An algebra of distributions related to a star product with separation of variables
aa r X i v : . [ m a t h . QA ] A ug AN ALGEBRA OF DISTRIBUTIONS RELATED TO ASTAR PRODUCT WITH SEPARATION OF VARIABLES
ALEXANDER KARABEGOV
Abstract.
Given a star product with separation of variables ‹ on a pseudo-K¨ahler manifold M and a point x P M , we constructan associative algebra of formal distributions supported at x . Weuse this algebra to express the formal oscillatory exponents of afamily of formal oscillatory integrals related to the star product ‹ . Introduction
Berezin and Berezin-Toeplitz quantizations on a K¨ahler manifold M which depend on a certain small numerical parameter h produce defor-mation quantizations with separation of variables on M of the anti-Wick and Wick type, respectively, via an asymptotic procedure as h Ñ h is replaced withthe formal parameter ν . Deformation quantizations with separationof variables exist on arbitrary pseudo-K¨ahler manifolds and admit abijective parametrization by ν -formal pseudo-K¨ahler forms.Both Berezin and Berezin-Toeplitz quantizations are based upon anintegral operator, the Berezin transform, which maps contravariantsymbols to the corresponding covariant symbols. The h -dependentBerezin transform admits an asymptotic expansion as h Ñ ν -formal differential operator on M , the formal Berezin trans-form. Any deformation quantization with separation of variables hasthe corresponding formal Berezin transform from which it can be com-pletely recovered. A formal Berezin transform can be expressed interms of what we call a formal oscillatory integral.It was shown in [14] and [13] that there exists a formal algebraiccounterpart of an oscillatory integral with a complex phase function ona manifold M . It is called a formal oscillatory integral (FOI). A FOI isgiven by a fixed point x P M and a formal oscillatory integral kernel,(1) exp p ϕ q ¨ ρ, Mathematics Subject Classification.
Key words and phrases. deformation quantization, formal oscillatory integral. where ϕ “ ν ´ ϕ ´ ` ϕ ` . . . is a ν -formal complex phase function on M such that x is a nondegenerate critical point of ϕ ´ with zero criticalvalue, ϕ ´ p x q “
0, and ρ “ ρ ` νρ ` . . . is a ν -formal complex volumeform on M such that ρ does not vanish at x . We call p ϕ, ρ q a phase-volume form pair at x . A FOI associated with the formal oscillatorykernel (1) is a ν -formal distribution Λ supported at x which actuallydepends only on the jet of (1) at x . A FOI is described by algebraicaxioms in terms of its oscillatory kernel. Heuristically, the ν -formaldistribution Λ gives an interpretation of the formal expressionΛ p f q “ ν ´ n ż M e ϕ f ρ, where n “ dim M and f is an amplitude supported near x .Let ‹ be a star product of the anti-Wick type on a pseudo-K¨ahlermanifold M , I be its formal Berezin transform, µ be its formal tracedensity, and x be a point in M . It was shown in [14] and [13] that the ν -formal distribution(2) K p l q p f b . . . b f l q : “ p If ‹ . . . ‹ If l qp x q on M l is a FOI at the diagonal point p x q l : “ p x , . . . , x q P M l . Itsformal oscillatory kernel is(3) exp ` F p l q ˘ ¨ µ b l , where the jet of F p l q at p x q l is expressed via what we call a cyclicformal p l ` q -point Calabi function of the star product ‹ (see detailsin the main body of the paper).In this paper we describe an associative algebra of ν -formal distribu-tions supported at x . For each l ě
1, the jet of the oscillatory exponentexp F p l q at p x q l is naturally expressed in terms of this algebra.Many constructions in this paper depend on jets of functions at agiven point but are stated in terms of functions which represent thesejets. These representatives exist by Borel’s theorem.2. Formal oscillatory integrals
Formal oscillatory integrals were introduced in [14] and developedfurther in [13]. Given a vector space V , we denote by V pp ν qq the spaceof ν -formal vectors(4) v “ ν r v r ` ν r ` v r ` ` . . . , All jets of functions considered in this paper are jets of infinite order given bythe full Taylor series.
HE ALGEBRA OF DISTRIBUTIONS 3 where r P Z and v k P V for all k ě r . The subspace V rr ν ss Ă V pp ν qq consists of the vectors (4) with r “ M be a manifold, x be a fixed point in M , and p ϕ, ρ q be a phase-volume form pair at x . Two pairs, p ϕ, ρ q and p ˆ ϕ, ˆ ρ q , at x are called equivalent if there exists a formal function u “ u ` νu ` . . . on a neighborhood of x such thatˆ ϕ “ ϕ ` u and ˆ ρ “ e ´ u ρ. Hence, it is natural to write the equivalence class of a pair p ϕ, ρ q as (1).Given a pair p ϕ, ρ q and a ν -formal volume form ˆ ρ “ ˆ ρ ` ν ˆ ρ ` . . . such that ˆ ρ does not vanish at x , there exists a formal phase functionˆ ϕ such that the pairs p ϕ, ρ q and p ˆ ϕ, ˆ ρ q are equivalent. Definition 2.1.
Given a pair p ϕ, ρ q on a manifold M at x P M , aformal distribution Λ “ Λ ` ν Λ ` . . . on M supported at x is calleda formal oscillatory integral (FOI) associated with the pair p ϕ, ρ q if Λ ‰ and (5) Λ p vf ` p vϕ ` div ρ v q f q “ for any vector field v and any function f on M . Here div ρ v “ L v ρ { ρ is the divergence of the vector field v with respectto ρ and L v is the Lie derivative with respect to v . As shown in [13],Λ “ αδ x , where α is a nonzero complex constant and δ x is the Dirac distributionat x , δ x p f q “ f p x q . For any pair p ϕ, ρ q there exists an associatedFOI which is determined up to a formal multiplicative constant c p ν q “ c ` νc ` . . . , where c ‰
0. In particular, there is a unique such FOI Λfor which Λ p q “
1. If a FOI is associated with a pair p ϕ, ρ q , then it isassociated with any equivalent pair. If Λ is a FOI at x associated witha pair p ϕ, ρ q and t x i u are local coordinates on a coordinate chart U containing x , then the pair p ϕ, ρ q is equivalent to some pair p ψ, dx q on U , where dx “ dx ^ . . . ^ dx n . In terms of the pair p ψ, dx q , condition(5) can be stated as follows,(6) Λ ˆ B f B x i ` B ψ B x i f ˙ “ i and any function f , because div dx pB{B x i q “ x associated with a pair p ϕ, ρ q depends only on the jets of ϕ and ρ at x . It was shown in [13]that if a FOI Λ at x is associated with pairs p ϕ, ρ q and p ˆ ϕ, ρ q with thesame volume form ρ , then the jet of ˆ ϕ ´ ϕ at x is a ν -formal constant. ALEXANDER KARABEGOV
This result is based on the following important statement. Given aFOI Λ at x , consider a pairing on C p M qrr ν ss given by the formula(7) p f, g q Λ : “ Λ p f ¨ g q . This pairing depends only on the jets of f and g at x and therefore itinduces a pairing on the space F of ν -formal jets at x . The inducedpairing will be denoted by the same notation p¨ , ¨q Λ . Lemma 2.1.
