DEFORMATION QUANTIZATIONA mini lecture

Martin BordemannLaboratoire des Mathematiques et Applications

Faculte des Sciences et TechniquesUniversite de Haute Alsace, Mulhouse

4, rue des Freres Lumiere68093 Mulhouse, France

e–mail: Martin.Bordemann@uha.fr

Contents

Introduction 1

1 Classical mechanics, symplectic geometry and Poisson geom-etry 31.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Symplectic Geometry I . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Cotangent bundles . . . . . . . . . . . . . . . . . . . . 61.2.2 Complex Projective Space . . . . . . . . . . . . . . . . 7

1.3 Poisson Geometry I . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Schouten bracket . . . . . . . . . . . . . . . . . . . . . 91.3.2 Poisson Structures . . . . . . . . . . . . . . . . . . . . 101.3.3 The dual space of a Lie algebra . . . . . . . . . . . . . 11

2 Quantum Mechanics 13

3 Symbol Calculus and Elementary Star-Products 163.1 Standard Ordering . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Weyl-Moyal ordering prescription . . . . . . . . . . . . . . . . 173.3 Wick Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Formal Deformations 224.1 Formal Power series . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Formal Deformations of Associative Algebras . . . . . . . . . . 234.3 Gerstenhaber’s Formula . . . . . . . . . . . . . . . . . . . . . 264.4 Standard, Weyl-Moyal and Wick ordering in R2n . . . . . . . . 274.5 Standard symbols of multidifferential operators . . . . . . . . 28

5 Star-produits 305.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Symplectic manifolds . . . . . . . . . . . . . . . . . . . 325.2.2 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . 335.2.3 Semi-Kahler manifolds . . . . . . . . . . . . . . . . . . 345.2.4 Even symplectic supermanifolds . . . . . . . . . . . . . 34

5.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3.1 Symplectic manifolds . . . . . . . . . . . . . . . . . . . 355.3.2 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . 35

6 Explicit Examples 366.1 Cotangent bundle of Sn . . . . . . . . . . . . . . . . . . . . . 366.2 Complex projective space . . . . . . . . . . . . . . . . . . . . . 386.3 The dual space of a Lie algebra . . . . . . . . . . . . . . . . . 386.4 L’espace dual d’une algebre associative . . . . . . . . . . . . . 40

7 Representations des star-produits I 417.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Representations GNS . . . . . . . . . . . . . . . . . . . . . . . 417.3 Fibres cotangent . . . . . . . . . . . . . . . . . . . . . . . . . 43

8 Geometrie de Poisson II 458.1 Applications de Poisson . . . . . . . . . . . . . . . . . . . . . 45

8.1.1 Applications moment . . . . . . . . . . . . . . . . . . . 468.1.2 Systemes integrables . . . . . . . . . . . . . . . . . . . 46

8.2 Sous-varietes et applications coısotropes . . . . . . . . . . . . 488.3 Reduction symplectique . . . . . . . . . . . . . . . . . . . . . 51

9 Quantification des applications de Poisson et des plonge-ments coısotropes? 529.1 Homomorphismes de star-produits . . . . . . . . . . . . . . . . 52

9.1.1 Applications moment quantiques et systemes integrablesquantiques . . . . . . . . . . . . . . . . . . . . . . . . . 53

9.2 Representations de star-produits II . . . . . . . . . . . . . . . 549.2.1 Lien entre homomorphismes et representations de star-

produits . . . . . . . . . . . . . . . . . . . . . . . . . . 569.2.2 Representation de star-produits quand l’espace reduit

existe . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Introduction

Since the seminal article by Bayen, Flato, Frønsdal, Lichnerowicz et Stern-heimer in 1978 [7] deformation quantization has become a large research areacovering several algebraic theories like the formal deformation theoriy of asso-ciative algebras and the more recent theory of operades, as well as geometrictheories like the theory of symplectic and (more generally) Poisson mani-folds), and of physical theories like string theory and noncommutative gaugetheory. In this theory, the noncommutative associative multiplication of oper-ators in quantum mechanics is considered as a formal associative deformationof the pointwise multiplication of the ‘algebra of symbols of these operators’:in physical terms, this means the ‘algebra of classical quantities’ which aregiven by the algebra of all complex-valued C∞-functions on a Poisson man-ifold, the ‘phase space’ of classical mechanics. The formal parameter is aninterpretation of Planck’s constant ~ in convergent situations. The advantageof this method is its universality: according to a theorem by Kontsevich [63]this construction is possible for any Poisson manifold. Moreover, geometricintuition is quite useful in concrete situations since everything is formulatedin geometrical terms on a differentiable manifold in contrast to the usualformulation of quantum mechanics whre one has to specify a Hilbert space.The price to pay is the fact that complex numbers are replaced by the ringof all formal complex power series whose convergence is a case-by-case study.

The main objective of this mini-lecture is a more pedagogical introductionto this subject: I have not included in detail all the existence and classifica-tion proofs which are quite technical. Neither do I speak about the theoryof operades which has become the algebraic framework of this theory sinceKontsevich. I’d rather want to underline some motivations from physics,discuss concrete examples and talk -at the end- about the still open theoryof modules and reduction.

In the first two chapters, I have given a little sketch of the relations be-tween classical mechanics and symplectic and Poisson geometry and usualquantum mechanics. The central motivation for the deformed multiplica-tion, the star-product, will be the symbol calculus of differential operators inChapter 3 for several ordering prescriptions used in quantum physics. Chap-ter 4 is devoted to fix the notation of formal power series and to presentGerstenhaber’s deformation formula which gives a more uniform view on thestar-products introduced in Chapter 3. The definition and existence and clas-sification theorems are given in Chapter 5, whereas certain explicit examplesother than R2n are discussed in Chapter 6. A preliminary discussion of rep-resentations of star-products is done in Chapter 7 where the analogues withC∗-algebras (positive linear functionals, GNS construction) are discussed.

1

After a second discussion of Poisson geometry, namely Poisson maps andco-isotropic maps, in Chapter 8 I have included some new results on thedeformation of certain Poisson maps and coistropic (in physics: first class)submanifolds.

Je me dois excuser d’avance pour la liste des references: je n’avais pasbeaucoup de temps pour profondement rechercher tous les articles impor-tants dans le domaine, alors je les ai ramasses d’une facon hative, meme pasbrassensienne (les copains d’abord), et je prie le lecteur de me pardonner desomissions et de regarder des resumes comme [81], [75] ou les comptes rendusde la conference Moshe Flato 1999 [35] pour plus de references.

Acknowledgements

I’d like to thank the organizers of the summer school in Villa de Leyva forthe friendly acceuil and the perfect organization.

2

1 Classical mechanics, symplectic geometry

and Poisson geometry

1.1 Classical Mechanics

Classical Mechanics is governed by Newton’s equations: the motion of a par-ticle of strictly positive mass m ∈ R in an open set of Rn is described by thefollowing system of second order differential equations:

md2xi

dt2= Fi(x1, . . . , xn) (1.1.1)

where x := (x1, . . . , xn) denote the n coordinates of the particle and F =(F1, . . . , Fn) : U → Rn is a C2-map, the force field. In case there are severalparticles of masses m1, . . . , mN the system (1.1.1) is generalized in an obviousfashion to Nn coordinates, namely (x11, . . . , xnN where the force field is nowa C2-map RNn ⊃ U → RNn with U an open set in RNn. The canonicalexample is the solar system with 1 particles, namely the sun and the nineplanets, where the open set U is given by R30 minus the union of all the vectorsubspaces describing all collision situations where any two positions in R3 oftwo distinct particles coincide, and where F is the Newtonian gravitationalfield. The latter is an example of a conservative field, i.e. for which there isa C∞-function V : U → R, the potential energy, such that

Fi = −∂V

∂xi

for all 1 ≤ i ≤ n. (1.1.2)

The principal idea of Hamiltonian mechanics is the transformation of the sec-ond ordre system in n variables (1.1.1) to a first order system in 2n variables

(q, p) := (q1, . . . , qn, p1, . . . , pn) := (x1, . . . , xn,mdx1

dt, . . . ,m

dxn

dt); (1.1.3)

upon introducing the Hamilton function (sum of kinetic and potential energy)

H(q, p) :=n∑

i=1

pi2

2m+ V (q), (1.1.4)

such that the system (1.1.1) is rewritten in the following way (1 ≤ i ≤ n):

dqi

dt=

pi

m=

∂H

∂pi

(q, p) =: XHqi(q, p)

dpi

dt= −∂V

∂qi(q) = −∂H

∂qi(q, p) =: XHpi

(q, p)

(1.1.5)

3

where XH := (XHq, XHp) := (XHq1 , . . . , XHqn , XHp1, . . . , XHpn

) : U × Rn →R2n is called the Hamitlonian vector field associated to the function H. Moregenerally such a vector field may be associated to an arbitrary C∞-functionH : U → R. It is clear that the Hamilton function H is always a first integralof the equation (1.1.5), i.e. H

(q(t), p(t)

)= H

(q(0), p(0)

), which expresses

the conservation of energy. The vector space C∞(U,R) := {f : U → R|f C∞}is called the set of classical observables, whereas the set of all points (q, p) ∈U × Rn is called the phase space or equivalently the set of pure states of thesystem.

1.2 Symplectic Geometry I

Symplectic geometry is the direct differential geometric generalization of thelocal concepts of the preceding subsection to the to differentiable manifolds.

Let M be a differentiable manifold. A 2-forme ω is called symplectic iffit is closed (i.e. dω = 0) and nondegenerate (i.e. for all m ∈ M the fact thatωm(X,Y ) = 0 for all Y ∈ TmM implies that X = 0), and the pair (M, ω) iscalled a symplectic manifold. The standard example is the vector space R2n

with coordinates (q, p) := (q1, . . . , qn, p1, . . . , pn)) equipped with the 2-form

ω :=n∑

i=1

dqi ∧ dpi. (1.2.1)

Other examples are all orientable manifolds of dimension 2 (e.g. all theorientable Riemann surfaces). In general, for a C∞-function H : M → R onedefines its Hamiltonian vector field XH by

dH =: ω(XH , ) (1.2.2)

which is well-defined thanks to the nondegenerescence of ω. Moreover

LXHω = diXH

ω + iXHdω = ddω + 0 = 0, (1.2.3)

and each Hamiltonian vector field is an infinitesimal symmetry of ω. Thetriple (M,ω, H) is called a Hamiltonian system and the function H is calledthe Hamiltonian function of the system. The first-order equation

dc

dt(t) = XH

(c(t)

). (1.2.4)

is called the Hamiltonian equations (of motion) corresponding to H. Dar-boux’s Theorem ensures the existence of particular coordinates (q, p) :=(q1, . . . , qn, p1, . . . , pn) around each point of M in which the symplectic form

4

takes the standard form (1.2.1), see e.g.[1], p.175, thm 3.2.2. for a proof. Thefact that the symplectic form is antisymmetric immediately implies that theHamiltonian function is always a first integral for ist Hamiltonian system, i.e.

dH(c(t)

)

dt= dH

(c(t)

)XH

(c(t)

)= ωc(t)

(XH

(c(t)

), XH

(c(t)

))= 0.

which again is the conservation of energy in physics.We recall the definition of the Poisson bracket for two C∞-functions f, g :

M → R:{f, g} := ω(Xf , Xg) = df Xg = −dg Xf , (1.2.5)

and we have the following

Proposition 1.1 Let f, g, h : M → R be three C∞-functions on a symplecticmanifold (M,ω). Then:

1. {f, g} = −{g, f} (antisymmetry).

2. {f, gh} = {f, g}h + g{f, h} (Leibniz rule).

3. {{f, g}, h}+ {{g, h}, f}+ {{h, f}, g} = 0 (Jacobi identity).

4. [Xf , Xg] = −X{f,g}

Proof: The first two assertions are trivial and the third follows from the fourthupon aplying it to h. The foruth statement is due to the fact that dω = 0, i.e.

i[Xf ,Xg ]ω = [LXf, iXg ]ω

eqn(1.2.3)= LXf

iXgω = diXfiXgω + iXf

diXgω

dω=0= d{g, f}+ iXf(diXg + iXgd)ω

eqn(1.2.3)= −iX{f,g}ω + 0.

2

It follows that the Poisson bracket equips the space of classical observablesC∞(M) with the structure of a Lie algebra.

Definition 1.1 Let (M, ω) symplectic manifold. A connection ∇ in the tan-gent bundle TM is called symplectic iff

∇Xω = 0 for each vector field X on M.

The following Proposition (due to Heß, Lichnerowicz, Tondeur, see e.g.[55]) is an analogue of the Levi-Civita Theorem in (semi)riemannian geometryalthough the uniqueness of the symplectic connection does no longer hold:

5

Theorem 1.1 Let (M,ω) be a symplectic manifold and ∇ a torsionfree con-nection in the tangent bundle TM of M .1. Then the following formula defines a symplectic torsionfree connection ∇in TM :

ω(∇XY, Z) := ω(∇XY, Z) +1

3(∇Xω)(Y, Z) +

1

3(∇Y ω)(X,Z)

for all vector fields X, Y, Z on M .2. The difference between two torsionfree symplectic connections ∇ and ∇′

defines a totally symmetric tensor field S of rank 3 in Γ(M, S3T ∗M):

ω(∇′XY, Z)− ω(∇XY, Z) =: S(X,Y, Z),

for all vector fields X, Y, Z on M .3. For any totally symmetric tensor field S of rank 3 in Γ(M,S3T ∗M) andevery torsion free symplectic connection ∇ in TM the connection ∇′ definedby

ω(∇′XY, Z) := ω(∇XY, Z) + S(X, Y, Z), ∀ X, Y, Z ∈ Γ(M,TM)

is a torsion free symplectic connection in TM .

The proof is a direct verification.Remark: there are topological obstructions for compact manifolds to

admit a symplectic form ω: the class of each of the following closed 2k-formsω∧k, 1 ≤ k ≤ n := dim M/2 has to be non zero: in fact, if there was ak − 1-form θ with ω∧k = dθ, the the volume form ω∧n would be equal todθ ∧ ω∧(n−k) = d(θ ∧ ω∧(n−k)) which would be absurd since the total volume∫

Mω∧n of M would be zero by Stokes’s Theorem. For example, the spheres

S2n do not admit any symplectic structure for all n ≥ 2.

1.2.1 Cotangent bundles

Let Q be a differentiable manifold, T ∗Q its cotangent bundle, and τ ∗Q :T ∗Q → Q the canonical bundle projection. The canonical 1-form θ0 onthe manifold T ∗Q is defined in the following manner: let q ∈ Q, α ∈ TqQ

∗,and Wα ∈ TαT ∗Q, then

θ0(α)(Wα

):= α

(Tατ ∗Q Wα

). (1.2.6)

Let((U, (q1, . . . , qn)

)be a chart of Q, and

(T ∗U, (q1, . . . , qn, p1, . . . , pn)

)the

corresponding canonical chart of T ∗Q (i.e. qk(α) := qk(τ ∗Q(α)

)and pl(α) :=

α(∂/∂ql)), then θ0 takes the form

θ0 :=∑

k=1

pkdqk (1.2.7)

6

whence the fact that the canonical 2-form

ω0 := −dθ0 (1.2.8)

is nondegenerate, hence a symplectic form on T ∗Q. The cotangent bundlesgeneralize the phase spaces in physics where Q is a configuration space andthe fibres represent the cojugate momenta.

