the projector
defining them, and the Szegö projector, are not uniquely determined by their
symbols. However, if E, E0 are two equivariant Toeplitz bundles with the same
symbol, there exists an equivariant elliptic Toeplitz operator U : E → E0 with
quasi-inverse V (i.e. V U ∼ 1E , U V ∼ 1E0 ). This may be used to transport
equivariant Toeplitz operators from E to E0 : P 7→ Q = U P V . Then if P ∼ 0 on
Z, Q = U P V and V U P have the same G-trace, and since P ∼ V U P , we have
G
∞
TrG
P − TrQ ∈ C (G).
Definition 72 We define the asymptotic G-trace of P as the singularity of TrG
P
∞
(i.e. TrG
P mod C (G)).
The asymptotic trace vanishes if and only if the sequence of Fourier coefficients
−m
for all m where cα is the eigenvalue of
of TrG
P is of rapid decrease, i.e. O(cα )
DG in the representation α. This is the case if P is of degree −∞.
Definition 73 We will say that a system P of Toeplitz operators is G-elliptic
(relatively elliptic in [5]) if it is elliptic on char g. When this is the case, the
asymptotic G-index (or IePG ) is defined as the singularity of IPG . (We will still
denote it by IPG if there is no risk of confusion.)
We denote by K G (X − Z) the equivariant K-theory with compact support.
G
By the excision theorem K G (X − Z) is the same as KX−Z
(X), the equivariant
K-theory of X with compact support in X − Z, i.e. the group of stable classes
of triples d(E, F, u) where E, F are equivariant G-bundles on X, u an equivariant isomorphism E → F defined near the set Z (the equivalence relation is:
d(E, F, a) ∼ 0 if a is stably homotopic (near Z) to an isomorphism on the whole
of X). The asymptotic index is also defined for equivariant Toeplitz complexes,
exact near char g.
If u : E → F is a G-elliptic Toeplitz system or complex, its principal symbol
defines is a homogeneous linear map on Σ, invertible on char g. Its restriction
to any equivariant section of Σ defines a K-theoretic element [u] ∈ K G (X −
Z) (in case of a complex, u defines the same K-theoretic element as u + u∗ :
Eeven → Eodd ). The asymptotic index depends only on the homotopy class of
the principal symbol σ(P ), and since it is obviously additive we get:
86
Theorem 74 The asymptotic index of u only depends on the K-theoretic element [u]. It defines an additive map from K G (X − Z) to C −∞ (G)/C ∞ (G),
where Z is, as above, the base of char g.
Note that the sequence of Fourier coefficients trdαPα is in any case of polynomial growth with respect to the eigenvalues of D or DX ; if P ∼ 0, it is of rapid
decrease. The coefficients dIαα of the asymptotic index are integers, so they are
completely determined, except for a finite number of them, by the asymptotic
index.
Remark: if V is a finite dimensional representation of G and V ⊗ P or V ⊗ d
is defined in the obvious way, we have IVG⊗P = χV IPG (i.e. Index (V ⊗ P )α =
(V ⊗ Index P )α , except at a finite number of places).
E.g. Let G = SU2 acting on the sphere X of V = C2 in the usual manner,
and E = S m V the m-th symmetric power . Then E × X is a G bundle with
the action g(v, x) = (gv, gx). The CR structure on the sphere gives rise to
a first Szegö projector S1 (v · f ) = v · S(f ), where S is the canonical Szegö
projector on holomorphic functions. On the other hand since X is a free orbit
of G, the bundle E × X is isomorphic to the trivial bundle E0 × X where E0
is some fiber (i.e. the vector space of homogeneous polynomials of degree m,
with trivial action of G). This gives rise to a second Szegö projector S0 , not
equal to the P
first, but giving the
P same asymptotic index; we recover the fact
that S m V ⊗ S k V ∼ (m + 1) S k V (= in degree ≥ m).
10.2.3
E-modules
For the sequel, it will be convenient to use the language of E-modules. In the C ∞
category, E is not coherent; general E-module theory is therefore not practical
and not usefully related to topological K-theory. We will just stick to the two
useful cases below (elliptic complexes or “good” modules).26 . Note also that the
notion of ellipticity is slightly ambiguous; more precisely: a system of Toeplitz
operators (or pseudo- differential operators) is obviously invertible mod C ∞ if
its principal symbol is, but the converse is not true. The notion of “good”
system below partly compensates for this; it is in fact indispensable for a good
relation between elliptic systems and K-theory.
If M is an E-module (resp. a complex of E modules), it corresponds to the
system of pseudo-differential (resp. Toeplitz) operators whose sheaf of solutions
is Hom (M, µH); e.g. a locally free complex of (L, d) of E-modules defines the
Toeplitz complex (E, D) = Hom (L, H).
More generally we will say that an E-module
M is “good” if it isTfinitely
S
generated, equipped with a filtration M = Mk (i.e. Ep Mq = Mp+q , Mk =
0) such that the symbol gr M has a finite locally free resolution. We write
σ(M) = M0 /M−1 , which is a sheaf of C ∞ modules on the basis X; since there
exist global elliptic sections of E, gr M is completely determined by the symbol,
as is the resolution.
26
Things work better in the analytic category.
87
A resolution of σ(M) lifts to a “good resolution” of M, i.e. a finite locally
free resolution 27 of M.
It is standard that two resolutions of σ(M) are homotopic, and if σ(M)
has locally finite locally free resolutions it also has a global one (because we
are working in the C ∞ category on a compact manifold or cone with compact
support, and dispose of partitions of unity); this lifts to a global good resolution
of M.
If M is “good”, it defines a K-theoretic element [M] ∈ KY (X) (where Y is
the support of σ(M)), viz. the K-theoretic element defined by the symbol of
any good resolution (this does not depend on the resolution since any two such
are homotopic).
All this works just as well in presence of a G-action (if the filtration etc. is
invariant).
As above (§10.1.2), the asymptotic G-trace TrG
A [using subscripts as before]
is well defined if A is an endomorphism of a good locally free complex of Toeplitz
modules. The same holds for a good module M: the asymptotic trace of A ∈
End E(M ) vanishing near char g is the asymptotic trace of any lifting of A to a
good resolution of M. (Such a lifting, vanishing near char g, exists and is unique
up to homotopy, i.e. modulo supercommutators.) Likewise, the asymptotic Gindex of a locally free complex exact on Z, or of a good E- module with support
outside of Z, is defined: it is the asymptotic G-trace of the identity.
Definition 73 of the asymptotic index (or Euler characteristic) extends in
an obvious manner to good complexes of locally free E-modules or to good
E-modules. The asymptotic G-index of such an object, when it is G-elliptic,
depends only on the K-theoretic element which it defines on the base.
Let us note that the asymptotic trace and index are still well defined for
locally free complexes or modules with a locally free resolution, not necessarily
good; in that case, what no longer works is the relation to topological K-theory
on the base.
10.2.4
Embedding
If M is a manifold, Σ ⊂ T • M a symplectic subcone, the Toeplitz space H is
the space of solutions of a pseudodifferential system mimicking ∂¯b . If I ⊂ E
is the ideal generated by these operators (mod C ∞ ), and M = E/I, we have
µH = Hom E (M, µ) (as a sheaf: f ∈ Hom (M, µ) 7→ f (1); here as above µ
denotes the sheaf of microfunctions). E.g. in the holomorphic situation, I is the
ideal generated by the components of ∂¯b .
We have End E (M) = [I : I], the set of pseudo-differential operators a such
that Ia ⊂ I, acting on the right: if a ∈ [I : I], the corresponding endomorphism
27 The converse is not true: if d is a locally free resolution of M, its symbol is not necessarily
a resolution of the symbol of M – if only because filtrations must be defined to define the
88
of M takes f (mod I) to f a (mod I); this vanishes if and only if a ∈ I. The
map which takes a ∈ [I : I] to the endomorphism f 7→ af of H defines an
isomorphism from End E (M) to the algebra of Toeplitz operators mod C ∞ . M
is thus an ET • M −EΣ bimodule (where EΣ ' End M denotes the sheaf of Toeplitz
operators mod C ∞ ).
This extends immediately to the case where T • M is replaced by an arbitrary
symplectic cone Σ00 with base X”28 . The small Toeplitz sheaf µH can be realized
as Hom E 00 (M, µH00 ), where M = E 00 /I and I ⊂ E 00 is the annihilator of the
Szegö projector S of Σ (i.e. the null-sheaf of I in Hom E 00 (M, H00 ) = µH). If
P is a (good) E-module, the transferred module is M ⊗E P, which has the
same solution sheaf (Hom E” (M ⊗ P, H00 ) = Hom E” (P, Hom E (M, H00 )) and
Hom E” (M, H00 ) = H). Thus the transfer preserves traces and indices.
The module M = E 00 /I is generated by 1 (mod I) and has a natural filtration,
which is a good filtration: in the holomorphic case, the good resolution is dual
to the complex ∂¯b on (0, ∗) forms.
In general it always has a good locally free resolution, well defined up to
homotopy equivalence. In a small tubular neighborhood
of Σ one can choose
V
this so that its symbol is the Koszul complex on N 0 , where N 0 is the dual of the
normal tangent bundle of Σ equipped with a positive complex structure (as in
G
(X 00 )
the holomorphic case). The corresponding K-theoretic element [M] ∈ KX
is precisely the element used to define the Bott isomorphism (with support
Y ⊂ Σ) KYG (Σ) → KYG (Σ00 ). (Here, Y is some set containing the support
of σ(M) and the map is the product map: [E] 7→ [M][E], where the virtual
bundle [E] on Σ is extended arbitrarily to some neighborhood of Σ in Σ00 . 29 )
For example if Σ00 is CN \ {0}, with Liouville form Im z̄ · dz and base the
unit sphere X 00 = S2N −1 , H00 is the space of boundary values of holomorphic
functions, Σ ⊂ Σ00 consists of the nonzero vectors in the subspace z1 = · · · =
zk = 0, and X ⊂ X 00 is the corresponding subsphere, then H consists of the
functions independent of z1 , . . . , zk , and I is the ideal spanned by the Toeplitz
operators T∂1 , . . . T∂k . In this example the ideal I is generated by z̄1 , . . . , z̄k , or
by Tz̄j , j = 1 . . . k (On the sphere we have T∂j = (A+N )Tz̄j with A = TPN
).
1 zj ∂j
The E-module M itself has a global resolution with symbol the Koszul complex
constructed on z̄1 , . . . , z̄k .
What precedes works exactly as well in the presence of a compact group
action. If P is a good module with support outside of Z (or a complex with
symbol exact on Z), the transferred module has the same property (Z ⊂ Z 00 ),
and it has the same G-index (the G-index of the complex Hom E (M, H) '
Hom E 00 (M00 , H00 )).
28 We use a double prime here because, eventually, we will be embedding two cones in a
third one.
29 Toeplitz operators (mod C ∞ ) live on Σ and their principal symbols are homogeneous
functions on Σ. However the K-theoretic element [u] ∈ K G (X − Z) of a G-elliptic element
lives on the base X, so as the support of “good” E-modules or complexes - in contrast with
what happens for pseudodifferential operators.
89
If X, X 00 are (compact) contact G-manifolds, f : X → X 00 an equivariant
embedding, P a good G − E-module with support outside of Z (the base of
char g in Σ), or a Toeplitz complex, exact on Z, the transferred module on X is
f+ P = M ⊗f∗ E 0 f∗ P 0 . This is exact outside of f (Σ) and has the same G- index
as P; its K-theoretic invariant [P] is the image of [P] by the equivariant Bott
G
homomorphism. The K- theoretic element [f+ P] ∈ KX−Z
(X) is the image of
[P] by the Bott homomorphism (it is well defined since f (Z) ⊂ Z 00 ). Thus
Theorem 75 Let f : X → X 00 be an equivariant embedding. The Bott homoG
G
00
morphism KX−Z
(X) → KX
00 −Z 00 (X ) commutes with the asymptotic G index.
30
It is always possible to embed a compact contact manifold in a canonical
contact sphere with linear G-action. In fact, it is easier to work with the corresponding cones, as follows:
Proposition 76 Let Σ be a G-cone (with compact base), λ a horizontal 1form, homogeneous of degree 1, i.e. ρyλ = 0 and Lρ λ = λ, where ρ is the
radial vector field, generating homotheties. Then there exists a homogeneous
embedding x 7→ z(x) of Σ in a unitary representation space V c of G such that
λ = Im z̄ · dz.
In the proposition, z(x) must be homogeneous of degree 12 . This applies of
course if Σ is a symplectic cone, λ its Liouville form. (The symplectic form is
ω = dλ and λ = ρyω).
We first choose a smooth equivariant function y = (yj ), homogeneous of
degree 12 , realizing an equivariant embedding of Σ in V − {0}, where V is a real
unitary G-vector space (this always exists if the base is compact; (the coordinates zj on V are homogeneous of degree 12 so that the canonical form Im z̄ · dz
is of degree 1)). Then there exists a smooth function x = (xj ) homogeneous
of degree 21 such that λ = 2x · dy. We can suppose x equivariant, replacing
it by its G-mean if need be. Since y is of degree 12 we have 2ρydy = y hence
x · y = ρyλ = 0. Finally we get
λ = Im z̄ · dz
10.3
with z = x + iy.
Relative index
As indicated in the introduction, we are considering the index of the Fredholm
map E0 : u 7→ S 0 (u◦f0−1 ) from H0 to H00 , where X0 , X00 are the boundaries of two
smooth strictly pseudoconvex Stein manifolds Ω, Ω0 , H, H0 the spaces CR distributions (ker ∂¯b , equal to space of boundary values of holomorphic functions),
S, S 0 the Szegö projectors, and f0 a contact isomorphism X0 → X00 .
30 As mentioned above the interplay between the Bott isomorphism and embeddings of
systems of differential or pseudodifferential operators lies at the root of Atiyah-Singer’s proof
of the index theorem; it is described in M.F. Atiyah’s works [2, 3, 4, 5], cf also [29] in the
context of holomorphic D-modules, close to the Toeplitz context.
90
As announced we modify the problem and move to the larger boundaries
X, X 0 of “balls” |t|2 < φ, |t0 |2 < φ0 in C × Ω, C × Ω0 , on which the circle group
acts (t 7→ eiλ t) (§10.3.1). We will see (§10.3.2) that the Toeplitz FIO E0 defines almost canonically an equivariant extension F which is U (1)- elliptic, and
Index (F |Hk ) = Index (E0 ) for all k (Hk ⊂ H(X) is the subspace of functions
f = tk g(x)), so that our relative index Index (E0 ) appears as an asymptotic
equivariant index, easier to handle in the framework of Toeplitz operators.
In §10.3.3 we will show that the whole situation can be embedded in a large
sphere, with action of U (1) as in the examples above. In the final result (section
10.3.4) the relative index appears as the asymptotic index of an equivariant
U (1)-elliptic Toeplitz complex on this large sphere. In general the equivariant
index (asymptotic or not) is rather complicated to compute, but in our case the
U (1)-action is quite simple 31 , it reduces naturally to the standard Atiyah-Singer
K-theoretic index formula on a symplectic ball. The result is better stated in
terms of K-theory anyway, but it can be translated via the Chern character in
terms of cohomology or integrals. We give here a (rather clumsy) cohomologicalintegral translation, essentially equivalent to the result conjectured in [114].
We will also see below (§10.3.2) that f0 has an almost canonical extension f
near the boundary, well defined up to isotopy, not holomorphic but symplectic.
We can then define a space Y by gluing together Y+ , Y− by means of f . Y is not
a Hausdorff manifold, but it is symplectic and both Y+ , Y− carry orientations
which agree on their intersection (as do the symplectic structures). We can
further choose differential forms ν± representatives of the Todd classes of Y± so
that they are equal near the boundary X0 (the symplectic structures agree, not
the complex structures, but they define the same Todd classes).
R
Theorem 77 The relative index (index of E0 ) is the integral Y (ν+ − ν− ),
where ν± are representatives of T odd(Y± ) as above, so that the difference has
compact support in Y − X0 .
This will be explained in more detail below (§10.3.4). This formula is related
to the Atiyah-Singer index formula on the glued space Y , but is not quite the
same since Y is not a symplectic manifold.
To prove the index theorem we will give an equivalent equivariant description
of the situation, where the index of E0 is repeated infinitely many times, and
embed everything in a large sphere where the index is given by the K- theoretic
index character (§10.3.4).
10.3.1
Holomorphic setting
Let Ω be a strictly pseudoconvex domain (or Stein manifold), with smooth
e ⊂ C × Ω̄ the
boundary X0 (Ω̄ = Ω ∪ X0 is assumed to be compact); we write Ω
ball |t|2 < φ, where φ is a defining function (φ = 0, dφ 6= 0 on X0 , φ > 0 inside),
31
it is free on the support of the K-theoretic symbol of our complex.
91
e is strictly pseudoconvex, i.e. Log 1 is
chosen so that the boundary X = ∂ Ω
φ
¯ 1 0).
strictly plurisubharmonic (i.e. Im ∂∂
φ
The circle group U (1) acts on X by (t, x) 7→ (eiλ t, x). We choose as volume
element on X the density dθ dv where dv is a smooth positive density on Ω
(t = eiθ |t|): this is a smooth positive density on X; it is invariant by the action
of U (1), so as the Szegö projector S and its range H, the space of boundary
values of holomorphic functions.
The infinitesimal generator of the action of U (1) is ∂θ , and we denote by D
the restriction to H of 1i ∂θ , which is a self-adjoint, ≥ 0, Toeplitz operator. D is
the restriction of Tt T∂t .
The model case is the sphere S2N +1 ⊂ CN +1 with the action
(t = z0 , z = (z1 , . . . , zN )) 7→ (eiθ t, z).
The Fourier decomposition of H
b k≥0 Hk
H=⊕
(Hk = ker (D − k) )
corresponds P
to the Taylor expansion of holomorphic functions: the k-th component of f =
fk (x)tk ∈ H is fk tk .
H0 identifies with the set of holomorphic functions on X0 (it is the set of
boundary values of holomorphic functions on Ω with moderate growth at the
boundary, i.e. |f | ≤ cst d(·, X0 )−N for some N , where d(·, X0 ) is the distance
to the boundary).
Remark:
norm is
If f = tk g(x) with g continuous, in particular if f ∈ Hk , its L2 (X)
Z
π
φk+1 |g(x)|2 dv
kf kL2 (X) =
k+1 Ω
where as above dv is the chosen smooth volume element on Ω. The restriction
of the Szegö projector to functions of the form tk g(x) is thus identified with
the orthogonal projector on holomorphic functions in L2 (Ω, φk+1 dv). Such sequences of projectors were considered by F.A. Berezin [10] and further exploited
by M. Englis [49, 50, 51], whose presentation is closely related to the one used
here.
For the sequel, it will be convenient to modify the factorisation D = t∂t . We
begin with the easy following result.
Lemma 78 Let D = P Q be any factorisation where P, Q are Toeplitz operators
and [D, P ] = P . Then there exists a (unique) invariant invertible Toeplitz
operator U such that P = tU, Q = U −1 ∂t .
Indeed it is immediate that any homogeneous function a on σ such that
= ±a is a multiple mt of t (resp. of t̄), with m invariant. For the
1
i ∂θ a
92
same reason (or by successive approximations) a Toeplitz operator A such that
[D, A] = ±A is a multiple of Tt M (or M 0 Tt ) Tt with M or M 0 invariant (resp.
of T∂t , on the right or on the left) . Thus in the lemma above we have P =
Tt U, Q = U 0 T∂t , where U, U 0 are Toeplitz operators which necessarily commute
with D, and are elliptic and inverse of each other.
Note that D = P Q, [D, P ] = P is equivalent to D = P Q, [Q, P ] = 1.
In particular, since D = D∗ = T∂∗t Tt∗ , there exists a Toeplitz operator A
such that T∂t = ATt∗ . A is elliptic of degree 1 (in fact invertible), positive since
D = Tt ATt∗ is self-adjoint ≥ 0; it is also invariant: [D, A] = 0.
1
Definition 79 We will set T = Tt A 2 ; its symbol is denoted by σ(T ) = τ .
Note that τ is homogeneous of degree 21 , and T is of degree 12 , so it is not a
Toeplitz operator in our strict sense, but for multiplications and automorphisms
P 7→ U P U −1 it is just as good. We have
1
T ∗ = A 2 Tt∗ , [D, T ] = T . D = T T ∗
(54)
(for any other such factorisation D = BB ∗ with [D, B] = B, B is of degree 12 ,
and we have B = T U with U invariant and unitary. T is the unique Toeplitz
operator giving such a factorisation and such that T = Tt A0 with A0 a Toeplitz
operator of degree 21 , A0 0).
In what precedes, all = signs can be replaced by ∼ (= mod C ∞ ); we then
get local statements.