For any FOI Λ at x the pairing (7) on F is nondegen-erate. We will give a shorter and more conceptual proof of this lemma thanthe one given in [13].
Proof.
Let Λ be a FOI at x . Fix a coordinate chart U containing x with local coordinates t x i u . The FOI Λ is associated with some pair p ψ, dx q on U . For any functions f, g , we have from (6) that ˆ B f B x i , g ˙ Λ ` ˆ f, B g B x i ` B ψ B x i g ˙ Λ “ Λ ˆ B f B x i g ` f B g B x i ` B ψ B x i f g ˙ “ Λ ˆ Bp f g qB x i ` B ψ B x i f g ˙ “ . It means that the transpose of the operator
B{B x i with respect to thepairing (7) is ˆ BB x i ˙ : “ ´ BB x i ´ B ψ B x i . The transpose of the multiplication operator by a function f with re-spect to the pairing (7) is the same operator, f : “ f . We see thatany formal differential operator of finite order A on U has a trans-pose A : with respect to this pairing. Suppose that a formal function f P C p M qrr ν ss lies in the kernel of the pairing (7), that is, p f, g q Λ “ g . For any differential operator A on U we have(8) Λ p Af q “ Λ p Af ¨ q “ p Af, q Λ “ p f, A : q Λ “ . Assume that the jet of f “ f ` νf ` . . . at x is nonzero. Let r bethe least nonnegative integer such that the jet of f r at x is nonzero.Since Λ “ αδ x , where α is a nonzero constant, we see from (8) that p Af r qp x q “ A which does not depend on ν . It contra-dicts the assumption that the jet of f r at x is nonzero. Therefore, thepairing (7) induced on F is nondegenerate. (cid:3) HE ALGEBRA OF DISTRIBUTIONS 5
Formal oscillatory integrals should naturally appear in the frameworkof deformation quantization because many star products are obtainedfrom asymptotic expansions of oscillatory integrals. In this paper weare concerned with the family (2) of FOIs related to a star productwith separation of variables.3.
Deformation quantization
Let M be a Poisson manifold equipped with a Poisson bracket t¨ , ¨u .A formal deformation quantization on M is given by a ν -linear asso-ciative product on the space C p M qrr ν ss of formal functions,(9) f ‹ g “ f g ` ÿ r “ ν r C r p f, g q , where C r are bidifferential operators on M and C p f, g q ´ C p g, f q “ i t f, g u . The product ‹ is called a star product. It is assumed that the unitconstant is the identity for a star product, f ‹ “ ‹ f “ f for any f .The product (9) naturally extends to the space C p M qpp ν qq .Two star products ‹ and ˜ ‹ on a Poisson manifold p M, t¨ , ¨uq are calledequivalent if there exists a formal differential operator T “ ` νT ` . . . on M such that f ˜ ‹ g “ T ´ p T f ‹ T g q . The operator T is called an equivalence operator between the star prod-ucts ‹ and ˜ ‹ .If a star product ‹ on a manifold M is fixed, we denote by L f and R f the left and the right star multiplication operators by a function f ,respectively, so that L f g “ f ‹ g “ R g f . It follows from the associativityof the star product that r L f , R g s “ f, g .Since a star product ‹ on M is given by bidifferential operators, itcan be restricted to any open subset of M . Moreover, it induces aproduct on the space of formal jets at a given point. We will retain thesame notation ‹ for these induced products.If M is a symplectic manifold, then for each star product ‹ on M there exists a ν -formal trace density µ globally defined on M such that ż M f ‹ g µ “ ż M g ‹ f µ if f or g is compactly supported (see [17]).The concept of deformation quantization was introduced in [1]. Kont-sevich showed in [15] that star products exist on arbitrary Poisson man-ifolds and gave an explicit parametrization of their equivalence classes. ALEXANDER KARABEGOV
On symplectic manifolds Fedosov constructed star products in eachequivalence class in [7] and [8].A star product (9) is called natural in [9] if, for every r , the bidifferen-tial operator C r is of order not greater than r in each argument. Manyimportant star products are natural, e.g., the Fedosov’s star products(see [16]).We call a formal differential operator N “ N ` νN ` . . . natural ifthe order of the differential operator N r is not greater than r for r ě ‹ on M is natural if and only if the operators L f and R f are natural for every f P C p M qrr ν ss . We denote by N the space ofnatural operators on M . It is an associative algebra. It is also a Liealgebra with the operation A, B ÞÑ ν ´ r A, B s . Alternatively, ν ´ N isa Lie algebra with respect to the usual commutator A, B
ÞÑ r
A, B s .Denote by E the group of formal differential operators on M of theform exp ` ν ´ N ˘ , where N “ ν N ` ν N ` . . . P N . Observe thatexp ` ν ´ N ˘ “ ` νN p mod ν q . We call the operators from E the operators of exponential type . Lemma 3.1. If S P E and A P N , then SAS ´ P N .Proof. If S “ exp p ν ´ N q and N “ ν N ` . . . P N , then SAS ´ “ exp p ad p ν ´ N qq A “ ÿ r “ n ! ` ν ´ ad p N q ˘ r A, where the series converges in the ν -adic topology. Since ν ´ ad p N q leaves N invariant, we see that SAS ´ P N . (cid:3) In [9] the following important theorem was proved.
