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19
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
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Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,
Moduli of objects in dgcategories
, 2006
"... To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness ..."
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To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −stack MT is locally geometric (i.e. union of open and geometric substacks). As a consequence we prove the algebraicity of the group of autoequivalences of saturated dgcategories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as
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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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
A note on Chern character, loop spaces and derived algebraic geometry
, 2008
"... In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of Omodules on schemes, as well as its quasicoherent and p ..."
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In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of Omodules on schemes, as well as its quasicoherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.
The theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface
, 2002
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DIFFERENTIAL CHARACTERS AS STACKS AND PREQUANTIZATION
, 2008
"... We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlasindependent and provides a bijection between isomorphism classes of principal S 1bundles (with or without connections) and second cohomology groups of certain chain complexes. ..."
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We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlasindependent and provides a bijection between isomorphism classes of principal S 1bundles (with or without connections) and second cohomology groups of certain chain complexes.
A SURVEY OF (∞, 1)CATEGORIES
, 2006
"... Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy ..."
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Abstract. In this paper we give a summary of the comparisons between different definitions of socalled (∞,1)categories, which are considered to be models for ∞categories whose nmorphisms are all invertible for n> 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasicategories. 1.
The Character Theory of a Complex Group
"... Abstract. We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. We first establish good functoriality properties for ordinary and equivariant Dmodules on schemes, in particular showing that the integral transforms studied in ..."
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Abstract. We apply the ideas of derived algebraic geometry and topological field theory to the representation theory of reductive groups. We first establish good functoriality properties for ordinary and equivariant Dmodules on schemes, in particular showing that the integral transforms studied in algebraic analysis give all continuous functors on Dmodules. We then focus on the categorified Hecke algebra HG of Borel biequivariant Dmodules on a complex reductive group G. We show that its monoidal center and abelianization (Hochschild cohomology and homology categories) coincide and are identified through the Springer correspondence with the derived version of Lusztig’s character sheaves. We further show that HG is a categorified CalabiYau algebra, and thus satisfies the strong dualizability conditions of Lurie’s proof of the cobordism hypothesis. This implies that HG defines (the (0,1,2)dimensional part of) a threedimensional topological field theory which we call the character theory χG. It organizes much of the representation theory associated to G. For example, categories of Lie algebra representations and Harish Chandra modules for G and its real forms give natural boundary conditions in the theory. In particular, they have characters (or charges) as Hecke modules which are character sheaves. The Koszul duality for Hecke categories provides an equivalence between character theories for Langlands dual groups, and in particular a duality of character sheaves. It can be viewed as a dimensionally reduced version of the geometric Langlands correspondence, or as Sduality for a generically twisted maximally supersymmetric gauge theory in three dimensions.
Loop spaces and Langlands parameters
, 2007
"... Abstract. We apply the technique of S 1equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free l ..."
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Abstract. We apply the technique of S 1equivariant localization to sheaves on loop spaces in derived algebraic geometry, and obtain a fundamental link between two families of categories at the heart of geometric representation theory. Namely, we categorify the well known relationship between free loop spaces, cyclic homology and de Rham cohomology to
Holomorphic generating functions for invariants
, 2006
"... counting coherent sheaves on Calabi–Yau 3folds ..."