A Note on the Koszul Complex in Deformation Quantization (original) (raw)

Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Advances in Mathematics, 2010

Let α be a quadratic Poisson bivector on a vector space V. Then one can also consider α as a quadratic Poisson bivector on the vector space V * [1]. Fixed a universal deformation quantization (prediction of some complex weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras S(V *) and Λ(V). These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on α, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [T1].

On the representation theory of deformation quantization

Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, May 31 - June 2, 2001 / Rencontre entre physiciens théoriciens et mathématiciens, Strasbourg, 31 mai - 2 juin 2001

In this contribution to the proceedings of the 68è me Rencontre entre Physiciens Théoriciens et Mathématiciens on Deformation Quantization I shall report on some recent joint work with Henrique Bursztyn on the representation theory of *-algebras arising from deformation quantization as presented in my talk.

Almost-Kähler deformation quantization

Letters in Mathematical Physics, 2001

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure to construct a Fedosov-type deformation quantization on this manifold. 1 2 A.V. KARABEGOV AND M. SCHLICHENMAIER parameterized by the elements of the affine vector space (1/iν)[ω] + H 2 (M, C)[[ν]] via the mapping * → cl( * ).

On Deformation Theory and Quantization

2008

Deformation theory requires solving Maurer-Cartan equation (MCE) associated to an DGLA (L-infinity algebra). The universal solution of [HS] is obtained iteratively, as a fixed point of a contraction, analogous to the Picard method. The role of the Kuranishi functor in this construction is emphasized. The parallel with Lie theory suggests that deformation theory is a higher “dimensional” version. The deformation determined by the solution of the Maurer-Cartan equation associated to a contraction, splits the epimorphism, leading to a “doubling and gluing” interpretation. The *-operator associated to a contraction is introduced, and the connection with Hodge structures and generalized complex structures ( dd∗ -lemma) is established. The relations with bialgebra deformation quantization on one hand and ConnesKreimer renormalization on the other, are suggested.

Almost Kaehler deformation quantization

Eprint Arxiv Math 0102169, 2001

Bimodules and branes in deformation quantization

Compositio Mathematica, 2010

We prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector spaceX. The result implies, in particular, that the Kontsevich deformation quantizations of S(X*) and(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.

A ] 1 1 Ju n 20 07 Deformation Quantization and Reduction

2007

This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and prePoisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞and A∞-algebras, and bimodule structures are recalled. As an application, an “almost” functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson–Lie groups.

Variations on deformation quantization

2000

I have chosen, in this presentation of Deformation Quantization, to focus on 3 points: the uniqueness --up to equivalence-- of a universal star product (universal in the sense of Kontsevich) on the dual of a Lie algebra, the cohomology classes introduced by Deligne for equivalence classes of differential star products on a symplectic manifold and the construction of some convergent star products on Hermitian symmetric spaces. Those subjects will appear in a promenade through the history of existence and equivalence in deformation quantization.

A remark on formal KMS states in deformation quantization

1998

In the framework of deformation quantization we define formal KMS states on the deformed algebra of power series of functions with compact support in phase space as C[[λ]]-linear functionals obeying a formal variant of the usual KMS condition known in the theory of C * -algebras. We show that for each temperature KMS states always exist and are up to a normalization equal to the trace of the argument multiplied by a formal analogue of the usual Boltzmann factor, a certain formal star exponential.

The character map in deformation quantization

Advances in Mathematics, 2011

The third author recently proved that the Shoikhet-Dolgushev L∞-morphism from Hochschild chains of the algebra of smooth functions on a manifold to differential forms extends to cyclic chains. Localization at a solution of the Maurer-Cartan equation gives an isomorphism, which we call character map, from the periodic cyclic homology of a formal associative deformation of the algebra of functions to de Rham cohomology. We prove that the character map is compatible with the Gauss-Manin connection, extending a result of Calaque and Rossi on the compatibility with the cap product. As a consequence, the image of the periodic cyclic cycle 1 is independent of the deformation parameter and we compute it to be the A-roof genus of the manifold. Our results also imply the Tamarkin-Tsygan index Theorem.

Deformation quantization and reduction

Contemporary Mathematics, 2008

This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and pre-Poisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞-and A∞-algebras, and bimodule structures are recalled. As an application, an "almost" functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson-Lie groups.

Deformation Quantization of A-infinity Equivalences

We show that Deformation Quantization of quadratic Poisson structures preserves the A ∞-Morita equivalence of a given pair of Koszul dual A ∞-algebras. Contents 1. Introduction 1 2. Acknowledgments 3 3. Notation and Conventions 3 4. A ∞-structures 3 5. The triple (A, K, B) 9 6. A ∞-Morita theory 11 7. Deformation Quantization of A ∞-structures 19 8. Topological A ∞-structures 20 9. Main result 31 Appendix A. Proof of prop. 6 31 Appendix B. Proof of thm. 7 33 Appendix C. Proof of thm. 9 40 Appendix D. On triangulated categories 46 References 48

Identification of Berezin-Toeplitz deformation quantization

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kähler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szegö kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjöstrand. 1 2 A.V. KARABEGOV AND M. SCHLICHENMAIER 4 A.V. KARABEGOV AND M. SCHLICHENMAIER IDENTIFICATION OF BEREZIN-TOEPLITZ DEFORMATION QUANTIZATION 5

Graph Complexes in Deformation Quantization

Letters in Mathematical Physics, 2005

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an L∞-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.

Deformation quantization for almost-Kähler

2016

On an arbitrary almost-Kähler manifold, starting from a natural affine connection with nontrivial torsion which respects the almost-Kähler structure, in joint work with A. Karabegov a Fedosov-type deformation quantization on this manifold was con-structed. This contribution reports on the result and supplies an overview of the essential steps in the construction. On this way Fedosov’s geometric method is ex-plained. 1

Q A ] 1 7 Ju l 2 00 5 Irreducible highest-weight modules and equivariant quantization

2005

The notion of deformation quantization, motivated by ideas coming from both physics and mathematics, was introduced in classical papers [2, 7, 8]. Roughly speaking, a deformation quantization of a Poisson manifold (P, { , }) is a formal associative product on (Fun P)[[]] given by f 1 ⋆ f 2 = f 1 f 2 + c(f 1 , f 2) + O(2) for any f 1 , f 2 ∈ Fun P , where the skew-symmetric part of c is equal to { , }, and the coefficients of the series for f 1 ⋆ f 2 should be given by bi-differential operators. The fact that any Poisson manifold can be quantized in this sense was proved by Kontzevich in [15]. However, finding exact formulas for specific cases of Pois-son brackets is an interesting separate problem. There are several well-known examples of such explicit formulas. One of the first was the Moyal product quan-tizing the standard symplectic structure on R 2n. Another one is the standard quantization of the Kirilov-Kostant-Souriau bracket on the dual space g * to a Lie algebra g (see [10]...

Deformation quantization of Poisson algebras

Proceedings of the Japan Academy. Series A, Mathematical sciences, 1992

O. Introduction. Let M be a C Poisson manifOld, and C(M) the set .of all C-valued C functions on M. In what follows, we put a=C-(M) for simplicity. By definition of Poisson manifolds, there exists a bilinear map. { }'a a-a, called the Poisson bracket, with the following properties" For anyf, g, hea,