Theorem 3.1. (S. Gutt and J. Rawnsley)Any equivalence operator between two equivalent natural star productsis of exponential type. The algebra B In what follows we will use functions on formal neighborhoods ofembedded submanifolds. Let Y be an embedded submanifold of amanifold X and let I Y be the vanishing ideal of Y in C p X q . We call C p X, Y q : “ C p X q{ X k “ p I Y q k the space of functions on the formal neighborhood of Y in X . HE ALGEBRA OF DISTRIBUTIONS 7
Given a manifold M , we identify the diagonal of M l with M (thus as-suming that M Ă M l for any l ). An l -differential operator C p f , . . . , f l q on M defines a mapping C : C p M l , M q Ñ C p M q . Let A : “ p C p M qrr ν ss , ‹q be a star algebra on a Poisson manifold M with natural star product ‹ . Denote by Ă M a copy of M with theopposite Poisson structure. The opposite product f ‹ opp g : “ g ‹ f is astar product on Ă M . The product d : “ ‹ b ‹ opp is a natural star product on M ˆ Ă M . For f, g, u, v P A we have p f b g q d p u b v q “ p f ‹ u q b p v ‹ g q . Here b is a tensor product over the ring C rr ν ss . The product d inducesa product on C p M ˆ M, M qrr ν ss which will be denoted by the samesymbol. We introduce the algebra B : “ p C p M ˆ M, M qrr ν ss , dq .We call an element of B factorizable if it is induced by a formalfunction f b g P A b A , and use the same notation f b g for thiselement. There exists a homomorphism F ÞÑ N F from B to N givenon the factorizable elements by N f b g “ L f R g . We will prove that if M is symplectic, then this mapping is an iso-morphism. To this end, we need to recall several definitions and factsfrom [11].If A is a differential operator of order r on a manifold M , thenits principal symbol Symb r p A q is a fiberwise polynomial function ofdegree r on the cotangent bundle T ˚ M . Given a natural operator N “ N ` νN ` . . . on M , we call the formal series σ p N q : “ ÿ r “ Symb r p N r q the sigma symbol of N . It can be interpreted as a function on theformal neighborhood of the zero section Z of T ˚ M , σ p N q P C p T ˚ M, Z q . The mapping N ÞÑ σ p N q is a surjective homomorphism from N onto C p T ˚ M, Z q whose kernel is ν N . It follows that the sigma symbol σ p N F q of F “ F ` νF ` . . . P C p M ˆ M, M qrr ν ss depends onlyon F . It was proved in [11] that if M is symplectic, then the mapping C p M ˆ M, M q Q F ÞÑ σ p N F q ALEXANDER KARABEGOV is an isomorphism of C p M ˆ M, M q onto C p T ˚ M, Z q . Theorem 4.1. If ‹ is a natural star product on a symplectic mani-fold M , then the mapping F ÞÑ N F is an isomorphism of the algebra B onto N .Proof. We will construct the inverse mapping of the mapping F ÞÑ N F .Let N be an arbitrary natural operator on M . There exists a uniqueelement F P C p M ˆ M, M q such that σ p N F q “ σ p N q . Then ν ´ p N ´ N F q P N . Let F denote the unique element of C p M ˆ M, M q such that σ p N F q “ σ p ν ´ p N ´ N F qq . Hence, ν ´ p N ´ N F ´ νN F q P N . Continuing this process, we producea unique element F “ F ` νF ` . . . P B such that N “ N F . (cid:3) We say that a formal distribution Λ “ Λ ` ν Λ ` . . . on M supportedat a point x P M is natural if the order of the distribution Λ r is notgreater than r for every r . We denote the set of all such distributionsby N . Lemma 4.1.
A formal distribution Λ supported at x is natural if andonly if there exists a natural operator N P N such that Λ “ δ x ˝ N, i.e., Λ p f q “ p N f qp x q for any function f .Proof. It is clear that if N P N , then Λ “ δ x ˝ N P N . Conversely,given Λ P N , one can fix local coordinates around x and find theunique formal differential operator with constant coefficients C suchthat Λ “ δ x ˝ C . Then C is natural. It can be extended to a naturaloperator on M by multiplying it by an appropriate cutoff function. (cid:3) Denote by τ the involution on B such that τ p f b g q “ g b f . It is anantiautomorphism of B . The algebra B acts on N so that an element F P B maps Λ P N to Λ ˝ N τ p F q P N . Given F P B and x P M , we setΛ F : “ δ x ˝ N τ p F q . Lemma 4.2.
For f, g P A we have Λ f b g p h q “ p g ‹ h ‹ f qp x q .Proof. Λ f b g p h q “ p N τ p f b g q h qp x q “ p N g b f h qp x q “p L g R f h qp x q “ p g ‹ h ‹ f qp x q . (cid:3) HE ALGEBRA OF DISTRIBUTIONS 9
The formal distribution Λ F depends only on the jet of F at thediagonal point p x , x q P M ˆ M . We denote by F p q the space of ν -formal jets on M ˆ M at p x , x q . Lemma 4.3. If ‹ is a star product on a symplectic manifold M , thenthe corresponding mapping F ÞÑ Λ F induces a surjective mapping λ : F p q Ñ N . Proof.
This statement follows from Theorem 4.1 and Lemma 4.1. (cid:3)
Given a factorizable element f b g P F p q , we get from Lemma 4.2that(10) x λ p f b g q , h y “ p g ‹ h ‹ f qp x q . Star products with separation of variables
Berezin described in [2] and [3] a quantization procedure on K¨ahlermanifolds which leads to star products with the property of separationof variables (see, e.g., [4], [5], [6], [10], [14]). It is natural to considerthe star products with this property on pseudo-K¨ahler manifolds. Re-call that an almost-K¨ahler manifold is a complex manifold equippedwith a real symplectic form of type p , q with respect to the complexstructure. Definition 5.1.