1.2.2 Complex Projective Space

Apart from the tori of even dimension, the complex projective spaces are thesimplest compact symplectic manifolds:

Consider the complex manifold Cn+1 \ {0} equipped with complex coor-dinates z := (z1 := q1 + ip1, . . . , zn+1 := qn+1 + ipn+1) and with the standardsymplectic form

ω0 :=i

2

n+1∑

k=1

dzk ∧ dzk =n+1∑

k=1

dqk ∧ dpk. (1.2.9)

Comnplex projective space CP n is defiend by the following equivalence rela-tion

z ∼ z′ si et seulement si ∃α ∈ C \ {0} such that z′ = αz, (1.2.10)

and CP n := Cn+1 \ {0}/ ∼. Let

π : Cn+1 \ {0} → CP n : z 7→ [z] (1.2.11)

be the canonical projection canonique whose fibres obviously are the complexlines in Cn+1\{0} passing through the origine There are n+1 complex charts(Uk, v) defined by

Uk := {[z] ∈ CP n|zk 6= 0}v :=

(v1 :=

z1

zk

, . . . , vk−1 :=zk−1

zk

, vk+1 :=zk+1

zk

, . . . , vn+1 :=zn+1

zk

).

the Fubini-Study 2-form ω is defined in each chart (Uk, v) (where we set|v|2 :=

∑n+1l=1,

l 6=k|vl|2):

ω|Uk:=

i

2(1 + |v|2)

n+1∑l=1,

l 6=k

dvl ∧ dvl − 1

(1 + |v|2)n+1∑

l,l′=1,

l,l′ 6=k

vldvl ∧ vl′dvl′

(1.2.12)

7

It can be shown that these locally defined closed two-forms ω|Ukare well-

behaved under the change of charts and thus define a global 2-form ω. More-over, the map

Φ : CP 1 → S2 : [z1, z2] 7→ 1

|z1|2 + |z2|2(z1z2 + z1z2,−i(z1z2 − z1z2), |z1|2 − |z2|2

)

is easily computed to be a diffeomorphism.

1.3 Poisson Geometry I

Poisson geometry is a generalization of symplectic geometry in the followingsens:

Let M a differentiable manifold and H : M → R a C∞-function. If onelikes to defines a vector field associated to H which depends in a C∞(M,R)-linear way on dH and for which H is a first integral, one needs a bivec-tor field P , i.e. a C∞-section of the fibre bundle Λ2TM . In a local chart(U, (x1, . . . , xn)

)this bivector field P takes the form

P =1

2

n∑i,j=1

P ij ∂

∂xi∧ ∂

∂xj. (1.3.1)

The rank of a bivector field P in m ∈ M is defined by the rank of the matrixP ij(m) in an arbitrary chart.

Each bivector field can canonically be considered as an antisymmetricbilinear form on the cotangent bundle T ∗M by using the natural pairing: letα, β two 1-forms on M

P (α, β) = iβiαP =n∑

i,j=1

P ijαiβj (1.3.2)

Using P in this way, one defines the Hamiltonian vector field XH in a fashionanalogous to (1.2.2):

XH := P ( , dH) =n∑

i,j=1

P ij ∂H

∂xj

∂

∂xi. (1.3.3)

Since dH(XH) = P (dH, dH) = 0 it is clear that H is always a first integralof the dynamical system defined by XH . Moreover, one may define a Poissonbracket for f, g ∈ C∞(M,R) in a way analogous to (1.2.5) by

{f, g} := P (df, dg) = df Xg = −dg Xf . (1.3.4)

8

In general, the analogue of Proposition 1.1 is no longer true, concerningthe Jacobi identity and has to be demanded as an additional condition onP . In order to study that condition one introduces the following algebraicstructure on the space of C∞-sections of the vector bundle ΛTM , the spaceof multivector fields:

1.3.1 Schouten bracket

Definition 1.2 Let X1, . . . , Xk, Y1, . . . , Yl vector fields onf the differentiablemanifold M and f, g ∈ C∞(M,R). The Schouten bracket1 on M by:

[X1 ∧ · · · ∧Xk, Y1 ∧ · · · ∧ Yl]S :=k∑

i=1

l∑j=1

(−1)i+j[Xi, Yj] ∧X1 ∧ · · · ∧Xi−1 ∧Xi+1 ∧ · · · ∧Xk

∧ Y1 ∧ · · · ∧ Yj−1 ∧ Yj+1 ∧ · · · ∧ Yl,

[X1 ∧ · · · ∧Xk, f ]S :=k∑

i=1

(−1)k−iXi(f)X1 ∧ · · · ∧Xi−1 ∧Xi+1 ∧ · · · ∧Xk

=: −(−1)k−1[f, X1 ∧ · · · ∧Xk]S,

[f, g]S := 0

By means of the following proposition one easily sees that this bracket doesnot depend on the decomposition of a multivector field in exterior productsof vector fields:

Proposition 1.2 Let ∇ be a torsion-free connection in the tangent bundleTM of M , let X ∈ Γ(ΛkTM) and Y ∈ Γ(ΛlTM) be multivector fields, andlet

(U, (x1, . . . , xn)

)be a local chart.

Then the Schouten bracket of X and Y can be computed by the followingformula

[X, Y ]S =n∑

i=1

(−1)k−1idxi(X) ∧∇ ∂

∂xiY

− (−1)(k−1)(l−1)

n∑i=1

(−1)l−1idxi(Y ) ∧∇ ∂∂xi

X

which does not depend on the chosen connection nor on the chosen chart.

1The pronunciation is “S-khaouten”.

9

The space of all smooth multivector fields, Γ(M, ΛTM), equipped with theSchouten bracket, is a Lie superalgebra, i.e.:

Proposition 1.3 Let X ∈ Γ(ΛkTM), Y ∈ Γ(ΛlTM) and Z ∈ Γ(ΛrTM).Then:

1. [X, Y ]S = −(−1)(k−1)(l−1)[Y, X]S (graded antisymmetry).

2. [X, Y ∧Z] = [X,Y ]∧Z +(−1)(k−1)lY ∧ [X, Z] (graded Leibniz identity).

3. (−1)(k−1)(r−1)[[X, Y ], Z] + (−1)(l−1)(k−1)[[Y, Z], X]+(−1)(r−1)(l−1)[[Z,X], Y ] = 0.

(graded Jacobi identity).

The proof is a direct computation. Note that the Z-grading for the Schoutenbracket [ , ]S is shifted by −1 to the usual grading of the exterior multipli-cation ∧, e.g. vector fields are of degree 1 in the Grassmann algebra and ofdegree 0 in Schouten Lie superalgebra. In general, a Z-graded vectorspaceg = ⊕k∈Zgk equipped with a graded commutative multiplication and a gradedLie bracket on the shifted space g[1] (where g[1]k := gk+1) such that both arecompatible by the graded Leibniz identity, is called a Gerstenhaber algebra.

1.3.2 Poisson Structures

Definition 1.3 A bivector field P on a differentiabl manifold M is called aPoisson structure iff

[P, P ] = 0 orn∑

r=1

(∂P ij

∂xrP rk +

∂P jk

∂xrP ri +

∂P ki

∂xrP rj

)= 0.

If this condition is satisfied the pair (M,P ) is called a Poisson manifold.

The following two propositions are proved by a direct computation:

Proposition 1.4 Let P be a bivector field on the differentiable mamnifoldM . The Poisson bracket of two real-valued smooth functions on M definedby (1.3.4) satisfies the Jacobi identity iff P is a Poisson structure.

Proposition 1.5 Let (M,P ) and (M ′, P ′) two Poisson manifolds and s, t ∈R. By means of the idsentification T(m,m′)(M × M ′) = TmM × Tm′M ′ forall m ∈ M and m′ ∈ M ′ and the canonical injections i(m,m′) : TmM →TmM×Tm′M ′ : v 7→ (v, 0) and i′(m,m′) : Tm′M ′ → TmM×Tm′M ′ : w 7→ (0, w)

one writes P(1)(m,m′) := (i(m,m′)⊗ i(m,m′))(Pm) and P ′(2)(m,m′) := (i′(m,m′)⊗

i′(m,m′))(P′m′).

Then sP(1) + tP ′(2) is a Poisson structure on the product manifold M ×M ′.

10

The case P(1) − P ′(2) is sometimes denoted by M1 ×M2.

The canonical example is R2n equipped with the symplectic Poisson struc-ture

P :=n∑

k=1

∂

∂qk∧ ∂

∂pk

. (1.3.5)

More generally, any symplectic manifold (M, ω) is a Poisson manifold: usingthe musical isomorphism ω[ : TM → T ∗M given by X 7→ ω(X, ) the bivectorfield P := ω

((ω[)−1, (ω[)−1

)is a Poisson structure coinciding with (1.3.5) in

any Darboux chart.As opposed to the case of a symplectic manifold where the existence of a

symplectic structure can restrict the topology of the manifold, it is no longertrue that the existence of a nonzero Poisson structure can influence the globalnature of the manifold:

Theorem 1.2 : Let M be a differentiable manifold of dimension n ≥ 2, letp a point in M and k a nonnegative integer with k ≤ [n/2], the integer partof n/2. Then there always exists a Poisson structure P on M whose rank atp is equal to 2k.

Proof: (C.Nowak, J.Schirmer, 1996). Note first that on Rn the following vec-tor fields Z1, . . . , Zn are independent at the origin, have compact support andcommute.

Zj(x) := φj(x1) · · ·φj(xj−1)φ1(xj)φj+1(xj+1)φj+2(xj+2) · · ·φn(xn)∂

∂xj

∀ 1 ≤ j ≤ n

where φj : R→ R is a nonnegative C∞-function being zero for |x| ≥ j and equal to1 for |x| ≤ j−(1/2). By means of a chart (U,ψ) of M with p ∈ U and ψ(p) = 0 theZj are pulled back and extended by 0 outside U . These vector fields still commute.One then chooses 2k vector fields X1, . . . , Xk, Y1, . . . , Yk among them such thatthey are independant p = ψ−1(0). The following bivector field

P :=k∑

i=1

Xi ∧ Yi

obviously is Poisson structure of rank 2k at p. 2

1.3.3 The dual space of a Lie algebra

We obtain another very important example of a Poisson manifold in thefollowing way: let (g, [ , ]) be an n-dimensional real Lie algebra and M := g∗

11

its dual space. Let e1, . . . , en be a base of g, let e1, . . . , en be the dual base,and ck

lm := ek([el, em]) the structure constants of g. Then for all ξ ∈ g∗ onedefines on M the linear Poisson structure corresponding to [ , ]:

Pg(ξ) := ξ([ , ]) =1

2

n∑

k,l,m=1

ξkcklm

∂

∂ξk

∧ ∂

∂ξl

. (1.3.6)

The Jacobi identity for this Poisson structure is a direct consequence of theJacobi identity for the Lie bracket [ , ] of g.

The Lie algebra g = so(3) ∼= R3 of all real 3× 3 antisymmetric matriceswith the bracket [e1, e2] = e3, [e2, e3] = e1, [e3, e1] = e2 coming from the vectorproduct is an important example for the dynamics of a freely spinning top:let Θ be a positive definite 3×3-matrix (the inertia tensor), and H : R3 → Rthe real valued function H(L) = 1

2

∑3i,j=1 Li(Θ

−1)ijLj. Then the dynamicalsystem corresponding to H is the Euler equation of a freely spinning top

dL

dt= [Θ−1L,L]

where L is the angular momentum and Θ−1L the angular velocity of the top.

12

2 Quantum Mechanics

In qunatum mechanics, the classical picture of observables, states and dy-namical laws is replaced by more complicated structures along the followingphilosophy: according to de Broglie every particle (like e.g. an electron)always has a ‘wave aspect’, and to its energy E he associates Planck’s for-mula E = ~ω (where ω is 2π times the frequency of the wave and ~ denotesPlanck’s constant) and to its momentum p he associates ~k where the lengthof the vector k is given by 2π/λ, λ being the wave length of the wave. A freeparticle (for which the potential energy is zero) is described by a plane wave

(t, q) 7→ ψ(t, q) = exp(−iωt + ik · q) (2.0.7)

where k · q :=∑n

j=1 kjqj. The wave function ψ obviously is a solution to theequation

Eψ = i~∂ψ

∂t= − ~

2

2m∆ψ =

p2

2mψ

where ∆ :=∑n

j=1∂2

∂q2j

is the Laplace operator and p2 := p · p. Erwin

Schrodinger has generalized this equation to include forces by his Schrodingerequation

i~∂ψ

∂t(t, q) = − ~

2

2m∆ψ(t, q) + V (q)ψ(t, q) =:

(Hψ

)(t, q) (2.0.8)

where the differential operator H is called the Hamilton operator of the sys-tem by its evident analogy with a Hamilton function. Schrodinger’s de-scription had a big success for the nonrelativistic hydrogene atom for whichV (q) = −α/|q|, α being a constant: the set of eigenvalues of H exactlymatched the experimental spectrum. Here H is considered to be a selfadjointoperator on a dense domain D(H) in the Hilbert space H := L2(R3, d3q). The(classes of) non zero wave functions in H –up to a nonzero complex multiple–are interpreted as pure states of the quantum system, i.e. those which givea complete description of the system. the square of the absolute value of ψ,|ψ|2, is then interpreted as a probability density for the position in case thenorm of ψ is equal to 1. Of course, the wave function for the free particle isnot an element of H: seen as a tempered distribution (in the sense of LaurentSchwartz) it is an approximation (in the sense of distributions) by elementsin H (which is dense in that distribution space). More generally, as alreadyin classical mechanics, one may consider other dans la mecanique classique,on peut considerer d’autres quantum observables like for instance position

Qk : ψ 7→ (q 7→ qkψ(q)

)(2.0.9)

13

or momentum (which is proportional to the speed for nonrelativistic systems)

Pl : ψ 7→ ~i

∂ψ

∂ql

, (2.0.10)

and in general all self-adjoint operators A defined on a dense domain D(A) ⊂H, i.e. for which one has a good definition of its spetrum. From the mathe-matical point of view it is more convenient to consider B(H), the C∗-algebraof all bounded operators : H → H since there are no domains other than H:the spectral projections of an unbounded self-adjoint operator are always inB(H). Heisenberg observed that the dispersion effect of a wave translatesinto his famous uncertainty relation between the measurement of positionand momentum. He deduced that the only experimental values of an ob-servable which can sharply be measured are its eigenvalues (in general itsspectral values).

There is the following synopsis

Elements CLASSICAL MECHANICS QUANTUM MECHANICS

{pure states} Poisson (or symplectic) man-ifold (M, P )

projective space of a complexHilbert space H

{observables} C∞(M,R) {selfadjoint operators in H}Algebraicstructure ofobservables

Poisson algebra associative algebra B(H)

Generator ofa dynamicalsystem

Hamilton functionH : M → R

Hamilton operatorH : D(H) ⊂ H → H

Equation ofmotion

Hamilton’s equation:dcdt

= XH(c)Schrodinger equation:i~∂ψ

∂t= Hψ

Value of anobservable ina state

value of the function f in m,f(m)

mean value of A in ψ,〈A〉ψ := 〈ψ, Aψ〉/〈ψ, ψ〉

Exactly mea-surable value

value of the function f in m,f(m)

spectral value of A

14

Two systems:{states}

product manifold(M1 ×M2, P(1) + P(2))

projective space of the tensorproductH1 ⊗ H2

Two systems:{observables}

C∞(M1,R) ⊗ C∞(M2,R) B(H1) ⊗ B(H2)

The interpretation problems of quantum mechanics take their origin inthe fact that the combination of the state spaces of two systems (for instancethe measured system and the measuring system in a measurement), namelythe projective space of the tensor product of two Hilbert spaces, contains‘more’ states than the so-called separated states which are the elements of thecartesian product of the two projective spaces: for example, if H1 = Cm+1

and H2 = Cn+1, then the set of separated states, CP (m) × CP (n), is a‘tiny’ submanifold of real dimension 2(m + n) of 2(m + n + mn)-dimensionalprojective space CP

((m + 1)(n + 1)− 1

)(the Segre embedding in algebraic

geometry). These ‘extra’ states are unavoidable for each nontrivial inter-action between the two systems and had been called entangled states (inGerman verschrankte Zustande) by Schrodinger. If the combined system isin an entangled state it is no longer possible to say that “system 1 is in state1 and system 2 is in state 2” like in classical mechanics where every combinedstate is separated since the combined state space is always a cartesian prod-uct. Until now there is no commonly accepted satisfactory interpretation ofthese entangled states.

