Dmitry Tamarkin

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.

Cyclic Formality and Index Theorems

The Letter announces the following results (the proofs will appear elsewhere). An operad acting on Hochschild chains and cochains of an associative algebra is constructed. This operad is formal. In the case when this algebra is the algebra of smooth function on a smooth manifold, the action of this operad on the corresponding Hochschild chains and cochains is formal. The induced map on the (periodic) cyclic homology is given by the formula involving the A-genus. The index theorem for degenerate Poisson structures follows from the latter fact.

What do dg-categories form?

We introduce a homotopy 2-category structure on the collection of dg-categories, dgfunctors, and their derived transformations. This construction provides for a conceptual proof of Deligne’s conjecture on Hochschild cochains.

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.

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.

Topological invariants of connections on symplectic manifolds

Let M 2n be a 2n-dimensional symplectic manifold, let E --~ M be a vector bundle whose structure group is a connected semisimple Lie group G, let V M be a symplectic connection on M , and let V E be a connectipn on E . ~ This paper is devoted to the problem of finding all closed differential forms on M that can be written in local .Darboux coordinates as polynomials in finite-order derivatives of the coefficients of the connections ~7E and V M on condition that the cohomology class of the manifold M defined by such a form is preserved under the deformations of the connections. It is required that this differential form could be well defined on M . This is possible only for the case in which the dependence of this form on the connection coefficients is preserved under the transformations of Darboux coordinates. These forms are said to be invariant. A similar problem for Riemannian manifolds was solved by Abramov [1] (also see Gilkey [2]). For a more detailed statement of this problem and its solution see Atiyah, Bott, and Patody [3]. By analogy with the case of Riemannian manifolds, every invariant form on M is a polynomial in Pontryagin classes of the manifold M , characteristic classes of the bundle E , and the sympleetic form w (up to the so-called trivial forrXs whose eohomology classes are always trivial). The author thanks B. V. Fedosov for the statement of the problem and constant at tention and B. L. Feigin for valuable advice.

A FORMALISM FOR THE RENORMALIZATION PROCEDURE

We describe a formalism underlying the renormalization procedure and Batalin-Vilkoviski formalism. In the framework of this formalism, we give a mathematical definition of OPE-algebra and describe an additional natural structure which produces a *-Lie algebra structure on the (cohomologically shifted) D-module of observables, whence an

Generalized Topological Index

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Another proof of M. Kontsevich formality theorem

algebra structure on the space of global sections of the de Rham complex of this D-module. Given a Maurer-Cartan element in this

Formality of Chain Operad of Little Discs

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