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Abstract. We study categorial properties of the operadic twisting functor Tw . In particular, we show that Tw is a comonad. Coalgebras of this comonad are operads for which a natural notion of twisting by Maurer-Cartan elements exists. We give a large class of examples, including the classical cases of the Lie, associative and Gerstenhaber operads, and their infinity-counterparts Lie∞, As∞, Ger∞. We also show that Tw is well behaved with respect to the homotopy theory of operads. As an application we show that every solution of Deligne's conjecture is homotopic to a solution that is compatible with twisting.

Abstract. The solution of Deligne's 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. (2000). 53D55. Mathematics Subject Classification

We consider L∞-quasi-isomorphisms for Hochschild cochains whose structure maps admit "graphical expansion". We introduce the notion of stable formality quasi-isomorphism which formalizes such an L∞quasi-isomorphism. We define a homotopy equivalence on the set of stable formality quasi-isomorphisms and prove that the set of homotopy classes of stable formality quasi-isomorphisms form a torsor for the group corresponding to the zeroth cohomology of the full (directed) graph complex. This result may be interpreted as a complete description of homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the "stable setting".

We prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R[[y 1 , . . . , y d ]] proposed in paper math.QA/0010321 by Shoikhet. This result allows us to describe traces on the quantum algebra of functions on an arbitrary Poisson manifold.MSC-class: 16E45; 53C15; 18G55.

The Van den Bergh duality and the modular symmetry of a Poisson varietyVasiliy Dolgushev AbstractWe consider a smooth Poisson affine variety with the trivial canonical bundle over C. For such a variety the deformation quantization algebra A enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A intrinsic to A . We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra A is free if and only if the corresponding Poisson structure is unimodular. IntroductionIn In this paper we show that the deformation quantization incarnation of the modular symmetry of Poisson manifolds is related to the Van den Bergh duality [38] between Hochschild homology and Hochschild cohomology. This relationship can be described in terms of Bursztyn-Waldmann bimodule quantization [8]. More precisely, a modular vector field of a Poisson structure gives us a flat contravariant connection on the A-bimodule A, where A is the algebra of functions on the Poisson variety. This contravariant connection can be quantized to a bimodule of the deformation quantization algebra A . The main result of the paper (see Theorem 2) states that the resulting bimodule is the Van den Bergh dualizing bimodule A . In this paper we also show that the Van den Bergh dualizing module of A is isomorphic to A if and only if the corresponding Poisson structure π is unimodular.The organization of the paper is as follows. In the second section we go over the notation and recall some required results. At the end of this section we describe a construction which produces out of a Poisson vector field a derivation of the deformation quantization algebra. The third section is devoted to the definition of the modular class of a Poisson structure in the algebraic setting. In section 4 we define the modular automorphism 1 of a deformation quantization algebra, formulate our main result (Theorem 2), and prove a useful technical proposition which we need in the proof of Theorem 2. Section 5 is devoted to criterion of unimodularity and section 6 is devoted to the proof of Theorem 2. In the concluding section we discuss some results in literature related to Theorems 2 and 3. In the Appendix we discuss properties of Poisson, Hamiltonian and log-Hamiltonian vector fields.

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 Kontsevich's and Fedosov's characteristic classes of star products.

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