As opposed to the set of states there is no longer a conceptual differencebetween the structures of the observable spaces of combined systems: in bothcases it is the (topological) tensor product of the two algebras. Therefore itseems to be more reasonable to consider the observables as more fundamentalobjects than the states –in contrast to the historical and intuitive physicalapproaches, but in agreement with the mathematics of C∗-algebras. For adetailed discussion of the interpretation problems of quantum mechanics seefor example the book by d’Espagnat [30].

15

3 Symbol Calculus and Elementary Star-Products

In order to describe a quantum system it is necessary to know its Hamiltonoperator H. In practise, the source of inspiration is the Hamilton function ofthe corresponding system of classical mechanics, and any ‘reasonable’ recipeof translating classical observables to quantum observables is called quanti-zation.

According to P.A.M. Dirac, all quantizations should satisfy a classicallimit condition, i.e. for all classical observables f, g

f g = fg + o(~), (3.0.11)

f g − gf = i~{f, g}+ o(~2). (3.0.12)

In this section we shall discuss several possible quantizations in one degreeof freedom which are used in quantum physics. We recall the differentialoperators Q (2.0.9) et P (2.0.10) in case n = 1: (Qψ)(q) := qψ(q) andP := (~/i)∂/∂q.

In the following section we denote by C[s1, . . . , sN ] the space of complexpolynomials in N variables s1, . . . , sN . Moreover, the symbol Diffoppoly(R)denotes the space of all differential operators with polynomial coefficients inthe space C∞(R,C), i.e. an element D takes the following general form

N∑

k=0

fk∂k/∂qk (3.0.13)

where f1, . . . , fN ∈ C[q].

3.1 Standard Ordering

We shall consider the following linear map ρs of the space of all complexpolynomials of two variables C[q, p] in the space Diffoppoly(R):

1 7→ ρs(1) := 1 (3.1.1)

q 7→ ρs(q) := Q (3.1.2)

p 7→ ρs(p) := P (3.1.3)

qmpn 7→ ρs(qmpn) := QmP n (3.1.4)

Since every differential operator in Diffoppoly(R) takes the form (3.0.13)is is obvious that this linear map is a bijection. The principal idea of star-products is to pull back the (noncommutative) associative multiplication ofdifferential operators by the map ρs:

16

Proposition 3.1 Let f, g be in C[q, p] and φ ∈ C∞(R,C). Then

ρs(f)(φ)

=∞∑

r=0

(~/i)r

r!

∂rf

∂pr

∣∣∣∣p=0

∂rφ

∂qr. (3.1.5)

Moreover

f ∗s g := ρ−1s

(ρs(f)ρs(g)

)=

∞∑r=0

(~/i)r

r!

∂rf

∂pr

∂rg

∂qr

is well-defined associative noncommutative multiplicatiion on the space C[q, p]which satisfies the classical limit

f ∗s g = fg − i~∂f

∂p

∂g

∂q+ o(~2).

Proof: The proof is direct computation; since ∗s is obviously isomorphic to theassociative multiplication of differential operators, it is clear that ∗s is also asso-ciative. 2

Note that for two given polynomials f, g the series in ~ is always a fi-nite sum. Moreover, every term in that series is a bidifferential operator(1/i)r

r!∂rf∂pr

∂rg∂qr .

3.2 Weyl-Moyal ordering prescription

From the point of view of physics, standard ordering is not satisfactory: whenconsidering the pre-Hilbert space

D(R) := {f : R→ C|f is C∞ and supp(f) is compact} (3.2.1)

equipped with the scalar product que (for all φ, ψ ∈ D(R)):

〈φ, ψ〉 :=

∫dq φ(q)ψ(q) (3.2.2)

we quickly see that the two real-valued functions q and p correspond tosymmetric operators, i.e. for A = Q or A = P

〈φ,Aψ〉 = 〈Aφ, ψ〉, (3.2.3)

whereas the real-valued function qp corresponds to the operator QP whoseadjoint in D(R) is equal to PQ = QP − i~1: hence ρs(qp) is no longer sym-metric which would be necessary to make it into a self-adjoint operator in the

17

completion L2(R, dq) of D(R). In order to avoid these problems, the Weyl-Moyal ordering prescription had been introduced: this uses a symmetrizationof the monomials in Q and P .

We consider the following linear map ρw of the space of all complex poly-nomials of two variables C[q, p] in the space Diffoppoly(R):

1 7→ ρw(1) := 1 (3.2.4)

q 7→ ρw(q) := Q (3.2.5)

p 7→ ρw(p) := P (3.2.6)

qmpn 7→ ρs(qmpn) :=

1

(m + n)!

∑σ∈Sm+n

Aσ(1) · · ·Aσ(m+n) (3.2.7)

where the operators A1, · · · , Am+n are given by

Ak :=

{Q if 1 ≤ k ≤ mP if m + 1 ≤ k ≤ m + n

For example, ρw(qp) = (QP +PQ)/2 and ρw(q2p) = (Q2P +QPQ+PQ2)/3.By definition, the operator ρw(f) is symmetric if f est reel, since you easilycompute that

ρw(f)† = ρw(f)

where A† is the adjoint operator of A in(D(R)), 〈 , 〉).

For two formal parameters α, β the exponential function exp(αq + βp)is mapped to ρw

(exp(αq + βp)

)= exp(αQ + βP ). Using the fact that

ρs

(exp(αq + βp)

)= exp(αQ) exp(βP ), the fact that [Q,P ] = i~1,and the

Baker-Campbell-Hausdorff formula, we compute

e(αQ+βP ) = e~αβ2i eαQeβP .

Since the exponential function exp(αq + βp) is a generating function for allpolynomials in q, p one realizes the following fundamental relation betweenstandard and Weyl-Moyal ordering:

ρw(f) = ρs(Nf) (3.2.8)

where the map N : C[q, p] → C[q, p] is defined by

N := e~2i

∂2

∂q∂p . (3.2.9)

It is clear that N is well-defined and invertible, and one deduces that ρw :C[q, p] → Diffoppoly(R) isa a linear bijection. There is the following analogueto Proposition 3.1:

18

Proposition 3.2 Let f, g be in C[q, p]. Then

f ∗w g := ρ−1w

(ρw(f)ρw(g)

)=

∞∑r=0

(i~/2)r

r!

r∑a=0

(r

a

)(−1)r−a ∂rf

∂qapr−a

∂rg

∂qr−apa

(3.2.10)is a well-defined noncommutative associative multiplication on the space C[q, p]satisfying the classical limit

f ∗w g = fg +i~2{f, g}+ o(~2),

and is isomorphic to ∗s via N :

N(f ∗w g) = (Nf) ∗s (Ng).

Proof: The proof is a direct computation using the operator N . 2

Again, it is easily seen that ∗w is a series of bidifferential operators.

3.3 Wick Ordering

There is a third quantization related to the harmonic oscillator which is veryoften used in quantum field theory: firstly, one forms the following complexvariable variable

z := q + ip. (3.3.1)

On the complex vector space

O(C) := {φ : C→ C|φ antiholomorphe }one defines the following scalar product

〈φ, ψ〉 :=1

4π~

∫dzdze−

|z|22~ φ(z)ψ(z) (3.3.2)

(which may still diverge), and finally the HIlbert space of all square integrableantiholomorphic functions

H := {φ ∈ O(C)|〈φ, φ〉 < ∞}, (3.3.3)

which is closed subspace of the Hilbert space L2(R2, e−(q2+p2)/(2~)dqdp/(2π~)

).

the space of polynomials in the variable z, C[z], is a dense subspace of H.Partial integration yields the fact that the operator A which multiplies bythe variable z induces the annihilation operator

A := 2~∂

∂z(3.3.4)

19

on C[z]. By a second partial integration C[z] we see that its adjoint A† (thecreation operator) is the operator

(A†φ)(z) := zφ(z). (3.3.5)

It follows that we can –in a manner completely analogous to standardordering– consider the following linear map ρwick of the space of all complexpolynomials in two variables C[z, z] in the space Diffoppoly(z) of all differ-ential operators having polynomial coefficients and acting in the space ofpolynomials C[z]:

1 7→ ρwick(1) := 1 (3.3.6)

z 7→ ρwick(z) := A (3.3.7)

z 7→ ρwick(z) := A† (3.3.8)

zmzn 7→ ρwick(zmzn) := A†mAn (3.3.9)

It is obvious that this linear map is a bijection.

Proposition 3.3 Let f, g be in C[q, p] and φ ∈ C[z]. Then

ρwick(f)(φ)

=∞∑

r=0

(2~)r

r!

∂rf

∂zr

∣∣∣∣z=0

∂rφ

∂zr.

Moreover

f ∗wick g := ρ−1wick

(ρwick(f)ρwick(g)

)=

∞∑r=0

(2~)r

r!

∂rf

∂zr

∂rg

∂zr

is a well-defined noncommutative associative multiplication on the space C[q, p]satisfying the classical limit

f ∗s g = fg + 2~∂f

∂z

∂g

∂z+ o(~2).

Proof: The proof is completely analogous to the proof of Proposition 3.1. Notethat

∂

∂q∧ ∂

∂p=

2i

∂

∂z∧ ∂

∂z.

2

As for the relation between standard ordering and Weyl ordering there isalso an analogue of the operator N (3.2.9): one defines

∆′ :=∂2

∂q2+

∂2

∂p2

20

andN ′ := e

~4∆′ .

Then, for any f, g ∈ C[z, z]:

N ′(f ∗w g) = (N ′f) ∗wick (N ′g).

Remark: The use of antiholomorphic functions instead of holomorphicfunctions is a quantum mechanical tradition: the creation (i.e. increase ofthe degree of the polynomial) is historical related to A†.

21

4 Formal Deformations

4.1 Formal Power series

In this section I shall recall several elementary about formal power serieswhich I shall need later on, for more details see e.g. the book by Ruiz [74].Let R be a ring (always with unit element, for instance a field) and M aleft module over R (for example R-vector space). We shall write a mapa : N→ M in the form of a formal power series with coefficients in M

a =:∞∑

r=0

λrar

where ar := a(r) is called the rth component of a, and the symbol λ is calledthe formal parameter. The set of all formal power series with coefficients inM is denoted by M [[λ]]. The sets M [[λ]] and R[[λ]] carry the structure of anabelian group in the canonical way (b =

∑∞r=0 λrbr where br ∈ M):

a + b :=∞∑

r=0

λr(ar + br).

Furthermore R[[λ]] carries the structure of ring via (α =∑∞

r=0 λrαr, β =∑∞r=0 λrβr, αr, βr ∈ R)

αβ :=∞∑

r=0

λr

r∑s=0

αsβr−s,

and M [[λ]] becomes a left R[[λ]]-module via

αb :=∞∑

r=0

λr

r∑s=0

αsbr−s.

The order of a power series a, o(a), is defined by the minimum of the set ofall nonnegative integers r such that ar 6= 0 in case a 6= 0 and is defined to be+∞ in case a = 0. It can be shown that the function

d : M [[λ]]×M [[λ]] → R : (a, b) 7→ d(a, b) :=

{2−o(a−b si a 6= b

0 si a = b

defines a metric on M [[λ]] qui induces a Hausdorff topology called the λ-adictopology of M [[λ]].

The following Lemma turns out to be useful:

22

Lemma 4.1 Let M1 and M2 two R-modules and Φ : M1[[λ]] → M2[[λ]] anR[[λ]]-linear map.Then for each nonnegative integer r there is a unique R-linear map Φr :M1 → M2 such that

Φ(a) =∞∑

r=0

λr

r∑s=0

Φs(ar−s) (4.1.1)

for all a =∑∞

r=0 λrar ∈ M1[[λ]].

Proof: The restriction of Φ to M1 is a R-linear map into M2[[λ]]. The componentsof this map are R-linear maps Φr : M1 → M2. The right hand side of (4.1.1) –which we shall call Φ is a R[[λ]]-linear map of M1[[λ]] in M2[[λ]]. By its definition,hte difference Φ − Φ is zero on all those formal series whose nonzero componentsform a finite set. Let us suppose that there is formal series a ∈ M1[[λ]] such thatque b := (Φ− Φ)(a) is not equal to 0. Let k be the order of b. Since b is also givenby b = (Φ− Φ)(a−a0−λ1a1−· · ·−λkak) and a−a0−λ1a1−· · ·−λkak =: λk+1a′

it would follow that the order of b soud be grater or equal to k+1 which is absurd.Hence Φ = Φ, and the Lemma is proved. 2

In case the ring R is commutative, one can easily generalize this Lemmato the case of R-multilinear maps.

4.2 Formal Deformations of Associative Algebras

Let (A0, µ0) be an associative algebra with unit 1 over a commutative ringR.

Definition 4.1 A formal associative deformation of the associative algebrawith unit 1, (A0, µ0), is given by a sequence of R-bilinear maps µ1, µ2, ...:A0 ×A0 → A0 such that

1.r∑

s=0

(µs

(µr−s(a, b), c

)− µs

(a, µr−s(b, c)

))= 0 (4.2.1)

for all r ∈ N and a, b, c ∈ A0.

2. µr(1, a) = 0 = µr(a, 1) for all r ∈ N, r ≥ 1 and a ∈ A0.

The following Proposition is obvious:

Proposition 4.1 The space A := A0[[λ]] equipped with the R[[λ]]-bilinearmultiplication µ :=

∑∞r=0 λrµr, i.e.

µ(a, b) :=∞∑

r=0

λr∑

s+t+u=0

µs(at, bu)

23

for all a =∑∞

t=0 λtat and b =∑∞

u=0 λubu dans A, is an associative algebrawith unit element 1 over the ring R[[λ]].

For the case r = 1 of equation (4.2.1) wer get (writing µ0(a, b) =: ab):

0 = aµ1(b, c)− µ1(ab, c) + µ1(a, bc)− µ1(a, b)c =: (δHµ1)(a, b, c

)

where δH is the Hochschild coboundary operator defined on Kest l’operateurcobord de Hochschild defini sur

C(A0,A0) :=∞⊕

k=0

Ck(A0,A0) :=∞⊕

k=0

HomR(A0 ⊗R · · · ⊗R A0,A0)

by

(δHf)(a1 ⊗ · · · ⊗ ak+1) := a1f(a2 ⊗ · · · ⊗ ak+1)

+k∑

r=1

(−1)rf(a1 ⊗ · · · ⊗ ar−1 ⊗ arar+1 ⊗ · · · ⊗ ak+1)

+(−1)k+1f(a1 ⊗ · · · ⊗ ak)ak+1.

It is well-known that δ2H = 0, hence that operator defines a cohomology

theory called the Hochschild cohomology:

Zk(A0,A0) := Ker(δH : Ck(A0,A0) → Ck+1(A0,A0)

)

Bk(A0,A0) := Im(δH : Ck−1(A0,A0) → Ck(A0,A0)

)

HHk(A0,A0) := Zk(A0,A0)/Bk(A0,A0)

The elements of Zk(A0,A0) are called Hochschild k-cocycles of A0, the ele-ments Bk(A0,A0) are called HOchschild k-coboundaries ofA0 (where B0(A0,A0) :=0), and HHk(A0,A0) is called the keme Hochschild cohomology group of A0

(with values in A0).It follows that for any formal deformation the term µ1 is always a HOchschild-

2-cocycle . In the more general case where µ is not necessarily associative itis easily computed that the associator of µ

A(a, b, c) := µ(µ(a, b), c

)− µ(a, µ(b, c)

)

satisfies the following identity:

0 = µ(a,A(b, c, d)

)− A(µ(a, b), c, d

)+ A

(a, µ(b, c), d

)

−A(a, b, µ(c, d)

)+ µ

(A(a, b, c), d

)

24

for all a, b, c, d ∈ A0. For an associative formal deformation we demand thatA =

∑∞r=0 λrAr = 0. Let us suppose that A0, A1, . . . , Ak are all zero. Thanks

to the preceding identity we get at order r + 1 in λ:

δHAr+1 = 0.