The symbol τ = σ(T ) is the unique homogeneous function of degree 12 such
that σ(D) = |τ |2 , ∂θ τ = iτ, τt > 0.
We also have the following (easy) local result:
Lemma 80 Given any Toeplitz operator K (mod C ∞ ) on H such that D ∼
KK∗ , [D, K] = K near the boundary, there exists a unique unitary equivariant
Toeplitz FIO F such that F |H0 ∼ Id , F T ∼ KF .
The geometric counterpart is: given any function k on Σ homogeneous of degree
1
2 such that σ(D) = k k̄ there exists a unique germ of homogeneous symplectic
isomorphism f such that f |Σ0 = Id , k ◦ f = τ . This is immediate because the
two hamiltonian pairs Hτ , Hτ̄ , Hk , Hk̄ define real 2-dimensional foliations, and
an isomorphism Σ ∼ Σ0 × C near Σ0 . Note that this would not work if we
replaced k, k̄ by two functions a, b such that σ(D) = ab, ∂θ a = ia but not b = ā,
because then the ’foliation’ defined by the Hamiltonian vector fields Ha , Hb ,
although it is formally integrable, is not real.
The operator statement follows, e.g. by successive approximations. Note
that F is completely determined by its restriction F0 if it commutes with T .
(In fact in EΣ , the commutator sheaf of T and T ∗ identifies with the inverse
image of EΣ0 - at least as far as the leaves of the Hamiltonian fields HT , HT ∗
define a fibration over Σ0 : EΣ is the (completed) tensor product of the Toeplitz
algebra Toepl(T , T ∗ ) generated by T and T ∗ and this commutator: EΣ ∼ EΣ0 ⊗
93
Toepl(T , T ∗ ) (in a neighborhood of Σ0 ). In this statement, (T , T ∗ ) cannot be
replaced by (Tt , T∂t ) whose commutator sheaf is only defined in the algebra of
jets of infinite order along Σ0 , because the Hamiltonian leaves are complex, no
longer real.)
Note that, in our case, the base of char g is the boundary X0 (the set of fixed
points), outside of which D is elliptic.
10.3.2
Collar isomorphisms
Let now Ω0 be another strictly pseudoconvex domain (or Stein manifold) with
smooth boundary X 0 . We do the similar constructions Ω̃0 , H0 , and D0 , . . . as in
the previous subsection. Let f0 : X0 → X00 be a contact isomorphism.
We define the Fourier Toeplitz operator E0 : u 7→ S 0 (u ◦ f0−1 ) : H → H0 ,
which is a Fredholm operator. It will be convenient to replace E0 by F0 =
1
(E0 E0∗ )− 2 E0 , which has the same index and is ∼ unitary (E0 E0∗ is an elliptic
1
≥ 0 Toeplitz operator on H0 ); (E0 E0∗ )− 2 is defined to be 0 on ker E0∗ (mod C ∞
e we construct a Toeplitz operator T 0 such
would be quite enough). As for Ω,
−1 0
0
0 0∗
0 0
0
that D = T T , [D T ] = T , Tt T 0.
Exactly as in Lemma 10.3.2, there exists a unique (unitary) Toeplitz FIO
F , defined near the boundary X0 and mod C ∞ , elliptic, such that F |H0 = F0 ,
and F T ∼ T 0 F near the boundary (mod C ∞ ).
The geometric counterpart is: there exists a unique equivariant germ of
contact isomorphism f : X → X 0 (defined and invertible near the boundary)
such that f |X0 = f0 , τ 0 = τ ◦ f .
We may extend F , using an invariant cut off Toeplitz operator, so that
it vanishes (mod C ∞ ) away from the boundary. There is an invariant FIO
parametrix F 0 , i.e. F 0 F ∼ 1H , F F 0 ∼ 1H0 , near the boundary.
Proposition 81 For any k, Fk = F |Hk has an index, equal to Index F0 .
Proof: both F 0 F and F F 0 are invertible on the boundary, so have a G-index;
the index of Fk = F |Hk is tr (1 − F 0 F )k − tr (1 − F F 0 )k . Now T , resp. T 0 is
an isomorphism Hk → Hk+1 , resp. H0k → H0k+1 , and we have Index (Fk+1 A) =
Index (A0 Fk ), so Index Fk+1 = Index Fk , i.e. the index does not depend on k
and is equal to Index E0 .32
The asymptotic index is stable by embedding; here the index is constant,
and the asymptotic index of F (which is essentially a Toeplitz invariant) gives
the index of F0 itself.
10.3.3
Embedding
Theorem 82 Let f : X → X 0 be a collar isomorphism defined in some invariant neighborhood of X0 in X. Then for large N there exists equivariant contact
32 For a more general situation where P is a Toeplitz operator elliptic on X , or where the
0
canonical Szegö projector is replaced by some other general equivariant one, we would only
get that the index Index (Pk ) is constant for k 0. Here the fact that Index Pk = Index P0
is obvious but important.
94
embeddings U : X → S2N +1 , U 0 : X 0 → S2N +1 such that U = U 0 ◦ f near the
boundary, and tX , t0X 0 map to positive multiples of tS2N +1 (as above the contact
sphere S2N +1 is equipped with the U (1)-action (t, z) 7→ (eiθ t, z)).
As usual, it will be more comfortable to work with the symplectic cones. The
symplectic cone of X is Σ = R+ × X, where we choose the radial coordinate
invariant.
The symbol of D is τ̄ τ with τ /t > 0 as in Definition 79. The Liouville form
is Im (τ̄ dτ ) + λ0 where λ0 is a horizontal form, i.e. the pull-back of a form on
Σb = U (1)\Σ ' R+ × Ω̄ (equivalently: ∂θ yλ0 = L∂θ λ0 = 0).
0
Lemma 129 provides an embedding x 7→ zb (x) of Σb in CN − {0} (with
the trivial action of U (1)). We now choose real functions ψ1 , ψ2 invariant,
homogeneous of degree 0, such that ψ12 + ψ22 = 1, with supp ψ1 contained in the
domain of definition of f and ψ2 vanishing near the boundary, and we construct
a new embedding z in 3 pieces: z = (z1 , z2 , z3 ) with z1 = ψ1 z0 , z2 = ψ2 z0 , z3 = 0
00
in CN , N 00 to be defined below.
Since Im z̄j zj ψj dψj = 0 (z̄j zj ψj dψj is real) we still have Im (z̄1 · dz1 + z¯2 ·
dz2 ) = (ψ12 + ψ22 )Im z¯0 · dz0 = Im z¯0 · dz0 inducing λ0 . The first embedding
U = (τ, v) : Σ → C1+N (N = 2N 0 + N 00 ).
00
Similarly there exists an embedding x0 7→ z00 (x0 ) of Σ0b in CN − {0} (with
the trivial action of U (1)).
We replace this by z 0 = (z10 , z20 , z30 ) with z10 = ψ10 z1 ◦ f −1 , z20 = 0, z30 = ψ30 z00
2
2
where ψ10 , ψ30 again are invariant, homogeneous of degree 0, ψ 0 1 + ψ 0 3 = 1, and
0
−1
0
supp ψ1 is contained in the domain of definition of f , ψ3 vanishes near the
boundary. This also defines an embedding U 0 = (a0 , z 0 ) : Σ0 → CN +1 ; we have
U = U 0 ◦ f near the boundary since ψ2 , ψ30 vanish there.
10.3.4
Index
We are now reduced to the case where both U (1)-manifolds X, X 0 sit in a large
sphere S = S2N +1 and coincide near the set of fixed points S0 .
As in the preceding section, we can embed the U (1) sheaves µHX , µHX 0 as
sheaves of solutions of two good equivariant ES - modules MX , MX 0 , and the
identification F gives an equivariant Toeplitz isomorphism Fe near X0 (we can
make the construction so that MX = MX 0 , Fe = Id near X0 ).
The asymptotic index then only depends on the difference element
d([MX ], [MX 0 ], σ(Fe)) ∈ K U (1) (S − S0 ).
Now U (1) acts freely on S − S0 , with quotient space U (1)\(S − S0 ) the open
unit ball B2N ⊂ CN . We have
Proposition 83 The pull back map is an isomorphism K(B) → K U (1) (S − S0 ).
We have K(B) ∼ Z, with generator the symbol of the Koszul complex kx at
the origin (or any point of the interior), whose index is 1.
95
U (1)
Its pull-back is the generator of KS−S0 (S): the symbol is the same, but now
P∞
acting on H(S). Its index is 0 J k , where (as in (53)) J is the tautological
character of U (1): J(eiλ ) = eiλ .
The first assertion is immediate (cf. [5]): if G is a compact group acting freely
on a space Y , the pull back defines an equivalence from the category of vector
bundles on G\Y to that of G-vector bundles on Y (an inverse equivalence is
given by E 7→ G\E), and this induces a bijection on K-theory (with supports).
The fact that kx defines the generator of K(B)(= K0 (B)) is just a restatement Bott’s periodicity theorem. Its pullback is then the generator of
K U (1) (S − S0 ): the corresponding complex of Toeplitz operators is then the
standard Koszul complex, acting on holomorphic functions, whose index is the
space of holomorphic functions of z0 = t alone.
P∞
Thus if [u] ∈ K U (1) (S − S0 ), its asymptotic index is m k=0 J k , where the
integer m is the value of the K-theoretic character K(B) on the element [uB ]
whose pull-back is [u].
Let us now come back to our index problem: we have constructed the difference bundle d([MX ], [MX 0 ], σ(Fe)). We may replace MX , MX 0 by good resolutions in small equivariant tubular neighborhoods of X, resp. X 0 , whose
K-theoretic symbol is the Bott element - the Koszul complex for a positive complex structure on the normal symplectic bundle of X, resp. X 0 . Fe lifts to the
resolutions (uniquely up to homotopy), and the symbol of the lifting u is an
isomorphism near X0 (we can make the construction so that u = Id near X0 ),
so our K-theoretic element is [u] = d(βX , βX 0 , u) (the equivariant K-theoretic
element attached to the double complex defined by u).
Theorem 84 Let m be the index of E0 we are investigating. Then, notations
and embeddings being as above,
1) the asymptotic index of our equivariant extension Fe is the asymptotic
0
index of the difference element [u] = d(βX , βX
, u) ∈ K U (1) (S − S0 ), where u is
e
the symbol of F (i.e. the identity map near S0 , where X and X 0 coincide).
2) the index m itself is the value of the index character of K(B) on the
element [uB ] = d(βΩ , βΩ0 , ū).
The first part has just been proved. The asymptotic index is ∼ m(1 − J)−1 for
some integer m.
To prove the second we go down to B2N . The bases of X, X 0 are the embeddings Y+ , Y− of Ω, Ω0 in B, which coincide near the boundary, and as above
G
the pullback is an isomorphism KY± (B) → KX
(S − S0 ). The Bott complexes
±
βX± descend as Bott elements βY± on B, realized as Koszul complexes of positive complex structure of the normal symplectic bundle 33 ; u descends as an
isomorphism near the boundary.
The index m we are looking for is the K-theoretic index character of the
difference element d(βY+ , βY− , u). This can be as usual translated in terms of
33 note that Y
± are symplectic submanifolds, not complex; but all positive complex structures are homotopic.
96
cohomology, or as an integral:
Z
m=
ω
B
where ω is a differential form with compact support, representative of the Chern
character of our difference element d(βY+ , βY− , u).
We can push this down further. The construction can be made so that
u = Id near the boundary, choose differential forms ω± with support in small
tubular neighborhoods of Y± so that they coincide near the boundary (so as the
tubular neighborhoods), so that ω is the difference ω+ − ω− .
The integral ν± of ω± over the fibers of the respective tubular neighborhoods
is then a representative of the Todd class of Y± ; ν+ and ν− coincide near the
boundary, so that the difference ν+ − ν−R has compact support in Y = Y+ ∪ Y− .
Finally the index m is the integral Y (ν+ − ν− ) as announced in Theorem
126.
R
R
The integral can also be thought of as the constant limit Y+, ν+ − Y−, ν− ,
where the subscript means that we have deleted the neighborhood φ < in
Y+ and the corresponding image in Y− .
10.4
Appendix
In this section we show how various symplectic extensions of f0 are related. It
is a little intriguing that, although in our proof, the extension f must be chosen
rather carefully so that the asymptotic index of the corresponding Toeplitz FIO
E is (asymptotically) the index of E0 , the final result, expressed as an integral
on the bases glued together by means of f near their boundaries, depends only
on the isotopy class of f , which is unique.
10.4.1
Contact isomorphisms and base symplectomorphisms
Let X be as above, with X0 the fixed point set of codimension 2. Near the
boundary, X is identified with X = X0 × C and the base √
U (1)\X ∼ Ω identifies
with X0 ×R+ ; we have φ = tt̄ and the C-coordinate is t = φ eiθ (it is smooth on
X). The contact form is λX = Im (t̄dt − ∂φ) = φ dθ + λΩ , where λΩ = −Im ∂φ
is a smooth basic form. The connection form is γ = dθ − λφΩ , and the base
Ω = X0 × R+ is equipped with the (basic) symplectic curvature form
µ = dγ
(with γ =
λΩ
,
φ
λΩ = −Im ∂φ) .
We will still use the symplectic cone of X: this is Σ = char g ' R+ × X, with
Liouville form aλX and symplectic form its derivative, with the R+ coordinate
a defined below:√with the notation of Lemma 79, we have a = σ(A), i.e. σ(D) =
aφ = τ τ̄ , τ = t a (as above D = 1i T∂θ denotes the infinitesimal√generator of
rotations). We will also write in polar coordinates τ = ρ eiθ (ρ = φ a).
Let F be a homogeneous equivariant symplectic transformation of Σ: then F
preserves σ(D) = τ τ̄ , so we have necessarily F∗ τ = u τ , with u invariant, |u| = 1.
97
F is then completely determined by its restriction to the boundary, since it
commutes with the two real commuting hamiltonian vector fields Re Hτ , Im Hτ ,
which are linearly independent and transversal to Σ0 .
Thus there is a one to one correspondence between unitary functions on the
base Ω and germs near Σ0 = char g of equivariant symplectomorphisms inducing
Id on char g - or equivalently of contact automorphisms of X inducing Id on
X0 .
If F is such a contact automorphism, the base map FΩ is obviously a diffeomorphism of Ω which induces Id on the boundary X0 and preserves the
symplectic form µ.
The converse is not true. If FΩ is a smooth symplectomorphism of Ω inducing
the identity on X0 , we have FΩ∗ ( λφΩ ) = λφΩ + α with α a closed form. It is
elementary that α = c dφ
φ + β where c is a constant and β is smooth on the
boundary. Locally on X0 , FΩ lifts to X or Σ: the lifting is F : (x, τ ) 7→ (x0 , τ 0 =
τ eiψ ) (θ0 = θ + ψ) where ψ is a primitive of α (this is not smooth at the
boundary, only continuous). It is immediate that conversely any α of the form
above gives rise to such a contact isomorphism with smooth base map. (on Σ
the horizontal (invariant) coordinates satisfy Hτ eiψ f = 0; the horizontal part
of the Hamiltonian Hτ eiψ is −iτ eiψ (∂ρ − Hψ0 ) (with Hψ0 = ψξj ∂xj − ψxj ∂ξj );
finally ∂ρ − Hψ0 is smooth so the horizontal coordinates (x0 , ξ) are determined
by smooth differential equations.) Summing up:
Theorem 85 The map which to a germ of contact isomorphism F (near X0 )
assigns the invariant unitary smooth function u such that F ∗ τ = τ u is one to
one (and continuous). In particular the homotopy class of F is determined by
that of u (an element of H 1 (X, Z)).
The map which to a smooth germ of symplectomorphism FΩ (near X0 ) assigns the closed one-form α = c dφ
φ + smooth is one to one, the group of such
symplectomorphisms is contractible. The contact lifting (which exists locally,
and globally if α is exact) is smooth on X0 if and only if c = 0.
The fact that this group is contractible (connected) simplifies the final result,
namely: in the proof of Theorem 84 it was essential that the base map FΩ have
a smooth symplectic extension preserving τ > 0; for Theorem 126 however any
symplectic FΩ will do since these are all isotopic.
10.4.2
Example
(A smooth symplectic automorphism of the base does not lift to a smooth
equivariant contact automorphism of the sphere.)
Let S be the unit sphere in CN +1 , with coordinates x0 = t, x1 , . . . , xN .
N
U (1) acts by t 7→ eiθ t. The base is B = S/U (1), the
Punit ball of C .
The contact form is Im t̄dt + λ = φdθ + λ with λ =
x̄j dxj , φ = t̄t = 1 − x̄x.
The connection form is γ = dθ + φλ , its curvature is the symplectic form
µ = d φλ (on the interior of B).
98
Let FB be the diffeomorphism of B defined by x 7→ x0 = FB (x) = eciφ x, c a
constant; this preserves φ and the inverse is x = e−ciφ x0 . We have
FB∗ λ = Im (x̄(dx + cix dφ)) = λ + c(1 − φ)dφ
∗
Since d(1 − φ) dφ
φ = 0, FB is symplectic (FB µ = µ).
But FB does not lift to a smooth equivariant contact automorphism of S:
such a lifting F must preserve the connection form, so it is of the form
t 7→ e−iα t
(θ 7→ θ − α)
with α = cLog φ − φ + cst
(dα = c(1 − φ) dφ
φ ), and this is not smooth at the boundary t = 0 if c 6= 0.
Of course the reverse works: if F is a smooth equivariant contact automorphism of the sphere S (or a germ of such near the fixed diameter S0 ), the base
map FB is a smooth symplectomorphism of the ball B (up to the boundary).
10.4.3
Final remarks
1) The preceding construction applies in particular to the following situation:
let V, W be two compact manifolds, and f0 a contact isomorphism S ∗ V → S ∗ W .
We may suppose V real analytic; then S ∗ V is contact isomorphic to the
boundary of small tubular neighborhoods of V in its complexification. For
example if V is equipped with an analytic Riemannian metric, and (x, v) 7→
ex (v) denotes the geodesic exponential map, the map (x, v) 7→ ex (iv) is well
defined for small v and for small it realizes a contact isomorphism of the
tangent (or cotangent) sphere of radius to the boundary of the complex tubular
neighborhood of radius (cf. [16]).
The corresponding FIO’s can be described as follows: as above there exists
a complex phase (as in [102, 101] function φ on T ∗ W × T ∗ V 0 such that 1) φ
vanishes on the graph of f0 and dφ = ξ.dx − η.dy there, 2) Im φ 0 i.e. it is
positive outside of the graph and the transversal hessian is 0. φ is then a
global phase function for FIO associated to f0 (φ is not unique, but obviously
the set of such functions is convex, hence contractible).
The elliptic FIO’s we are interested in are those that can be defined by a
positive symbol (or a symbol isotopic to 1):
Z
f 7→ g(x) = eiφ a(x, ξ, y, η)f (y)dydηdξ with a > 0 on the graph .
The degree of such operators depends on the degree of a, but they all have the
same index, given by the formula above.
2) The formula above extends also to vector bundle cases: if E, E 0 are holomorphic vector bundles (or complexes of such) on Ω, Ω0 , f0 a contact isomorphism (∂Ω → ∂Ω0 ) as above, and A a smooth (not holomorphic) isomorphism
f0∗ E → E 0 on the boundaries, the Toeplitz operator a 7→ S 0 (Af0∗ a) is Fredholm
99
and its index is given by the same construction as above. For this construction
f0 only needs to be defined where the complexes are not exact.
In particular let Ω, Ω0 have singularities (isolated singularities, since we still
want smooth boundaries): we can embed Ω, Ω0 in smooth strictly pseudoconvex
e Ω
e 0 of the same (higher) dimension; the contact isomorphism extends
domains Ω,
e The coherent sheaves OΩ , OΩ0
at least in a small neighborhood of ∂Ω in ∂ Ω.
e Ω
e 0 ; near the boundary
have global locally free holomorphic resolutions on Ω,
these are homotopy equivalent to a Koszul complex, hence equivalent.
The theorem above shows that the relative index is the K-theoretical character of the difference virtual bundle d([OΩ ], [OΩ0 ]) (vanishing near the boundary).
e with
Note however that the virtual bundles [OΩ ], [OΩ0 ] lie in the K-theory of Ω
support in Ω. This can be readily described in terms of cohomology classes
e etc. with support in Ω, not on Ω itself (the relation between coherent
on Ω
holomorphic modules and topological K-theory, or K-theory and cohomology, is
not good enough when there are singularities).
100
11
Complex Star Algebras.