A star product (9) on a pseudo-K¨ahler manifold M has the property of separation of variables of the Wick type if the opera-tors C r , r ě , differentiate the first argument in holomorphic directionsand the second argument in antiholomorphic ones. A star product is ofthe anti-Wick type if C r , r ě , differentiate the first argument in anti-holomorphic directions and the second argument in holomorphic ones.Remark. Observe that if ‹ is a star product of the anti-Wick type ona pseudo-K¨ahler manifold M , then the opposite product f ‹ opp g : “ g ‹ f is a product of the Wick type on the manifold M with the samecomplex structure but with the opposite symplectic structure. Also, ‹ is a product of the Wick type on the manifold Ď M , which is a copy of M with the opposite complex structure but with the same symplecticstructure.Let ‹ be a product of the anti-Wick type on M . If a is a holomorphicfunction and b is an antiholomorphic function locally defined on M ,then for any function f we have a ‹ f “ af and f ‹ b “ bf, i.e., L a “ a and R b “ b are pointwise multiplication operators. Throughout this paper we will denote the pointwise multiplicationoperator by a function f by the same symbol f .Let ω ´ be a pseudo-K¨ahler form on M (which determines a sym-plectic structure on M ). In [10] it was shown that the star products ofthe anti-Wick type on M are bijectively parametrized (not only up toequivalence) by the formal closed (1,1)-forms ω “ ν ´ ω ´ ` ω ` νω ` . . . on M . We will briefly recall this parametrization.Suppose that ω is fixed. Let U be a contractible coordinate chart on M with holomorphic coordinates t z k u . There exists a formal potentialΦ “ ν ´ Φ ´ ` Φ ` ν Φ ` . . . of ω on U , so that ω “ i B ¯ B Φ. As shown in [10], there exists a unique starproduct of the anti-Wick type ‹ on M such that on every contractiblechart U and for any potential Φ of ω on U , L B Φ B zk “ B Φ B z k ` BB z k and R B Φ B ¯ zl “ B Φ B ¯ z l ` BB ¯ z l . The formal form ω is called the classifying form of the star product ‹ . Every star product of the anti-Wick type has a unique classifying form.Given a star product ‹ of the anti-Wick type on M , there exists a ν -formal differential operator I “ ` νI ` ν I ` . . . globally defined on M such that for any local holomorphic function a and local antiholomorphic function b , I p ab q “ b ‹ a. It is called the formal Berezin transform of the star product ‹ . Observethat Ia “ a and Ib “ b . It is proved in [12] that(11) L b “ I ˝ b ˝ I ´ and R a “ I ˝ a ˝ I ´ . One can recover the product ‹ from the operator I using that p ab q ‹ p a b q “ aI p a b q b , where the functions a, a are local holomorphic and b, b are local anti-holomorphic. The equivalent star product(12) f ‹ g : “ I ´ p If ‹ Ig q on M is a star product with separation of variables of the Wick type (see [12]). Lemma 5.1.
The formal Berezin transform I of a star product of theanti-Wick type ‹ is of exponential type. HE ALGEBRA OF DISTRIBUTIONS 11
Proof.
The star products with separation of variables ‹ and ‹ are nat-ural (see [16]). Since I is an equivalence operator between the products ‹ and ‹ , it is of exponential type according to Theorem 3.1. (cid:3) It was shown in [14] and [13] that for any point x P M and anyinteger l ě K p l q p f , . . . , f l q “ I p f ‹ . . . ‹ f l qp x q “ p If ‹ . . . ‹ If l qp x q on M l is a FOI at p x q l P M l . Below we give a phase-volume form pairassociated with K p l q , which was found in [14] and [13].Let U be a contractible neighborhood in M and Φ be a local potentialof the classifying form ω of the product ‹ on U . Let s U denote a copyof U equipped with the opposite complex structure. One can find afunction ˜Φ p x, y q on U ˆ s U such that ˜Φ p x, x q “ Φ p x q and¯ B U ˆ s U ˜Φhas zero of infinite order at every point of the diagonal of U ˆ s U . Thefunction ˜Φ p x, y q is called an almost analytic extension of Φ (see detailsin [13] ). For each l ě G p l q on U l by theformula G p l q p x , . . . x l q : “ ˜Φ p x , x q ` ˜Φ p x , x q ` . . . ` ˜Φ p x l , x q´p Φ p x q ` Φ p x q ` . . . ` Φ p x l qq . This function defines an element of ν ´ C p U l , U qrr ν ss , where U is iden-tified with the diagonal of U l . This element does not depend on thechoice of the potential Φ and of the almost analytic extension of Φ.Thus, taking such functions for every contractible neighborhood in M ,we get a global element of ν ´ C p M l , M qrr ν ss . We call it a cyclicformal l -point Calabi function of the classifying form ω . Now suppose that x P U and consider the function(13) F p l q p x , . . . , x l q : “ G p l ` q p x , x , . . . , x l q on U l . The jet of F p l q at p x q l P U l is determined by the jet of G p l ` q at p x q l ` P U l ` , which is the jet of the formal p l ` q -point Calabifunction of ω at p x q l ` .It was shown in [14] and [13] that the FOI K p l q at p x q l is associatedwith the pair p F p l q , µ b l q on U l , where µ is a trace density of the starproduct ‹ . One can check that ˜Φ defines a function on the formal neighborhood of thediagonal of U ˆ s U which does not depend on the choice of the almost analyticextension of Φ. The main goal of this paper is to develop an algebraic frameworkwhich will incorporate the jet of the formal oscillatory exponent exp G p l q at p x q l P M l for every l ě The algebra C Let ‹ be a star product of the anti-Wick type on a pseudo-K¨ahlermanifold M , I be its formal Berezin transform, and x be a fixed pointin M . We choose a coordinate chart U containing x with coordinates t z k , ¯ z l u such that z k p x q “ ¯ z l p x q “ k, l . We consider variousjet spaces on M at the point x and on M ˆ M at the point p x , x q . Weidentify these spaces with spaces of formal series in local coordinates.Denote by F “ C rr ν, z, ¯ z ss the space of ν -formal jets on M at x andby A “ p F , ‹q the algebra on F with the induced product ‹ . Denoteby F p q “ C rr ν, z, ¯ z, w, ¯ w ss the space of ν -formal jets on M ˆ M at p x , x q , where t z k , ¯ z l u and t w k , ¯ w l u are the coordinates on the firstand the second factors of the chart U ˆ U . For the involutive mapping τ : F p q Ñ F p q such that τ p f b g q “ g b f for f, g P F , one has τ p z k q “ w k and τ p ¯ z l q “ ¯ w l .Since ‹ is a natural star product on M and M is symplectic, onecan construct a bijection F ÞÑ N F from F p q onto the space N ofnatural operators on M as in Section 4. For a factorizable element f b g P F p q , N f b g “ L f R g . There is a mapping λ : F p q Ñ N to thespace N of natural distributions at x , λ p F q “ δ x ˝ N τ p F q , which is surjective by Lemma 4.3. On factorizable elements f b g P F p q the mapping λ is given by formula (10).The space J “ C rr z, ¯ z ss of jets on M at x has a descending filtration J “ F J Ą F J Ą . . . , where F r J is the space of jets which have zeroof order at least r at x . We assume that F r J “ J for r ă
0. Weintroduce a filtration F “ F F Ą F F Ą . . . on the space of formal jets F “ J rr ν ss which agrees with the filtrationon J and for which the filtration degree of ν is 2, F r F “ F r J ` νF r ´ J ` ν F r ´ J ` . . . . We call it the standard filtration . Observe that F { F r F is a finite di-mensional vector space over C . One can check that F p q “ lim ÐÝ r p F b F q{ F r p F b F q , HE ALGEBRA OF DISTRIBUTIONS 13 where the subspaces F r p F b F q : “ ÿ i ` j “ r F i F b F j F form the standard filtration on F b F . Here b is the tensor productover the ring C rr ν ss . Lemma 6.1.