SinceAr+1 = δHµr+1 + A′

r+1

where the rest A′r+1 contains only the terms µ0, · · · , µr it follows that

We have : δHA′r+1 = 0 =⇒ A′

r+1 ∈ Z3(A0,A0)

We want : A′r+1

!= −δHµr+1 =⇒ A′

r+1

!∈ B3(A0,A0)

Consequently, the recursive obstructions to continue the construction of theterm µr+1 of a formal associative deformation of µ0 are contained at eachstage in

HH3(A0,A0).

For the very important particular case where A0 is given by = C∞(M,C)(equipped with the pointwise multiplication. Here one considers Hochschildcochains which are given by multidifferential operators. This subspace of theHOchschild complex C

(C∞(M,C), C∞(M,C)

)(denoted by by Cdiff

(C∞(M,C), C∞(M,C)

))

is a subcomplex with respcet to the Hochschild coboundary. Its cohomol-ogy is called the differential Hochschild cohomology of C∞(M,C) and is de-noted by HHdiff

(C∞(M,C), C∞(M,C)

). The computation of this cohomol-

ogy is due Hochschild-Kostant-Rosenberg [56], Cahen-DeWilde-Gutt [22] etDeWilde-Lecomte [31]:

Theorem 4.1

HHdiff

(C∞(M,C), C∞(M,C)

) ∼= Γ(M, ΛTM).

A generalization of this result had been obtained by A.Connes in 1985 (see[27], p.207-210) who had replaced the differential cochains by cochains whichare continuous with respect to the standard Frechet topology of this space.Pflaum [72] and Nadaud [64] have shown that on may drop Connes’s hypoth-esis that the Euler characteristic of the manifold is zero. In this cases theresulting Hochschild cohomology is isomorphic to the right hand side of theHKR-theorem 4.1, i.e. the space of all multivector fields on M .

25

4.3 Gerstenhaber’s Formula

The explicit formulas for the associative multiplications ∗s, ∗w et ∗wick have acommon algebraic feature: the following theorem is due to M.Gerstenhaber[47], p.13, Thm.8:

Theorem 4.2 Let (A, µ0) be an associative algebra with unit 1 over a com-mutative ring k which contains the rationals Q where µ0 : A⊗A → A denotesthe (not necessarily commutative) multiplication of A. Let D1, . . . , Dn, E1, . . . , En

2n be derivations of (A, µ0) which all pairwise commute, i.e. Dkµ0 = µ0(Dk⊗1 + 1⊗Dk), Elµ0 = µ0(El ⊗ 1 + 1⊗ El), DkDl = DlDk, DkEl = ElDk andEkEl = ElEk for all integers 1 ≤ k, l ≤ n. Let r :=

∑nk=1 Dk ⊗Ek. Then on

the k[[λ]]-module A[[λ]] there is a k[[λ]]-bilinear associative multiplication µdefined by

µ := µ0 ◦ eλr. (4.3.1)

Proof: The following elegant reasoning has been found by A.Dimakis and F.Muller-Heussen in [34] for a particular case: one defines the following three linear maps:A⊗A⊗A → A⊗A⊗A (where 1 denotes the identity map A → A) r12 := r⊗ 1,r23 := 1⊗ r and r13 :=

∑nk=1 Dk⊗1⊗Ek. Since the derivations commute we have

[r12, r13] = 0, [r12, r23] = 0 and [r13, r23] = 0. Thanks to the derivation identity itfollows that

r µ0 ⊗ 1 = µ0 ⊗ 1 (r13 + r23) andr 1⊗ µ0 = 1⊗ µ0 (r12 + r13),

hence

eλr µ0 ⊗ 1 = µ0 ⊗ 1 e(r13+r23) andeλr 1⊗ µ0 = 1⊗ µ0 e(r12+r13),

therefore, as the rij commute:

µ µ⊗ 1 = µ0 eλr µ0 ⊗ 1 eλr12 = µ0 µ0 ⊗ 1 eλ(r12+r13+r23).

Analogously:

µ 1⊗ µ = µ0 eλr 1⊗ µ0 eλr23 = µ0 1⊗ µ0 eλ(r12+r13+r23).

Since µ0 is associative we have µ0 µ0 ⊗ 1 = µ0 1⊗ µ0, whence the associativity ofthe multiplication µ. 2

26

4.4 Standard, Weyl-Moyal and Wick ordering in R2n

It is easily seen that the multiplications ∗s et ∗w are particular cases ofthe Gerstenhaber formula if the real number ~ is replaced by the formalparameter λ and if we set

rs :=1

i

∂

∂p⊗ ∂

∂q

to find ∗s,

rw :=i

2

(∂

∂q⊗ ∂

∂p− ∂

∂p⊗ ∂

∂q

)

for ∗w and

rwick := 2∂

∂z⊗ ∂

∂zfor ∗wick. In this fashion we obtain the associative multiplications on theC[[λ]]-module C∞(R2,C)[[λ]]: in this framework of formal power series themultiplications ∗s et ∗w remain well-defined although we can no longer setλ = ~ because the formal series do no longer converge for general smoothfunctions.

The generalisations of these formulas ∗s, ∗w et ∗wick to the case R2n = Cn

are now completely clear:

rs :=1

i

n∑

k=1

∂

∂pk

⊗ ∂

∂qk

(4.4.1)

rw :=i

2

n∑

k=1

(∂

∂qk

⊗ ∂

∂pk

− ∂

∂pk

⊗ ∂

∂qk

)(4.4.2)

rwick := 2n∑

k=1

∂

∂zk

⊗ ∂

∂zk

(4.4.3)

and we get

f ∗s g =∞∑

r=0

(λ/i)r

r!

n∑

k1,...,kr=1

∂rf

∂pk1 · · · ∂pkr

∂rg

∂qk1 · · · ∂qkr

(4.4.4)

for the standard product, and (writing∑n

k=1∂

∂qk∧ ∂

∂pk= 1

2

∑2nk,l=1 P kl ∂

∂xk∧ ∂

∂xl

with (q, p) = x) for the Weyl-Moyal product:

f ∗w g =∞∑

r=0

(iλ/2)r

r!

2n∑

k1,...,kr,l1,...,lr=1

P k1l1 · · ·P krlr∂rf

∂xk1 · · · ∂xkr

∂rg

∂xl1 · · · ∂xlr

.

(4.4.5)

27

and

f ∗wick g =∞∑

r=0

(2λ)r

r!

n∑

k1,...,kr=1

∂rf

∂zk1 · · · ∂zkr

∂rg

∂zk1 · · · ∂zkr

. (4.4.6)

for the Wick product.

4.5 Standard symbols of multidifferential operators

The formulas of ∗w et ∗s of the preceding section converge in the variable λin the case where the functions f and g are polynomials in the momentumvariables p1, . . . , pn with coefficients in C∞(Rn,C), and we call this spaceC∞pp(R2n,C). For α ∈ R2n∗ let eα be the exponential function

eα(x) := eα(x).

Let D be a differential operator in C∞(Rn,C), i.e. there is a nonnegativeinteger N and functions Da;i1...ia ∈ C∞(Rn,C) such that

D =N∑

a=0

n∑i1,...,ia=1

Da;i1...ia∂a

∂xi1 · · · ∂xia. (4.5.1)

Its standard symbol is defined by

D(q, α) := (Deα)(q)e−α(q) =N∑

a=0

n∑i1,...,ia=1

Da;i1...iaαi1 · · ·αik , (4.5.2)

and it is immediate that

ρs(f)(q,ip

~) = f(q, p). (4.5.3)

It is also not hard to see that the standard symbol defines a linear bijectionbetween the space Diffop(Rn) of all differential operators in Rn and the spaceC∞pp(R2n,C).

Let k be a nonnegative integer and D be a k-differential operator, i.e.there is a nonnegative integer N and functions D(a1,...,ak),I1,...,Ik where I1 :=(i11, . . . , i1a1), I2 := (i21, . . . , i2a2),. . .,Ik := (ik1, . . . , ikak

) are blocks of indices(the ibc always vary between 1 et n) such that

D(f1, . . . , fk) =N∑

a1,...,ak=1

∑I1,...,Ik

D(a1,...,ak),I1,...,Ik∂a1f1

∂xI1· · · ∂

akfk

∂xIk(4.5.4)

28

where for example ∂a1/∂xI1 is short for ∂a1/(∂xi11 · · · ∂xi1a1 ). One defines thestandard symbol of a k-differential operator D for α(1), . . . , α(k) ∈ Rn∗ by

D(q, α(1), . . . , α(k)) :=(D(eα(1) , . . . , eα(k))

)(q)e−(α(1)+···+α(k))(q). (4.5.5)

As in the case of differential operators this is equivalent to saying thatone replaces the partial derivatives in (4.5.4) by the nk additional variablesα(1), . . . , α(k), on which D depends in a polynomial way.The following Lemma is obvious:

Lemma 4.2 Each k-differential operator D in Rn est uniquely determined byits standard symbol D or, equivalently, by its values on exponential functions.

29

5 Star-produits

In the preceding chapter we have seen that on can construct noncommu-tative or “quantum” associative multiplications ∗ on C[q, p] and even onC∞(R2n,C)[[λ]] by using symbol calculus, i.e. by using a linear bijection be-tween C[q, p] and an already given associative algebra, namely the algebraof all differential operators with polynomial coefficients acting on C∞(R,C).The principal idea of star products is to construct such an associative multi-plication ∗ directly on the space of classical observables, i.e. on C∞(M,C)[[λ]](where M is a given Poisson manifold) without referring to a ‘representation’in a differential operator algebra: for most of the Poisson manifolds it is notat all clear how such a differential operator algebra could be chosen. Fromthe point of view of physics this means that the construction of the the quan-tum system starts with the observable algebra (unlike the usual approach),whereas the construction of the Hilbert space is postponed.

5.1 Definition

The following definition had been given by F. Bayen, M. Flato, C. Frønsdal,A. Lichnerowicz et D. Sternheimer in 1978 [7]:

Definition 5.1 Let (M,P ) be a Poisson manifold. The structure of a star-produit on M or a deformation quantization on M is defined by the followingsequence of C-bilinear maps

Cr : C∞(M,C)× C∞(M,C) → C∞(M,C)

for all r ∈ N subject to the following conditions (f, g, h ∈ C∞(M,C)):

1. Every Cr is a bidifferential operator, i.e. there is a nonnegative integerNr such that in each chart

(U, (x1, . . . , xn)

)there are smooth functions

C(a,b),i1···ia,j1···jbr : U → C such that Cr takes the following form in U :

Cr(f, g) =Nr∑

a,b=0

n∑i1,...,ia,j1,...,jb=1

C(a,b),i1···ia,j1···jbr

∂af

∂xi1 · · · ∂xia

∂bg

∂xj1 · · · ∂xjb.

2. C0(f, g) = fg (classical limit).

3. C1(f, g)− C1(g, f) = i{f, g} := iP (df, dg) (classical limit).

4. Cr(1, g) = 0 = Cr(f, 1) for all r ≥ 1 (the constant function 1 remainsthe unit element).

30

5.∑r

s=0

(Cs(Cr−s(f, g), h

)=

∑rs=0

(Cs(f, Cr−s(g, h)

)for all r ∈ N (asso-

ciativity).

The formal series

∗ :=∞∑

r=0

λrCr

is called a star-product on M .Moreover, if for all r ∈ N and f, g ∈ C∞(M,C)

Cr(f, g) = Cr(g, f)

(where ¯ denotes pointwise complex conjugation) the star-product is calledsymmetric or hermitean.

For example ∗w and ∗wick are hermitean.The following corollary is obvious:

Corollary 5.1 Let ∗ be a star-product on the Poisson manifold (M,P ).Then the C[[λ]]-module C∞(M,C)[[λ]] becomes an associative algebra overthe ring C[[λ]] via (F =

∑∞r=0 λrFr, G =

∑∞r=0 λrGr ∈ C∞(M,C)[[λ]])

F ∗G :=∞∑

r=0

λr∑

s+t+u=0

Cs(Ft, Gu).

If moreover the star-product is hermitean, then the pointwise complex con-jugation ¯ becomes an antiautomorphism of the algebra (C∞(M,C)[[λ]], ∗),i.e.

F ∗G = G ∗ F.

We list further properties of star-products:

Definition 5.2 A hermitean star-product is called of Weyl-Moyal type iff

Cr(g, f) = (−1)rCr(f, g).

If the order Nr of the bidifferential operator Cr is equal to r the star-product iscalled natural by S.Gutt and J.Rawnsley. Finally, if (M, P ) is a semi-Kahlermainifold (i.e. (M,ω) is symplectic and admits a complex structure J (i.e.J ∈ Γ(M,Hom(TM, TM)), J2 = −1, and J having vanishing Nijenhuistorsion) such that ω(JX, JY ) = ω(X,Y ) for any vector field X, Y ) the star-product is called of type Wick or admitting separation of variables iff in eachcomplex chart

(U, (z1, . . . , zn)

)there are C∞-functions C

(a,b),i1···ia,j1···jbr : U →

C such that in U :

Cr(f, g) =Nr∑

a,b=0

n∑i1,...,ia,j1,...,jb=1

C(a,b),i1···ia,j1···jbr

∂af

∂zi1 · · · ∂zia

∂bg

∂zj1 · · · ∂zjb.

31

For example, ∗s, ∗w, and ∗wick are natural, ∗w are ∗Wick are hermitean, ∗w isof Weyl-Moyal type, and ∗wick is of Wick type.

For two star-products ∗ and ∗′ there is the following notion of formalisomorphy which had already encountered for ∗s and ∗w:

Definition 5.3 Let (M, P ) a Poisson manifold and ∗, ∗′ two star-products.We say that ∗ is equivalent to ∗′ iff there is a formal series of differentialoperators, called an equivalence transformation

S = id +∞∑

r=1

λrSr

(where Sr : C∞(M,C) → C∞(M,C) are differential operators vanishing onthe constants) such that

F ∗′ G = S−1((SF ) ∗ (SG)

)

for all F, G ∈ C∞(M,C)[[λ]].

Since the operator series N (see eq. (3.2.9)) takes the form 1 + λ2i

∂∂q

∂∂p

+

o(λ2), it defines an equivalence transformation between the star-products ∗w

et ∗s.

5.2 Existence

5.2.1 Symplectic manifolds

After some important results for special cases (like symplectic manifoldswhose third de Rham-cohomology group vanishes [65] and cotangent bun-dles of parallelisable manifolds [23]) the first complete existence result hadbeen shown by M.DeWilde et P.Lecomte in 1983, [32]:

Theorem 5.1 (DeWilde,Lecomte 1983) On any symplectic manifold (M,ω)there is a star-product.

The proof was based on explicit computations of the differential Hochschildcohomology of

(C∞(M,C), ·) and of the second and third Chevalley-Eilenbergcohomology of the Lie algebra C∞(M,C) equipped with the Poisson bracket{ , }, see [33] and the use of a local homogeneity by means of local Eulerfield in a Darboux chart.

Independantly of this result, B.Fedosov has given a proof of Thorem 5.1in 1985, [40]. HIs proof is remakable since it rather uses symplectic connec-tions than local charts: therefore his method allows to construct directly intensorial terms, which sometimes is more adapted to the implementation ofsymmetries.

32

5.2.2 Poisson manifolds

The main obstacle to translate (even locally) the methods of the previoussection to a general Poisson manifold was the fact that there is in general noconnection in the tangent bundle leaving invariant the Poisson structure (ifthis is the case it must have constant rank). In the deformation communityit came out as a big sensational surprise in 1997 when the following resultwas announced by Maxim Kontsevitch:

Theorem 5.2 (Kontsevitch 1997) On any Poisson manifold (M,P ) thereis a star-product.

For the algebraic framework of operades and L∞ structures, see the origi-nal article [63] and the article [3] for more details. Cattaneo and Felder haveretraced the quuntum-fieldtheoretic roots of Kontsevitch’s construction inthe theory of Poisson-Sigma models, see [25], and have given a globalisationin the spirit of Fedosov in [26].