In this chapter we describe a classification of star algebras on the cotangent
bundle of a complex manifold, locally isomorphic to the algebra of pseudodifferential operators ; this requires a slight extension of the usual definition of
star algebras. We show that in dimension ≥ 3 these are essentially trivial and
come from algebras of differential operators on X ; in dimension 1 and 2 there
are many more, which we describe. 34
11.1
Introduction
Let us first recall what a star-product is (detailed definitions are given in section
b denote the algebra of formal series
2) : let X be a manifold and let O
X
fk hk
f=
k≥k0
where the fk are smooth functions on X and h is a “small” formal parameter.
b for which the unit is 1 and
A star product on X is a unitary algebra law on O
the product is local, i.e. given by a formula :
X
f, g → B(f, g) = f g +
hk Bk (f, g)
k≥k0
where
the Bk are bidifferential operators on X : in local coordinates Bk (f, g) =
P
aαβ ∂ α f ∂ β g with smooth coefficients aαβ (it is further required that the unit
is 1, i.e. B0 (f, g) = f g and Bk (1, f ) = Bk (f, 1) = 0 for any k > 0 and any
b A star product can be thought
f ; the addition law is the usual addition of O.
of as a non-commutative deformation of the usual product. The leading term
of commutators {f, g} = hB1 (f, g) − hB1 (g, f ) defines a Poisson bracket on X
(star products are also called “deformation quantization of Poisson manifolds”).
In this paper I will use a slightly extended definition, where star products live
on cones. A cone Σ with basis BΣ = X is the complement of the zero section
in a line bundle L → X (a complex line bundle if X is a complex manifold,
and preferably a half-line bundle if X is real) ; in the semi-classical case above
1
Σ = X × R×
+ and h = r if r denotes the fiber variable (the small “Planck
constant” plays the role of the inverse of a large frequency). In this context
b is the set of formal series f = P
O
k≤k0 fk where for each k, fk is a function
homogeneous of degree k on Σ and, locally, a star product is defined as above
b
as a bidifferential product law on O
X
f, g → B(f, g) = f g +
Bk (f, g)
k≤k0
where Bk is now a bidifferential operator on Σ, homogeneous of degree k → −∞
with respect to fiber homotheties. The Bk may involve derivations in any direction, so there is no longer a distinguished “Planck constant” commuting with
34 Mathematical
Physics, Analysis and Geometry 00: 1-27, 1999.
101
the rest 35 . The associated Poisson bracket now lives on Σ and is homogeneous
of degree −1. This definition includes the algebras of pseudo-differential operators or Toeplitz operators, which are after all among the most important and
belong to the same formalism.
Complex star algebras arrive naturally and unavoidably in many problems
concerning differential operators, whose symbols are polynomials and always
live on a complex manifold. So it is important to study them, and to study
their relations with “polynomial” objects associated to differential operators.
In his paper [91] M. Kontsevitch has shown that any homogeneous Poisson
bracket on a real manifold comes from a star product. His proofs extend without
changing a word to star -products on a cone. Kontsevitch’s formula giving a
star product from a Poisson bracket on an affine space also works without any
modification in the complex case (i.e. Σ = Cn × C× ). But the argument used to
go from local to global does not work for complex manifolds, because it uses in an
unavoidable manner partitions of unity and tubular neighborhoods. In general
I do not know if a global star product exists for a given Poisson bracket, even in
the symplectic case, nor do I know what the classification of such algebras looks
like (see however [88], where it is shown that even if such an algebra E may not
exist, the category of sheaves of E-modules can be defined up to equivalence).
In this paper I investigate those star algebras which live on a complex cotangent cone T ∗ X − {0} deprived of its zero section, equipped with its canonical
symplectic Poisson bracket. All star algebras associated to this Poisson bracket
are locally isomorphic, and there exists a global such algebra, viz. the algebra
of pseudo-differential operators ; so there is at least a starting point for the
classification. This will turn out to be essentially trivial in dimension n ≥ 3
(Theorem100), but instructively not in dimension 2. More precisely algebras
over a manifold X of dimension 2 or ≥ 2 are described in section 4, and compared to D-algebras, i.e. sheaves of algebras over X locally isomorphic to E, the
algebra of pseudo-differential operators coming from differential operators on X.
It turns out that if dimX ≥ 3 we get nothing new : the functor which takes a Dalgebra to the associated star-algebra is an equivalence. If dimX = 2 the same
functor is fully faithful, i.e. two D-algebras are isomorphic if and only if the
associated star-algebras are isomorphic, and an isomorphism between such staralgebras comes from a unique isomorphism between the original D-algebras ;
however there are in general many more “exotic” star-algebras which do not
come from a D-algebra.
If X is of dimension 1 the classification depends on whether X is open, of
genus ≥ 2, of genus 1 or of genus 0.
An inner automorphism of the algebra E of pseudo-differential operators on
X (U : P → AP A−1 ) has a symbol σ(U ) = d Log σ(A), which is a section of
the sheaf ω (on the “basis” BΣ = Σ/C× of closed forms homogeneous of degree
0 on Σ, and an exponent which is the degree of A ; we will see in section 2 that
any automorphism U of E has likewise a symbol and an exponent ∈ C. Similarly
35 There is absolutely no reason that the Planck constant should commute with the rest,
especially when it is a parameter without physical significance
102
a star algebra has a symbol σ(A) ∈ H 1 (BΣ, ω) and an exponent ∈ H 1 (BΣ, C).
We will see in section 3 that if X is an open curve or a curve of genus ≥ 1, star
algebras on Σ are completely determined by their exponent. The classification
is more subtle when X is closed of genus 1 or 0.
The techniques used in this paper are a mixture of non-commutative cohomology, holomorphic cohomology, and the relation between the cohomology of
a sheaf with a filtration and the cohomology of the associated graded sheaf.
This contains nothing really new or difficult, but the mixture can be somewhat
muddling.
As far as I know the questions studied here have not been investigated before
and the results are new.
In sections 2 and 3 we recall the definition of star algebras, and some classification principles.
In section 4 we describe the classification when dim X ≥ 2.
In section 5 we describe the case where X is a curve (dim X = 1) : results
are substantially different if X is open, X = P1 , X is of genus 1, or X is of
genus g ≥ 2.
11.2
11.2.1
Star Algebras
Cones
Definition 86 A real (resp. complex) cone is a C ∞ (resp. holomorphic) prin×
×
cipal bundle Σ with group R×
+ (resp. C ). The basis is BΣ = Σ/R+ (resp.
×
Σ/C ).
36
A complex cone is
A real cone is isomorphic to a product cone BΣ × R×
+.
isomorphic to L − {0} (L deprived of its zero section) where L is a complex line
bundle over BΣ. L will usually not be a trivial bundle.
Definition 87 (i) We denote O(m) the sheaf on BΣ of homogeneous functions
of degree m of Σ (holomorphic in the complex case).
b the sheaf on BΣ of formal symbols (“asymptotic expansions”
(ii) We denote O
for ξ → ∞ in Σ) :
X
b if f =
f ∈O
fm with fm ∈ O(m)
(55)
m≤m0
(m an integer, m → −∞).
36 at least if we are dealing with paracompact manifolds, which will always be the case in
this article.
103
b
Definition 88 For an integer k ≥ 1 we denote
P Dk the sheaf (on BΣ) of formal
k-differential operators : P (f1 , . . . , fk ) = m≤m0 Pm (f1 , . . . , fk ) with Pm a klinear differential operator homogeneous of degree m with respect to homotheties
(m an integer, m → −∞).
b
If k = 1 we will just write D.
Locally Σ is a product cone and we may choose homogeneous coordinates
(real or complex) xj of degree 0 on the basis, and r of degree 1 on the fiber.
Then Pm (f1 , . . . , fk ) is a sum of monomials
ϕ(x) rm ∂xα1 (r∂r )m1 (f1 ) . . . ∂xαk (r∂r )mk (fk ).
There is no restriction on the order of Pm .
The presence of two “degrees” is confusing so in what follows degree will
always refer to the degree with respect to homotheties, and order refers to the
bk each term Pm of degree m is of
degree as a differential operator; thus if P ∈ D
finite order, although the resulting infinite sum P may be of infinite order.
b× ⊂ D
b the sheaf of invertible formal differential operators :
We
D
Pwill denote
×
b
P =
Pk ∈ D is invertible iff its leading term σ(P ) = Pm0 is invertible, i.e.
Pm0 is of order 0, the multiplication by a nonvanishing function homogeneous
b × the subsheaf of those invertible P such that
of degree m0 . We denote by D
−
P (1) = 1, i.e. P is of degree 0, its leading term is P0 = 1 and terms of lower
degree have no constant term : Pm (1) = 0 if m < 0.
Remark 1 Sheaves are of course useless in the real case but must be used in
the complex case where global sections do not necessarily exist.
Remark 2 For analytic cones there is also a notion of convergent symbol
(introduced by the author in [13] to define analytic pseudodifferential operators).
These are in fact the more important and for many questions it is essential to
use convergent rather than formal symbols.37 However for the classification
results below, there is no significant qualitative difference between formal and
convergent symbols, so we will stick to formal symbols and avoid convergence
technicalities.
11.2.2
Star Products on a Real or Complex Cone.
Definition 89 A star product on Σ is a sheaf A on the basis BΣ, locally isob as a sheaf of vector spaces (the structural sheaf of groups is demorphic to O
scribed below), equipped with an associative unitary algebra law whose product
(star product) f ∗ g = B(f, g) is locally a formal bidifferential operator.
37 e.g. convergent rather than formal symbols are essential in the finiteness theorems of
T. Kawai and M. Kashiwara [91], or for going from E-modules to D-modules in the thesis
of D. Meyer [103], and probably in most problems involving a comparison between E and
D-modules.
104
P
Locally f ∗ g =
Bm (f, g) with Bm a bidifferential operator homogeneous
of degree m → −∞, B0 = 1. The first idea is that the structural sheaf of groups
b × (on BΣ) of invertible
used to patch together local frames of A is the sheaf D
formal differential operators, but there is a unit that we can choose equal to 1
b× .
in all local frames so this obviously reduces to D
−
b × . However
Note that homotheties (hence degrees) are not respected by D
−
b × , f and P f have the same leading term ; so P respects the filtration
if P ∈ D
−
bm if f = P
b
defined
by degrees (f ∈ O
j≤m fj ) and gr P is the identity on gr O =
L
O(m).
In the semi-classical definition, Σ is a product cone Σ = BΣ × L (L = R×
+
b and does not involve vertical
or C× ), the star product law is defined on O
derivatives, so the “Planck constant” h = r−1 plays the role of a constant.
The definition above includes the “semi-classical” case and also the algebras of
pseudodifferential or Toeplitz operators. This conic framework for star products
was described in [20].
In the real case, using partitions of unity, it is immediate to see that A
b as a sheaf (“there exists a global total symbolic
is always isomorphic to O
calculus”). This is no longer true in the complex case, and in particular it is
not true in the most simple and natural examples as we will see below, so the
sheaf theoretic presentation cannot be avoided.
11.2.3
Associated Poisson bracket
If A is a star algebra on Σ it has a canonical filtration coming from the filtration
b by homogeneity degrees, and there is a canonical isomorphism :
of O
b
gr A ' gr O
b × induces the identity on gr O).
b The
because the structural sheaf of groups D
−
b i.e. the leading
commutator law then defines a Poisson structure on gr A = gr O
term of the commutator law
{f, g} = B1 (f, g) − B1 (g, f )
is a Poisson bracket on Σ, homogeneous of degree −1. This means that it is a
skew-symmetric bivector field
{f, g} = −{g, f },
{f, gh} = {f, g}h + g{f, h}
satisfying the Jacobi identity (i.e. it is a Lie bracket) :
{f {g, h}} = {{f, g}h} + {g{f, h}}
and it is homogeneous of degree −1 with respect to homotheties
deg {f, g} = deg f + deg g − 1
105
if f, g are homogeneous.
Existence of a global star-algebra on a real symplectic cone Σ was proved
by V. Guillemin and myself in [27] (see also [17]), and by M. De Wilde and
P. Lecomte ([38],[39]) in the semiclassical symplectic case (cf. also the nice
deformation proof of B.V. Fedosov [60]).
In [91] M. Kontsevitch proved that any Poisson bracket comes from a starproduct in the real semiclassical case. More precisely he proves that there is
a one to one correspondence between isomorphic classes of star-products and
isomorphic classes of formal families of Poisson brackets depending on the “small
parameter” h. His result extends, without changing a word, to star-products
on a real cone with the definition above ; families of Poisson brackets should be
replaced by formal Poisson brackets on Σ :
X
c=
cm .
(56)
k≤−1
Kontsevitch’s formula giving a star product from a Poisson bracket on an affine
space also works without any modification in the complex case (i.e. Σ = Cn ×
C× ). But as mentioned above the argument used to go from local to global does
not work for complex manifolds, and in general I do not know if a global star
product exists for a given Poisson bracket, even in the symplectic case, nor do I
know what the classification of such algebras looks like (see however [88], where
it is shown that even if E may not exist, the category of sheaves of E-modules
is defined up to equivalence).
In the rest of the paper we investigate a special class of star algebras, i.e.
those which live on a cotangent bundle Σ = T ∗ X − {0}, X a complex manifold,
equipped with its canonical Poisson bracket. In this case there is a canonical
global star-algebra, viz. the algebra E of pseudo-differential operators, which
is the “microlocalization” of the sheaf DX of differential operators on X. It
is known and easy (cf. below) that any two star algebras with the same symplectic Poisson bracket are locally isomorphic, so our algebras are classified by
H 1 (BΣ, Aut E). It is also interesting to compare these with algebras of differential operators, locally isomorphic to DX on X hence classified by H 1 (X, Aut D) :
this is done in the next three sections.
11.3
11.3.1
Pseudo-differential Algebras
E-algebras
Let Σ = T ∗ X − {0} be the cotangent bundle (minus the zero section) of a
complex manifold X, equipped with its canonical symplectic structure. The
basis is BΣ = Σ/C× = P X, the projective cotangent bundle. There is a
canonical star algebra on Σ, viz. the algebra of pseudo-differential operators,
microlocalization of the algebra of differential operators on X, whose Poisson
bracket is the standard Poisson bracket of T ∗ X. If we choose local coordinates
x = (x1 , . . . , xn ) on X and the dual cotangent coordinates ξ = (ξ1 , . . . , ξn ) on
106
the fibers, the pseudodifferential product is given by the Leibniz rule for symbols
b:
f, g ∈ O
X 1
f ∗g =
∂ α f ∂xα g.
(57)
α! ξ
The patching cocycle is the cocycle defined by changes of coordinates : this is
a cocycle because it does patch together total symbols of differential operators
(locally : polynomials in ξ), to give the global sheaf DX of differential operators.
We are interested in star algebras on Σ associated to the canonical Poisson
bracket : we will call E-algebra such an algebra.
Proposition 90 Any E-algebras is locally isomorphic to E through an operator
b× .
P ∈D
−
This result is well known and we just give an indication of the proof : locally
the pseudo-differential algebra E has (topological) generators xi , ξi satisfying
the canonical relations
[xi , xj ] = [ξi , ξj ] = [ξi , xj ] − δij = 0.
If A is a star algebra with the same Poisson bracket, one can construct by
successive approximations symbols Xi , Ξi with the same principal part as xi , ξi
and satisfying the same canonical relations
[Xi , Xj ]A = [Ξi , Ξj ]A = [Ξi , Xj ]A − δij = 0.
Now there is a unique isomorphism U : E → A which takes xi to Xi and ξi to
b× .
Ξi and this is always a differential operator U ∈ D
−
Remark 3 The construction also works globally over any open subcone U ⊂
T ∗ Cn which is Stein and contractible (e.g. the set {ξi 6= 0} ⊂ T ∗ B, B a ball in
Cn , or a Stein contractible set). Over such a set, any E-algebra A is isomorphic
to E, and any section α of O(m) is the symbol of a section of Am .
Thus one obtains all E-algebras by gluing together models of E over a covering of Σ by open conic subsets Σi , using automorphisms of E on the intersections.
The following proposition sums up what was said above :
Proposition 91 Star algebras on Σ = T ∗ X − {0} are locally isomorphic to the
pseudo-differential algebra E. The set Alg E of isomorphy classes is canonically
isomorphic to H 1 (P X, Aut E).
Aut E denotes the sheaf of automorphisms of E ; the noncommutative cohomology H 1 (P X, Aut E) is described below in section 3.4.
107
11.3.2
Differential Operators and D-algebras
If X is a complex manifold, the sheaf DX of differential operators on X is
well defined. If U is an automorphism of DX preserving symbols, it fixes the
subalgebra OX ⊂ DX of operators of order 0, (because it fixes symbols and
preserves invertible operators, which are necessarily of order 0). It follows that
U is locally an inner automorphism of the form Int ef (f holomorphic). We
have Int ef = Id iff f is (locally) constant, so the automorphism sheaf is
×
Aut DX ' OX
/C× ' OX /C.
(58)
We will call D-algebra a sheaf of algebras on X locally isomorphic to DX (such
algebras appear in [7] where they are called “twisted algebras of differential
operators”). The set Alg D of isomorphic classes of these algebras is canonically
isomorphic to H 1 (X, OX /C).
A D-algebra obviously also defines a star-algebra on P X, and it is natural
to compare the two sets Alg D and Alg E .
11.3.3
Automorphisms and Symbols of Automorphisms
To understand how local E-algebras can be patched together to make global
objects, we have to know what automorphisms of E look like.
b × be an automorphism of E : U preserves symbols and the unit 1,
Let U ∈ D
−
P
so U − 1 is of degree ≤ −1 and the logarithm D = Log U = − n≥1 − n1 (1 − U )n
is well defined ; it is a derivation of degree ≤ −1 of E.
Now if D is a derivation of degree ≤ k its symbol δ = σk (D) is a homogeneous
b i.e. a symplectic vector field on
derivation of degree k of the Poisson algebra O,
Σ, homogeneous of degree k. This corresponds, via the symplectic structure of
Σ, to a closed differential form α, homogeneous of degree k + 1.
Let ρ P
denote the radial vector field, infinitesimal generator of the action of
C× (ρ =
ξj ∂ξj in local coordinates on X, T ∗ X as above) : the associated Lie
derivation is Lρ = iρ d + diρ (iρ denotes the interior product) so
diρ α = (k + 1) α.
Hence α is exact (the differential of a homogeneous function) if k + 1 6= 0. If
k + 1 = 0, s = iρ α is locally constant, and α is locally the differential of a
homogeneous function of degree 0 iff s = 0.
By successive approximations, it follows that locally any derivation D of E
is of the form s ad(Log P1 ) + adQ with P1 elliptic of degree 1, Q ∈ E, and any
automorphism of E is locally of the form
U = (Int P1 )s Int Q0
with P1 elliptic of degree 1, Q0 elliptic of degree 0.
automorphism a → P a P −1 .
38
(59)
38
Int P denotes the inner
as usual in the context of pseudodifferential operators, elliptic = invertible.
108
If U is an automorphism of E, we define its symbol σ(U ) as the closed 1-form
on Σ homogeneous of degree 0 corresponding to the leading term of Log U .
We have σ(U ) = dLog σ(P ) if U = Int P .global section of ω (this is a closed
1-form on Σ). If σ(U ) = 0 (Log U of degree ≤ 2) there exists a unique P ∈ E ×
of degree 0 and symbol 1 such that U = Int P . Summing up we have proved :
Proposition 92 There is an exact sequence of sheaves of groups on P X:
×
0 → E−
→ Aut E → ω → 0
(60)
×
where E−
denotes the multiplicative sheaf of groups on BΣ of sections of E of
symbol 1, and ω is the sheaf on P X of closed 1-forms homogeneous of degree 0
on Σ.
If A ∈ Alg E ' H 1 (P X, Aut E) its symbol σ(A) ∈ H 1 (P X, ω) is defined as
the image cocycle.
Remark 4 If U is an automorphism of A, it defines a one parameter group
U s = exp sLog U , s ∈ C. This is polynomial in s mod.An for any n < 0.
11.3.4
Non Commutative Cohomology Classes
In this section we recall the elementary resuls of noncommutative cohomology
that we will use (for more information see [69]). Let Y be a space and G a sheaf
of groups on Y . We denote H 0 (Y, G) = Γ(Y, G) the set of global sections of G
over Y : this is a group.
We denote H 1 (Y, G) the set of equivalence classes of cocycles
uij ∈ Γ(Yij = Yi ∩ Yj , G) such that uij ujk = uik
S
associated to open coverings Y = Yi ; two cocycles are equivalent if, after
a suitable refinement of the covering, we have uij = ui u0ij u−1
for some family
j
ui ∈ Γ(Yi , G).
H 1 (Y, G) classifies the set of isomorphy classes of G principal homogeneous
right G sheaves, i.e. sheaves α on Y , equipped with a right action of G, locally
isomorphic to G considered as a right G-sheaf.