The algebra A “ p F , ‹q is a filtered algebra with respectto the standard filtration.Proof. Since ‹ is a natural star product (see [16]), the bidifferentialoperator C r in (9) is of order not greater than r in each argument.Therefore, if f P F i J and g P F j J , then C r p f, g q P F i ` j ´ r J and ν r C r p f, g q P F i ` j F , whence the lemma follows. (cid:3) Lemma 6.1 allows to extend various mappings of the space F b F to its completion F p q “ F ˆ b F with respect to the topology associ-ated with the standard filtration. We will tacitly assume that theseextensions can be justified with the use of this lemma.We define a filtered associative algebra C : “ p F p q , ˚q , where theproduct ˚ is given on the factorizable elements by the formula p g b h q ˚ p g b h q : “ p h ‹ g qp x q ¨ p g b h q . We introduce a trace on C given on the factorizable elements by theformula(14) tr p f b g q : “ p g ‹ f qp x q . One can check the trace property on factorizable elements,tr pp g b h q ˚ p g b h qq “ p h ‹ g qp x q ¨ p h b g qp x q “ tr pp g b h q ˚ p g b h qq . Lemma 6.2.
For F P C , the following identity holds, tr F “ x λ p F q , y . Proof.
Given a factorizable element f b g P F p q , we get from for-mula (10) that tr p f b g q “ p g ‹ f qp x q “ x λ p f b g q , y , whence the lemma follows. (cid:3) We introduce a splitting of C ,(15) C “ G ‘ H , where G “ C rr ν, z, ¯ w ss and H is generated by ¯ z l and w k for all k, l , i.e.,any H P H can be represented as H “ ¯ z l A l ` w k B k for some A l , B k P C . This splitting does not depend on the choice oflocal holomorphic coordinates used in its definition. We will show thatin the splitting (15) the subspace G is a subalgebra of C and H is atwo-sided ideal of C . Lemma 6.3.
The subspace H Ă C is a two-sided ideal of the algebra C which lies in the kernel of the mapping λ .Proof. It suffices to check the statement of the lemma on the generators U l “ p ¯ z l u q b v “ p u ‹ ¯ z l q b v and V k “ u b p z k v q “ u b p z k ‹ v q of H and factorizable F “ f b g P C , where u, v, f, g P F are arbitrary.We have F ˚ U l “ p f b g q ˚ pp u ‹ ¯ z l q b v q “p g ‹ u ‹ ¯ z l qp x q ¨ p f b v q “ pp g ‹ u q ¯ z l qp x q ¨ p f b v q “ , because ¯ z l p x q “
0. Then we see that U l ˚ F “ pp ¯ z l u q b v q ˚ p f b g q “ p v ‹ f qp x q ¨ pp ¯ z l u q b g q P H . One can check similarly that F ‹ V k P H and V k ‹ F “
0. It followsthat H is a two-sided ideal of C . We get from formula (10) that for any h P F , x λ p U l q , h y “ p v ‹ h ‹ u ‹ ¯ z l qp x q “ pp v ‹ h ‹ u q ¯ z l qp x q “ , because ¯ z l p x q “
0. Thus, λ p U l q “
0. One can similarly check that λ p V k q “
0, which implies the second statement of the lemma. (cid:3)
Lemma 6.4.
The subspace G Ă C is a subalgebra of C isomorphic tothe algebra C { H .Proof. The space G is topologically generated by the elements a b b ,where a P C rr ν, z ss is formally holomorphic and b P C rr ν, ¯ z ss is anti-holomorphic. We have p a b b q ˚ p a b b q “ p b ‹ a qp x q ¨ p a b b q P G . Therefore, G is a subalgebra of C . Clearly, it is isomorphic to thealgebra C { H . (cid:3) Let α : C p U qrr ν ss Ñ F be the mapping that maps f to its jet at x .It is surjective by Borel’s theorem. We define a mapping γ : C p U qrr ν ss Ñ G HE ALGEBRA OF DISTRIBUTIONS 15 as follows. Given f P C p U qrr ν ss , let ˜ f P C p U ˆ s U qrr ν ss be analmost analytic extension of f . We set γ p f q equal to the jet of ˜ f at p x , x q . This jet lies in G and does not depend on the choice of thealmost analytic extension of f . The mapping γ is surjective. There isa bijection β : F Ñ G such that γ “ β ˝ α . In coordinates, β : f p z, ¯ z q ÞÑ f p z, ¯ w q . Lemma 6.5.
Given g P C p U qrr ν ss , the following formula holds, λ p γ p g qq “ δ x ˝ I ˝ g ˝ I ´ . Proof.
Let g “ ab , where a is a holomorphic and b is an antiholomorphicfunction on U . Then γ p g q “ a b b . Using formula (11), we get that λ p γ p g qq “ δ x ˝ N τ p a b b q “ δ x ˝ N b b a “ δ x ˝ p R a L b q “ δ x ˝ I ˝ p ab q ˝ I ´ “ δ x ˝ I ˝ g ˝ I ´ . For a generic g P C p U qrr ν ss , the distribution δ x ˝ I ˝ g ˝ I ´ dependsonly on the jet of g at x and the space F is topologically generated bythe elements α p ab q . Therefore, the lemma follows from the calculationabove. (cid:3) Lemma 6.6.
The restriction of the mapping λ to G , λ | G : G Ñ N , isinjective.Proof. Let G be an arbitrary element of G which lies in the kernelof λ . There exists g P C p U qrr ν ss such that G “ γ p g q . Then for any h P C p U qrr ν ss we have from Lemma 6.5 that(16) I p g ¨ I ´ h qp x q “ x λ p γ p g qq , h y “ x λ p G q , h y “ . It was proved in [14] that the distribution f ÞÑ p If qp x q is a FOI at x .By Lemma 2.1, the pairing u, v ÞÑ I p u ¨ v qp x q on C p U qrr ν ss inducesa nondegenerate pairing on F . Since I ´ h is an arbitrary element of C p U qrr ν ss , we see from (16) that the jet of g at x is zero. Therefore, G “
0, whence the lemma follows. (cid:3)
Corollary 6.1.