Pour the Poisson manifold (Rn, 12

∑na,b=1 P ab∂a∧ ∂b) Kontsevitch uses the

following ansatz for the bidifferential operators Cr of the star-product: letf, g ∈ C∞(Rn,C), let 2r = n1+· · ·+nr+M+N a partition of the nonnegativeinteger 2r as a sum of nonnegative integers, and let σ be permutation of{1, 2, . . . , 2r}. Let us denote the pair

((n1, . . . , nr,M, N), σ

)=: Γr, and one

defines the bidifferential operator

CΓr(f, g) :=n∑

a1,...,a2r=1

(∂n1P aσ(1)aσ(2)

∂xa1 · · · ∂xan1

· · · ∂nrP aσ(2r−1)aσ(2r)

∂xan1+···+nr−1+1 · · · ∂xan1+···+nr

∂Mf

∂xan1+···+nr+1 · · · ∂xan1+···+nr+M

∂Ng

∂xan1+···+nr+M+1· · · ∂xa2r

).

(5.2.1)

The operator Cr is obtained by a particular linear combination of the preced-ing operators parametrised by all possible pairs Γr with weights wΓ which areat the heart of its construction: Kontsevitch represents Γr by graphs havingr + 2 vertices (corresponding to r Poisson structures and two functions) and2r edges (corresponding to 2r partial derivatives) in the upper half plane,and the weights wΓr are obtained by an integration related to the geometricimage of the graph:

Cr(f, g) =∑Γr

wΓrCΓr(f, g). (5.2.2)

33

5.2.3 Semi-Kahler manifolds

As symplectic manifolds semi-Kahler manifolds are equipped with star-products.A more interesting question is whether the star-product can always be chosento be of Wick type, which has to be proved separately:

Theorem 5.3 On any semi-Kahler manifold (M, ω, I) there is a star-productof Wick type.

This Theorem is due to A.Karabegov [57] and later – but independantly–by S.Waldmann and the author [18]. Karabegov has glued local differentialoperators on local holomorphic functions while Bordemann-Waldmann wereusing an evident modification of Fedosov’s method.

5.2.4 Even symplectic supermanifolds

Let τ : E → M be a real vector bundle over a symplectic mainifold (M,ω)equipped with a fibre metric q and a connection ∇E in E compatible withq. The space of all C∞-sections of the bundle ΛE∗, C0 := Γ(ΛE∗) is a gradedcommutative associative algebra with respect to the point-wise exterior mul-tiplication. C0 is called the algebra of superfunctions. This structure is calleda split supermanifold, see [5], [38, 39], [49] for maore details. There is a gradedPoisson bracket { , }R : C0 × C0 → C0 on this algebra due to M.Rothstein[73] and is defined in the following way: let U be the domain of a chart of Mwhich trivializes the vector bundle E where: (x1, . . . , xm) denotes the chartof M , ∂1, . . . , ∂m denotes a local base of vector fields on M corresponding tothat chart, let P kl (1 ≤ k, l ≤ m) be the components of the Poisson structureon M in that chart, let e1, . . . , en be a base of local sections of E, e1, . . . , en

its dual base, let qAB (1 ≤ A,B ≤ m) be the components of the fibre metricq with respect to the preceding base, let qAB (the inverse matrix of qAB) thecomponents of the fibre metric q−1 on the dual bundle E∗ induced by q, and

let R(E)A

Bkl be the components of the curvature tensor of the connection ∇E.One first forms the tensor field R(E) ∈ Γ

(Hom(TM)⊗ Λ2E∗) by

(R(E)

)k

lAB:= −1

2

m∑r=1

n∑C=1

qACP krR(E)C

Brl. (5.2.3)

The space of smooth sections Γ(Hom(TM)⊗ΛE∗) is an associative graded

algebra in the natural way, and the element R(E) is obviously nilpotent.Hence the geometric series

(1− 2R(E))−1 :=m∑

a=0

(2R(E)

)a(5.2.4)

34

is well-defined. Let φ ∈ Γ(ΛsE∗) and ψ ∈ Γ(ΛE∗) be two superfunctions.Hence the Rothstein superbracket is defined in the following way:

{φ, ψ} :=m∑

i,j,k=1

P ij((1− 2R(E))−1

)k

i∧∇E

∂kφ ∧∇E

∂jψ

+n∑

A,B=1

qAB(−1)s−1(ieaφ) ∧ (ieBψ). (5.2.5)

Upon using some elements of Fedosov’s construction I have shown in [21](voir aussi [10]) the following result:

Theorem 5.4 The graded commutative associative algebra C0 admets a gradedformal associative deformation such that the term of order 1 is equal to iλ

2

times the Rothstein superbracket.

R.Eckel has formulated a full Fedosov construction in the framework of su-permanifolds in his PhD-thesis [39].

5.3 Equivalence

5.3.1 Symplectic manifolds

Twelve years after the first star-product existence proof the classificationof equivalence classes of star-products on a symplectic manifold has beenachieved my Deligne [29], Nest-Tsygan [66, 67] and Bertelson-Cahen-Gutt[9]:

Theorem 5.5 Let (M, ω) a symplectic manifold. Hence the equivalenceclasses of star-products on (M, ω) are in bijection with the formal serieshaving coefficients in H2

dR(M), the second de Rham cohomology group on themanifold M .

The above bijection is given explicitly and is called the Deligne class [∗]of ∗ (see e.g. the excellent review [53]). In Fedosov’s construction one couldequally well introduce a formal series of closed 2-forms: Neumaier has shownthat they coincide with the representatives of the Deligne class, see [68].

5.3.2 Poisson manifolds

In the case of a Poisson manifold the classification result proved to be muchmore difficult and had also been done by Kontsevitch [63]: a formal Poisson

35

structure P on a differentiable manifold M is a formal series P =∑∞

r=0 λrPr

where the coefficients Pr are bivector fields in Γ(Λ2TM) such that

[P, P ]S = 0 ⇐⇒r∑

a=0

[Pa, Pr−a]S = 0 quel que soit r ∈ N

where [ , ]S denotes the Schouten bracket. Likewise, a formal vector field Xis a formal series X =

∑∞r=0 λrXr where the coefficiens are vector fields on

M . The Lie derivative of P with respect to X is defined in a natural wayvia the Schouten bracket

LX(P ) := [X,P ]S =∞∑

r=0

λr

r∑a=0

[Xa, Pr−a].

Two formal Poisson structures P and P ′ are said to be formally diffeomorphiciff there exists a formal vector field X sucht that

P ′ = eλLX (P ),

where equivalence is seen upon using the Baker-Campbell-Hausdorff series.By means of these structures the star-products are calssified as follows:

Theorem 5.6 (Kontsevitch 1997) Let (M, P0) be a Poisson manifold. Hencethe equivalence classes of star-products on (M,P0) are in bijection with theformal diffeomorphism classes of formal Poisson structures whose zeroth or-der terem is equal to P0.

6 Explicit Examples

6.1 Cotangent bundle of Sn

This example is due to F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz etD. Sternheimer [7]:

Consider the symplectic manifold M ′ := T ∗(Rn+1 \{0}) = (Rn+1 \ {0})×Rn+1 equipped with canonical coordinates (q, p) and the canonical symplecticform

∑n+1k=1 dqk ∧ dpk. The following two functions

H1(q, p) :=n+1∑

k=1

qkpk =: q · p (6.1.1)

H2(q, p) :=n+1∑

k=1

(qk)2 =: |q|2 (6.1.2)

36

span the two-dimensional non abelian Lie algebra (with respect to the Poissonbracket, and we have

{H1, H2} = −2H2. (6.1.3)

Moreover, the flows of H1 and of H2 take the form

Φ1s(q, p) = (esq, e−sp) (6.1.4)

Φ2t (q, p) = (q, p− 2tq) (6.1.5)

and they generate the action of the Lie group

G := {(α, t) ∈ R2|α > 0} (6.1.6)

on M ′ given by(α, t).(q, p) := (αq,−2tq + α−1p). (6.1.7)

Let T ∗Sn be defined by

M := T ∗Sn := {(q, p) ∈ M ′|q · p = 0 et |q|2 = 1}. (6.1.8)

It is easily seen that this definition gives the tangent bundle of the n-sphereSn which is isomophic to its cotangent bundle via the canonical metric onSn. There is a projection

π : M ′ → M : (q, p) 7→(

q

|q| , |q|p−q · p|q| q

)(6.1.9)

which is a surjective submersion. The fibres of the projection are the orbitsof the group G. Therefore there is the following

Lemma 6.1 Let F ∈ C∞(M ′,C). Hence there is a function f ∈ C∞(M,C)such that F = f ◦ π if and only if F is G-invariant, i.e F

((α, t).(q, p)

)= F

(q, p

).

Since G is connected, it follows that F is G-invariant iff

{F, H1} = 0 = {F,H2}. (6.1.10)

Using the star-product ∗w (4.4.5) on M ′ one sees that for every quadraticpolynomial F and every F ∈ C∞(M ′,C) there is the important formula

F ∗w F − F ∗w F = iλ{F, F} (6.1.11)

where the terms of higher order vanish. If this formula is applied to F = H1

or F = H2 one directly sees –using (6.1.10)– that a function F is G-invariantiff it commutes with H1 and H2 with respect to ∗w. It follows that the space ofall G-invariant functions is an associative subalgebra of (C∞(M ′,C)[[λ]], ∗w).Therefore one has the following

37

Theorem 6.1 There exists a star-product ∗BFFLS on M for which one hasthe following explicit formula:

f ∗BFFLS g(π(q, p)

)= (π∗f) ∗w (π∗g)(q, p).

6.2 Complex projective space

The following explicit formula described further down for a star-product oncomplex projective space CP n has been found in [20] where one may find thedetails of its deduction:

Letπ : Cn+1 \ {0} → CP n (6.2.1)

be the canonical projection whose fibres are the complex lines in Cn+1 \ {0}passing through the origin. As in the preceding example, the fibres areobtained by the action of a two-dimensional Lie group, namely the multi-plicatif group of all nonzero complex numbers. Unfortunately, this groupdoes no longer preserve the star-product of Wick type on Cn+1 \ {0} whichrenders the deduction more difficult. By means of the complex coordinatesz := (z1, . . . , zn+1) on Cn+1 \ {0} we define

x :=n+1∑

k=1

|zk|2. (6.2.2)

By modifying the usual star-product of Wick type ∗Wick on Cn+1 \ {0} by anexplicit equivalence transformation we get the following

Theorem 6.2 Let f, g ∈ C∞(CP n,C). Hence the following formula definesa star-produit ∗ of Wick type on the Kahler mainifold CP n:

π∗(f ∗ g)(z) := π∗(fg)(z)

+∞∑

r=1

(2λ)r

r!

xr

(1 + λ) · · · (1 + rλ)

n+1∑

k1,...,kr=1

∂rπ∗f∂zk1 · · · ∂zkr

(z)∂rπ∗g

∂zk1 · · · ∂zkr

(z).

In [19] we have shown that this star-product converges on all represen-tative functions for the canonical action of the unitary group U(n + 1) forcertain values of λ.

6.3 The dual space of a Lie algebra

This very important star-product has been found independantly by V.Drinfel’det S.Gutt, [36], [52], in 1983:

38

Let (g, [ , ] be a finite-dimensional real Lie algebra and g∗ its dual spacewhich is a Poisson manifold (see 1.3.6)). Here we use the formal parameterν := iλ. Let H : g× g → g[[ν]] be the formal group law by Baker-Campbell-Hausdorff:

H(x, y) := x + y

+∞∑

n=1

(−1)n

n + 1

∑k1,...,kn≥0l1,...,ln≥0ki+li≥1

νPn

i=1 ki+li

(ad(x)

)k1(ad(y)

)l1 · · · (ad(x))kn

(ad(y)

)ln

(k1 + · · ·+ kn + 1)k1! · · · kn!l1! · · · ln!x.

(6.3.1)

It is easily seen that one can extend H to g[[ν]] × g[[ν]]. By its definition,H is equal to the logarithm of a product of two exponential functions in thecompleted free algebra generated by the two letters x and y,

H(H(x, y), z

)= H

(x,H(y, z)

) ∀x, y, z ∈ g. (6.3.2)

One defines the standard symbol of ∗: x, y ∈ g ∼= g∗∗ by

ex ∗ ey := eH(x,y) (6.3.3)

Since it is evident from (6.3.1) that H(x, y) − x − y is a multiple of ν thestandard symbol of ∗ is a formal series in the parameter ν. Moreover, foreach power of ν there is only a finite number of summands in H(x, y) (6.3.1):this implies that the standard symbol of ∗ is polynomial in (x, y) for eachpower of ν. Therefore the formula (6.3.3) is well-defined. The star-productis associative because (x, y, z ∈ g)

(ex ∗ ey) ∗ ez = eH(x,y) ∗ ez = eH(

H(x,y),z)

= eH(

x,H(y,z)) = ex ∗ eH(y,z) = ex ∗ (ey ∗ ez)

thanks to (6.3.2). Hence the two tridifferential operators defined by theirstandard symbols as (ex ∗ ey) ∗ ez and ex ∗ (ey ∗ ez) coincide, hence they areequal thanks to Lemma 4.2. The formal series of the standard symbol of ∗has the following terms of order zero and one:

∗(ξ, x, y) = eξ(H(x,y)−x−y) = 1 +ν

2ξ([x, y]) + o(ν2)

whence the classical limit of ∗ is readily deduced. Finally, the Euler-likeoperator

ν∂

∂ν+

n∑

k=1

ξk∂

∂ξk

39

which counts the sum of the degree in ν and the degree in ξ ∈ g∗ is a deriva-tion of de ∗ by (6.3.1), hence the bidifferential operator Cr of ∗ (which hasdegree r in ν) has at leasts r partial derivatives with respect to ξ distributedon the two functions f et g. It follows that f ∗ g is a polynomial in ν if fand g are polynomials on g∗. Hence one may set ν = 1 on polynomials. Thislatter complex associative algebra is isomorphic to the complexified universalenvelopping algebra Ug of g (see [52]).

Theorem 6.3 Let g be a finite-dimensional real Lie algebra. Then thereexists a star-product ∗ (called BCH (Baker-Campbell-Hausdorff) defined by(6.3.1) on the Poisson manifold (g∗, Pg) which converges (in ν) on the sub-space of all polnomials on g∗ where the induced multiplication is ismorphicto the complexified universal envelopping algebra of g.In particular, for ξ, η ∈ g it follows that for the two linear functions ξ and ηdefined on g∗ by ξ(α) := 〈α, ξ〉:

ξ ∗ η − η ∗ ξ = iλ ˜[ξ, η].

6.4 L’espace dual d’une algebre associative

Cet exemple est une version simplifiee de l’exemple precedent qui est due al’auteur.

Soit A une algebre associative reelle de dimension finie n. Dans unebase e1, . . . , en de A (et la base duale e1, . . . , en de A∗) on peut exprimer lesconstantes de structure

mijk := ei(ejek) ∈ R (6.4.1)

On definit le star-produit ∗ suivant sur A∗ (f, g ∈ C∞(A∗,C)):

f ∗ g (ξ) :=∞∑

r=0

νr

r!

∑1≤i1,...,ir≤n1≤j1,...,jr≤n1≤k1,...,kr≤n

mi1j1k1

· · ·mirjrkr

ξi1 · · · ξir

∂rf

∂ξj1 · · · ∂ξjr

(ξ)∂rg

∂ξk1 · · · ∂ξkr

(ξ)

(6.4.2)

Pour verifier l’associativite on calcule le symbole standard de ∗: soientx, y, z ∈ A ∼= A∗∗, alors

ex ∗ ey = ex+y+νxy

et par consequent

(ex ∗ ey) ∗ ez = ex+y+νxy ∗ ez = ex+y+z+ν(xy+xz+yz)+ν2xyz et

ex ∗ (ey ∗ ez) = ex ∗ ey+z+νyz = ex+y+z+ν(xy+xz+yz)+ν2xyz,

40

ce qui prouve l’associativite. On peut demontrer le

Theorem 6.4 Soit A une algebre associative reelle de dimension finie. Alorsil existe un star-produit ∗ defini par (6.4.2) sur la variete de Poisson A∗ (mu-nie de la structure (1.3.6) pour le crochet de Lie (x, y) 7→ [x, y] := xy − yx)qui converge (en ν) sur le sous-espace des polynomes sur A∗ ou la multipli-cation est isomorphe a l’algebre enveloppante complexifiee de (A, [ , ]).

