Vasiliy Dolgushev

The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal

The solution of Delignes conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C � .A/ of a regular commutative algebra A over a field K of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky Gelfand-Fuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold.

Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology

We prove the additive version of the conjecture proposed by Ginzburg and Kaledin. This conjecture states that if X/G is an orbifold modeled on a quotient of a smooth affine symplectic variety X (over C) by a finite group G\subset Aut(X) and A is a G-stable quantum algebra of functions on X then the graded vector space HH(A^G) of the Hochschild cohomology of the algebra A^G of invariants is isomorphic to the graded vector space H_{CR}(X/G)((h)) of the Chen-Ruan (stringy) cohomology of the orbifold X/G.

Formality theorems for Hochschild chains in the Lie algebroid setting

Abstract In this paper we prove Lie algebroid versions of Tsygans formality conjecture for Hochschild chains both in the smooth and holomorphic settings. Our result in the holomorphic setting implies a version of Tsygans formality conjecture for Hochschild chains of the structure sheaf of any complex manifold. The proofs are based on the use of Kontsevichs quasi-isomorphism for Hochschild cochains of ℝ[[y 1, …, yd ]], Shoikhets quasi-isomorphism for Hochschild chains of ℝ[[y 1, …, yd ]], and Fedosovs resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid. In the smooth setting we discuss an application of our result to the description of quantum traces for a Poisson Lie algebroid.

Morita equivalence and characteristic classes of star products

Abstract This paper deals with two aspects of the theory of characteristic classes of star products: first, on an arbitrary Poisson manifold, we describe Morita equivalent star products in terms of their Kontsevich classes; second, on symplectic manifolds, we describe the relationship between Kontsevichs and Fedosovs characteristic classes of star products.

Wick type deformation quantization of Fedosov manifolds

Abstract A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The geometry of the symplectic manifolds admitting the symbol construction is explored and a certain analogue of the Newlander–Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the Kahler manifolds this class is shown to be proportional to the first Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed.

Formality theorems for hochschild complexes and their applications

We give a popular introduction to formality theorems for Hochschild complexes and their applications. We review some of the recent results and prove that the truncated Hochschild cochain complex of a polynomial algebra is non-formal.

An algebraic index theorem for Poisson manifolds

Abstract The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map.

Hochschild cohomology versus de Rham cohomology without formality theorems

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of any version of the formality theorem. We also show that the Gerstenhaber bracket induced on H(M,C((h))) via the isomorphism is vanishing. We discuss equivariant properties of the isomorphism and propose an analogue of this statement in an algebraic geometric setting.

Noncommutative Calculus and the Gauss–Manin Connection

After an overview of noncommutative differential calculus, we construct parts of it explicitly and explain why this construction agrees with a fuller version obtained from the theory of operads.

R-Matrix Structure of Hitchin System in Tyurin Parameterization

Abstract: We present a classical r-matrix for the Hitchin system without marked points on an arbitrary non-degenerate algebraic curve of genus g≥2 using Tyurin parameterization of holomorphic vector bundles.