The ideal H is the kernel of the mapping λ and themapping λ | G : G Ñ N is bijective.Proof. The mapping λ is surjective by Lemma 4.3. It was proved inLemma 6.3 that H lies in the kernel of λ . The corollary follows fromthe splitting (15) and Lemma 6.6. (cid:3) The algebra of distributions
Corollary 6.1 implies that one can transfer the product ˚ from thealgebra C to N . We denote the resulting product on N by ‚ . Theorem 7.1.
The algebra p N , ‚q is isomorphic to the algebra p G , ˚q – C { H . The mapping (17) N Q u ÞÑ x u, y is a trace on the algebra p N , ‚q . Its pullback via the mapping λ is thetrace tr on C .Proof. The theorem follows from Lemmas 6.4 and 6.2. (cid:3)
In the rest of the paper we will express the trace of the product of l elements of the algebra p N , ‚q in terms of the formal l -point Calabifunction of the star product ‹ .The standard filtration on F induces a filtration on the formal differ-ential operators on F , which we also call standard. If A is a differentialoperator of order r which does not depend on ν , its filtration degree isat least ´ r . We denote by N x the algebra of natural operators on F .These operators are induced by the operators from N . Observe that if N “ N ` νN ` . . . is a natural operator, then the filtration degree of ν r N r is at least r .In the remainder of this section ϕ “ ν ´ ϕ ´ ` ϕ ` . . . is a formalfunction on M such that x is a critical point of ϕ ´ with zero criticalvalue, ϕ ´ p x q “
0. We do not assume that the critical point x isnondegenerate. Observe that the filtration degree of ϕ is a least zero. Lemma 7.1. If N P N x , then e ´ ϕ N e ϕ P N x .Proof. Assume that N “ N ` νN ` . . . P N x . Then for each r ě e ´ ϕ p ν r N r q e ϕ “ r ÿ k “ k ! p´ ad ϕ q k p ν r N r q is of order not greater than r . The operator (18) is natural and its ν -filtration degree is at least zero. Its standard filtration degree is atleast r . Therefore, the series e ´ ϕ N e ϕ “ ÿ r “ e ´ ϕ p ν r N r q e ϕ converges in the topology associated with the standard filtration to anelement of N x . (cid:3) HE ALGEBRA OF DISTRIBUTIONS 17
Below we define an action e ϕ : u ÞÑ u ˝ e ϕ on N which behaveslike a composition. However, the multiplication operator by the formaloscillatory exponent e ϕ is not a natural operator, because the Taylorseries of e ϕ at x contains negative powers of ν .Given u P N , there exists N P N x such that u “ δ x ˝ N . We set u ˝ e ϕ : “ e ϕ p x q δ x ˝ p e ´ ϕ N e ϕ q . Since ϕ ´ p x q “
0, we see that e ϕ p x q P C rr ν ss . By Lemma 7.1, u ˝ e ϕ isan element of N . We will show that it does not depend on the choiceof N . Lemma 7.2. If u has two different representations u “ δ x ˝ N “ δ x ˝ ˜ N for N, ˜ N P N x , then e ϕ p x q δ x ˝ p e ´ ϕ N e ϕ q “ e ϕ p x q δ x ˝ p e ´ ϕ ˜ N e ϕ q .Proof. We have δ x ˝ p N ´ ˜ N q “
0. Therefore, in coordinates, one canwrite N ´ ˜ N “ z k A k ` ¯ z l B l for some A k , B l P N x . We need to showthat e ϕ p x q δ x ˝ p e ´ ϕ p N ´ ˜ N q e ϕ q “ , which follows from the observation that e ´ ϕ p N ´ ˜ N q e ϕ “ z k e ´ ϕ A k e ϕ ` ¯ z l e ´ ϕ B l e ϕ and the fact that z k p x q “ ¯ z l p x q for all k, l . (cid:3) Lemma 7.3.
Let ϕ “ ν ´ ϕ ´ ` ϕ ` . . . and ψ “ ν ´ ψ ´ ` ψ ` . . . beformal functions on M such that x is a critical point of ϕ ´ and ψ ´ with zero critical value, ϕ ´ p x q “ ψ ´ p x q “ . Then for any u P N one has p u ˝ e ϕ q ˝ e ψ “ u ˝ e ϕ ` ψ . Proof.
Let N P N x be such that u “ δ x ˝ N . Then p u ˝ e ϕ q ˝ e ψ “ p e ϕ p x q δ x ˝ p e ´ ϕ N e ϕ qq ˝ e ψ “ e ϕ p x q` ψ p x q δ x ˝ p e ´ ψ e ´ ϕ N e ϕ e ψ q “ u ˝ e ϕ ` ψ . (cid:3) We introduce a ν -linear functional K : N Ñ C rr ν ss , K p u q : “ x u ˝ e ϕ , y . If A is a differential operator on M , we denote by A t its transpose thatacts on a distribution u as A t u : “ u ˝ A . Let v be a vector field on M .Since νv and νvϕ are natural operators, then for u P N we get that p νv ´ νvϕ q t u P N . Lemma 7.4.
For any u P N , pp νv ´ νvϕ q t u q ˝ e ϕ “ p u ˝ e ϕ q ˝ p νv q . Proof.
Assume that u “ δ x ˝ N for some N P N x . Then p νv ´ νvϕ q t u “ δ x ˝ N ˝ p νv ´ νvϕ q . Therefore, pp νv ´ νvϕ q t u q ˝ e ϕ “ e ϕ p x q δ x ˝ p e ´ ϕ p N ˝ p νv ´ νvϕ qq e ϕ q “ e ϕ p x q δ x ˝ p e ´ ϕ N e ϕ q ˝ p νv q “ p u ˝ e ϕ q ˝ p νv q , because e ´ ϕ ˝ p v ´ vϕ q ˝ e ϕ “ v . (cid:3) Corollary 7.1.
For any u P N , K pp νv ´ νvϕ q t u q “ .Proof. We have by Lemma 7.4 that K pp νv ´ νvϕ q t u q “ xpp νv ´ νvϕ q t u q ˝ e ϕ , y “xp u ˝ e ϕ q ˝ p νv q , y “ x u ˝ e ϕ , p νv q y “ . (cid:3) Theorem 7.2.