7 Representations des star-produits I

7.1 Definition

Soit (M,P ) une variete de Poisson et C une variete differentiable. Onconsidere l’espace Diffop(C) de tous les operateurs differentiels sur l’espaceC∞(C,C). Ceci est une algebre associative munie de la multiplication usuelled’operateurs differentiels. Il en est de meme avec Diffop(C)[[λ]]. Soit ∗ unstar-produit sur M .

Definition 7.1 Une representation du star-produit ∗ dans C est un homo-morphisme d’algebres associatives ρ : C∞(M,C)[[λ]] → Diffop(C)[[λ]] surC[[λ]].

Les applications ρs et ρw de la section ?? (etendues aux series formelles)sont des exemples des representations des star-produits. On obtient un autreexemple en choisissant M = C et en definissant ρ(f) := Lf : g 7→ f ∗g comme la multiplication gauche. On va etudier d’autres proprietes desrepresentations en paragraphe 9.2.

7.2 Representations GNS

Une classe de representations particulieres s’obtient par un procede analoguea celui qu’on utilise pour les representations de Gel’fand, Naimark et Segal(GNS) des algebres stellaires et a ete etudiee par [17], [78]:

Supposons que le star-produit ∗ sur la variete de Poisson (M, P ) soitsymetrique, c.-a-d. f ∗ g = g ∗f quelles que soient f, g ∈ A := C∞(M,C)[[λ]]ou ( ) designe la conjugaison point-par-point. L’anneau C := C[[λ]] s’ecritvisiblement de facon C = R ⊕ iR ou R := R[[λ]]. On observe que R est unanneau commutatif unitaire ordonne: ceci veut dire que R se decompose entrois parties R = R+ ∪ {0} ∪ (−R+) ou R+ designe l’ensemble des elements

41

strictement positifs dont la definition est la suivante:

α =∞∑

r=0

λrαr

> 0si αo(α) > 0

< 0si αo(α) < 0

(7.2.1)

et on a R+ + R+ ⊂ R+ et R+R+ ⊂ R+. On a la conjugaison complexer1 + ir2 = r1 − ir2 dans C et on ecrit |c|2 pour cc. Soit maintenant A′ unideal bilatere de A stable par la conjugaison complexe (par exemple A′ =A ou A′ := C∞0 (M,C)[[λ]], l’espace des series formelles a coefficients dansl’espace des fonctions de classe C∞ a valeurs complexes a support compact),et ω : A′ → C une fonction C-lineaire. ω est dite reelle ssi ω(f) = ω(f) etpositive ssi

ω(f ∗ f) ≥ 0 quel que soit f ∈ A′ (7.2.2)

ou la relation ≥ est celle dans R ⊂ C. Grace a l’inegalite de Cauchy-Schwarzpour une forme lineaire reelle positive,

ω(f ∗ g) ω(f ∗ g) ≤ ω(f ∗ f) ω(g ∗ g), (7.2.3)

il s’ensuit que l’ideal de Gel’fand,

Iω := {f ∈ A′ | ω(f ∗ f) = 0} (7.2.4)

est un ideal gauche de A. L’espace quotient

Hω := A′/Iω (7.2.5)

(pour lequel on note f 7→ ψf la projection canonique) est un A-modulegauche de facon naturelle

ρω(f)ψg := ψf∗g. (7.2.6)

En outre Hω est muni d’un produit scalaire 〈 , 〉 a valeurs dans C defini par

〈ψf , ψg〉 := ω(f ∗ g) (7.2.7)

avec les proprietes de sequilinearite et positivite (〈ψf , ψf〉 > 0 quel que soitψf 6= 0), et la representation ρω satisfait

〈ρω(f)ψg, ψh〉 = 〈ψg, ρω(f)ψh〉. (7.2.8)

Alors Hω peut etre regarde comme un espace prehilbertien sur l’anneau C etl’algebre A se represente dans Hω. La construction precedente (qui s’appelle

42

la construction GNS dans le cadre des algebres stellaires) est independantede la nature de C = R⊕iR: les seules choses importantes sont les faits que Rest un anneau commutatif unitaire ordonne, que C = R⊕ iR avec i2 = −1 etque (A, ∗) est une algebre associative sur C munie d’un antihomomorphismeantilineaire involutif f 7→ f (c.-a-d.: ∀α, β ∈ C et ∀f, g ∈ A on a αf + βg =αf + βg et f ∗ g = g ∗ f).

Un exemple simple est donne par la variete symplectique (R2n,∑n

k=1 dqk∧dpk) (ou j : Q = Rn → M : (q1, . . . , qn) 7→ (q1, . . . , qn, 0, . . . , 0) designel’espace des configurations), A := (C∞(R2n,C)[[λ]], ∗w),

A′ := {∞∑

r=0

λrfr | fr ∈ C∞(R2n,C) et supp(fr) ∩Q est compact} (7.2.9)

et la fonction lineaire ω

ω : A′ → C[[λ]] : f 7→∫

Q

dnq f(j(q)

). (7.2.10)

On peut montrer que ω est positive et que Hω est isomorphe a l’espaceC∞0 (Rn,C) muni du produit scalaire L2 standard, c.-a-d.

〈φ1, φ2〉 =

∫

Q

dnq φ1(q)φ2(q)

et la representation ρω est egale a ρw (voir (3.2.8) dans le cas n = 1) cequi est une espece de representation de Schrodinger sur les fonctions d’ondedefinies sur l’espace des configurations (voir [17] pour plus de details).

Il y a beaucoup d’autres exemples de representations de physique quise formulent dans le cadre des representations GNS formelles precedentescomme la representation de Schrodinger pour un fibre cotangent T ∗Q d’unevariete differentiable arbitraire Q (voir le paragraphe prochain et [14] et[13] pour plus de details ou se trouve egalement le cas de la representationsemiclassique WKB) et la representation dans un fibre en droites complexesholomorphe pour des monopoles magnetiques (voir [15]). Dans ce domaine,surtout Stefan Waldmann a continue la recherche, voir par exemple [78].

7.3 Fibres cotangent

Soit Q une variete differentiable arbitraire de dimension n, τ ∗Q : T ∗Q → Qson fibre cotangent, i : Q → T ∗Q la section nulle et ∇Q une connectionsans torsion dans le fibre tangent de Q. La generalisation directe de la

43

representation standard ρs dans Diffop(R)[[λ]] (voir eqn (3.1.5) est la suivanteou f ∈ C∞(T ∗Q,C)[[λ]] et φ ∈ C∞(Q,C)[[λ]]

ρs(f)φ :=∞∑

r=0

(λ/i)r

r!

n∑i1,...,ir=1

i∗( ∂f

∂pi1 · · · ∂pir

)(∇Q)

(r)(∂

∂qi1,..., ∂

∂qir

)φ, (7.3.1)

ou on a utilise une carte (q, p) pour rendre l’expression moins encombranteet (∇Q)(r) designe la rme derivee covariante de φ, c.-a-d. quels que soient leschamps de vecteurs X1, . . . , Xr+1 ∈ Γ(TQ):

(∇Q)(1)X1

φ := LX1φ = X1φ

(∇Q)(r+1)(X1,...,Xr+1)

ψ := (∇Q)(1)X1

((∇Q)

(r)(X2,...,Xr+1)

φ)

−r+1∑

k=2

(∇Q)(r)

(X2,...,Xk−1,∇QX1

Xk,Xk+1,...,Xr+1)φ.

On peut montrer a l’aide de la construction de Fedosov qu’il existe un star-produit ∗s sur la variete symplectique (T ∗Q, ω0) (voir paragraphe 1.2.1) –quiest d’une certaine maniere defini par eqn (7.3.1), voir [14],[13]–, tel que ρs

est une representation de(C∞(T ∗Q,C), ∗s)[[λ]], ∗s

)dans Diffop(Q)[[λ]].

Il y a aussi une possibilite (nonunique) de definir un analogue au star-produit Weyl-Moyal ∗w (voir (3.2.10)) a l’aide d’une serie operateurs diffe-rentiels, N , due a N.Neumaier, qui generalise l’application N (eqn (3.2.9))du paragraphe 3.2:

Soit RQ le tenseur de courbure de ∇Q. On fixe un champ de densitespositives µ sur Q. Alors il y a une 1-forme unique α sur Q definie par ∇Qµ =:αµ, et l’on calcule dα = −traceRQ. On considere l’operateur differentiel

∆ :=n∑

k=1

∂2

∂qk∂pk

+n∑

j,k,l=1

(τ ∗Q)∗Γjklpj

∂2

∂pk∂pl

+n∑

k,l=1

(τ ∗Q)∗Γkkl

∂

∂pl

+n∑

k=1

αk∂

∂pk

(7.3.2)

(ou α =∑n

k=1 αkdqk et Γjkl := dqj(∇Q

∂

∂qk

∂∂qk designent les symboles de Christof-

fel de la connection ∇Q) qui ne depend pas de la carte choisie (en fait, lasomme des premiers trois termes constituent le Laplacien de la metrique semi-riemannienne sur T ∗Q obtenue par l’accouplement naturel entre les champsde vecteurs horizontaux (definis par la connection ∇Q) et les champs devecteurs verticaux) et l’on pose

N := eλ2i

∆. (7.3.3)

44

Alors le star-produit

f ∗w g := N−1((Nf) ∗s (Ng)

)(7.3.4)

est symetrique et est represente par

ρw(f) := ρs(Nf). (7.3.5)

Cette representation est GNS: on definie la fonction lineaire positive ωµ surl’ideal bilatere C∞

0 (T ∗Q,C)[[λ]] de C∞(T ∗Q,C)[[λ]]:

ωµ(f) :=

∫

Q

µ(i∗f) ∀f ∈ C∞0 (T ∗Q,C)[[λ]] . (7.3.6)

On a montre [14, 13] que ωµ est positive, que l’ideal de Gel’fand est donnepar l’image par N−1 du sous-espace de toutes les fonctions appartenant aC∞

0 (T ∗Q,C)[[λ]] qui s’annulent sur Q, et que l’espace prehilbertien Hωµ dela construction GNS est isometrique en tant que C∞(T ∗Q)[[λ]]-module aC∞

0 (Q)[[λ]] (muni du produit scalaire L2 moyennant µ) via ψf 7→ i∗(Nf)quelle que soit f ∈ C∞

0 (T ∗Q,C)[[λ]]. La representation ρw mentionnee ci-dessus coıncide avec la representation GNS.

8 Geometrie de Poisson II

Pour preparer la discussion de la deformation (ou quantification) des mor-phismes de Poisson, je rappelle quelques proprietes des varietes de Poisson etdes applications interessantes entre elles (voir [76], [61], [80], [11], [24], [37],[43], [50], [54], [60], [79]).

8.1 Applications de Poisson

Definition 8.1 Soient (M, P ) et (M ′, P ′) deux varietes de Poisson. Uneapplication Φ : M → M ′ de classe C∞ s’appelle application de Poisson ssi Pet P ′ sont Φ-liees, c.-a-d.

TmΦ⊗ TmΦ(Pm

)= P ′

Φ(m) quel que soit m ∈ M.

La proposition suivante est une consequence directe de la definition:

Proposition 8.1 Soit Φ : (M, P ) → (M ′, P ′) une application de Poissonentre deux varietes de Poisson.Alors l’application Φ∗ : C∞(M ′,C) → C∞(M,C) : g 7→ g ◦ Φ est un homo-morphisme d’algebres de Poisson, c.-a-d.:

Φ∗(g1g2) = (Φ∗g1)(Φ∗g2)

Φ∗{g1, g2}′ = {Φ∗g1, Φ∗g2}

45

quelles que soient g1, g2 ∈ C∞(M ′,C).

8.1.1 Applications moment

L’outil principal pour la description des symetries dans le cadre des varietesde Poisson et celui des application moment:Soit (g, [ , ]) une algebre de Lie reelle de dimension finie et g∗ son espacedual.

Definition 8.2 Soit (M, P ) une variete de Poisson. Une application J :M → g∗ s’appelle application moment (pour l’algebre de Lie g) ssi

{〈J, x〉, 〈J, y〉} = 〈J, [x, y]〉 quels que soient x, y ∈ g

Cette definition entraıne evidemment la proposition suivante:

Proposition 8.2 Toute application moment est une application de Poisson(M, P ) → (g∗, Pg).

La definition classique d’une application moment par J.-M. Souriau com-mence par une action gauche d’une groupe de Lie G sur M , G×M → M :(g, m) 7→ gm =: Φg(m) telle que 1. l’algebre de Lie de G soit egale a g,2. l’action preserve la structure de Poisson, c.-a.-d. toutes les Φg sont desapplication de Poisson, 3. il existe une application moment J : M → g∗

comme dans definition 8.2, 4. les champs hamiltoniens X〈J,x〉 coıncident avecles generateurs infinitesimaux xM(m) := d/dt

(exp(tx)m

)|t=0 et 5. J estG-equivariante: J(gm) = Ad∗(g)

(m

) ∀ g ∈ G. Ici, proprietes 4. et 5.impliquent propriete 3.

8.1.2 Systemes integrables

Definition 8.3 Soit (M,ω, H) un systeme hamiltonien sur une variete sym-plectique (M, ω) de dimension 2n. Il est dit completement integrable (dansle sens de Liouville) ss’il existe n fonctions de classe C∞ F1, . . . , Fn : M → Rtelles que

1. Les fonctions F1, . . . , Fn sont des integrales premieres, c.-a-d. {H, Fi} =0 quel que soit 1 ≤ i ≤ n,

2. les fonctions F1, . . . , Fn sont en involution, c.-a-d. {Fi, Fj} = 0 quelsque soient 1 ≤ i, j ≤ n,

3. F1, . . . , Fn sont independantes, c.-a-d. la mesure (par rapport a laforme de volume ω∧n) de l’ensemble singulier S := {m ∈ M | dF1(m)∧· · · ∧ dFn(m) = 0} s’annule, et

46

4. il existe une fonction h : Rn → R de classe C∞ telle que H(m) =h(F1(m), . . . , Fn(m)

)quel que soit m ∈ M .

Le nom ‘integrabilite’ provient du fait qu’il y a une procedure algebrique duea Liouville de trouver des coordonnees locales (Q1, . . . , Qn) autour de toutpoint regulier de l’application F := (F1, . . . , Fn) telles que les coordonnees(Q1, . . . , Qn, F1, . . . , Fn) forment une carte de Darboux: les solutions desequations d’Hamilton se simplifient drastiquement:

dQk

dt=

∂h

∂Fk

(F ) =: αk(F )

dFk

dt= 0

alors (Q(t), F (t)

)=

(Q(0) + tα(F (0)), F (0)

).

Les sous-varietes F−1(µ) pour des valeurs regulieres µ ∈ Rn sont invariantespar le flot de H, et au cas ou elles sont compactes et connexes elles sontdiffeomorphes au tore S1×· · ·×S1 (Theoreme de Liouville-Arnol’d, voir [4],p.271-285, ou [1], p.392-400).