Proposition 93 Let
0→A→B→C→0
(61)
be an exact sequence of sheaves of groups on Y , with A normal in B. Then
there is a long cohomology sequence ;
0 → H 0 (Y, A) → H 0 (Y, B) → H 0 (Y, C) →
→ H 1 (Y, A) → H 1 (Y, B) → H 1 (Y, C).
This is “exact” in the sense that
109
(62)
i) it is exact at the first three places (the H 0 are groups, the H 1 are pointed
sets).
ii) The group H 0 (Y, C) acts on the set H 1 (Y, A), and its orbits are the fibers
of the map H 1 (Y, A) → H 1 (Y, B) (the action is given by c · (aij ) = (bi aij b−1
j )
if c is a global section of B, and bi ∈ Γ(Yi , B) a lifting of c to B over a fine
enough covering Yi ).
iii) If β ∈ H 1 (Y, B) it defines twisted sheaves of groups Aβ ⊂ Bβ (where Bβ
is the sheaf of B-automorphisms of the principal B-sheaf β), and the fiber of
the map H 1 (Y, B) → H 1 (Y, C) is the image of H 1 (Y, Aβ ) in H 1 (Y, C).
More explicitly if β, β 0 are two principal B-sheaves, then γ = Hom B (β, β 0 )
is a principal Bβ -sheaf. If β, β 0 have the same image in H 1 (Y, C) then γ/Aβ =
Hom C (β/A, β 0 /A) has a global section, i.e. is trivial, so γ is the image of a
sheaf α ∈ H 1 (Y, Aβ ). Finally β 0 ∼ α ×Aβ β is in the image of H 1 (Y, Aβ ).
In this paper the noncommutative cohomology sequence stops there, and we
will not use higher cohomology H j , j ≥ 2 whose definition is more elaborate
(the substitutes are more complicated objects sometimes described by means of
“stacks”). Exact sequences concerning torsors as above were introduced by J.
Frenkel [65, 66]. Of course if A, B, C are commutative, the higher cohomology
groups H j , j ≥ 0 are well defined commutative groups, and we will occasionally
use the long cohomology exact sequence in that case up to order j = 2.
11.3.5
Symbols
If A ∈ Alg E ' H 1 (P X, Aut E) we have defined its symbol as the image of its
defining cocycle in H 1 (P X, ω). To compute H 0 and H 1 for automorphisms, it
will be useful to compute them first for symbols.
The following exact sequences of sheaves are also useful to handle ω :
0 → OP X /C → ω → C → 0
(63)
0 → C → OP X → OP X /C → 0
(64)
These give rise to long exact cohomology sequences. We will call “exponent
map” the cohomology maps coming from the map ω → C in (63).
With slight abuse we will call “Chern maps”
39
the maps :
ch : H j (Y, O/C) → H j+1 (Y, C).
(65)
in the long exact cohomology sequence derived from (64).
The sheaf O/C (Y = X or P X) identifies with the sheaf of closed holomorphic 1-forms on Y . If Y is a Stein manifold we have H j (Y, O) = 0 for j ≥ 1 so
the Chern map H j (Y, O/C) → H j+1 (Y, C) is an isomorphism for j ≥ 1.
39
The standard Chern map : H 1 (Y, O× ) → H 2 (Y, C) factors through H 1 (Y, O/C).
110
If Y is a compact Kähler manifold, the long exact cohomology sequence from
(64) splits into a sequence of short split exact sequences :
0 → H j−1 (Y, O/C) → H j (Y, C) → H j (Y, O) → 0
(j ≥ 0)
and for j ≥ 0 we have an isomorphism
X
H j (Y, O/C) =
H pq
(66)
p+q=j+1,p>0
where (here, and whenever possible) H pq denotes the space of harmonic forms
of type p, q on Y .
Proposition 94 (i) If n = dim X ≥ 2, or if X is a closed curve of genus 6= 1,
the map H 0 (X, O/C) → H 0 (P X, ω) is an isomorphism.
(ii) If X is an open curve or a closed curve of genus 1, then ω is split and
H 0 (P X, ω) ' H 0 (X, O/C) ⊕ H 0 (X, C).
Proof : A global section of ω is a closed 1-form on T ∗ X − {0}, homogeneous
of degree 0. Locally on X such a form α reads
X
α=
αk dxk + βk dξk
(67)
where the coefficients αk resp. βk are of degree 0 resp. −1. If n ≥ 2 this implies
βk = 0 so the αk only depend on x. Hence (i) for n ≥ 2.
If X is a closed curve of genus 6= 1 (n = 1 so P X = X), then the Chern map
H 0 (X, C) ' C → H 1 (X, O/C) = C is injective : it maps s ∈ C to s ch O(1)
(where as above O(1) denotes the sheaf of homogeneous functions of degree 1 on
T ∗ X) and ch O(1) 6= 0 if g 6= 1.40 So the exponent map H 0 (X, ω) → H 0 (X, C)
vanishes, and the map H 0 (X, O/C) → H 0 (X, ω) is an isomorphism, hence (i)
in this case.
If n = 1 and X is open or of genus 1, there exists a global nonvanishing
vector field, so ω is split : ω = O/C ⊕ C hence (ii).
Proposition 95 (i) If n = dim X ≥ 2 the map H 1 (X, O/C) → H 1 (P X, ω) is
an isomorphism.
(ii) If n = dim X = 1 (P X = X) and X is open or closed of genus 1 (ω
split), then H 1 (X, ω) = H 1 (X, O/C) ⊕ H 1 (X, C).
(iii) If X is a closed curved of genus g 6= 1 the exponent map H 1 (X, ω) →
H (X, C) ' C2g is an isomorphism.
1
40
The corresponding cocycle is dLog ( ξξi ) if ξi is the symbol of a nonvanishing vector field
j
on a covering Xi of X , whose image in H 1 (X, ω) is
111
dξi
ξi
−
dξj
ξj
, obviously a coboundary.
This should be complemented as follows in case (ii) : if X is an open curve,
H 1 (X, O/C) = 0 so H 1 (X, ω) ' H 1 (X, C).
If X is closed of genus 1, then H 20 = 0 so H 1 (X, O/C) ' H 20 + H 11 '
11
H ' C, and H 1 (X, ω) ' H 11 + H 1 (X, C) ' C3 .
Lemma 96 If X is a ball (or more generally Stein contractible space), we have
H 1 (P X, ω) = 0.
Proof : We have P X ' X ×Pn−1 , so H 1 (P X, C) = 0 (P X is simply connected)
and the map H 1 (P X, O/C) → H 1 (P X, ω) is onto (if n = 1 we are finished).
Next we wave H j (P X, O) = 0 for any j > 0 (O has no cohomology on
Pn−1 ) so the Chern map H 1 (P X, O/C) → H 2 (P X, C) ' C is one to one.
Now, as above for curves of genus 6= 1, H 2 (P X, C) ' C is generated by the
Chern class of O(1), corresponding to the cocycle d Log ( ξξji ) for ξi an elliptic
symbol of degree 1 over a covering Ui of P X. This is also precisely the image
of 1 ∈ H 0 (P X, C) ' C by the exponent map H 0 (P X, C) → H 1 (P X, O/C), so
the exponent map is onto and the map H 1 (P X, O) → H 1 (P X, ω) vanishes.This
proves the lemma.
Proof of Proposition 95. (i) Let α be a principal
S ω-sheaf on P X corresponding to a cocycle in H 1 (P X, ω), and let X = Xi be a covering of X
by complex balls (or Stein contractible open sets). Then αi = α|Xi is trivial. The patching isomorphism uij : αj → αi is the translation by a section
uij ∈ H 0 (P Xi ∩ P Xj , ω) = H 0 (Xi ∩ Xj , O/C) ; thus α is defined by a cocycle
(uij ) ∈ H 1 (X, O/C). If n ≥ 2 and if (uij ) = (αi − αj ) ∼ 0 in H 1 (P X, ω) then
again αi ∈ H 0 (Xi , O/C) by Proposition 94, so (uij ) ∼ 0 in H 1 (X, O/C). This
proves (i).
If n = 1 (P X = X) and X is open or of genus 1, ω is split so H 1 (X, ω) =
H (X, O/C) ⊕ H 1 (X, C).
If X is open then H 1 (X, O/C) = 0 because in the long exact cohomology sequence from (64) we have H 1 (X, O) = H 2 (X, C) = 0, so H 1 (X, ω) ' H 1 (X, C).
If X is of genus g = 1, we have H 1 (X, O/C) = H 20 + H 11 = C and
1
H (X, ω) ' H 11 + H 1 (X, C) ' C3 .
1
If X is a closed curve of genus g 6= 1 we have seen that the map H 0 (X, C) →
H 1 (X, O/C) is one to one, and H 2 (X, O/C) = H 30 + H 21 + H 12 = 0 so from
the long exact cohomology sequence from (63)
· · · → H 0 (X, C) → H 1 (X, O/C) → H 1 (X, ω) → H 1 (X, C) → . . .
we see that the map H 1 (X, ω) → H 1 (X, C) is one to one.
This proves Proposition 95 and its complement. Note that if X is a curve,
the only case where H 1 (X, ω) = 0 is when X is simply connected.
11.3.6
Filtrations
×
As mentioned above Aut E has a natural filtration (as well as E−
⊂ Aut E) :
any a ∈ Aut E is of degree ≤ 0 and a is of degree n < 0 if a = Ad (1 + b), b ∈ En .
112
The corresponding graded sheaf is
M
M
gr Aut E =
(Aut E)k /(Aut E)k−1 ' ω +
O(k).
k≤0
(68)
k<0
It is commutative, and this will help to extract more information. This also
works for any E-algebra A because the filtration above, and the leading terms,
are by definition invariant by automorphisms so gr Aut A ' gr Aut E.
Proposition 97 Let A be an E-algebra.
(i) The natural map gr H 0 (P X, Aut A) → H 0 (P X, gr Aut E) is injective. If
H (P X, gr Aut E) = 0 it is one to one.
1
(ii) If H 1 (P X, gr Aut E) = 0 then H 1 (P X, A× ) = 0.
(iii) If H 2 (P X, gr Aut E) = 0 the symbol map induces a surjective map
H 1 (P X, gr Aut E) → gr H 1 (P X, Aut A).
(69)
Proof : (i) The map gr H 0 (P X, Aut A) → H 0 (P X, gr Aut E) takes any U of
degree m ≤ 0 to its symbol σm (U ) which is a section of ω if m = 0 or of O(m)
if m < 0. σm (U ) = 0 means that U is really of degree ≤ m − 1 so the resulting
graded map is injective.
Conversely let Xi be a covering of P X, and ai ∈ H 0 (Xi , Aut A) be such
that ai a−1
is of degree m < 0 (this is true if σ(ai ) = a, a global section of
j
gr m+1 (P X, Aut E)). Then σm (ai a−1
j ) is a 1-cocycle with coefficients in O(m).
×
) = 0 this is a coboundary i.e. of the form σm (bi ) − σm (bj ),
If H 1 (P X, E−
bi ∈ Am so the Int (1 + bi )−1 ai Int (1 + bj ) are equal to ai mod. Aut Am and
patch together mod. (Aut A)m−1 . Note that if the Xi and their intersections are
Stein it is not necessary to shrink the covering, so by successive approximations
we get a cocycle a ∈ H 0 (P X; Aut A) equal to (ai ) mod. (Aut A)m .
×
(ii) If H 1 (P X, gr E−
) = 0 and if (aij ) is a cocycle of degree n < 0 with
×
coefficients in A , then σn (aij ) is a 1-cocycle with coefficients in O(n), hence a
coboundary σn (bi ) − σn (bj ), bi ∈ An . So aij is equivalent to the cocycle
(1 + bi )−1 aij (1 + bj )
which is of degree n − 1. Again we do not need to shrink the covering if it has
been chosen as above (Stein, contractible), so by successive approximations we
get a ∼ 0.
(iii) If bij is a cocycle with coefficients in Aut m A (m ≤ 0) its symbol σ(bij )
is a cocycle with coefficients in grm Aut E but in general this does not give rise
to a map gr H 1 (P X, Aut A) → H 1 (P X, gr Aut E) nor the other way round, at
best an ill-defined “noncommutative spectral sequence”.
However if H 2 (P X, gr Aut E) = 0, the same argument as above shows that
if a cochain aij ∈ H 0 (Ui ∩ Uj , Aut A) is a cocycle mod. (Aut
P A)m , m < 0,
i.e. aij ajk aki ∈ (Aut A)m then σm (aij ajk aki ) is a coboundary
σm (bjk ) with
113
coefficients in O(m), and again by successive approximations there exists a
cocycle a0ij with coefficients in Aut A equal to aij mod. Am .
In particular, by successive approximations, we see that any cocycle a with
coefficients in gr m Aut E (m ≤ 0) is the symbol of a cocycle b ∈ H 1 (P X, Aut m A)
which is well defined mod. H 1 (P X, Aut m−1 A) and vanishes if a is a coboundary. Thus our map is well defined and onto (if H 0 (P X, Aut A) 6= 0 it may not
be injective because two cocycles of degree m with coefficients in Aut A can
then be equivalent although their symbols are not).
11.4
11.4.1
E-Algebras on T ∗ X, dim X ≥ 2
General Results.
We first point out the following results (which will also be useful in section 5) :
Lemma 98 (Global automorphisms of E) If A is an E-algebra A on X and
n = dim X ≥ 2 the symbol map
H 0 (P X, Aut A) → H 0 (P X, ω) ' H 0 (X, OX /C).
is injective. It is bijective if A = E.
Proof : If n ≥ 2, A and Aut A have no global section of degree < 0 so the symbol
map u → σ(u) is injective. More generally if A, A0 are two E-algebras and u, v
two isomorphisms A → A0 the difference symbol σ(u−1 v) ∈ H 0 (P X, ω) is well
defined and completely determines v (given u) (note that we have σ(u−1 v) =
σ(vu−1 )).
On the other hand if A = E (more generally if A comes from a D-algebra) the
symbol map is onto because, by Proposition 94, H 0 (P X, ω) ' H 0 (X, O/C) '
Aut D, and this obviously lifts to Aut E.
If X is a ball of Cn or more generally a Stein contractible domain, we have
×
H 1 (P X, ω) = 0 (Lemma 96) so H 1 (P X, E−
) → H 1 (P X, Aut E) is onto, i.e. any
×
E-algebra can be defined by a cocycle with coefficients inE−
.
×
Now let A, A0 be two algebras defined by cocycles a = (aij ), a0 = (a0ij ) ∈ E−
0
and let u : A → A be an isomorphism, i.e. a family (ui ) ∈ Aut E such that
ui a0ij = aij uj . Then the symbols σ(ui ) patch together since σ(aij ) = σ(a0ij ) = 0,
and the resulting symbol σaa0 (u) is well defined. It only depends on the classes
×
×
) and
of a, a0 in H 1 (P X, E−
) (however it does depend on a, a0 ∈ H 1 (P X, E−
1
not just on their images in H (P X, Aut E) : any other representatives are of
the form α · a, α0 · a0 with α, α0 ∈ H 0 (P X, ω) for the action of H 0 (P X, ω) of
Proposition 93, and we get σα·a,α0 ·a0 (u)) = σ(u) + α − α0 .
If X ⊂ Cn , n ≥ 2, is a Stein contractible domain, the exponent of σaa0 (u)
vanishes : σaa0 (u) ∈ H 0 (X, O/C), and again σaa0 (u) completely determines u.
S
Let now A ∈ Alg E . There exists a covering X =
Xi where all finite
intersections are isomorphic to Stein contractible domains of Cn . Then Ai =
114
×
A|Xi can be defined by a cocycle (ai ) with coefficients in E−
; this being so
the patching isomorphisms uij all have exponent 0 and are determined by their
symbols σai aj (uij ) (for fixed Ai ). In particular we have proved :
Proposition 99 If dim X ≥ 2 any E-algebra A has exponent 0 (the image of
σ(A) ∈ H 1 (P X, ω) in H 1 (P X, C) by the exponent map is zero), so A can be
defined by a cocycle with coefficients in Int E0 = E0× /C× (E0 is the sheaf of
pseudo-differential operators of degree ≤ 0).
11.4.2
The case dim X ≥ 3
If X is a ball and dim X ≥ 3 we have H 1 (P X, O(−k)) = 0 for all k > 0, i.e.
×
H 1 (P X, gr E−
) = 0 (this is also true if X is a Stein manifold).
×
It follows that we have H 1 (P X, E−
) = 0, and more generally for any Ealgebra A we have H 1 (P X, A× ) = 0.
Hence if A is an E-algebra, it is built by patching together models of E over
a covering Xi of X, where the patching cocycle belongs to H 1 (X, OX /C).
Moreover if A, B are two such algebras, any isomorphism B → A comes
from a ϕ ∈ H 0 (X, OX /C) i.e. comes locally from an inner automorphism P →
ϕ P ϕ−1 . Summing up we have proved :
Theorem 100 If dim X ≥ 3 the functor which to a D-algebra associates the
corresponding E-algebra is an equivalence.
This result is closely related to the result of [37] on microlocally free D-modules
in dimension ≥ 3.
11.4.3
The case dim X = 2
If dim X = 2 what was said above remains true, in particular any symbol
α ∈ H 1 (P X, ω) is the symbol of an E-algebra (in fact of a D-algebra). However
×
the picture changes considerably because H 1 (P X, E−
) is usually very large. The
following examples show what can happen, and also how, in global situations
on compact manifolds, things can nevertheless at least partially cancel out.
Example 1. Let X be the unit ball of C2 (or more generally a Stein contractible manifold). 41
Then H 1 (P X, ω) = H 1 (X, O/C) = 0 so H 1 (P X, Aut E) is the quotient of
×
1
) by the action of H 0 (P X, Aut E) = H 0 (X, O/C).
H (P X, E−
Now P X is the union of the two Stein subcones Ui = {ξi 6= 0} (i = 1, 2) so
×
a cocycle is represented by just one section a12 ∈ E−
(U1 ∩ U2 ). It is elementary
×
1
that any a ∈ H (P X, E− ) has a unique normalized representative of the form
X
a12 =
apq (x)ξ1p ξ2q
(70)
p,q<0
41 what is used is H 1 (X, O/C) = 0 and the fact that T ∗ X is a trivial holomorphic vector
bundle.
115
i.e. with no holomorphic term in ξ1 or ξ2 (this is obvious for the additive
×
cohomology H 1 (P X, gr E−1 ) and follows by successive approximation for E−
).
1
So H (P X, Aut E) is the set of conjugate classes of normalized symbols a12 as
above, with a12 ∼ ϕ(x)a12 ϕ(x)−1 for ϕ a nonvanishing function on X. This set
is still very large ; on the other hand such algebras tend to have very few global
sections or automorphisms.
The analysis of these algebras is closely related to that of “microlocally” free
D-modules in dimension 2, made by M. Carette [32].
For global compact manifolds, some things may cancel out.
Example 2.
Let X = P2 (C) be the complex projective plane : then
P X is isomorphic to the incidence manifold {x · ξ = 0} ⊂ X × X ∗ (X ∗
the dual projective space). T ∗ X itself is the quotient of the incidence cone
Γ = {x.ξ = 0} ⊂ C − {0} × C by the group action (x, ξ) ∼ (λx, λ1 ξ). The sheaf
OP X (n) of homogeneous functions of degree n on T ∗ X identifies with the sheaf
of restrictions to Γ of functions f (x, ξ) such that f (λx, ξ) = f (x, λξ) = λn f (x, ξ)
i.e. OP X (n) = OX (n) ⊗ OX ∗ (n) (where exceptionally here OX (n) denotes the
×
canonical sheaf of the projective space). It follows easily that H 1 (P X, gr E−
)=
1
×
0 so H (P X, A ) = 0 for any E-algebra A.
The symbol map H 1 (P X, Aut E) → H 1 (P X, ω) = H 1 (X, O/C) is one to
one, and again, as in dimension ≥ 3, the correspondence D-algebras → Ealgebras is an equivalence.
Note that in this case we have H 1 (P X, O/C) ' H 20 + H 11 ' H 11 = C, and
E-algebras ∼ D-algebras are parameterized by H 11 = C.
Example 3 Let X be a holomorphic complex torus of dimension 2 (a torus
C2 /Γ with Γ ' Z4 acting by translations).
The group of automorphisms of E or D is
H 0 (P X, Aut E) = H 0 (X, O/C) = H 10 = C2
(71)
and any automorphism comes from an inner automorphism of E and D on C2
of the form :
P = P (x, d) → ea.x P e−a.x
(x → x, d → d − a).
(72)
Any E- or D-algebra on X lifts as the trivial algebra EC2 on the universal cover C2 , and is the quotient of EC2 by a group of isomorphisms over the
translation group of periods Γ.
By Proposition 95 we have
H 1 (P X, ω) = H 1 (X, O/C) = H 20 + H 11 = C5 .
More precisely an element α ∈ H 20 + H 11 is represented by a harmonic form
X
α = a dz1 dz2 +
aij dz̄i dzj .
