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44
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 190 (4 self)
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Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, DuggerShipley,..., Toën and ToënVaquié.
Two kinds of derived categories, Koszul duality, and comodulecontramodule correspondence
, 2009
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Twisted differential String and Fivebrane structures
, 2009
"... Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten an ..."
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Cited by 26 (21 self)
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Abelian differential generalized cohomology as developed by Hopkins and Singer has been shown by Freed to formalize the global description of anomaly cancellation problems in String theory, such as notably the GreenSchwarz mechanism. On the other hand, this mechanism, as well as the FreedWitten anomaly cancellation, are fundamentally governed by the cohomology classes represented by the relevant nonabelian O(n) and U(n)principal bundles underlying the tangent and the gauge bundle on target space. In this article we unify the picture by describing nonabelian differential cohomology and twisted nonabelian differential cohomology and apply it to these situations. We demonstrate that the FreedWitten mechanism for the Bfield, the GreenSchwarz mechanism for the H3field, as well as its magnetic dual version for the H7field define cocycles in twisted nonabelian differential cohomology that may be addressed, respectively, as twisted Spin(n), twisted String(n) and twisted Fivebrane(n)structures on target space, where the twist in each case is provided by the obstruction to lifting the gauge bundle through a higher connected cover of U(n). We work out the (nonabelian) L∞algebra valued connection data provided by the differential refinements of these twisted cocycles and demonstrate that this reproduces locally the differential form data with the twisted Bianchi identities as known from the
Deformation quantization of gerbes, I
"... This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as MaurerCartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic man ..."
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Cited by 24 (7 self)
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This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as MaurerCartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformationtheoretic interpretation of the first RozanskyWitten class. 1.
The Van den Bergh duality and the modular symmetry of a Poisson variety
"... We consider a smooth Poisson affine variety with the trivial canonical bundle over C. For such a variety the deformation quantization algebra A � enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A � intrinsic ..."
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Cited by 21 (1 self)
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We consider a smooth Poisson affine variety with the trivial canonical bundle over C. For such a variety the deformation quantization algebra A � enjoys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of A � intrinsic to A �. We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra A � is free if and only if the corresponding Poisson structure is unimodular. 1
Threedimensional topological field theory and symplectic algebraic geometry
 I, Nuclear Physics B
"... To our parents Abstract. Motivated by the path integral analysis [KRS09] of boundary conditions in a 3dimensional topological sigmamodel, we suggest a definition of the 2category ¨ L(X) associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of ¨ L(X) ar ..."
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Cited by 19 (6 self)
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To our parents Abstract. Motivated by the path integral analysis [KRS09] of boundary conditions in a 3dimensional topological sigmamodel, we suggest a definition of the 2category ¨ L(X) associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of ¨ L(X) are holomorphic lagrangian submanifolds Y ⊂ X. We pay special attention to the case when X is the total space of the cotangent bundle of a complex manifold U or a deformation thereof. In the latter case the endomorphism category of the zero section is
Poisson geometry and deformation quantization near a strictly pseudoconvex boundary
, 2006
"... Let X be a complex manifold with strongly pseudoconvex boundary M. If ψ is a defining function for M, then −log ψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2form σ = i∂∂( − log ψ) is a symplectic structure on the complement of M in a neighborhood in X of M; it blows up along M ..."
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Cited by 10 (2 self)
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Let X be a complex manifold with strongly pseudoconvex boundary M. If ψ is a defining function for M, then −log ψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2form σ = i∂∂( − log ψ) is a symplectic structure on the complement of M in a neighborhood in X of M; it blows up along M. The Poisson structure obtained by inverting σ extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure near M is completely determined up to isomorphism by the contact structure on M. In addition, when −log ψ is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Engliˇs for the BerezinToeplitz deformation quantization of X are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on M, along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.
Twisting Derived Equivalences
, 2006
"... I would like to thank my parents, Arie and Elayne, and my sister Daniella, for their support throughout my life and for being helpful to me whenever I had any problems. Thank you to all my fellow graduate students at (or visiting) the University of Pennsylvania, for friendship and interesting mathem ..."
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Cited by 9 (2 self)
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I would like to thank my parents, Arie and Elayne, and my sister Daniella, for their support throughout my life and for being helpful to me whenever I had any problems. Thank you to all my fellow graduate students at (or visiting) the University of Pennsylvania, for friendship and interesting mathematical discussions. Thanks to
The integration problem for complex Lie algebroids
, 2006
"... A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic con ..."
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Cited by 8 (1 self)
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A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse examples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in complex geometry. This suggests that the object integrating a complex Lie algebroid should be a holomorphic stack.