On verifie rapidement l’integrabilite des systemes hamiltoniens (M, ω, H)importants suivants:

1. La particule libre dans Rn: (M, ω) = (R2n,∑n

k=1 dqk ∧ dpk) et

H(q, p) :=1

2

n∑

k1

p2k (8.1.1)

Fk(q, p) := pk ∀1 ≤ k ≤ n. (8.1.2)

2. L’oscillateur harmonique dans Rn: (M, ω) = (R2n,∑n

k=1 dqk ∧ dpk) et

H(q, p) :=1

2

n∑

k1

(p2

k + q2k

)(8.1.3)

Fk(q, p) :=1

2

(p2

k + q2k

) ∀1 ≤ k ≤ n. (8.1.4)

47

3. Le flot geodesique sur la sphere Sn (voir section 6.1): on commence par(M, ω) = (Rn+1 \ {0} × Rn+1,

∑n+1k=1 dqk ∧ dpk) et

H(q, p) :=1

2

n+1∑

l,l′=1

(q2l p

2l′ − qlplql′pl′

)(8.1.5)

F1(q, p) :=n+1∑

l=1

qlpl (8.1.6)

Fk(q, p) :=1

2

k∑

l,l′=1

(q2l p

2l′ − qlplql′pl′

) ∀2 ≤ k ≤ n + 1. (8.1.7)

Etant invariantes par le groupe G de dimension 2 (voir (6.1.6)) ces fonc-tions se restreignent bien sur la sous-variete symplectique T ∗Sn (donneepar F1(q, p) = 0 et

∑n+1l=1 q2

l = 1) ou F2, . . . , Fn+1 = H definissent unsysteme integrable qui represente le flot geodesique sur Sn dont lessolutions sont des grands cercles parametres avec leurs vitesses.

Il est clair que chaque systeme integrable est un cas particulier d’uneapplication moment

J : M → Rn∗ : m 7→ (F1(m), . . . , Fn(m)

)(8.1.8)

ou Rn est consideree comme une algebre de Lie abelienne (c.-a-d. ou tous lescrochets s’annulent).

8.2 Sous-varietes et applications coısotropes

Soit M une variete differentiable. Soit E un sous-espace vectoriel de l’espacetangent TmM au point m ∈ M . On note

Eann := {α ∈ TmM∗ | α(v) = 0 ∀v ∈ E} (8.2.1)

l’espace annihilateur de E.

Definition 8.4 Soient (M, P ) une variete de Poisson et C une variete quel-conque et Φ : C → M une application de classe C∞.

1. Φ s’appelle coısotrope ssi

PΦ(c)(α, β) = 0 quels que soient c ∈ C; α, β ∈ (TcΦ TcC)ann.

2. En particulier, si Φ est l’injection canonique d’une sous-variete fermeeC de M , alors C s’appelle sous-variete coısotrope quand Φ est coısotrope.

48

Il est immediat que l’application identique d’une variete de Poisson est uneapplication coısotrope.

Pour une variete symplectique (M, ω) et un sous-espace E d’un espacetangent TmM il y a la notion du sous-espace ω-orthogonal

Eω := {w ∈ TmM | ωm(v, w) = 0 ∀v ∈ E}, (8.2.2)

et l’on en deduit aisement la

Proposition 8.3 Soit (M, ω) une variete symplectique et C une sous-varietefermee de M . Alors C est coısotrope ssi

TcCω ⊂ TcC quel que soit c ∈ C.

Remarque: Si une sous-variete coısotrope C d’une variete symplectique (M,ω)est telle que

TcCω = TcC quel que soit c ∈ C. (8.2.3)

elle est appelee sous-variete lagrangienne. Les sous-varietes lagrangiennesjouent un role principal dans la theorie du developpement semiclassique desoperateurs differentiels, voir par exemple [6].

Proposition 8.4 Soient (M, P ) et (M ′, P ′) des varietes de Poisson, Φ :M → M ′ une application de Poisson et C ′ une sous-variete coısotrope de M ′

qui soit transverse a Φ, c.-a-d. TmΦ(TmM) + TΦ(m)C′ = TΦ(m)M

′ quel quesoit m ∈ M .Alors l’image reciproque C := Φ−1(C ′) est une sous-variete coısotrope de M .

Proof: Grace a la transversalite de Φ et C ′ il s’ensuit que C est une sous-varietede M qui a la meme codimension que C ′. Soit c ∈ C et α′, β′ ∈ TΦ(c)C

′ann. Alors,puisque TcΦ v ∈ TΦ(c)C

′ quel que soit v ∈ TcC il s’ensuit que α := α′ ◦ TcΦ etβ := β′◦TcΦ sont des elements de TcC

ann. Si α = 0 alors α′ s’annule sur TcΦ(TcM)et sur TΦ(c)C

′, donc α′ = 0 grace au fait que Φ et C ′ sont transverses. Alors leselements de TcC

ann sont tous de la forme α = α′ ◦ TcΦ. On calcule

Pc(α, β) = Pc(α′ ◦ TcΦ, α′ ◦ TcΦ) = (TcΦ⊗ TcΦ)(Pc)(α′, β′) = P ′Φ(c)(α

′, β′) = 0,

et C est coısotrope. 2

Proposition 8.5 Soient (M, P ) et (M ′, P ′) deux varietes de Poisson et Cune variete quelconque. Soit Ψ : C → (M, P ) une application coısotrope etΦ : (M, P ) → (M ′, P ′) une application de Poisson.Alors la composee Φ ◦Ψ : C → M ′ est une application coısotrope.En particulier, le cas C = M et Ψ = 1M montre que toute application dePoisson est une application coısotrope.

49

Proof: Soient c ∈ C et α′, β′ ∈ TΦ(Ψ(c))M′∗ telles que α′ ◦ Tc(Φ ◦ Ψ) = 0 et

β′ ◦ Tc(Φ ◦ Ψ) = 0. Alors α ◦ TcΨ :=(α′ ◦ TΨ(c)Φ

) ◦ TcΨ = 0 et β ◦ TcΨ :=(β′ ◦ TΨ(c)Φ

) ◦ TcΨ = 0. Puisque Ψ est coısotrope il s’ensuit:

0 = PΨ(c)(α, β) = PΨ(c)(α′ ◦ TΨ(c)Φ, β′ ◦ TΨ(c)Φ)

=(TΨ(c)Φ⊗ TΨ(c)Φ

)(PΨ(c))(α

′, β′) = P ′Φ(Ψ(c))(α

′, β′)

ce qui prouve que Φ ◦Ψ est coısotrope. 2

Soit (M,P ) une variete de Poisson. On a les exemples des sous-varietescoısotropes suivants:

1. La variete M elle-meme.

2. Si m0 ∈ M tel que Pm0 = 0, alors C := {m0} est coısotrope.

3. Soit (g, [ , ]) une algebre de Lie reelle de dimension finie et i : h → g

une sous-algebre. Alors la restriction i∗ : (g∗, Pg) → (h∗, Ph) est uneapplication de Poisson et une submersion surjective, {0} ⊂ h∗ est unesous-variete coisotrope de (h∗, Ph), alors

hann := {ξ ∈ g∗ | ξ(y) = 0 ∀y ∈ h} (8.2.4)

est une sous-variete coısotrope de (g∗, Pg) selon proposition 8.4.

4. Soit J : (M, P ) → (g∗, Pg) une application moment dont 0 ∈ g∗ est unevaleur reguliere. Alors C := J−1(0) est une sous-variete coısotrope de(M, P ) d’apres proposition 8.4.

5. Dans la variete de Poisson (M×M, P(1)−P(2)) (voir proposition 1.5) ladiagonale ∆(M) := {(m,m) | m ∈ M} est une sous-variete coısotrope.

6. Soit Φ : (M, P ) → (M ′, P ′) une application de Poisson. Alors songraphe

C := {(Φ(m),m) ∈ M ′ ×M | m ∈ M} (8.2.5)

est une sous-variete coısotrope de la variete de Poisson (M ′×M, P ′(1)−

P(2)): en fait, idM ′×Φ : (M ′×M, P ′(1)−P(2)) → (M ′×M ′, P ′

(1)−P ′(2)) :

(m′, m) 7→ (m′, Φ(m)) est une application de Poisson, et C = (idM ′ ×

Φ)−1(∆(M ′)

)(A.Weinstein [80]).

On note que toute structure de Poisson definit un homomorphisme de fibresvectoriels

P ] : T ∗M → TM : αp 7→ P ]p(αp) := Pp(αp, ) (8.2.6)

avec l’identification naturelle TpM∗∗ = TpM .

50

Proposition 8.6 Soit C une sous-variete coısotrope d’une variete de Pois-son (M,P ).Alors la distribution E := ∪c∈CP ]

c

(TcC

ann)

est lisse et involutive, c.-a-d. sideux champs de vecteurs X, Y sur C prennent leur valeurs dans E, alors ilen est de meme pour leur crochet [X, Y ].

On obtient la caracterisation algebrique des sous-varietes coısotropes suiv-ante:

Proposition 8.7 Soit (M, P ) une variete de Poisson et C une sous-varietefermee de M . Soit IC l’ideal annulateur de C, c.-a-d.

IC := {f ∈ C∞(M,C) | f(c) = 0 ∀c ∈ C}.Alors C est une sous-variete coısotrope si et seulement si IC est une sous-algebre de Poisson, c.-a-d.:

Si f, g ∈ IC alors {f, g} ∈ C.

Voir [76], p.99, Prop.7.6 pour une demonstration.

8.3 Reduction symplectique

Dans ce sous-paragraphe on ne traite que les varietes symplectiques (pourdes generalisations voir [61]):

Soit i : C → M une sous-variete coısotrope d’une variete symplectique(M, ω). Ici la distribution E de proposition 8.6 est egal au sous-fibre TCω :=∪c∈CTcC

ω (voir proposition 8.3. Le feuilletage F correspondant au fibreintegrable E s’obtient a l’aide du theoreme classique de Frobenius, voir [59],p. 28, Thm.3.25. Supposons que l’espace des feuilles Mred := C/F est munid’une structure differentiables compatible avec la topologie quotient telle quela projection canonique π : C → Mred soit une submersion surjective. Alorson a le theoreme classique suivant:

Theorem 8.1 Avec les hypotheses mentionnees ci-dessus, l’espace quotientMred est muni d’une structure symplectique canonique, ωred, definie par

i∗ω =: π∗ωred.

La variete symplectique (Mred, ωred) s’appelle la variete symplectique reduite.

Voir [1], p. 416, Thm. 5.3.23, pour une demonstration.Un cas particulier important s’obtient par une application moment J :

M → g∗ pour laquelle 0 est une valeur reguliere dont l’image reciproque

51

C := J−1(0) n’est pas vide. Dans ce cas-la, C est une sous-variete coısotrope,et la variete reduite (au cas ou elle existe) s’obtient en tant qu’espace quotientdu groupe de Lie G (a algebre de Lie g) agissant de facon libre et propre sur C.Cette construction importante et extremement utile est appelee la reductionde Marsden-Weinstein [62]. Par exemple l’espace projectif complexe s’obtienten tant que variete symplectique reduite de M = R2n+2, ω =

∑n+1k=1 dqk ∧ dpk

a l’aide de l’application moment J(q, p) :=Pn+1

k=1 (q2k+p2

k)

2− 1

2pour l’action du

groupe U(1) sur Cn+1 = R2n+2.

9 Quantification des applications de Poisson

et des plongements coısotropes?

Dans ce paragraphe je voudrais bien discuter quelques questions –a ma con-naissance ouvertes– et quelques-uns de mes resultats au sujet de la quantifica-tion des applications de Poisson et des sous-varietes coısotropes des varietesde Poisson.

9.1 Homomorphismes de star-produits

Definition 9.1 Soient (M, P ) et (M ′, P ′) deux varietes de Poisson muniesdes star-produits ∗ et ∗′, respectivement.Une application C[[λ]]-lineaire Φ : C∞(M,C)[[λ]] → C∞(M ′,C)[[λ]] est ap-pelee homomorphisme de star-produits ssi Φ est un homorphisme d’algebresassociatives unitaires sur C[[λ]]:

Φ(F ∗G) =(Φ(F )

) ∗′ (Φ(G))

quels que soient F, G ∈ C∞(M,C)[[λ]].

Le lien avec les applications de Poisson est contenu dans le lemme suivant:

Lemma 9.1 Soit Φ =∑∞

r=0 λrΦr : C∞(M,C)[[λ]] → C∞(M ′,C)[[λ]] un ho-momorphisme de star-produits.Alors il existe une application de Poisson φ : (M ′, P ′) → (M, P ) telle queΦ0(f) = φ∗f := f ◦ φ quelle que soit f ∈ C∞(M,C).

Proof: Soient f, g ∈ C∞(M,C). La propriete d’homomorphisme de Φ s’ecrit al’ordre 0 de λ:

Φ0(fg) =(Φ0(f)

)(Φ0(g)

Alors Φ0 est un homomorphisme d’algebres commutatives associatives unitairesC∞(M,C) → C∞(M ′,C). D’apres l’exercice de Milnor (voir [59], p. 301, Cor.

52

35.9) il existe une application de classe C∞ φ : M ′ → M telle que Φ0(f) = φ∗f .Ensuite, le commutateur de la propriete d’homomorphismes a l’ordre 1 de λ s’ecrit

Φ0{f, g} = {Φ0(f), Φ0(g)}d’ou le fait que φ est une application de Poisson. 2

La question reciproque de savoir quand une application de Poisson donneese deforme dans un homomorphisme de star-produits est sans doute interes-sante:

Problem 9.1 Quelles sont les conditions sur une application de Poissonφ : (M ′, P ′) → (M,P ) pour qu’il existent des star-produits ∗′ et ∗ sur lesvarietes de Poisson (M ′, P ′) et (M, P ), respectivement, et des applicationslineaires Φ1, Φ2, . . . : C∞(M,C) → C∞(M ′,C) tels que

Φ := φ∗ +∞∑

r=1

λrΦr

soit un homomorphisme de star-produits?

9.1.1 Applications moment quantiques et systemes integrablesquantiques

Un cas particulier tres important est donne par les applications momentsJ : M → g∗ (voir paragraphe 8.1.1). Puisque J est une application dePoisson on peut specifier probleme 9.1 de facon suivante:

Problem 9.2 Quelles sont les conditions sur une application moment J :M → g∗ pour qu’il existent un star-produit ∗ sur la variete de Poisson (M, P )et des applications lineaires Φ1, Φ2, . . . : C∞(g∗,C) → C∞(M,C) tels que

Φ := J∗ +∞∑

r=1

λrΦr

soit un homomorphisme de star-produits si la variete de Poisson (g∗, Pg) (voireqn (1.3.6)) est munie du star-produit BCH (theoreme 6.3)?

Si l’on definit les applications Jr : M → g∗, r ∈ N par (ξ ∈ g):

J0 := J et 〈Jr, ξ〉 := Φr(ξ) (9.1.1)

ou ξ : g∗ → C : α 7→ 〈α, ξ〉, il resulte de la propriete d’homomorphismes de

Φ et du fait que ξ ∗ η − η ∗ ξ = iλ[ξ, η] (voir thm 6.3):

〈J, ξ〉 ∗ 〈J, η〉 − 〈J, η〉 ∗ 〈J, ξ〉 = iλ〈J, [ξ, η]〉. (9.1.2)

53

Une serie formelle de fonctions J ∈ g∗⊗C∞(M,C)[[λ]] satisfaisant eqn (9.1.2)s’appelle une application moment quantique d’apres Xu [82]. Si J = J = J0 lestar-produit ∗ s’appelle g-covariant d’apres Arnal, Cortet, Molin et Pinczon[2]. Plus particulierement, un star-produit ∗ s’appelle fortement g-invariantd’apres ces auteurs si

〈J, ξ〉 ∗ f − f ∗ 〈J, ξ〉 = iλ{〈J, ξ〉, f} (9.1.3)

quelle que soit la fonction f ∈ C∞(M,C)[[λ]. On a le critere suffisant suivantpour l’existence de ces star-produits:

Theorem 9.1 (Fedosov,1996) Soit (M,ω) une variete symplectique, J :M → g∗ une application moment et ∇ une connection dans le fibre tangenttelle que

0 =(LX〈J,ξ〉∇

)X

Y := [X〈J,ξ〉,∇XY ]−∇[X〈J,ξ〉,X]Y −∇X [X〈J,ξ〉, Y ].

quel que soit ξ dans l’algebre de Lie g. Alors il existe un star-produit forte-ment g-invariant ∗.Voir [42] pour la demonstration. Par exemple, si les flots des champs devecteurs X〈J,ξ〉 definissent l’action d’un groupe de Lie compacte ou plusgeneralement une action propre d’un groupe de Lie, il resulte d’un theoremeclassique de R.Palais que ces champs de vecteurs preservent une metrique rie-mannienne sur M , alors sa connection Levi-Civita, et le theoreme de Fedosovest applicable.