(73)
116
There is a unique corresponding D-algebra, which is isomorphic to the quotient
of DC2 by the lifting µ → Uµ of the group Γ of periods (acting by translations) :
Uµ :
x→x+µ
d → d + p(µ, µ̄)
(74)
where p = (p1 , p2 ) is a linear map C4 = C2 × C2 → C2 such that dp(z, z̄).dz =
dp1 dz1 + dp2 dz2 = α, where z, resp. z̄ denotes the variable in C2 resp. C2 ,
and we use the notations of differential calculus. Such a map p splits into
holomorphic and antiholomorphic parts : p = p0 (z) + p00 (z̄). They form an
8-dimensional space, but it is classical that maps which differ by a symmetric
holomorphic map (dp.dz = 0) define isomorphic algebras.
Remark 5 Cocycles coming from H 11 are related to holomorphic line bundles
on X : if L is a line bundle, DL the sheaf of differential operators on L, the
corresponding cocycle is the image in H 1 (X, O/C) of the multiplicative cocycle
with coefficients in O× defining L ; the corresponding harmonic form is an
integral form in H 11 , and such forms generate H 11 if X is algebraic.
The cocycle associated to dz1 dz2 ∈ H 20 corresponds to the group of isomorphisms
Uµ : z → z + µ, d1 → d1 , d2 → d2 + µ1
(75)
This corresponds to the 1-form p(x) · dx = x1 dx2 (which could be replaced by
any holomorphic primitive of dx1 dx2 ). It never appears in a context of line
bundles.
We may now classify E-algebras. The map H 1 (P X, Aut E) → H 1 (P X, ω)
is onto, and for α ∈ H 1 (P X, ω) ' H 20 + H 11 the fiber σ −1 (α) is the image of
H 1 (P X, A× ) for A the unique D-algebra as above with this symbol.
Let us examine H 1 (P X, A× ) : by (72) two elements of H 1 (P X, A× ) give
the same E-algebra iff there is a translation ξ → ξ + a which transforms one to
×
the other. An a ∈ H 1 (P X, A× ) lifts to an element ã ∈ H 1 (C2 , E−
) invariant by
the Uµ , so the normalized representative (70) is invariant:
a(x, ξ) = a(x + µ, ξ + p0 (µ) + p00 (µ̄))
(76)
where as above p0 : C2 → C2 , resp. p00 : C2 → C2 denote the holomorphic and
antiholomorphic parts of p, which correspond to the H 20 , H 11 components of
the symbol α. Equivalently the symbol b(x, ξ) = a(x, ξ − p0 (x)) satisfies
b(x, ξ) = b(x + µ, ξ + p00 (µ̄)) =
X1 γ
! ∂ b(x + µ, ξ)(p00 (µ̄))γ .
γ ξ
(77)
If p00 = 0 this means that b does not depend on x (it is periodic hence constant).
If p00 6= 0, the periodicity condition implies b = 0 : for if b is of degree n ≤ −1,
its leading term is periodic in x hence independent of x : bn = bn (ξ) ; the next
term satisfies
bn−1 (x, ξ) − bn−1 (x + µ, ξ) = b0n (ξ) · p00 (µ̄)
(78)
117
so it is linear in x : bn−1 = β(ξ) + γ(ξ) · x with p00 (µ̄) = −γ(ξ).µ. Since p00 is
antiholomorphic this implies b0n = 0 so bn = 0 since its degree is negative.
Summing up we have proved :
Proposition 101 If X is a torus (C2 /Γ, Γ ' Z4 ), we have
H 1 (P X, ω) ' H 1 (X, O/C) ' H 20 (X) + H 11 (X) ' C5 .
(79)
Any symbol α ∈ H 1 (P X, ω) is the symbol of a unique D-algebra on X.
If the H 11 component of α is 6= 0 there is no other E-algebra with this symbol.
If α ∈ H 20 the E-algebras with symbol α can be defined by a mormalized cocycle
X
(80)
bpq ξ1p ξ2q (bpq ∈ C)
b(ξ) =
p,q<0
whose coefficients are translation invariant (independant of x). Two such cocycles b, b0 define the same E-algebra iff b0 (ξ) ' b(ξ + a) for some constant vector
a.
P1 α α
(this is an asymptotic relation between symbols : b(ξ + a) =
α !∂ξ b a )
11.5
E-Algebras over Curves (dim X = 1)
We now describe E-algebras, and compare them to D-algebras, when X is a
curve (dim X = 1). In this case P X = X. The general method is the same but
as we will see the classification is strikingly different depending on whether X
is an open curve, or a closed curve of genus g = 0, 1 or ≥ 2.
11.5.1
Open curves
If X is an open curve, the exponent map H 1 (X, ω) → H 1 (X, C) is an isomorphism (Proposition 95). Also X is Stein, so H j (X, O(n)) = 0 for j > 0 and for
×
×
all n, j ≥ 1, so H j (X, gr E−
) = 0 for j = 1, 2, and H 1 (X, E−
) = 0 (Proposition
97). Finally we have
H 1 (X; Aut E) ' H 1 (X, ω) ' H 1 (X, C).
(81)
Typically if (sij ) is a cocycle with coefficients in C, the corresponding algebra
is defined by a cocycle with symbol (Int ξ)sij , ξ a global nonvanishing vector
field.
×
These algebras have many sections because we have H 1 (X, E−
) = 0 so by
0
−1
0
Proposition 97 the map H (X, gr E) ' O(X)[ξ, ξ ] → gr H (X, A) is one
to one. They also have many automorphisms, because the sequence 0 →
H 0 (X, A× ) → H 0 (X, Aut A) → H 0 (X, ω) → 0 is exact.
D-algebras are classified by H 1 (X, O/C) = 0 and all give isomorphic Ealgebras. All non trivial E-algebras come from the exponent map. 42
42 the fact that such “exotic” algebras exist is related to the fact that coherent D-modules
do not always possess global good filtrations.
118
11.5.2
Curves of genus g ≥ 2
Note that in any case OP X (1) identifies with the sheaf of sections of T X (vector
fields) and the dual OP X (−1) identifies with the sheaf of sections of T ∗ X. If X
is of genus ≥ 2, we have H 1 (X, O(−k)) = 0 if k > 1, but H 1 (X, O(−1)) = C.43
Proposition 102 If X is a closed curve of genus g > 1, E-algebras on X are
classified by H 1 (X, Aut E) = C⊕C2g ; D-algebras are classified by H 1 (X, O/C) =
H 0 (X, C) = C and give isomorphic E-algebras.
We have a split exact sequence of sheaves of groups
×
0 → E−1 → Int E0 → OX
/C× → 0
It follows that we have H 1 (X, Int E) = C ⊕ C : the second factor comes from
×
H 1 (X, OX
/C× ) ∼ H 1 (X, O(−1)) = C; it classifies D-algebras.
The first factor is the image of H 1 (X, E−1 ) ∼ H 1 (X, O(−1)) = C (E−k
denotes the multiplicative L
group of elements (1 + a), deg a ≤ −k ; the graded
sheaf associated to E−1 is k≤−1 O(k), and O(k) is cohomologically trivial for
k < −1).
Consider the cohomology exact sequence (where the four last terms are commutative groups, even though Int E0 and Aut E are not):
0 → H 0 (X, Int E0 ) → H0 (X, Aut E) → H0 (X, C) →
→ H 1 (X, Int E0 ) → H1 (X, Aut E) → H1 (X, C) → 0.
The second factor C2g in prop.10 lifts H 1 (X, C) ; it classifies“exotic” algebras
as for open curves. Such an algebra can be defined by a cocycle Ad ξ sij where ξ
is a nonvanishing
vector field over X minus one point, subordinate to a covering
S
X = Xj where all Xj except one avoid the point.
In the long exact sequence, the unit 1 ∈ H 0 (X, C) maps to the cocycle
×
(ξi ξj −1 ) ∈ H 1 (X, OX
/C× ) ⊂ H 1 (X, Int E0 ), where (ξi ) is a family on nonvanishing vector fields over a covering (Xi ) of X. This is is not zero since the
Chern class of T X is not zero ; it is obviously killed by the map H 1 (X, Int E0 ) →
H1 (X, Aut E) (the E-algebra defined by a D-algebra is always trivial). In prop.
10 the first factor, range of the map H 1 (X, Int E) → H1 (X, Aut E), is isomorphic
to H 1 (X, E−1 ) ∼ C.
Here again E-algebras on X have many sections of negative degree and many
automorphisms.
11.5.3
Curves of genus 1
This is the most complicated of the cases examined here. Let X be a closed
curve of genus 1 : X = C/Γ where the group of periods is Γ ' Z2 acts by
translations.
43 the
original manuscript contained an error, corrected by P.Polesello.
119
We denote ξ the symbol of the constant vector field ∂/∂x on C.
Since T X is trivial, ω is split : ω = O/C + C. Also, for all n, we have
H 0 (X, O(n)) = H 00 ' H 1 (X, O(n)) = H 01 ' C, H 2 (X, O(n)) = 0.
We denote
G,
resp. G− ⊂ G
(82)
the group of automorphisms of E of the form Ad ξ s Ad (1 + a(ξ −1 )), resp. the
sub-group s = 0 : this is the commutant of ad ξ, it is a constant subsheaf of
Aut E.
For any a ∈ C we set ξa = eax ξ. This is only defined up to a multiplicative
constant eaµ , µ ∈ Γ, but the inner automorphism
Ad (eax ξ)
is well defined, as well as the corresponding commutator sheaf
Ga− ⊂ Ga
(83)
which is a locally constant subsheaf of Aut E.
Proposition 103 We have
H 0 (X, ω) = H 0 (X, O/C) + H 0 (X, C) = H 10 + H 00 ' C2
H 1 (X, ω) = H 1 (X, O/C) + H 1 (X, C) = H 11 + (H 10 + H 01 ) = C3 .
For the commutative locally constant sheaf Ga− we have
44
H j (X, Ga− ) = H j (X, C) ⊗ G− if a = 0, 0 if a 6= 0.
L
We have gr E− = n<0 O ξ n and with an obvious notation
×
) = gr G ' ξ −1 C[ξ −1 ]
H 0 (X, gr E−
×
H 1 (X, gr E−
) ' H 10 ⊗ ξ −1 C[ξ −1 ]
×
H 2 (X, gr E−
) = 0.
Theorem 104 If X is of genus 1, the symbol map Alg E → H 1 (X, ω) is onto.
We will denote σ(A) = α = (α11 , α10 , α01 ) ∈ H 11 × H 10 × H 01 the symbol of an
E-algebra A. Then
(i) Algebras such that α11 = 0 are characterized by the fact that they possess
a global section of degree 6= 0, or an automorphism of symbol dξ
ξ = σ(Int ξ). For
−1
such an algebra the set of global sections is C((ξ )) and except for E the group
of automorphisms is G.
44 the cohomology of the locally constant sheaf generated by eax ξ or enax ξ n vanishes if
na 6= 0, because eanµ cannot be identically 1 for µ ∈ Γ, so H ∗ (X, gr Ga− ) = 0.
120
E is distinguished by the fact that its symbol map H 0 (X, Aut E) → H 0 (X, ω)
is onto.
(ii) If α11 6= 0, A has no section of degree 6= 0 (H 0 (X, A) = C), and A is
completely determined by its symbol, in other words the image of H 1 (X, A× ) in
H 1 (X, Aut A) is reduced to a single point.
For such an algebra the group of automorphisms is a one parameter group
01
= 0, α11
with symbol C(a dx + b dξ
ξ ) for some (a, b) 6= 0, except in the case α
10
and α 6= 0, where there is no automorphism other than Id.
(iii) Among these, E-algebras associated to a nontrivial D-algebra are those
for which σ(A) = α11 ∈ H 11 (α10 = α01 = 0). They are characterized by the
fact that their group of automorphisms is a one parameter group with symbol
σ(Int etx ), (t ∈ C).
Thus for a torus X of genus 1, D-algebras which give isomorphic E-algebras
are already isomorphic as D-algebras, and E-automorphisms are the same as
D-automorphisms, except for the canonical algebra E.
Let A be an E-algebra. The symbol map Alg E → H 1 (X, ω) is onto because
×
H (X, gr E−
) = 0 so (Proposition 97) any cocycle with coefficients in ω is the
symbol of an E-algebra.
2
Next note that any star algebra on X lifts as the trivial E-algebra EC on C
with an action of the group Γ over the translation group :
µ → Tµ = Tµ Uµ
(84)
where Tµ is the translation (x → x + µ, ξ → ξ) and the Uµ are automorphisms
of EC , subjected to the cocycle condition expressing that µ → Tµ Uµ is a group
homomorphism.
Here are typical examples (models) :
Example 4 Let µ ∈ Γ → Uµ = α(ξ) ∈ G be an additive map. This defines
such a cocycle, because the Uµ commute with translations, hence an E-algebra,
obviously of the first type. since ξ is invariant. Typically the period group
µ → Uµ = (Int ξ)α
10
µ+α01 µ̄
defines such an E-algebra with symbol α10 + α01 (α11 = 0).
Example 5 Let α(µ) = α10 µ + α01 µ̄ be an additive map Γ → C and a ∈ C.
Then the automorphisms Int (exp α(µ)(ax+Log ξ)) commute and also commute
with translations (because the commutator [ξ, ax + Log ξ] = a is a constant so
exp s(ax + Log ξ) commutes with translations, mod. constant factors which give
trivial inner automorphisms). So the group homomorphism
µ → Uµ = exp α(µ)(ax + Log ξ)
121
defines an E-algebra, whose symbol is aα01 , α10 , α01 ∈ H 11 × H 10 × H 01 . If we
identify the H pq with spaces of differential forms, the symbol of A writes
σ(A) = (a dα(x, x̄)dx̄ , dα(x, x̄)).
Such an algebra admits the automorphisms Int exp s(ax + Log ξ), s ∈ C.
The only symbols we have missed are those for which
a = α11 6= 0, b = α10 6= 0, α01 = 0
As model for this case we can take the algebra defined by
Uµ = (Int ξ)bµ Int (eaµ̄x̃ )
with x̃ = x(1 + ξb )−1 so that σ(x̃) = x, [ξ + bLog ξ , x̃] = 1 : with this choice
the Tµ (Int ξ)bµ (symbolically exp µ(ξ + bLog ξ)) commute with the Int eaµ̄x̃ , so
again the Uµ define an E-algebra with symbol α11 = a , α10 = b , α01 = 0.
We now prove Theorem 104.
(i) First suppose that A has a nonzero section s of degree 6= 0. Then σ(s) = cξ k
for some constant c 6= 0 and integer k 6= 0; c−1 s has a unique k-th root with
symbol ξ (this is true locally because it works for pseudo-differential calculus;
the roots with symbol ξ are unique and patch together into a global section).
Similarly if a is an automorphism with symbol σ(Int ξ), locally there exists a
unique section s with symbol ξ such that a = Int s (a = Int b determines b
locally up to a constant, and σ(b) = ξ fixes the constant so again these patch
into a global section with symbol ξ).
If A has
P a section s with symbol ξ, then clearly all global sections of A are of
the form k≤k0 ck sk , and H 0 (X, Aut A) contains the group G (formula (82)).
Furthermore, again by elementary pseudo-differential calculus, any two sections, resp. automorphisms of E of symbol ξ are locally conjugate, so (A, a) is
locally isomorphic to (E, Int eξ ), and A can be defined by a cocycle with coefficients in G, the commutator of Int ξ (formula (82)). Hence α = σ(A) belongs
to the image of H 1 (X, G) in H 1 (X, ω) i.e. α ∈ H 1 (X, C) and α11 = 0.
Conversely let α ∈ H 1 (X, C) (α11 = 0). Example 4 gives an algebra A with
symbol α which has a section of symbol ξ.
Now any other algebra A0 with symbol α is defined by a cocycle (aij ) ∈
H (X, A× ). We know that there is a surjective map from H 1 (X, gr A× ) to
×
gr H 1 (X, A× ) and also that the map H 1 (X, G) → H 1 (X, E−
) = H 1 (X, A× )
×
is surjective. It follows that the embedding G− → A gives a surjective map
H 1 (X, G) → H 1 (X, A× ) (the symbol map (gr) is onto, and surjectivity follows
by successive approximations). Thus any E-algebra with symbol α can be defined from A by a cocycle with coefficients in G− , or from E with coefficients in
G; in particular it has a section with symbol ξ.
1
122
Lemma 105 If such an algebra A (α11 = 0) is not trivial, it has no other global
automorphism than those of G.
Proof : If A has two automorphisms a, b with independent symbols, we may
suppose that these symbols are σ(Int ξ), σ(Int ex ). So A has a section α with
symbol ξ such that a = Int α. The section Log b = β is locally well defined up to
an additive constant, so the section γ = [α, β] is globally defined and commutes
with α (as any global section of degree < 0).
Now the symbol of γ is 1, so γ is invertible, and replacing β by βγ −1 ,
we see that we can suppose [α, β] = 1 (or equivalently b−1 ξb = ξ + 1). It
follows again, by successive approximations, that A equipped with two such
automorphisms is locally isomorphic to E equipped with Int ξ and Int ex ; but
the only automorphisms which commute with both are obviously trivial (the
leading term is constant because it commutes with x and ξ), so A is isomorphic
to E.
(ii) Suppose now α11 6= 0. Then any any section is a constant (of degree 0)
and there is no global section of degree 6= 0. Let us choose an algebra A with
×
symbol α (one of the models above). Here again since H 2 (X, gr E−
) = 0 the
1
×
1
×
graded map H (X, gr A ) → gr H (X, A ) is surjective, and any cochain with
coefficients in A× which is a cocycle mod. An (n < 0) is equivalent mod. An to
a cocycle.
Lemma 106 We have H 1 (X, A× ) = C.
Proof
S : Let A be defined by a cocycle Uij relative to some Stein covering
X = Xi . We can choose Uij = Int eαij x Int ξ sij mod. lower order terms, with
constant αij , sij .
Then α11 = σ(A)11 ∈ H 11 corresponds to the (01) part of the cocycle αij ,
and does not vanish. Let a = (ai ) be the “constant” 1-cochain ai = ξ n (n ≤ 0)
on Xi . Its coboundary with coefficients in A× (computed in A(Xi )) is of degree
n − 1; more precisely on Xi we have
ai − αij (aj ) = ξ n − aij (ξ n ) = ξ n − (ξ − αij )n + · · · = nαij ξ n−1 + . . .
(85)
where the . . . are lower order terms, because Int ξ commutes with ξ and the rest
only contributes to terms of degree ≤ n − 2.
It follows by successive approximations that any cocycle of degree < −1
is equivalent to 0, and the same computation shows that two cocycles with
the same leading term are equivalent. Since the symbol map H 1 (X, A× ) →
H 1 (X, O(−1)) = H 01 is onto, we have H 1 (X, A× ) = C : any cocycle βij ξ −1 is
the symbol of a unique element of H 1 (X, A× ).
Lemma 107 The map H 1 (X, A× ) → H 1 (X, Aut A) is constant.
Proof : The fibers of this map are the orbits of the action of H 0 (X, ω) (Proposition 93). We will prove that this is transitive.
123
For this action σ(Intξ) acts by u = (uij ) → us = usij with Int usij =
Int ξis Int uij Int ξjs (mod. coboundary equivalence), where ξi ∈ A(Xi ) has symbol ξ, and multiplication is the multiplication of A. Now in the local frame on
Xi we have ξj = Uij (ξ) = x − αij + . . . so for leading terms we get
σ(usij ) = σ(uij )
ξ
= σ(u) (1 + αij ξ −1 )
ξ − αij
(86)
or with additive notation σ(us ) = σ(u) + sα11 .
This proves Lemma 107, and the other assertions of Theorem 104 are immediate consequences.
11.5.4
The projective line
Let X be the projective line (X = P1 (C)). It is the union of the two open sets
X0 = {z 6= ∞}, X∞ = {z 6= 0}, and since these are Stein, contractible (' C), E
or D-algebras are classified by cocycles reduced to one function on X0 ∩ X∞ .
D-algebras are classified by H 1 (X, O/C) = H 2 (X, C) = C. The D-algebra
Ds (s ∈ C = H 1 (X, O/C)) is defined by the cocycle (Int z)s .
Let us introduce homogeneous coordinates x, y (z = xy ). We make use of
the sheaf of homogeneous differential operators Dhom on C2 , i.e. differential
operators of x and y which commute with the generator of homotheties ρ =
x∂x + y∂y ; this algebra is generated by ρ and the operators
e = x∂y ,
h = x∂x − y∂y ,
f = y∂x
(87)
which satisfy the relations
[h, e] = 2e,
[h; f ] = −2f,
[e, f ] = h,
h2 + 2(ef + f e) = ρ(ρ + 2).
