Riemann-Roch via deformation quantization, II
We prove a Riemann-Roch formula for deformation quantization of complex manifolds and its corollary, an index theorem for elliptic pairs conjectured by Schapira and Schneiders.
math.KT↗
arXiv subjects
Boris Tsygan
Publications and source records attributed to Boris Tsygan.
Local formula for the index of a Fourier Integral Operator
We show that the index of an elliptic Fourier integral operator associated to a contact diffeomorphism $ϕ$ of cosphere bundles of two Riemannian manifolds X and Y is given by $\int_{B^*X}\hat{A}(T^*X)\expθ - \int_{B^*Y}\hat{A}(T^*Y)\expθ$. Here $B^*$ stands for the unit coball bundle and $θ$ is a certain characteristic class depending on the principal symbol of the Fourier integral operator. In the special case when X=Y we obtain a different proof of the theorem of Epstein and Melrose.
math.DG↗
Noncommutative differential calculus, homotopy BV algebras and formality conjectures
We define a notion of astrongly homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild and cyclic chains are formulated. We prove some partial results supporting these conjectures.
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Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems
Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold. In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures. For deformations of these structures we prove the classification theorems and a general a general index theorem.
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Formality conjecture for chains
We state the analogs of Kontsevich's formality conjecture for Hochschild and cyclic chains, as well as their
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On the Cohomology Ring of an Algebra
We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum multiplication on Floer cohomology of free loop spaces. We discuss some examples, as well as applications to index theorems, characteristic classes and deformations.
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Fukaya Type Categories for Associative Algebras
We define for an associative algebra an $A_{\infty}$ category whose objects are automorphisms of this algebra. This construction has some resemblance with Fukaya'a categories related to Floer cohomology.
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