Puisque les systemes hamiltoniens integrables constituent une sous-classedes applications moment (voir paragraphe 8.1.2 et eqn (8.1.8)) on peut ap-peler un systeme hamiltonien (M,ω, H) un systeme integrable quantiquess’il ya une application F =

∑∞r=0 λrFr ∈ Rn ⊗ C∞(M,C)[[λ]] telle que(

M, ω, H,F0 =: F =: (F1, . . . , Fn))

soit un systeme integrable classique et

Fk ∗ Fl − Fl ∗ Fk = 0 (9.1.4)

quels que soient 1 ≤ k, l ≤ n. La plupart des systemes hamiltoniens integrablesconnus sont aussi integrables quantiques, par exemple tous les exemples men-tionnes en paragraphe 8.1.2 si l’on choisit ∗ = ∗w et F = F ou les exemplesde [12].

9.2 Representations de star-produits II

On rappelle la definition d’une representation de star-produit du paragraphe7 dans Definition 7.1: ceci etait un homomorphisme d’algebres associatives

54

entre l’algebre (C∞(M,C)[[λ]], ∗) ((M,P ) etant une variete de Poisson munied’un star-produit ∗) et l’algebre d’operateurs differentiels sur une varietedifferentiable C.

Le lien entre les representations de star-produits et les sous-varietes coıso-tropes est contenu dans la proposition suivante:

Proposition 9.1 Soit (M,P ) une variete de Poisson munie d’un star-produit∗, C une variete differentiable et

ρ =∞∑

r=0

λrρr : C∞(M,C)[[λ]] → Diffop(C)[[λ]]

une representation de star-produits.Alors il existe une application de classe C∞, i : C → M telle que ρ0(f)(ψ) =(i∗f)ψ := (f ◦ i)ψ quelles que soient f ∈ C∞(M,C) et ψ ∈ C∞(C,C).Ensuite, au cas ou i est un plongement sur une sous-variete fermee i(C) deM , alors i(C) est une sous-variete coısotrope.

Proof: La propriete de representation s’ecrit a l’ordre 0: ρ0(f)ρ0(g) = ρ0(fg).Donc ρ0 est un homomorphisme de l’algebre associative commutative C∞(M,C)dans l’algebre associative Diffop(C). Puisque ρ0(1) = l’application identique, alorsρ0 envoie des fonctions qui ne s’annulent nulle part sur des operateurs differentielsinversibles. Si l’on regarde le symbole standard (voir paragraphe 4.5) d’un operateurdifferentiel inversible dans des coordonnees locales, on voit qu’il ne contient au-cune puissance strictement positive d’une derivee partielle. Alors, un tel operateurdifferentiel prend la forme ψ 7→ χψ ou χ ∈ C∞(C,C). Soit f ∈ C∞(M,R). Alors lafonction 1+f2 est un element inversible dans l’algebre C∞(M,C). Par consequent,il existe une fonction χ ∈ C∞(C,R) telle que

ψ + ρ0(f)2(ψ) = ρ0(1 + f2)(ψ) = χψ

quelle que soit ψ ∈ C∞(C,C). Il s’ensuit qu’il existe une fonction χ′ ∈ C∞(C,C)telle que ρ(f)(ψ) = χ′ψ. Pour une fonction a valeurs complexes on arrive a la memeconclusion tout en separant en partie reelle et partie imaginaire. Alors il existe unhomomorphisme d’algebres associatives commutatives ρ0 : C∞(M,C) → C∞(C,C)tel que ρ0(f)ψ = ρ0(f) ψ. D’apres l’exercice de Milnor (voir [59], p. 301, Corollary35.10) il existe une application de classe C∞ i : C → M telle que ρ0(f) = f ◦ i.Soit i maintenant un plongement. Tout en identifiant C et son image i(C) nousconsiderons deux fonctions f, g ∈ C∞(M, C) qui s’annulent sur C. Le commutateurde l’identite de representation a l’ordre 1 s’ecrit

[ρ1(f), ρ0(g)]− [ρ1(g), ρ0(f)] = ρ0({f, g})Puisque ρ0(f) = i∗f = 0 = i∗g = ρ0(g) il s’ensuit que {f, g} ◦ i = 0, alors C estcoısotrope selon proposition 8.7. 2

55

Encore une fois, la question reciproque de savoir quand l’injection canon-ique i d’une sous-variete coısotrope fermee C d’une variete de Poisson donneese deforme dans une representation de star-produits me semble aussi interes-sante:

Problem 9.3 Quelles sont les conditions sur l’injection canonique i : C →(M, P ) d’une sous-variete coısotrope fermee C d’une variete de Poisson(M, P ) pour qu’il existent un star-produit ∗ sur la variete de Poisson (M, P )et des applications lineaires ρ1, ρ2, . . . : C∞(M,C) → Diffop(C) tels que

ρ := i∗ +∞∑

r=1

λrρr

soit une representation de star-produits?

Ici j’ai utilise la notation simplifiee ρ0 = i∗ pour ρ0(f)(ψ) = (i∗f)ψ.

9.2.1 Lien entre homomorphismes et representations de star-pro-duits

Soient (M, P ) et (M ′, P ′) deux varietes de Poisson munies des star-produits∗ =

∑∞r=0 λrCr et ∗′ =

∑∞r=0 λrC ′

r, respectivement. Pour deux entiers posi-tifs s, t on definit dans une carte (U × U ′, x1, . . . , xn, y1, . . . , yn′) l’operateurbidifferentiel suivant

(Cs ⊗ C ′

t

)(F, G) :=

Ns∑

a,b=0

Nt∑

c,d=0

∑1≤i1,...,ia≤n1≤j1,...,jb≤n

∑

1≤i′1,...,i′c≤n′1≤j′1,...,j′d≤n′

C(a,b),i1···ia,j1···jbs C ′(c,d),i′1···i′c,j′1···j′d

t

∂a+cF

∂xi1 · · · ∂xia∂yi′1 · · · ∂yi′c

∂b+dG

∂xj1 · · · ∂xjb∂yj′1 · · · ∂yj′d.

quelles que soient F, G ∈ C∞(M ×M ′,C). La definition ne depend pas descartes choisies. On pose

∗ ⊗ ∗′ :=∞∑

r=0

λr∑

s+t=r

Cs ⊗ C ′t

ce qui definit evidemment un star-produit sur (M × M ′, P(1) + P ′(2)) (voir

Proposition 1.5). En outre, on definit la multiplication ∗opp

f ∗opp g := g ∗ f quelles que soient f, g ∈ C∞(M,C)[[λ]]

56

qui est evidemment un star-produit pour la variete de Poisson (M,−P ). Lelien entre la quantification des application de Poisson et la quantification dessous-varietes coısotropes se trouve dans la suivante

Proposition 9.2 (M.B. 2000) Soit φ : (M ′, P ′) → (M,P ) une applica-tion de Poisson entre deux varietes de Poisson. Soit i : C := M ′ → M ×M ′

le plongement canonique dans le graphe de φ, {(φ(p′), p′) := i(p′) ∈ M ×M ′ | p′ ∈ M ′} (ce qui est une sous-variete coısotrope de (M ×M ′, P(1)−P ′

(2)

d’apres le theoreme de Weinstein, voir exemple 6 dans paragraphe 8.2).On suppose qu’il y ait une representation de star-produits ρ de C∞(M ×M ′,C)[[λ]] munie du star-produit ∗⊗∗′opp dans Diffop(C)[[λ]] = Diffop(M ′)[[λ]]telle que ρ0 = i∗. Soit r : C∞(M ′,C)[[λ]] → C∞(M ′,C)[[λ]] l’applicationr(g) := ρ(1 ⊗ g)(1) et l : C∞(M,C)[[λ]] → C∞(M ′,C)[[λ]] l’applicationl(f) := ρ(f ⊗ 1)(1).Alors r est inversible et

Φ := r−1 ◦ l : C∞(M,C)[[λ]] → C∞(M ′,C)[[λ]]

est un homomorphism de star-produits tel que Φ0 = φ∗.

Proof: Soient f, f1, f2 ∈ C∞(M,C) et g, g1, g2 ∈ C∞(M ′,C). On a r0(g) =ρ0(1⊗g)(1) = i∗(1⊗g) = g, alors r0 est l’application identique ce qui entraıne quer est inversible. Ensuite, l0(f) = ρ0(f ⊗ 1)(1) = i∗(f ⊗ 1) = φ∗f , par consequentΦ0 = (r−1 ◦ l)0 = l0 = φ∗. On a

ρ(1⊗ g1)r(g2) = ρ(1⊗ g1)ρ(1⊗ g2)(1) = ρ(1⊗ (g1 ∗′opp g2)

)(1)

= r(g1 ∗′opp g2) = r(g2 ∗′ g1).

d’oug2 ∗′ g1 =

(r−1 ◦ ρ(1⊗ g1) ◦ r

)(g2)

En outre,(r−1 ◦ ρ(f ⊗ 1) ◦ r

)(g) = r−1

(ρ(f ⊗ 1)ρ(1⊗ g)(1)

)= r−1

(ρ(f ⊗ g)(1)

)

= r−1(ρ(1⊗ g)ρ(f ⊗ 1)(1)

)=

(r−1 ◦ ρ(1⊗ g) ◦ r

)(r−1(l(f))

)

= Φ(f) ∗′ gd’apres l’equation precedente. Par consequent

Φ(f1 ∗ f2) ∗′ g =(r−1 ◦ ρ((f1 ∗ f2)⊗ 1) ◦ r

)(g)

=(r−1 ◦ ρ(f1 ⊗ 1) ◦ r

) ((r−1 ◦ ρ(f2 ⊗ 1) ◦ r

)(g)

)

= Φ(f1) ∗′(Φ(f2) ∗′ g

)=

(Φ(f1) ∗′ Φ(f2)

) ∗′ g

ce qui montre que Φ est un homomorphisme de star-produits. 2

57

9.2.2 Representation de star-produits quand l’espace reduit existe

Voici un resultat positif simple:

Theorem 9.2 (M.B. 2001) Soit (M, ω) une variete symplectique munied’un star-produit ∗. On note [∗] sa classe de Deligne. Soit i : C → Mune sous-variete coısotrope fermee de M telle que la variete symplectiquereduite π : C → (Mred, ωred) (voir theoreme 8.1 en paragraphe 8.3) existe.Supposons en outre qu’il existe une serie formelle β a coefficients dans ledeuxieme groupe de cohomologie de de Rham de Mred telle que

i∗[∗] = π∗β.

Alors il existe une representation de star-produits ρ : (C∞(M,C)[[λ]], ∗) →Diffop(C)[[λ]]. En outre, il est toujours possible de choisir ρ, un star-produit∗red sur (Mred, ωred) de classe de Deligne [∗red] = β et une anti-representationρred : C∞(Mred,C)[[λ]], ∗red) → Diffop(C)[[λ]] ( ρred(g1∗redg2) = ρred(g2)ρred(g1)quelles que soient g1, g2 ∈ C∞(Mred,C)[[λ]]) de telle facon que

ρ(f)ρred(g) = ρred(g)ρ(f)

quelles que soient f ∈ C∞(M,C)[[λ]] et g ∈ C∞(Mred,C)[[λ]].

Proof: On considere la variete symplectique (M × Mred, ω(1) − ωred(2)). Graceau theoreme de classification 5.5 des star-produits symplectiques il existe un star-produit ∗red sur Mred tel que sa classe de Deligne [∗red] soit egale a β. On considerele star-produit ∗ := ∗ ⊗ ∗opp

red , et sa classe de Deligne vaut

[∗] = pr∗1[∗]− pr∗2[∗red]

ou pr1 : M × Mred → M et pr2 : M × Mred → Mred designent les projectionscanoniques. Grace a l’equation i∗ω = π∗ωred on voit que

j : C → M ×Mred : c 7→ (i(c), π(c)

)

est un plongement sur une variete lagrangienne L := j(C) (voir eqn 8.2.3) deM×Mred. D’apres un theoreme de Weinstein (voir [79] ou [1], p.411, thm 5.3.18) ilexiste un voisinage ouvert U ⊃ L dans M×Mred un voisinage ouvert L ⊂ V ⊂ T ∗Lde la section nulle du fibre cotangent de L et un diffeomorphisme symplectiqueφ : U → V dont la restriction a L donne l’identification usuelle de L avec la sectionnulle L → T ∗L. On peut choisir U de telle facon que V ∩ TlL

∗ est contractile quelque soit l ∈ L. Soit τ : U → C la submersion surjective induite par la projectiondu fibre τ∗L : T ∗L → L moyennant φ (c.-a-d.: τ∗L ◦ φ =: j ◦ τ) et soient iL : L → Uet iU : U → M les injections canoniques. U est une sous-variete ouverte, doncsymplectique de M ×Mred, et le star-produit ∗ se restreint a U , ∗|U . L’application

58

(j ◦ τ)∗ : Γ(ΛT ∗L) → Γ(ΛT ∗U) induit un isomorphisme des cohomologies de deRham (dont l’application reciproque est induite par i∗L : Γ(ΛT ∗U) → Γ(ΛT ∗L))puisque j ◦ τ est une retraction par deformation de U sur L. Par consequent, laclasse de Deligne de ∗|U vaut

[∗|U ] = (j ◦ τ)∗i∗L[∗|U ] = (j ◦ τ)∗i∗Li∗U [∗] = τ∗j∗i∗Lpr∗1[∗]− τ∗j∗i∗Lpr∗2[∗red]= τ∗

(i∗[∗]− π∗[∗red]

)= 0

puisque iU ◦ iL = iL, pr1 ◦ iL ◦ j = pr1 ◦ j = i et pr2 ◦ iL ◦ j = pr2 ◦ j =π. Dans [15] on a montre qu’un star-produit sur T ∗L ∼= T ∗C dont la classe deDeligne s’annule est toujours equivalent au star-produit ∗s (voir eqn 7.3.1). Parconsequent, en utilisant le symplectomorphisme φ on peut montrer qu’il existe uneserie d’operateur differentiels S = id +

∑∞r=1 λrSr sur U telle que

S(F ∗|UG) = (SF ) ∗s (SG) quelles que soient F, G ∈ C∞(U,C)[[λ]],

et il suit directement que la representation ρs : C∞(U,C)[[λ]] → Diffop(C)[[λ]]relative a une connection sans torsion ∇ sur C ∼= L (voir eqn 7.3.1) definit unerepresentation ρ pour l’algebre

(C∞(M ×Mred,C)[[λ]], ∗)) par

ρ(F ) := ρs

(S(F |U )

)

quelle que soit F ∈ C∞(M × Mred,C)[[λ]]. Evidemment, ρ0 = j∗. En partic-ulier, quand on restreint la representation ρ a la sous-algebre

(C∞(M,C)[[λ]], ∗)

de(C∞(M × Mred,C)[[λ]], ∗)) (c.-a-d.: ρ(f) := ρ(pr∗1f) = ρ(f ⊗ 1)) on obtient

la representation ρ souhaitee. En plus, ρ0 = ρ0 ◦ pr∗1 = j∗pr1∗ = i∗. Quand onrestreint la representation ρ a la sous-algebre

(C∞(Mred,C)[[λ]], ∗oppred

)on obtient

l’anti-representation souhaitee par un raisonnement analogue. Puisque les deuxsous-algebres commutent, il en est de meme pour ses representations. 2

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