(88)
Ds is isomorphic to the quotient sheaf Dhom /(ρ+s) and can be thought of as
the sheaf of differential operators on the virtual sheaf O(s) on X of homogeneous
functions of degree s of x, y (which really only exists s when s is an integer).
We now turn to E-algebras.
Lemma 108 If X = P1 is the projective line then
(i) H 0 (X, gr Aut E) = 0 hence H 0 (X, Aut A) = 0 for any E-algebra A.
×
(ii) H 1 (X, ω) = 0 and H 1 (X, E−
) → H 1 (X, Aut E) is one to one.
Proof : In homogeneous coordinates as above, O(n) is the sheaf of homogeneous functions of degree 2n of x, y and has no global section if n < 0 , hence
×
×
H 0 (X, gr E−
) = H 0 (X, E−
) = 0.
124
We have proved above (Proposition 94, 95) that for the projective line we
have H 0 (X, ω) = H 1 (X, ω) = 0, hence the lemma.
Note that any q ∈ H 1 (X, Aut E) has a unique “normalized” representative :
X
b 0 ∩ X∞ ).
q0∞ =
apq z p ζ q ∈ O(X
(89)
0>p>2q
This is because the two vector fields ∂0 = ∂z , ∂∞ = ∂1/z are globally holomorphic
and elliptic on X0 , resp. X∞ and their symbols are ζ, −z 2 ζ, so any cocycle can
uniquely be reduced to the form above, as for the additive cohomology group
×
H 1 (X, gr E−
).
To compare D-algebras and E-algebras it is convenient to use the following intermediate exact sequence ; let Int E0 ' E0× /C× be the group of inner
automorphisms of E0 ; we have an exact sequence :
0 → Int E0 → Aut E → C → 0
hence a surjection
H 1 (X, Int E0 ) → H 1 (X, Aut E)
0
(90)
1
whose fibers are the orbits of the action of C = H (X, C) on H (X, Int E)
(q0∞ → (Int ∂0 )s q0∞ (Int ∂∞ )−s , cf. Proposition 93).
Lemma 109 We have the following relation :
(Int z)−s−2 = (Int ∂0 )s+1 (Int z)s (Int ∂∞ )−s−1 .
(91)
Proof : If s = k is a positive integer we have
z −k−2 (z 2 ∂)k+1 = ∂ k+1 z k .
Indeed both are ordinary differential operators of order k + 1, with leading term
z s ∂ s+1 , which kill all monomials z j , 0 ≥ j ≥ −k.
Identity (91) for arbitrary s follows, because it is polynomial in s mod.
(Aut E)m , for any m < 0.
It follows that Ds resp. D−s−2 give isomorphic E-algebras, although they
are not isomorphic D-algebras. This is the only case where two D-algebras on
X = P1 give isomorphic E-algebras: the algebra of global sections is obviously
an invariant of an E-algebra, and in this the global sections e, h, f (with the
notations above) are well defined (up to an additive constant by their symbols,
and the commutation relations fix these constants). It follows that s(s + 2) =
h2 + 2(ef + f e) is an invariant of the E-algebra coming from Ds .
Note that D-algebras form a one-parameter family, so there are many Ealgebras which do not come from an D-algebra.
As last remark we turn to the following problem: does there exist a global
symbolic calculus, i.e. is the underlying sheaf of a given E-algebra isomorphic to
125
b This is always true for real E-algebras, where one can patch global symbols
O?
using a partition of the unity.
Let us examine what happens on X = P1 (C) . There is a canonical 2covering of T ∗ X − {0} by C2 − {0} : (u, v) → (z = u/v, ζ = 21 v 2 ). If A is a
E-algebra on Σ = T ∗ X its pull-back on Σ0 = C2 − {0} is a star-algebra for the
canonical Poisson bracket ({v, u} = 1), equipped with an involution above the
symmetry (u, v) → (−u, −v) (note that on Σ0 , u and v are of degree 12 ). If A
b 0 ).
has a global symbolic calculus, its pull-back defines a star-product on O(Σ
0
b
Now on O(Σ ) there is (up to isomorphism) only one star-algebra law for the
canonical Poisson bracket, generated by u, v withPthe relation [v,
Pu] = p1. qUp to
isomorphism this is given by the representation
apq up v q →
apq u ∂u . For
this law there are many global sections (i.e. all polynomials of u, v) : the global
sections e, h, f are necessarily
1
e = − u2 ,
2
1
h = 2u ∗ v + ,
2
f=
1 2
v
2
(92)
because their respective symbols are
1
σ(e) = −z 2 ζ ∼ − u2
2
σ(h) = 2zζ ∼ u v,
σ(f ) = ζ ∼
1 2
v
2
these determine e, h, f up to additive constants, and the commutation relations
(88) determine the constants as above.
For these constants we get
h2 + 2(ef + f e) = −
3
4
1
3
so s = − . or s = −
2
2
(93)
We have proved :
Proposition 110 The only D-algebras on P1 for which there is a global total
symbolic calculus are D−1/2 and D−3/2 . In particular there is no global total
symbolic calculus for D.
126
12
12.1
Related symplectic star algebras.
Geometric quantization
In [92] B. Kostant introduced pre-quantization or geometric quantization, in
order to describe some remarkable representations of semi-simple groups. We
describe this here in terms of Toeplitz operators, as in [27].
For this the data is
- a complex holomorphic cone Γ of complex dimension n in a numeric space
CN , smooth outside of the origin. Xe set Γ• = Γ − {0}; the complex base
Y = Γ• /C× is a smooth projective complex manifold Y ⊂ PN (C), and Γ•
identifies with L0 − the zero section, where L is an ample line bundle over Y .
- a smooth strictly pseudo-convex hermitian metric r on Γ, i.e. r is homogeneous of degree 1 (r(λz) = |λ|r(z)), > 0 and smooth outside of the origin,
and
P
the matrix of second derivatives ∂zp ∂z̄q is hermitian 0. E.g. r2 =
zj z¯j ; r
is given on Γ, but it can always be extended to CN ).
¯
Note that r is strictly pseudo-convex ⇔ rs is for any s > 0 ⇔ ∂ ∂Log
r defines
a Kähler metric on X.
Let X be the sphere r = 1 in Γ. The Szegö projector and Toeplitz operators
are well defined on X. The corresponding contact form is i∂r|X , the symplectic
cone, set of positive multiples of λ, can be canonically identified with Γ.
The circle group U (1) acts on the whole situation; the Szegö projector S is
invariant (provided we choose an invariant volume element to define it - in fact
there is a canonical one : dλ(λ)n−1 ).
We denote θ the infinitesimal generator; it is elementary that it is the hamiltonian field θ = Hr .
P
Let A be the Toeplitz operator iTθ : f 7→ iSθf = iθf =
zj ∂zj f . Then the
eigenspace Hk = ker H − k is the space of holomorphic functions homogeneous
of degree k, and O0 (X) is the Hilbert sum of the Hk .
Recall that Toeplitz operators are those of the form TP : f 7→ SP (f ) with P
a pseudo-differential operator on X. Modulo smoothing operators they define
a symplectic star algebra A on Σ.
A Toeplitz operator TP is invariant iff [A, TP ] = 0; it is always equal to a TQ
with Q invariant (replace P by its mean over U (1)); such an operator preserves
each Hk . Mod. smoothing invariant operators form a sub algebra B ⊂ A which
is a deformation algebra, iff we set the deformation parameter ~ = A−1 (A
is invertible mod. smoothing operators - in fact its kernel H0 is the space of
constants).
Remark 12 Any Toeplitz operator os degree 0 is of the form Tf mod. operators
of degree −1 (its symbol is homogeneous of degree 0 so it is a function f on X).
By successive approximations
we see that any Toeplitz operator ha a unique
P
formal expansion m≤m0 Tfk Ak . For an in variant Toeplitz operator, the fk
are invariant i.e. P
they are smooth (not holomorphic) functions on Y ; this can
be rewritten P ∼
k ≥ k0 Tfk ~k .
127
12.2
Homomorphisms between Star Algebras
If A, A0 are two star-algebras over cones Σ, Σ0 we have described in section4.1
homomorphisms U : A → A0 (preserving the filtrations); the symbol map is
f 7→ u∗ f = f ◦ u where u is a smooth homogeneous map Σ0c → Σc preserving
the Poisson brackets, i.e. u∗ {f, g} = {u∗ f, u∗ g}.
Below we will only consider algebras whose Poisson bracket is real (or pure
imaginary), and (positive) homomorphisms whose symbol map maps Σ0 to Σ
For instance the algebra of semi-classical pseudo-differential P (x, h∂, h) on
a manifold V is isomorphic to the algebra of pseudo-differential operators on
on the sub-cone τ = σ(∂t ) > 0 of T • (X × R) which do not depend on t) (i.e.
commute with ∂t ) : the map takes h to ∂t−1 . It is easy to see that any 1codimensional embedding of a symplectic deformation algebra to a symplectic
star algebra is locally isomorphic to the embedding above.
Using this example it is easy to embed a symplectic deformation algebra AX over
a symplectic manifold X in a symplectic star algebra AΣ where Σ is a disjoint union
of pieces as above. However without more information on the projection BΣ → X
this only gives very poor information: to reconstruct AX from AΣ one would need
to know how various components patch together, which requires further non trivial
information.
Definition 111 We will say that a symplectic deformation algebra B on X is
“related” to A if there exists an injective homomorphism B → A, where the
corresponding projection Y → X is a principal circle bundle (this definition will
be slightly refined below).
A typical example comes from the “geometric prequantization” above. We
will determine in the section which pairs A, B of symplectic star algebras and
deformation algebras can be related in that manner. We need first to investigate
how the circle group or more generally a compact group acts on a symplectic
algebra.
12.3
Action of a Compact Group
Let A be a symplectic star algebra on Σ (resp. a deformation algebra on X) as
above.
Theorem 112 Let G be a compact group acting on Σ by symplectic homogeneous isomorphisms. We suppose g ∗ A ∼ A for all g ∈ G, i.e. g ∗ RA − RA
is exact, with RA the Fedosov curvature of A (this is always true if G is connected).Then
1. the action of G lifts to A.
2. Any two liftings are conjugate through an automorphism of A.
The same result holds for symplectic deformation algebras.
128
The theorem follows from the fact that the group of global automorphisms
Aut A is a complete filtered group, and gr A is a G-vector space, so since G is
compact any continuous cocycle is a coboundary. Note that if G is connected,
it fixes all cohomology classes (because it fixes integral cohomology), so the
invariance condition is automatic.
The first assertion is also seen using Fedosov connections:
c resp. W
ch .
1. If G acts on Σ (resp. X), its action extends functorially to W
By hypothesis the cohomology class of A is invariant, so it has an invariant
representative R since G is compact. Then Fedosov’s construction yields an
invariant connection ∇ (we first choose ∇s invariant so the starting point ∇s −τ
is invariant), and G acts on ker ∇ ∼ A.
2. If U0 (g), denoted below g (f 7→ gf ∈ A), and U1 (g) = Ug are two liftings,
we set
ug = Ug g −1 ∈ Aut A
so that we have ugh = ug guh g −1
This means that σ(ug ) is a 1-G-cocycle with coefficients in gr Aut A (i.e. we
have σ(g · uh ) − σ(ugh ) + σ(ug ) = 0), so it is of the form σ(ug ) = g.σ(v) − σ(v)
(the cohomology vanishes in positive degree since G is compact). By successive
approximations we get v ∈ Aut A such that ug = v −−1 gvg −1 i.e. Ug = v −1 gV .
12.4
Circle Action
We suppose now that G is the circle group G = U (1). The action of G on Σ
has an infinitesimal generator θ, which is a symplectic vector field homogeneous
of degree 0, and a generating function a which is a homogeneous function of
degree 1 :
θ = Ha , with a = Iθ (λΣ )
(94)
P
where Iθ =
θj L ∂ denotes the interior product by θ, λΣ the Liouville form
∂xj
of Σ.
We will also denote
Lθ
c lifting θ
the vector field with coefficients in W
(95)
Let A be a symplectic star algebra on Σ. As noted above (theorem 112) the
circle group action lifts to A. The infinitesimal generator D of this action is well
defined; it is unique up to conjugation by an automorphism of A,45 and there
45 Here is an alternate proof of this: we first choose a derivation D of degree 0 in A such
0
that σ(D0 ) = ∂θ , for instance D0 = ad A0 where A0 ∈ A is any element with symbol a. Then
for t ∈ R, exp tD0 is a well defined group of isomorphisms above exp tθ. In particular e2πD0
is an automorphism of A (over exp 2πθ = Id ): it is of the form exp 2πδ with δ a derivation of
degree −1 which commutes with D0 . Then D = D0 − δ is the infinitesimal generator of an
action of U (1), lifting the action on Σ (Ut = exp tD).
If D1 = D + δ1 with δ1 a derivation of degree −1), we see by successive approximations
129
c , e.g. we choose an invariant repreexists an equivariant embedding u : A → W
sentative of the curvature R: the corresponding canonical Fedosov connection
∇ is then invariant:
[Lθ , ∇] = 0
(96)
c the image of A, and set
We denote Ae ⊂ W
∇θ = [Iθ , ∇],
Rθ = [Iθ , R]
(97)
here and everywhere else [, ] is the superbracket: [f, g] = (−1)f g (f ∗ g − g ∗ f ),
where f , g denotes the degree of f resp. g as differential forms. ∇θ is a vector
c , and we have
field with coefficients in W
Lθ = ∇θ + α
c)
(α ∈ W
(98)
We have α(x, 0) = 0 if ∇ “vanishes” on the zero section, in particular if ∇ is
the canonical connection associated to R.
We have Rθ = [Iθ , ∇2 ] = [[Iθ , ∇], ∇]. Since [Iθ , ∇] = ∇θ = Lθ − b and ∇ is
invariant, i.e. [Lθ , ∇] = 0, we have
Rθ = [Lθ − α, ∇] = ∇(α)
(99)
Lemma 113 Notations being as above, the infinitesimal generator D of the
action of U (1) on A is an inner derivation if and only if Rθ is exact (as a form
b
on Σ with coefficients in O).
e In other words the O-derivation
b
Proof : ad Lθ coincides with ad α on A.
of
c corresponding to D is ad α. If D is an inner derivation there exists b ∈ Ae
W
(∇b = 0) such that α − b is central, so Rθ = ∇α = d(α − b) is exact.
12.5
Elliptic Circle Action
Definition 114 An action on Σ of the circle group U (1) is elliptic if its generating function a is > 0 (there is an obviously symmetric case a < 0).
From now on we suppose that Σ is equipped with a free elliptic action of
U (1), with generating function a > 0 (θ = Ha , a ∈ OΣ (1)). We will use a as
privileged radial coordinate. The basis of Σ is identified with the“unit sphere”
Y = {a = 1} ⊂ Σ ∼ Σ/R×
+
(100)
Y is a principal U (1)-bundle, with basis X and connection form λY :
X = Y /U (1) ,
λY = λΣ |Y
(101)
that D1 is conjugate to D + δ2 where δ2 commutes with D (if U = 1 + u with u of degree
P −N ,
U −1 D1 U = [D,
uk , we
Pu] + v with v of degree −N − 1, and if the Fourier series of u is u =
have [D, u] =
ikuk ). Then if exp 2πD1 = Id we have exp 2πδ2 = Id i.e. δ/2 = 0 since δ2
is of degree < 0.
130
where as above λΣ denotes the Liouville form λΣ of Σ; we will also denote
λY = λaΣ the pull-back to Σ.
The basis X = Y /U (1) is a symplectic manifold with symplectic form
ωX
such that p∗ (ωX ) = dλY
(102)
cΣ , and as above Lθ denotes the vector
The U (1) action lifts canonically to W
cΣ defined by the infinitesimal generator θ.
field with coefficients in W
Let Γ = Σ/UP
(1). We have T Γ = T Σ/T U (1), where the tangent generator is
(obviously) the
θj ∂ξ∂ j ; it corresponds to a vector field τθ with coefficients in
c:
W
τθ = Iθ · τ
(103)
∂
c of the interior product by θ,
θj I( ∂x
) is the extension to Ω ⊗ W
j
c (def. 38).
and τ is the canonical 1-form of W
where Iθ =
P
cΓ is identified with the sub-algebra of W
cΣ of sections invariLemma 115 1) W
ant by the tangent group T U (1), i.e. killed both by Lθ and ad τθ .
ch (X) is identified with the quotient W
cΓ / (τθ = 0)
2) The Weyl algebra W
(if U (1) acts by translations (x1 , . . . , xn ) 7→ (x1 + u, x2 , . . . , xn ), which is always
true in a suitable set of local coordinates, then T U (1) acts by (x, ξ) 7→ (x1 +
u, . . . , xn , ξ1 + v, . . . , ξn )).
Let now A be a symplectic star algebra on Σ, equipped with an extension of
the action of U (1) (we have seen this exists, and is unique up to conjugation).
Up to equivalence, A is determined by the cohomology class of its Fedosov
curvature R. We may choose R invariant, so as the Fedosov connection (if we
start with an invariant torsionless symplectic connection, the canonical construction of §4.6 produces an invariant connection). There is a corresponding
equivariant embedding, which we will adjust further suitably below.
The curvature is R = ωΣ + r, with r closed, of weight w(r) ≥ 0. As any
closed 2-form on Σ, r is cohomologous to an invariant form homogeneous of
degree 0, so we may suppose:
r = µX + λY νX + (γX + cλY )
da
a
(104)
where µX , νX , γX are the pull-backs of 2 and 1-forms on X.
Since dr = 0, c is a constant, and dγX +cωX = 0, dνX = 0, dµX +ωX νX = 0.
We necessarily have c = 0 if ωΣ is not exact (on any component of X); this
is always the case if X is compact.
131
Since Iθ ωΣ = −da we have, with the notations above:
Rθ = Iθ R = −da + νX + c
da
a
(105)
We have set Lθ = ∇θ + α. Since ∇θ = [Iθ , ∇ and [Lθ, ∇] = 0 we get
Rθ = [Iθ , ∇2 ] = [∇θ , ∇] = −[α, ∇, i.e.
Rθ = ∇(α)
(106)
Le us further note that the leading term of α is τθ = Iθ (τ ).
c such that U αU −1 = τθ .
Lemma 116 There exists an invertible section U of W
This is immediate by successive approximations as above: if w(αn ) ≥ n2 there
P
θj ∂ξ∂ j = αn (there is a unique
exists u such that w(u) ≥ n+1
2 , [τθ , u] =
solution such that u(, ξ) = 0 if ξ belongs to the hyperplane orthogonal to the
Liouville form λ, which is transversal to ad τθ ). Then (1 + u)−1 τθ (1 + u) =
τθ + αn + (w ≥ n+1
2 ).
Let B = AU (1) ⊂ A be the invariant sub-algebra. The image Be ⊂ Ae is
Be = ker Lθ ∩ Ae = ker ∇ ∩ ker ad α
(107)
cΓ if α = τθ .
It is also contained in W
Lemma 117 B possesses a non-trivial central element iff the component νX of
the Fedosov curvature of A vanishes. Then B possesses a structure of related
symplectic deformation algebra.
Proof : Let B 6= 0 be a central element of B, of degree k 6= 0. The symbol of B
1
is c ak for some constant c 6= 0, and replacing B by ( Bc ) k ∈ B we may suppose
σ(B) = a = h−1 . Then the infinitesimal generator is necessarily of the form
D = ad (B 0 ),
with B 0 = B + c Log B +
∞
X
ck B −k
1
With the notations above, α0 = α − B̃ 0 is central, and Tθ = ∇α = dα0 . The
da
only term of B 0 not in B is cLog B with differential c da
a +exact, so Rθ − c a is
da
exact, i.e. νX ∼ 0 and c is the coefficient of a which appears in (104),(105).
12.6
End of description
We may now finish the analysis of related symplectic algebras. We will suppose
the basis X compact (or more generally that ωX is not exact, on each component
of X). We leave as an exercise the case where ωX is exact so the constant c
above can be 6= 0.
We refine definition 111 as follows:
132
Definition 118 We will say that a relating homomorphism B → A is good
(or that we have a good relation) if the operator A corresponding to h−1 is an
infinitesimal generator of U (1)
Theorem 119 We suppose X compact.
1) Let A be a symplectic star algebra over Σ with elliptic action of U (1) as
above, and Fedosov curvature RA , and let B be the invariant sub-algebra. Then
B possesses a structure of symplectic deformation algebra if and only if νX =
Iθ R ∼ 0. In this case the infinitesimal generator of the action of U (1) on A is an
inner derivation ad A and there exists a unique semiclassical structure
P on B such
that h = f (A) = A−1 + . . . for any formal series f (T ) = T −1 + k>1 fk T −k .
It is well-related to A iff h−1 = A + c, c ∈ C a constant.
