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14
A Proof of Tsygan’s formality conjecture for an arbitrary Smooth Manifold
, 2005
"... Proofs of Tsygan’s formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the AtiyahPatodiSinger index theorem and the RiemannRochHirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various ..."
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Cited by 37 (13 self)
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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 AtiyahPatodiSinger index theorem and the RiemannRochHirzebruch 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 quasiisomorphism for Hochschild cochains of R[[y 1,...,y d]] and Shoikhet’s formality quasiisomorphism for Hochschild chains of R[[y 1,...,y d]] I prove Tsygan’s formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasiisomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M, ∇), where M is the manifold and ∇ is an affine connection on the
Cyclic cocycles on deformation quantizations and higher index theorems for orbifolds
"... ABSTRACT. We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasiisomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation ..."
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Cited by 12 (7 self)
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ABSTRACT. We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasiisomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We prove an algebraic higher index theorem by computing the pairing between such cyclic cocycles and the Ktheory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes–Moscovici and its extension to orbifolds.
NONCOMMUTATIVE POISSON STRUCTURES ON ORBIFOLDS
, 2006
"... Abstract. In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of C ∞ (M)⋊Γ. Using this computation, we classify all the noncommutative Poisson structures on C ∞ (M) ⋊ Γ when M is a symplectic manifold. We provide examples of deformation quantizations of these noncommutat ..."
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Cited by 8 (2 self)
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Abstract. In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of C ∞ (M)⋊Γ. Using this computation, we classify all the noncommutative Poisson structures on C ∞ (M) ⋊ Γ when M is a symplectic manifold. We provide examples of deformation quantizations of these noncommutative Poisson structures. 1.
EQUIVARIANT LEFSCHETZ NUMBER OF DIFFERENTIAL OPERATORS
, 2007
"... Let G be a compact Lie group acting on a compact complex manifold M. We prove a trace density formula for the GLefschetz number of a differential operator on M. We generalize Engeli and Felder’s recent results to orbifolds. ..."
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Cited by 3 (1 self)
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Let G be a compact Lie group acting on a compact complex manifold M. We prove a trace density formula for the GLefschetz number of a differential operator on M. We generalize Engeli and Felder’s recent results to orbifolds.
Gerstenhaber and BatalinVilkovisky structures on modules over operads
, 2013
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BATALINVILKOVISKY ALGEBRA STRUCTURES ON pCoqTor AND POISSON BIALGEBROIDS
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ON THE ALGEBRAIC INDEX FOR RIEMANNIAN ÉTALE GROUPOIDS
, 2008
"... In this paper we construct an explicit quasiisomorphism to study the cyclic cohomology of a deformation quantization over a riemannian étale groupoid. Such a quasiisomorphism allows us to propose a general algebraic index problem for riemannian étale groupoids. We discuss solutions to that index ..."
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Cited by 1 (1 self)
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In this paper we construct an explicit quasiisomorphism to study the cyclic cohomology of a deformation quantization over a riemannian étale groupoid. Such a quasiisomorphism allows us to propose a general algebraic index problem for riemannian étale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3dim torus.
Orbifold cup products and . . .
, 2007
"... In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantiz ..."
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In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an S 1equivariant version of the Chen–Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology.