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63
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
Deformation Quantization in Algebraic Geometry
, 2003
"... We study deformation quantization of Poisson algebraic varieties. Using the universal deformation formulas of Kontsevich, and an algebrogeometric approach to the bundle of formal coordinate systems over a smooth variety X, we prove existence of deformation quantization of the sheaf OX (assuming t ..."
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Cited by 28 (7 self)
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We study deformation quantization of Poisson algebraic varieties. Using the universal deformation formulas of Kontsevich, and an algebrogeometric approach to the bundle of formal coordinate systems over a smooth variety X, we prove existence of deformation quantization of the sheaf OX (assuming the vanishing of certain cohomologies). Under slightly stronger assumptions we can classify all such deformations.
The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
 Jour. Noncommutative Geom
, 2007
"... 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 quasiisomorphisms of homo ..."
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Cited by 28 (11 self)
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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 quasiisomorphisms 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 GelfandFuchs 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.
THE BAR DERIVED CATEGORY OF A CURVED DG ALGEBRA
, 2008
"... Curved A∞algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A∞algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module ..."
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Cited by 18 (2 self)
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Curved A∞algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A∞algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.
LECTURES ON DUFLO ISOMORPHISMS IN LIE ALGEBRAS AND COMPLEX GEOMETRY
"... Abstract. For a complex manifold the HochschildKostantRosenberg map does not respect the cup product on cohomology, but one can modify it using the square root of the Todd class in such a way that it does. This phenomenon is very similar to what happens in Lie theory with the DufloKirillov modifi ..."
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Abstract. For a complex manifold the HochschildKostantRosenberg map does not respect the cup product on cohomology, but one can modify it using the square root of the Todd class in such a way that it does. This phenomenon is very similar to what happens in Lie theory with the DufloKirillov modification of the PoincaréBirkhoffWitt isomorphism. In these lecture notes (lectures were given by the first author at ETHZürich in fall 2007) we state and prove DufloKirillov theorem and its complex geometric analogue. We take this opportunity to introduce standard mathematical notions and tools from a very downtoearth viewpoint.
A Counterexample to the Quantizability of Modules
 Lett. Math. Phys
"... n Let π be a Poisson structure on vanishing at 0. It leads to a Kontsevich type star product ⋆π on C ∞ ( n)[[ǫ]]. We show that 1. The evaluation map at 0 ev0: C ∞ ( n → �) can in general not be quantized to a character of (C ∞ ( n)[[ǫ]], ⋆π). 2. A given Poisson structure π vanishing at zero can i ..."
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n Let π be a Poisson structure on vanishing at 0. It leads to a Kontsevich type star product ⋆π on C ∞ ( n)[[ǫ]]. We show that 1. The evaluation map at 0 ev0: C ∞ ( n → �) can in general not be quantized to a character of (C ∞ ( n)[[ǫ]], ⋆π). 2. A given Poisson structure π vanishing at zero can in general not be extended to a formal Poisson structure πǫ also vanishing at zero, such that ev0 can be quantized to a character of (C ∞ ( n)[[ǫ]], ⋆πǫ). We do not know whether the second claim remains true if one allows the higher order terms in ǫ to attain nonzero values at zero. How to read this paper in 2 minutes The busy reader can take the following shortcut: 1. Read Theorem 6 on page 3 for the main result. 2. Read Definition 2 if its statement is not clear. 3. Look at eqns. (13) and the preceding enumeration for the definition of the counterexample. 1
A.Sharapov, Quantizing nonLagrangian Gauge Theories: An Augmentation Method
 JHEP 0701 (2007) 047, hepth/0612086
"... Abstract. We discuss a recently proposed method of quantizing general nonLagrangian gauge theories. The method can be implemented in many different ways, in particular, it can employ a conversion procedure that turns an original nonLagrangian field theory in d dimensions into an equivalent Lagrang ..."
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Abstract. We discuss a recently proposed method of quantizing general nonLagrangian gauge theories. The method can be implemented in many different ways, in particular, it can employ a conversion procedure that turns an original nonLagrangian field theory in d dimensions into an equivalent Lagrangian, topological field theory in d+1 dimensions. The method involves, besides the classical equations of motion, one more geometric ingredient called the Lagrange anchor. Different Lagrange anchors result in different quantizations of one and the same classical theory. Given the classical equations of motion and Lagrange anchor as input data, a new procedure, called the augmentation, is proposed to quantize nonLagrangian dynamics. Within the augmentation procedure, the originally nonLagrangian theory is absorbed by a wider Lagrangian theory on the same spacetime manifold. The augmented theory is not generally equivalent to the original one as it has more physical degrees of freedom than the original theory. However, the extra degrees of freedom are factorized out in a certain regular way both at classical and quantum levels. The general techniques are exemplified by quantizing two nonLagrangian models of physical interest.
Lagrange structure and quantization
 JHEP
"... Abstract. A pathintegral quantization method is proposed for dynamical systems whose classical equations of motion do not necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange structure which is more general than the Lagrangian formalism in ..."
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Abstract. A pathintegral quantization method is proposed for dynamical systems whose classical equations of motion do not necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange structure which is more general than the Lagrangian formalism in the same sense as Poisson geometry is more general than the symplectic one. The Lagrange structure is shown to admit a natural BRST description which is used to construct an AKSZtype topological sigmamodel. The dynamics of this sigmamodel in d+1 dimensions, being localized on the boundary, are proved to be equivalent to the original theory in d dimensions. As the topological sigmamodel has a well defined action, it is pathintegral quantized in the usual way that results in quantization of the original (not necessarily Lagrangian) theory. When the original equations of motion come from the action principle, the standard BV pathintegral is explicitly deduced from the proposed quantization scheme. The general quantization scheme is exemplified by several models including the ones whose classical dynamics are not variational. 1.
From Topological Field Theory to Deformation Quantization and Reduction
 Proceedings of ICM 2006, Vol. III, 339365 (European Mathematical Society
, 2006
"... Abstract. This note describes the functionalintegral quantization of twodimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the B ..."
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Abstract. This note describes the functionalintegral quantization of twodimensional topological field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded manifolds and to the Batalin–Vilkovisky formalism is included.