2) Let B be a semiclassical algebra over X = Y /U (1) as above. Then B has a
related symplectic star algebra if and only if its Fedosov curvature RB is constant
mod. ωX , i.e. of the form RB = h−1 ωX + constant + ϕ(h)ωX . It is well-related
A iff RB = h−1 ωX + µ, µ a constant 2-form on X (independent of h).
∗
The Fedosov curvature of A is then RA = p∗ RB + da
a p γX for some γX ∈
1
H (X). In particular there is a unique (up to isomorphism) “non exotic” such
A (γX = 0).
Proof : Let A be a symplectic star algebra on Σ, and choose R, ∇ as above, so
Lθ = ∇θ + α, α = τθ . Since X is compact we have c = 0 anyway. If A contains
a symplectic deformation algebra, we have νX ∼ 0 so we may as well suppose
νX = 0, R = ωΣ + µX + γX da
a .
This being so we have Rθ = ∇α = −da so ∇(a + α) = 0. The infinitesimal
generator is ad A where A is the element with total symbol a (Ã = a + α).
Moreover with this choice Ã is a polynomial of degree 1 with respect to ξ so
Ã ∗ f = Ãf + 12 {Ã, f }. It follows in particular that Ã ∗ f = Ãf (or a ∗ f = af in
A for the star product and total symbol defined by this choice of embedding) if
A commutes with f , and we get a well related semiclassical algebra by setting
h = A−1 .
If V is a vector field on X, we denote Ṽ the unique vector field on Σ which
projects on V , such that Iθ Ṽ = Iρ Ṽ = 0, with ρ the generator of homotheties
(i.e. Ṽ is orthogonal to the conic rays and fiber circles and projects on V - in
particular it is homogeneous of degree 0 and rotation invariant). With our choice
of embedding (D̃ = ad α, α = τθ ), ∇Ṽ obviously preserves ΩΓ = ker Lθ ∩ker ad α,
and kills a and α so goes down to Ωh . The Fedosov connection ∇B of B is then
obtained by restriction:
∇B
V = the derivation of Ωh defined by ∇Ṽ |Ωh
(108)
The curvature RB is the induced by R: RB = µX . Note that it does
not depend on γX , and changing γX , which is equivalent to twisting A by a
cocycle Asij , (sij ) a cocycle on X with coefficients in C, does not change B, the
commutator of A. Among the symplectic star algebras related to B the only
one which is not “exotic” has curvature µX .
133
Any other related symplectic deformation structure
P∞ is obtained by replacing
h by ϕ(h) for some formal series ϕ(h) = h + 2 ϕk hk ; as seen in Prop.42,
the corresponding symplectic deformation algebras are pairwise non-isomorphic.
h
(ϕ(h)−1 =
Among them, are well-related to A those for which ϕ(h) = 1+γh
h−1 + γ).
Remark 13 If νX is not cohomologous to 0, the invariant sub-algebra Be is not
a semiclassical algebra, because its center is not big enough. However its lifting
e is; Be itself is obtained by gluing together local models
to the universal cover X
of symplectic deformation algebrasP
with isomorphisms which preserve symbols,
but do not fix h (h 7→ U (h) = h + k≥2 uk hk ).
Example 4 In the canonical model
e have SS = Cn − 0, a = |z|2 . The symP
plectic form is ωΣ = i∂∂a = i dzj dzj (twice the usual one - the coordinates
1
zj , z j are homogeneous of degree
P 2 so a is of degree 1); we have {z p , zq } = iδpq
(the Poisson bracket is c = i ∂zj ∧ ∂zj . The infinitesimal generator of homoP
theties is ρ = 12
zj ∂z∂ j + zj ∂z∂ j . Rotations are the usual ones: z 7→ eit z, ith
P
infinitesimal generator θ = i zj ∂z∂ j − zj ∂z∂ j = Ha .
A is the Weyl algebra, with product f ∗g = exp 12 c(∂x , ∂y ) f (x)g(y)|y=x . The
c
action of U (1) lifts to A, with generator ad a. The canonical embedding A → W
is f 7→ f (x+ξ), the corresponding connection is ∇ = d−τ , τ = i(dzj ζ j −dz j ζj ).
The invariant subalgebra B is a semiclassical algebra over X = Pn−1 (C), ith
h = a−1 .
Note that in this case, with the notations above, we have a+α = |z+ζ|2 6= τθ
(we still have to modify ∇ to get the connection used above).
134
13
Toeplitz operators and asymptotic
equivariant index.
This is an account of a lecture given at the 7th Issac congress (july 12-18 2009),
where I described a joint work with E. Leichtnam, X. Tang and A. Weinstein
giving a proof of the Atiyah-Weinstein index formula. This concerns the index
of an operator closely related to Toeplitz operators, for which analogues of the
Atiyah-Singer index formula does not make sense. Instead we used an equivariant asymptotic index formula, which does; it is an outgrowth of Atiyah and
Singers theory of equivariant index for transversally elliptic pseudodifferential
operators. 46
13.1
Szegö projectors, Toeplitz operators
We first describe generalized Szegö projectors and Toeplitz operators, which
generalize pseudo-differential operators on arbitrary contact manifolds. An important case arises from complex (CR) analysis.
Let M be a compact manifold, and Σ ⊂ T • M a symplectic subcone 47 .
Definition 120 A generalized Szegö projector associated to Σ (or Σ-Szegö projector) is a self adjoint elliptic Fourier integral projector S of degree 0 (S =
S ∗ = S 2 ), whose complex canonical relation C is 0, with real part the diagonal diag Σ (elliptic means that the principal symbol of S does not vanish on
Σ).
Specially useful examples are
1) Σ is the full cotangent bundle T • M , S is the identity operator.
2) M is the boundary of a strictly pseudoconvex bounded complex domain,
S is the Szegö projector (see below). More generally, M is a compact oriented
contact manifold, Σ ⊂ T • M is the set of positive multiples of the contact form
(a generalized Szegö projector always exists, see below).
13.1.1
Example 1: Microlocal model
The following example was described in [6]. It is universal in the sense that
any generalized Szegö projector is micolocally isomorphic to it, via some elliptic
Fourier integral transformation (with dim Σ = 2p, dim M = p + q).
Let (x, y) = (x1 , . . . , xp , y1 , . . . , yq ) denote the variable in Rp+q . Set D =
(Dj ), with
Dj = ∂yj + |Dx |yj (j = 1, . . . , q)
The Dj commute; the complex involutive variety char D is defined by the
complex equations ηj − i|ξ|yj = 0; it is 0, in the sense of [20, 101]. Its real
part is the symplectic manifold Σ : {ηj = yj = 0}.
46 MSC2010:
47 T •
19L47, 32A25, 53D10, 58J40.
denotes the cotangent bundle deprived of its zero section.
135
The kernel of D in L2 is the range of the Hermite operator H (in the sense
of [6]) defined by its partial Fourier transform:
f ∈ L2 (Rp ) 7→ Hf with Fx Hf (ξ, y) =
|ξ| q4 − 1 |ξ|y2 ˆ
e 2
f (ξ)
π
The orthogonal projector on ker D is S = HH ∗ :
Z
|ξ|
0
2
02
|ξ| q2
f 7→ (2π)−p
ei(<x−x ,ξ>+i 2 (y +y ) (
f (x0 , y 0 )dx0 dy 0 dξ
π
R2p+q
As H, it is a Fourier integral operator, whose complex canonical relation is 0,
with real part the graph of Id Σ .48
13.2
Example 2 : holomorphic model
Let X be the boundary of a strictly pseudoconvex Stein complex manifold (with
smooth boundary); the contact form of X is the form induced by Im ∂φ where
φ is any defining function (φ = 0, dφ 6= 0 on X, φ < 0 inside).
e.g. if X is the unit sphere bounding the unit ball of Cn , with defining
function z̄ · z − 1, the contact form is Im z̄ · dz|X .
The Szegö projector S is the orthogonal projector on the holomorphic
subspace H = ker ∂¯b boundary values of holomorphic functions - the fact that
S is Fourier integral operator as above was proved in [15]). The system of
(pseudo)differential operators playing the role of D is the tangential Cauchy
Riemann system ∂¯b .49
Remark: A basic example of Toeplitz structure is Σ = T • M (M a compact
manifold), S = Id : the Toeplitz algebra is the algebra of pseudodifferential operators acting on the sheaf of microfunctions on M . This is in fact a special
case of the holomorphic case - example 2.
13.3
Main properties
Cf. [27, 9, 26]
1) A Σ-Szegö projector S always exists. All such projectors have a unique
microlocally model (via some elliptic FIO transformation) depending only
on dimΣ, dimM .
2) Toeplitz operators defined by S are the operators on H of the form u ∈
H 7→ TP (u) = SP S(u) with P a pseudodifferential operator on M . They
48 Fourier integral operators are described in [82]. Fourier integral operators with complex
canonical relation are described in [20, 101]
P∞
49 at least if the dimension n is > 1 - if n = 1, S is the Hilbert projector
k
−∞ fk z 7→
P∞
k , it is a pseudodifferential projector
f
z
k
0
136
form an algebra EX (or EΣ or E 50 . Mod smoothing operators, they form
a sheaf acting on µH, locally isomorphic to the sheaf of pseudodifferential
operators acting on the sheaf of microfunctions (in p variables if dim Σ =
2p).
3) If S, S 0 are two Σ-Szegö projectors with range H, H0 , S 0 induces a quasi
isomorphism H → H0 (the restriction of SS 0 to H is a positive (≥ 0) elliptic
Toeplitz operator).
More generally, if Σ ⊂ T • M, Σ0 ⊂ T • M 0 are two symplectic cones and
f : Σ → Σ0 a homogeneous symplectic isomorphism, there always exists a
Fourier integral operator F from M to M 0 , inducing an “elliptic” Fredholm
map H → H0 (such elliptic FIO exist, they were called “adapted” in [27,
9]).
The pair EΣ , µH consisting of the sheaf of micro Toeplitz operators (i.e.
smoothing operators), acting on µH is well defined, up to (non unique) isomorphism: it only depends on the symplectic cone Σ, not on the embedding.
4) H is the set of solutions of a system (an ideal) of pseudo-differential equations described by a pseudo-differential complex DΣ mimicking the ∂¯b in
the holomorphic case (see below).
The K-theoretic element [DΣ ] ∈ KX (§∗ M ) it defines is precisely the Bott
element, defining the Bott periodicity isomorphism K(X) → KX (S ∗ M ).
5) All these constructions allow a compact group action.
We also use a vector bundle extension: an equivariant G-bundle is an invariant direct factor E of a trivial bundle G vector-bundle X × V , defined by
an invariant projector p (V a finite representation of G). The corresponding
Toeplitz space (or Toeplitz bundle) HE , with symbol E, is the range of an
equivariant Toeplitz projector P of degree 0 in H ⊗ V , with symbol p. Here
again HE is only defined up to a Fredholm map. Equivalently, H is defined by a
’good’ projective E module M, i.e. the range of Toeplitz projector P 0 of degree
0 in some free left-module E N : E = HomE (M, H).
If E, F are two equivariant Toeplitz bundles, there is an obvious notion of
Toeplitz operator P : E → F, and of its principal symbol σd (P ) if it is of degree
d, which is a homogeneous vector-bundle homomorphism E → F on Σ.
P is elliptic of degree d if its symbol is invertible; then it is a Fredholm
operator E(s) → F(s−d) and has an index (which does not depend on s)51 .
50 if
M is a manifold one writes EM for ES ∗ M
its space of Sobolev H s sections of E.
51 E(s)
137
13.4
Miscellaneous
Toeplitz-Fourier integral operators
The analogue of Fourier integral transformations is the following: let X, X 0
be two contact manifolds, S, S 0 generalized Szegö projectors, anf f : X → X 0 a
contact isomorphism. The pushforward map u 7→ u ◦ f −1 does not send H to
H0 : we correct it as for Toeplitz operators Tf (u) = S 0 (u ◦ u−1 ); this behaves as
an elliptic Fourier operator attached to the contact map f . Other analogues of
F.I.O attached to f are of the form u 7→ A0 Tf u, A0 a Toeplitz operator on X 0 .
Atiyah-Weinstein problem:
The Atiyah-Weinstein problem can be described as follows: If X is a compact
contact manifold, and S, S 0 to Szegö projectors defined by two embeddable CR
structures giving the same contact structure, then the restriction of S 0 to H
is a Fredholm operator H → H0 (SS 0 induces an elliptic Toeplitz operator on
H). In this case the spaces H, H0 and the index are well defined. The AtiyahWeinstein conjecture computes the index in terms of topological data of the
situation (topology of the holomorphic fillings of which X is the boundary).
13.5
Equivariant Toeplitz algebra
In the sequel we use the following notations:
R
G: a compact Lie group, with Haar measure dg ( dg = 1), Lie algebra g.
Σ: a G- symplectic cone, basis X (a compact oriented contact G-manifold).
ω its symplectic form, λ the Liouville form (ω = dλ) (G- invariant).
Σ is canonically identified with the set of positive multiples of λX in T ∗ X.)
L
S a G-invariant generalized Szegö projector, with range H = ˆ Hα
(where α runs over the set of irreducible representations, and Hα is the corresponding isotypic component of H).
13.6
Equivariant trace
The G-trace and G-index were introduced by M.F. Atiyah in [5] for equivariant
pseudo-differential operators on a G-manifold. The G-trace of P is a distribution
on G, describing tr (g ◦ P ). We adapt this to Toeplitz operators.
Any v ∈ g defines a vector field Lv on X and a Toeplitz operator Tv on H
(or any Toeplitz bundle E).
Definition 121 char g (characteristic set of g) denotes the closed subcone of Σ
where all symbols of infinitesimal operators Tv , ξ ∈ g vanish.
The base Z of char g is the set of points of X where all Lie generators
Lv , v ∈ g are orthogonal to the Liouville form λX . char g contains the fixed
point set ΣG , whose basis is the fixed point set X G because G is compact. Note
that ΣG is always a smooth symplectic cone and its base X G a smooth contact
manifold; char g and Z may be singular.
138
L
Let E be an equivariant Toeplitz bundle as above, E =
Eα its the decomposition in isotipic components. If P : E → E is a Toeplitz operator of
trace class (deg P < −n), the trace function TrG
P (g) = tr (g ◦ P ) is a continuous
function on G (it is smooth if P is of degree −∞), and we have
TrG
P (g) =
X 1
tr P |Hα χα
dα
α
(109)
where χα is the character of α, dα the dimension (the Fourier coefficient is
1
dα tr P |Eα ).
The following result is an immediate adaptation of the similar result of [5]
for pseudo-differential operators.
Theorem 122 Let P : E → E be a Toeplitz operator, with P ∼ 0 near char g.
Then TrG
P (g) = tr g ◦ P is defined as a distribution on G; P |Eα is of trace class
for each α and formula (109) holds.
G
We have TrG
P Q (g) = TrQ P (g) if one of the two operators is equivariant and
one ∼ 0 near char g; so TrG defines a trace map on the algebra of equivariant
Toeplitz operators.
Proof: this is true if P is of trace class. For the general case, we choose a
bi-invariant elliptic operator D of order m > 0 on G, e.g. the Casimir of a
faithful representation, with m = 2; it defines an invariant Toeplitz operator
DX : E → E, elliptic outside of char g. If P ∼ 0 near Σ, we can divide it
repeatedly by DX (mod. smoothing operators) and get for any N :
N
P = DX
Q+R
(with R ∼ 0)
G
G
N
Then TrG
P = D TrQ + TrR : this is well defined as a distribution since Q is of
trace class if N is large, and it does not depend on the choice of D, N, Q, R.
The series is convergent in distribution sense, i.e. the coefficients have at
most polynomial growth with respect to the eigenvalues of D.
More generally if we have let an equivariant Toeplitz complex of finite length:
(E, d) :
d
· · · → Ej −
→ Ei+1 → · · ·
i.e. E is a finite sequence Ek of equivariant Toeplitz bundles, d = (dk : Ek →
Ek+1 ) a sequence of Toeplitz operators such that d2 = 0. Then for a Toeplitz
operator
P : E → E, P ∼ 0 near char g, its equivariant supertrace TrG
P =
P
k
(−1) TrG
Pk is well defined; it vanishes if P is a supercommutator [A, B] where
A, B are equivariant, and one of them vanishes near char g.
139
13.7
Equivariant index
Let E0 , E1 be two equivariant Toeplitz bundles.
Definition 123 We will say that an equivariant Toeplitz operator P : E0 → E1
is G-elliptic (transversally elliptic in [5]) if it is elliptic on char g, i.e. the
principal symbol σ(P ), which is a homogeneous equivariant vector bundle homomorphism E0 → E1 , is invertible on char g.
If P is G-elliptic it has a G-parametrix Q, i.e. Q : F → E is equivariant, and
QP ∼ 1E , P Q ∼ 1F near char g.
The G-index Ind G
P is then defined as the distribution
G
G
Ind G
P = Tr1−QP − Tr1−P Q .
(110)
More generally, an equivariant complex (E, d) as above is G-elliptic if the
([d, s] = ds + sd). The index (Euler characteristic) is the super trace
X
G
I(E,d)
= supertr (1 − [d, s]) =
(−1)j TrG
(1−[d,s])j
If P is G-elliptic, the restriction Pα : E0,α → E1,α is a Fredholm operator for
any irreducible representation α. Its index Iα is finite (resp. more generally the
cohomology Hα∗ of d|Eα is finite dimensional), and we have
Ind G
P =
13.8
X 1
Iα χα
dα
(or Ind G
(E,d) =
X (−1) j
dα
dim Hjα χα )
(111)
Asymptotic index
The G-index Ind G
P is obviously invariant under compact perturbation and deformation, so for fixed Ej it only depends on the homotopy class of the symbol
σ(P ). But it does depend on the choice of Szegö projectors: the Toeplitz bundles Ej are known in practice only through their symbols Ej , and are only
determined up to a space of finite dimension, just as the Toeplitz spaces H.
However if E, E0 are two equivariant Toeplitz bundles with the same symbol,
there exists an equivariant elliptic Toeplitz operator U : E → E0 with quasiinverse V (i.e. V U ∼ 1E , U V ∼ 10E ). This may be used to transport equivariant
Toeplitz operators from E to E0 : P 7→ Q = U P V . Then if P ∼ 0 on X0 ,
Q = U P V and V U P have the same G-trace, and since P ∼ V U P , we have
TP − TQ ∈ C ∞ (G). Thus the equivariant G-trace or index are ultimately well
defined up to a smooth function on G.
Definition 124 We define the asymptotic G-trace AsTrG
P as the singularity of
G
∞
the distribution TrG
P (i.e. TrP mod. C (G)).
140
If P ∼ 0, we have TrG
P ∼ 0, i.e. the sequence of Fourier coefficients is of
rapid decrease, O(cα )−m for all m, where cα is the eigenvalue of DG in the
representation α.
Definition 125 If P is elliptic on char g, the asymptotic G-index AsInd G
P is
defined as the singularity of of Ind G
.
P
P
It can also be viewed as a virtual trace-class representation or character nα χα
of G, mod finite representations.
It only depends on the homotopy class of the principal symbol σ(A), and
since it is obviously additive we get :
Theorem 126 (Main theorem) 1) The asymptotic index defines an additive
G
map from AsInd : KX−Z
(X) to Sing(G) = C −∞ /C ∞ (G) (Z ⊂ X denotes the
basis of char g).
2) If u : X → X 0 is a contact map, the the asymptotic index map AsInd
commutes with the Bott periodicity map K(X − Z) → K(X 0 − u(Z))
The Bott periodicity map is described below.
G
KX−Z
(X) denotes the equivariant K-theory of X with compact support
in X − Z, i.e.the group of stable classes of triples (E, F, u) where E, F are
equivariant G-bundles on X, u an equivariant isomorphism E → F defined near
Z, with the usual equivalence relations ((E, F, a) ∼ 0 if a is stably homotopic
near Z to an isomorphism on the whole of X).
The asymptotic index is as well defined for equivariant Toeplitz complexes,
exact on Z.
Example : let Σ be a symplectic cone, with free positive elliptic action of U (1),
i.e. the Toeplitz generator A = 1i ∂θ is elliptic with positive symbol (this is the
situation studied in [27]). Then the algebra of invariant Toeplitz operators (mod.
C ∞ ) is a deformation star algebra, setting as “deformation parameter” ~ =
A−1 . char g is empty and the asymptotic trace or
index is always defined. The
P∞
asymptotic trace of any element A is the series −∞ ak ekiθ , ak = tr A|Hk , mod
smooth functions of θ, i.e. the sequence (ak ) is known mod rapidly decreasing
sequences. It is standard knowledge that the sequence (ak ) has an asymptotic
expansion in (negative) powers of k:
X
ak ∼
αj k j .