Let S : N Ñ C rr ν ss be a ν -linear functional such thatthe equality S ` p νv ´ νvϕ q t u ˘ “ holds for any vector field v and any u P N . Then there exists a formalconstant c p ν q P C rr ν ss such that S p u q “ c p ν qx u ˝ e ϕ , y . Proof.
Consider a functional T : N Ñ C rr ν ss given by the formula T p u q : “ S p u ˝ e ´ ϕ q . We will show that T pp νv q t u q “ v and any u P N .Let N P N x be such that u “ δ x ˝ N . Given a vector field v and u P N , we have T ` p νv q t u ˘ “ S ppp νv q t u q ˝ e ´ ϕ q “ S pp δ x ˝ p N ˝ p νv qq ˝ e ´ ϕ q “ S p e ´ ϕ p x q δ x ˝ p e ϕ p N ˝ p νv qq e ´ ϕ qq “ S p e ´ ϕ p x q δ x ˝ p e ϕ N e ´ ϕ ˝ p νv ´ νvϕ qqq “ S pp u ˝ e ´ ϕ q ˝ pp νv ´ νvϕ qqq “ S pp νv ´ νvϕ q t p u ˝ e ´ ϕ qq “ . In local coordinates one can write any operator N P N x as N “ f ` A p ˝ ˆ ν BB z p ˙ ` B q ˝ ˆ ν BB ¯ z q ˙ , HE ALGEBRA OF DISTRIBUTIONS 19 where f “ N P C rr ν, z, ¯ z ss and A p , B q P N x . Then for u “ δ x ˝ N we have T p u q “ T ˆ δ x ˝ ˆ f ` A p ˝ ˆ ν BB z p ˙ ` B q ˝ ˆ ν BB ¯ z q ˙˙˙ “ f p x q T p δ x q ` T ˜ˆ ν BB z p ˙ t p δ x ˝ A p q ¸ ` T ˜ˆ ν BB ¯ z q ˙ t p δ x ˝ B q q ¸ “ f p x q T p δ x q . It follows that T p u q “ T p δ x qx u, y . We set c p ν q : “ T p δ x q . UsingLemma 7.3, we get that S p u q “ T p u ˝ e ϕ q “ c p ν qx u ˝ e ϕ , y . (cid:3) Let x be a point in M , U be a contractible coordinate chart withcoordinates t z p , ¯ z q u such that z p p x q “ ¯ z q p x q “ p, q , and Φbe a potential of the classifying form ω of the star product ‹ on U . Wechoose an almost analytic extension ˜Φ of Φ on U ˆ s U . In Section 5 weintroduced the cyclic function G p l q p x , . . . , x l q “ ˜Φ p x , x q ` . . . ` ˜Φ p x l , x q ´ p Φ p x q ` . . . ` Φ p x l qq on the neighborhood U l of the diagonal point p x q l of M l . The jetof the function G p l q at p x q l P M l is given in local coordinates by theformula G p l q p z, ¯ z q “ Φ p z , ¯ z q ` Φ p z , ¯ z q ` . . . ` Φ p z l , ¯ z q´p Φ p z , ¯ z q ` . . . ` Φ p z l , ¯ z l qq P ν ´ C rr ν, z , ¯ z , . . . z l , ¯ z l ss , where we have used the notations z “ p z , . . . , z l q , z i “ p z i , . . . , z mi q , ¯ z “p ¯ z , . . . , ¯ z l q , ¯ z j “ p ¯ z j , . . . , ¯ z mj q , and m “ dim C M . This is the jet of theformal l -point Calabi function of ω at p x q l . Lemma 7.5.
The diagonal point p x q l P M l is a critical point of thefunction G p l q with zero critical value.Proof. Clearly, G p l q pp x q l q “
0. In local coordinates, B G p l q B z pi “ B Φ B z p p z i , ¯ z i ` q ´ B Φ B z p p z i , ¯ z i q and(19) B G p l q B ¯ z qj “ B Φ B ¯ z q p z j ´ , ¯ z j q ´ B Φ B ¯ z q p z j , ¯ z j q , where we identify ¯ z l ` with ¯ z and z with z l . Therefore, B G p l q B z pi pp x q l q “ B G p l q B ¯ z qj pp x q l q “ . (cid:3) Remark.
The point p x q l P M l is a degenerate critical point of the func-tion G p l q , but it is a nondegenerate critical point of the function (13). Theorem 7.3.
The following identity holds for any natural distribu-tions u , . . . , u m P N , (20) x u ‚ . . . ‚ u l , y “ xp u b . . . b u l q ˝ exp G p l q , y . Remark.
Observe that the left-hand side of (20) is the trace of theproduct u ‚ . . . ‚ u l in the algebra p N , ‚q , which agrees with the factthat G p l q is cyclic. The distribution u b . . . b u l on the right-hand sideis a natural distribution on M l supported at p x q l . Proof.
