A proof of Tsygan's formality conjecture for an arbitrary smooth manifold
Author(s)Dolgushev, Vasiliy A
Massachusetts Institute of Technology. Dept. of Mathematics.
Pavel Etingof and Dmitry Tamarkin.
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Proofs of Tsygan's formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevich's formality quasi-isomorphism for Hochschild cochains of R((y1, . . . , yd)) and Shoikhet's formality quasi-isomorphism for Hochschild chains of R((y1, . . . , yd)) I prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M,(vector differential)), where M is the manifold and (vector differential) is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 105-110).
DepartmentMassachusetts Institute of Technology. Dept. of Mathematics.
Massachusetts Institute of Technology