(112)
j≤j0
In this case the asymptotic trace is as well defined by this asymptotic expansion; it encodes the same thing as the residual trace, viewed as a power series
of ~ = k −1 .
Remark. For a general the circle group action, with generator A = eiθ , all
simple representations are powers of the identity representation, denoted T , and
141
all representations occurring as indices can be written as formal power series
with integral coefficients:
X
nk T k (mod. finite sums)
k∈Z
In fact, using the sphere embedding below, it can be seen that the positive and
negative parts of the series are “weakly periodic”, of the form
P± (T, T −1 )
(1 − T ±k )k
for suitable polynomials P± and some integer k, i.e. both the positive and
negative parts are the Taylor series of rational functions whose poles are roots
of 1; the asymptotic index corresponds to the polar parts.
13.9
It is convenient (even though not technically indispensable), in particular to
follow the index in an embedding (§129), to reformulate some constructions
above in terms of sheaves of Toeplitz algebras and modules. in the C ∞ category
E is not coherent and general E-module theory is not practical. We will just
stick to two useful examples 52
As above we use the following notation: for distributions, f ∼ g means that
f − g is C ∞ ; for operators, A ∼ B (or A = B mod. C ∞ ) means that A − B
is of degree −∞, i.e. has a smooth Schwartz kernel (if M is a manifold, T • M
denotes the cotangent bundle deprived of its zero section; it is a symplectic cone
with base the cotangent sphere S ∗ M = T • M/R+ ). As mentionned earlier, if Σ
is a G-symplectic cone, the sheaf EΣ of Toeplitz operators (mod C ∞ ) acting on
µH is well defined, with the action of G, up to isomorphism, independently of
G
any embedding Σ → T • M . The asymptotic trace AsTrG
P resp. index AsInd P
are well defined for a section P of EΣ vanishing (resp. invertible) near char g.
(If M is a G-manifold and X = S ∗ M (Σ = T • M ), EΣ identifies with the sheaf
of pseudodifferential operators acting on the sheaf µH of microfunctions on X;
even in that case the exact index problem does not make sense: a Toeplitz
bundle E corresponds to a vector bundle E on the cotangent sphere X = S ∗ M ,
not necessarily the pullback of a vector bundle on M , and E is in general at
best defined up to a space of finite dimension).
An E-module M, corresponds to a system of Toeplitz operators, whose sheaf
of micro-solutions is Hom E (M, µH); likewise a locally free complex (L, d) of Emodules defines a Toeplitz complex (E, D) = Hom (L, H).
We will say that the E-module
S M is “good” if it is finitelyT generated,
equipped with a filtration M =
Mk (i.e. Ep Mq = Mp+q ,
Mk = 0)
52 In the proof of the Atiyah-Weinstein conjecture we need to patch together two smooth
embedded manifolds near their boundaries: this cannot be done in the real analytic category,
even if things work slightly better there.
142
such that the symbol σ(M ) = M0 /M−1 has a finite locally free resolution (as
a C ∞ (X)-modul 53 ). A locally free resolution of σ(M) lifts to a “good resolution” of M (i.e. locally free and whose symbol is a resolution of σ(M)).54 Two
resolutions of of σ(M) are homotopic, and if σ(M) has locally finite locally free
resolutions it also has a global one (because on comact X (or on the cone Σ
with compact basis) we dispose of smooth (homogeneous) partitions of unity);
this lifts to a global good resolution of M.
Similarily we will say that a G-elliptic complex (E, d) is “good” if its symbol is
exact on char g. Note that “good” is not indispensable to define the asymptotic
index, but it is to define the K-theoretic element [(E, d)] ∈ K G (X − Z).
All this works just as well in presence of a G-action (one must choose invariant filtrations etc.).
The asymptotic trace and index extend in an obvious manner to endomorphisms of good complexes or modules:
• if M = E N is free, End E (M) identifies with the ring of N × N matrices
with coefficients in the opposite ring E op , and if A = (Aij vanishes near
P
char g we set AsTrG (A) = AsTrG (Ajj ).
• If M is isomorphic to the range P N of a projector P in a free module
N (this does not depend on the choice of N our if A ∈ End E(M ) we set
AsTrG (A) = AsTrG (P A).
• If (L, d) is a locally free complex and A isPa A = (Ak ) endomorphism,
vanishing near char g, we set AsTrG (A) = (−1)k AsTrG (Ak ) (the Euler
characteristic or super trace; if A, B are endomorphisms of opposite de2
grees m, −m, we have AsTrG [A, B] = 0, where [A, B] = AB − (−1)m BA
is the superbracket).
• If M is a good E-module, (L, d) a good locally free resolution of M,
e where A
e is any extension
A ∈ End E (M), we set AsTrG (A) = AsTrG (A,
of A to (L, d) (such an extension exists, and is unique up to homotopy i.e.
up to a supercommutator).
• Finally if M is a locally free complex with symbol exact on char g, or a
good E-module with support outside of char g, it defines a K-theoretical
element [M] ∈ KZG (X), and its asymptotic index (the supertrace of the
identity), is the image by the index map of theorem 126 of [M].
Remark. The equivariant trace or index are defined just as well for modules
admitting a projective resolution (projective meaning direct summand of some
53 The symbol map identifies E /E
∞
0
−1 with C (X); since there exist global elliptic sections
of E, gr M is completely determined by the symbol, same for the resolution.
54 the converse is not true: if d is a locally free resolution of M its symbol is not necessarily
a resolution of the symbol of M - if only because filtrations must be defined to define the
143
E N , with a projector not necessarily of degree 0). What does not work for these
more general objects is the relation to topological K-theory.
13.10
Embedding and transfer
Let Σ be a G-symplectic cone, embedded equivariantly in T • M with M a compact G-manifold, and S an equivariant Szegö projector. As recalled in §1, the
range µH of S is the sheaf of solutions of an ideal I ⊂ EM . The corresponding
EM -module is M = EM /I; it is “good”, as is obvious on the microlocal model
or the holomorphic model (for which a good resolution near Σ is ∂¯b ).
Endomorphisms of M are induced by right multiplications m 7→ ma where
aI ⊂ I (a ∈ [I : I], so E 0 = End Mop ' [I : I]/I. The map which to a ∈ [I : I]
associates the Toeplitz operator Ta gives an isomorphism from End E (M)op to
the Toeplitz algebra (mod C ∞ ). (this is is easily seen by successive approximations since the symbol of Ta is σ(a)|Σ , or because, as indicated in [27], any
Toeplitz operator is also of the form TP where P commutes with the Szegö
projector).
If P is a Toeplitz module, i.e. a left E 0 -module supported by Σ, the transferred module is M⊗E 0 P (also supported by Σ); it has the same solution sheaf as
P, since we have Hom M ⊗ P, H) = Hom (P, Hom (M, H)) and Hom (M, H) =
H0 . In this equality we can replace P by its global good resolution (i.e. replace
Hom by Rhom 0 , because this resolution is locally isomorphic to ∂¯b which has
no cohomology mod C ∞ near Σ in degree > 0. Thus the transfer preserves
asymptotic traces and indices.
This extends obviously to the case where Σ is embedded equivariantly in
another symplectic cone Σ ⊂ Σ0 : the Toeplitz sheaf µH is Hom EΣ (M, µH0 ),
with M = E/I and I ⊂ E is the annihilator of the Szegö projector S of Σ.
Theorem 127 Let X 0 , X be two compact contact G-manifolds and f : X →
X 0 be an equivariant embedding. Then the K-theoretical push-forward (Bott
0
G
G
(X) → KX
homomorphism) KX−Z
0 −Z 0 (X ) commutes with the asymptotic G
index of G-elliptic equivariant Toeplitz operators.
Let F : EΣ → EΣ0 be an equivariant embedding of the corresponding Toeplitz
algebras (over f ), and let M be the EΣ0 -module associated with the Szegö projector SΣ (transfer module). We have seen that transfer P 7→ M ⊗ P preserves
the asymptotic index.
Lemma 128 Notations being as above, the K-theoretical element (with support
in Σ) [M] ∈ KΣG (T • M ) is precisely the Bott element used to define the Bott
G
isomorphism K G (X) → KX
(X 0 ); [M ⊗ P] is the Bott image of [P].55
55 if f : X → Y is a map between manifolds (or suitable spaces), the K-theoretical pushforward is the topological translation of the Grothendieck direct image in K-theory (for algebraic or holomorphic coherent modules). Its definition requires a spinc structure on the
virtual normal bundle of f (cf [27], §1.3) and this always exists canonically if X, Y are almost
symplectic or almost complex, or as here if f is an immersion whose normal tangent bundle
is equipped with a symplectic or complex structure.
144
Proof: The tranfer module M is good: it has, locally (and globally), a good
resolution. Its symbol is a locally free resolution of σ(M) = C ∞ (X)/σ(I). We
may identify a small equivariant tubular neighborhood of Σ with the normal
tangent bundle N of Σ in Σ0 ; N is a symplectic bundle; the ideal I endows it
with a compatible positive complex structure N c (for which the first order jet of
elements of σ(I) are holomorphic in the fibers of N c ). In such a neighborhood a
good symbol resolution is homotopic to the Koszul complex of N (or the symbol
of ∂¯b in the holomorphic case): the K-theoretical element it defines is precisely
the Bott element.
Example : Let X = S 2N −1 be the unit sphere of CN , H the space of holomorphic functions (the symplectic cone Σ can be identified with CN ). Similarly
X 0 = C2k−1 and H0 . We can embed X 0 as a subsphere of X (equivariantly if we
are given suitable unitary group actions).
We can identify H0 with the subset of functions independent of zk+1 , · · · , zN .
The corresponding operators are the ∂zj , k < j ≤ N and the corresponding
complex of Toeplitz operators is the partial De Rham complex.
PN
Another way of relating the two is to identify H0 to H/ k+1 zj H, identifying
H0 with the cohomology of the Koszul complex.
PN
Note that we have ∂zm = (N + 1 zj ∂j )Tz̄m so up to a positive factor, the
De Rham complex is the adjoint of the Koszul complex, and both define the
same K-theoretical (equivariant) element.
Remark. It is always possible to embed equivariantly a compact contact manifold in a canonical contact sphere with linear G-action (this reduces the problem
of computing asymptotic indices to the case where the base space is a sphere but if G 6= 1 this is still complicated):
Lemma 129 Let Σ be a G cone (with compact base), λ a horizontal 1-form,
homogeneous of degree 1 (Lρ λ = λ, ρyλ = 0, where ρ is the radial vector field,
generating homotheties). Then there exists a homogeneous embedding x 7→ Z(x)
of Σ in a complex representation V c of G such that λ = Im Z̄.dZ
In this construction, Z is homogeneous of degree 12 as above. This applies
of course is Σ is a symplectic cone, λ its Liouville form (the symplectic form is
ω = dλ and λ = ρyω. We first choose a smooth equivariant function Y = (Yj ),
homogeneous of degree 12 , realizing an equivariant embedding of Σ in V − {0},
where V is a real unitary G-vector space (this always exists if the basis is
compact).
Then there exists a smooth function X = (Xj ) homogeneous of degree 21
such that
λ = 2X.dY . We can suppose X equivariant, replacing it by its
R
mean g.X(g −1 x) dg if need be. We have 2ρydY = Y (Y is of degree 12 ) so
X.Y = ρyX.dY = 0. Finally we get λ = Im Z̄.dZ with Z = X + iY (the
coordinates zj on V are homogeneous of degree 12 so that the canonical form
Im Z̄.dZ is of degree 1)
145
13.11
Relative index
Let Ω, Ω0 be two strictly pseudo convex Stein domains with smooth boundaries
X, X 0 . Let f be a smooth contact isomorphism X → X 0 . Then the holomorphic
push-forward
W : u ∈ H 7→ S 0 (u ◦ f −1 ) ∈ H0
(113)
is well defined, and is a (Toeplitz FIO) Fredholm map. The Atiyah-Weinstein
formula computes its index in terms of the geometrical data.
The original Atiyah question was : if M, M 0 are two smooth manifolds,
f : S ∗ M → S ∗ M 0 a contact isomorphism, F an elliptic FIO associated to f ,
then F has an index, which should be given by a similar formula.
This reduces to the former problem since ΨDO on M are the same thing as
Toeplitz operators on the boundary of a small tubular neighborhood of M in a
complexification M c (cf. [16]) 56
The main difficulty in this problem is that, with a fixed contact structure,
we are changing the CR structure, hence the Szegö projectors, and there is no
formula, using only the contact boundary data, telling how the index behaves.
To overcome this, we enlarge the spaces of holomorphic boundary values
in such a manner that the index is repeated infinitely many times and can be
interpreted as an asymptotic index, which can be handled geometrically.
13.12
Enlargement
Let Ω be as above, with defining function −φ (φ > 0 , I have changed the sign).
We denote the boundary by X0 rather than X).
e ⊂ C× Ω̄ denotes the ball |t|2 < φ. Its boundary X is strictly pseudoconvex,
Ω
provided that Log φ1 is strictly psh. (e.g. −φ strictly psh. on Ω̄ 57 ). We still
denote by Σ ⊃ Σ0 the symplectic cones.
e : (t, x) 7→ (eiθ t, x).
The circle group U (1) acts on X
e is dθ dv (smooth, positive, invariant) with dv a
The volume element on X
smooth positive density on Ω̄; S denotes the Szegö projector, H its range (space
of boundary values of holomorphic functions of moderate growth near X).
D denotes the Toeplitz operator defined by 1i ∂θ on H. It is self-adjoint, ≥ 0,
equal to Tt T∂t .
56 except one should also take into account the homotopy class (“winding number”) of the
principal symbol.
57 φ can always be chosen so.
146
The expansion of a function in the Fourier decomposition
X
H=
Hk (Hk = ker (D − k) )
k≥0
is equivalent to its Taylor expansion:
X
f=
fk (x)tk .
H0 identifies with the set of holomorphic distributions on X0 (set of boundary
values of holomorphic functions on Ω with moderate growth at ∂Ω).
Note that the L2 norm of a holomorphic function tk f (x) on X is
Z
Z
k 2
|t f | = 2π
φk |f 2 |dv
X
Ω
(because |t|2 = φ on X and the measure on X is dθ dv)
P
Sk , we get a seIf we decompose S in its equivariant components S =
quence closely related to that of Berezin (see [5, 49])
It will be convenient to replace the Toeplitz FIO operator W by a unitary
multiple
1
(114)
E0 = (W W ∗ )− 2 W : H0 → H00
1
with the convention that (W W ∗ )− 2 vanishes on the kernel of W ∗ ; E0 is in
any case unitary mod smoothing operators and has obviously the same index
as W . We are using the norm of H0 i.e. the L2 nor of X (or of Ω̄), which is not
1
the L2 norm of X0 (it is rather related to the Sobolev H − 2 norm) - but for the
index this makes no difference)
As mentioned above the Toeplitz operator corresponding to rotations is
D = t∂t
(=
1
∂θ ).
i
we have D = D∗ = T∂∗t Tt∗ ; it follows that
∂t∗ = tC, (t∗ = C −1 ∂t ).
for an invariant Toeplitz operator C > 0 (unique)
We set
58
1
τ = tC 2
(115)
1
2,
This is a Toeplitz operator of degree
not an integer, but for the commutation constructions below this does not matter
D = τ τ ∗ , [D, τ ] = τ, [τ ∗ , τ ] = 1
(116)
58 if we have a factorization D = P Q with [D, P ] = P , there exists a (unique) invariant
invertible Toeplitz operator U such that P = tU, Q = U −1 ∂t . Here we have D = tCt∗ , so
C = C ∗ > 0 since D is ≥ 0 and Tt injective.
147
τ is globally defined, a positive multiple of t , τ Hk = Hk+1
τ is uniquely defined by by these conditions.59
There is a similar construction for Ω0 .
Theorem 130 (embedding) There exists an equivariant Toeplitz FIO:
E : X → X 0 (with microsupport close to X0 ) such that (mod C ∞ )
1) E is unitary elliptic (mod C ∞ ) near X0
2) E induces E0 on H0 (mod smoothing operators)
3) Eτ = τ 0 E
Then the Ek = Hk → H0k all have the same index Index E0
If 2) holds, E is elliptic on X0 hence G-elliptic (because here G = U (1) acts
freely, with a positive action, on the “interior” X − X0 ). The last assertion
follows: we have E − τ 0 Ek = Ek+1 τ and since τ is a bijection Ek → Ek+1 (same
for τ 0 ), Ek , Ek+1 have same index.
The theorem replaces the relative index Index (E0 ) by the G-asymptotic
index AsInd (E).
13.13
Collar isomorphism
The geometric counterpart is: there is a (unique) equivariant homogeneous
symplectic isomorphism f of some equivariant neighborhood of Σ0 in Σ to (same
for Σ0 ) such that f |Σ0 = Id , and σ(τ ) ◦ f = σ(τ 0 ), i.e. f commutes with the
hamiltonians of the real and imaginary parts of τ, τ 0 . This works because the
hamiltonians of Re τ, Im τ commute.60
The operator statement follows from the geometric one in the usual manner.
Notice that E is at first only defined mod smoothing operators near X0 . We
extend it globally using any Toeplitz cut-off.
13.14
Embedding
We have mentioned that any G-contact manifold (compact) can be embedded
in a standard contact sphere with linear unitary action of G. Here we choose
embeddings more precisely.
e = S 2N +1 ⊂ CN +1 be a large sphere, with variables (T, Z).
Let X
The circle group G = U (1) acts by
(T, Z) 7→ (eiθ T, Z)
The base of char g is the diameter Z(T = 0); it is equal to the fixed point set.
59 in fact we need a little less than that: τ should be globally defined over Ω, and τ :
k
Hk → Hk+1 should have index zero; The the hamiltonians of the real and imaginary parts of
τ should commute.
60 This would not work if we replaced τ by t because the hamitonians of Re t, Im t do not
commute in general
148
Theorem 131 There exist equivariant contact embeddings F, F 0 of X, X 0 in
the sphere S 2N +1 (with U (1)-action as above) such that F = F 0 ◦ f near the
boundary X0 .
We are now reduced to the case where X, X 0 sit in a large sphere S and
coincide near the fixed points. The trivial bundle of X defines, via the transfer
e whose
homomorphism, a complex A of Toeplitz operators on the large sphereX,
G
K-theoretical element in KX
(S) is the equivariant Bott image. Same for X 0 .
The Toeplitz FIO E of theorem 130 provides a Toeplitz isomorphism A → A0
e whose asympnear the boundary X0 , thus defining a G-elliptic complex on X,
totic G-index is precisely what we want to compute.
13.15
Index
Now U (1) acts freely on S −S 0 and U (1)\(S −S 0 ) is the open unit ball B ⊂ CN ,
G
so the pull back is an isomorphism K0 (CN ) = Z → KS−S
(S) (the generator is
0
the symbol of the partial De Rham complex ∂X , or of the Koszul complex).
We may now go back to the original situation: Ω and Ω0 are complex manifolds, glued together by the symplectic map f0 the result Y is not a manifold,
but the K-theoretical index is well defined: χ : Kcomp (Y ) → Z:
Theorem 132 The relative index is χ(1Ω − 1Ω0 ); χ is the K-theoretical character defined by the Bott periodicity theorem; the two trivial bundles 1Ω − 1Ω0
are glued together along the boundary to give an element of compact support.
The K-theoretical element defined by the complex above is the difference
element between the K-theoretical (spinc ) images of Ω and Ω0 defined by F0 on
the boundary (or its extension near the boundary defined by F ; any symplectic
diffeomorphism near the boundary would do as well since these are all isotopic.
This can be readily translated in terms of cohomology, using the Chern characters and Todd class, as done in [28]; the Todd class appears when comparing
the Chern class of the Bott element with the Euler class used for integration
along fibers.
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159
Index
A× 29
A×
0 33
b9
D
bk 9
D
b0 10
D
∗
Hhom
33
O(m) 8
O8
e 8
Ω
τ 52
c 49
W
automorphisms 29, 32, 33, 35
morphisms 28
involutions 30
symbol, exponant 34
inner -29, 49
Cartan formula 12
cone 8
complexified cone 8
radial vector 8
connection 52
curvature 53
embedding 94
functional calculus 16
group star product 21
involution 30
automorphism preserving - 35
homomorphisms 28
Hochschild cohomology 37
isomorphism 40
Moyal star product 19, 20
non commutative cohomology 38
starproduct, star algebra 10
Poisson bracket 12
symplectic manifold 13
cotangent bundle 14, 14
pseudo-differential operators 23
formal – 22
Leibniz’ rule 10
oscillatory asymptotics 22
semi-classical – 25
soft (sheaf) 39
star algebra 10
subprincipal symbol 31
automorphism preserving - 35
Toeplitz operators 26
valuation 48
160

Star products, deformation quantization and Toeplitz operators

Transcription

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