We introduce a functional W p l q on the space of natural distri-butions on M l supported at the point p x q l by the formula W p l q p u b . . . b u l q : “ x u ‚ . . . ‚ u l , y . Suppose that u i “ λ p f i b g i q for 1 ď i ď l , where f i , g i P F are arbitrary.Then, by formula (10), W p l q p u b . . . b u l q “ p g ‹ f qp x q ¨ p g ‹ f qp x q ¨ . . . p g l ‹ f qp x q . Observe that δ x “ λ p b q and δ x ‚ δ x “ δ x . Clearly, W p l q ` δ p x q l ˘ “ x δ p x q l ˝ exp G p l q , y “ x δ p x q l , y “ , where we have used that δ p x q l “ δ x b . . . b δ x and G p l q pp x q l q “ i, j, p, q ,(21) W p l q ˝ ˆ ν BB z pi ´ ν B G p l q B z pi ˙ t “ W p l q ˝ ˆ ν BB ¯ z qj ´ ν B G p l q B ¯ z qj ˙ t “ . We will check the first equality on the elements u b . . . b u l with u i “ λ p f i b g i q , which topologically generate N . We use formula (19)to calculate the action of ˆ ν BB z pi ´ ν B G p l q B z pi ˙ t “ ˆ ν BB z pi ´ ν B Φ B z p p z i , ¯ z i ` q ` ν B Φ B z p p z i , ¯ z i q ˙ t on u b . . . b u l . The operator ˆ ν BB z pi ` ν B Φ B z p p z i , ¯ z i q ˙ t HE ALGEBRA OF DISTRIBUTIONS 21 acts only on the factor u i in u b . . . b u l . We have Cˆ ν BB z p ` ν B Φ B z p ˙ t u i , h G “ B u i , ν B Φ B z p ‹ h F “ ˆ g i ‹ ν B Φ B z p ‹ h ‹ f i ˙ p x q “ B λ ˆ f i b ˆ g i ‹ ν B Φ B z p ˙˙ , h F . Therefore,ˆ u i : “ ˆ ν BB z p ` ν B Φ B z p ˙ t u i “ λ ˆ f i b ˆ g i ‹ ν B Φ B z p ˙˙ . We get W p l q p u b . . . b ˆ u i b . . . b u l q “ x u ‚ . . . ‚ ˆ u i ‚ . . . ‚ u l , y “p g ‹ f qp x q ¨ p g ‹ f qp x q ¨ . . . ¨ p g i ´ ‹ f i qp x q ¨ ˆ g i ‹ ν B Φ B z p ‹ f i ` ˙ p x q ¨ p g i ` ‹ f i ` qp x q ¨ . . . ¨ p g l ‹ f qp x q . It remains to calculate W p l q ˜ˆ ν B Φ B z p p z i , ¯ z i ` q ˙ t p u b . . . b u l q ¸ . The jet ν B Φ B z p p z i , ¯ z i ` q can be expressed as the following series convergentin the topology associated with the standard filtration, ν B Φ B z p p z i , ¯ z i ` q “ ÿ α a α p z i q b α p ¯ z i ` q . We have Cˆ ν B Φ B z p p z i , ¯ z i ` q ˙ t p u b . . . b u l q , h b . . . b h l G “ B u b . . . b u l , ν B Φ B z p p z i , ¯ z i ` qp h b . . . b h l q F “ C u b . . . b u l , ˜ÿ α a α p z i q b α p ¯ z i ` q ¸ p h b . . . b h l q G “ ÿ α ´ p g ‹ h ‹ f qp x q ¨ . . . p g i ‹ a α p z i q ‹ h i ‹ f i qp x q ¨p g i ` ‹ h i ` ‹ b α p ¯ z i ` q ‹ f i ` qp x q ¨ . . . ¨ p g l ‹ h l ‹ f l qp x q ¯ “ Cÿ α u b . . . ˆ u iα b ˆ u i ` α b . . . b u l , h b . . . b h l G , where ˆ u iα “ λ p f i b p g i ‹ a α qq and ˆ u i ` α “ λ pp b α ‹ f i ` q b g i ` q . We havethus proved that ˆ ν B Φ B z p p z i , ¯ z i ` q ˙ t p u b . . . b u l q “ ÿ α u b . . . ˆ u iα b ˆ u i ` α b . . . b u l . Now, W p l q ˜ˆ ν B Φ B z p p z i , ¯ z i ` q ˙ t p u b . . . b u m q ¸ “ W p l q ˜ÿ α u b . . . ˆ u iα b ˆ u i ` α b . . . b u m ¸ “ ÿ α ` p g ‹ f qp x q ¨ . . . ¨ p g i ´ ‹ f i qp x q ¨ p g i ‹ a α ‹ b α ‹ f i ` qp x q ¨p g i ` ‹ f i ` qp x q ¨ . . . ¨ p g l ‹ f qp x q ˘ . We see that ÿ α p g i ‹ a α ‹ b α ‹ f i ` qp x q “ ˆ g i ‹ ν B Φ B z p ‹ f i ` ˙ p x q , because a α ‹ b α “ a α b α . Hence, W p l q ˜ˆ ν B Φ B z p p z i , ¯ z i ` q ˙ t p u b . . . b u l q ¸ “ W p l q ˜ˆ ν BB z pi ` ν B Φ B z p p z i , ¯ z i q ˙ t p u b . . . b u l q ¸ , which proves the first equality in (21). The second one can be checkedsimilarly. (cid:3) Formula (20) allows to express the jet of exp G p l q at p x q l in terms ofthe algebra p N , ‚q for every l ě References [1] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., and Sternheimer, D.:Deformation theory and quantization. I. Deformations of symplectic structures.
Ann. Physics (1978), no. 1, 61 – 110.[2] Berezin, F.A.: Quantization. Math. USSR-Izv. (1974), 1109–1165.[3] Berezin, F.A.: Quantization in complex symmetric spaces. Math. USSR-Izv. (1975), 341–379.[4] Bordemann, M., Waldmann, S.: A Fedosov star product of the Wick type forK¨ahler manifolds. Lett. Math. Phys. (3) (1997), 243 – 253.[5] Cahen, M., Gutt, S., and Rawnsley, J.: Quantization of K¨ahler manifolds. II, Trans. Amer. Math. Soc. (1993), 73 – 98.
HE ALGEBRA OF DISTRIBUTIONS 23 [6] Engliˇs, M.: A Forelli-Rudin construction and asymptotics of weighted Bergmankernels.
J. Funct. Anal. (2000), 257–281.[7] Fedosov, B.: A simple geometrical construction of deformation quantization.
J. Differential Geom. (1994), no. 2, 213–238.[8] Fedosov, B.: Deformation quantization and index theory . Mathematical Topics,9. Akademie Verlag, Berlin, 1996. 325 pp.[9] Gutt, S. and Rawnsley, J.: Equivalence of star products on a symplectic man-ifold.
J. Geom. Phys. (1999), 347 – 392.[10] Karabegov, A.: Deformation quantizations with separation of variables on aK¨ahler manifold. Commun. Math. Phys. (1996), no. 3, 745–755.[11] Karabegov A.V.: On the dequantization of Fedosov’s deformation quantiza-tion.
Lett. Math. Phys. (2003), 133 – 146.[12] Karabegov, A.: Formal symplectic groupoid of a deformation quantization.Comm. Math. Phys. (2005), 223 – 256.[13] Karabegov A.: Formal oscillatory integrals and deformation quantization. Lett.Math. Phys. (2019), 1907 – 1937.[14] Karabegov, A., Schlichenmaier, M.: Identification of Berezin-Toeplitz defor-mation quantization.
J. reine angew. Math. (2001), 49 – 76.[15] Kontsevich, M.: Deformation quantization of Poisson manifolds, I.
Lett. Math.Phys. (2003), 157 – 216.[16] Neumaier, N.: Universality of Fedosov’s construction for star products of Wicktype on pseudo-K¨ahler manifolds. Rep. Math. Phys. (2003), 43 – 80.[17] Nest, R., Tsygan, B.: Algebraic index theorem.
Comm. Math. Phys. (1995), 223 – 262.(Alexander Karabegov)
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