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Differentiable Stacks and Gerbes
, 2008
"... We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹ ..."
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Cited by 14 (3 self)
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We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study S¹bundles and S¹gerbes over differentiable stacks. In particular, we establish the relationship between S¹gerbes and groupoid S¹central extensions. We define connections and curvings for groupoid S¹central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for S¹gerbes over manifolds. We develop a ChernWeil theory of characteristic classes in this general setting by presenting a construction of Chern classes and DixmierDouady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both S¹bundles and S¹gerbes extending the wellknown result of Weil and Kostant. In particular, we give an explicit construction of S¹central extensions with prescribed curvaturelike data.
A GERBE FOR THE ELLIPTIC GAMMA FUNCTION
, 2006
"... The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3dimensional space. The language of stacks and gerbes gives a natural framework for a systematic descr ..."
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Cited by 14 (4 self)
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The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three (it is a stack). Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curve.
Integrating Poisson manifolds via stacks
, 2004
"... A symplectic groupoid G.: = (G1 ⇉ G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita ..."
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Cited by 2 (0 self)
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A symplectic groupoid G.: = (G1 ⇉ G0) determines a Poisson structure on G0. In this case, we call G. a symplectic groupoid of the Poisson manifold G0. However, not every Poisson manifold M has such a symplectic groupoid. This keeps us away from some desirable goals: for example, establishing Morita equivalence in the category of all Poisson manifolds. In this paper, we construct symplectic Weinstein groupoids which provide a solution to the above problem (Theorem 1.1). More precisely, we show that a symplectic Weinstein groupoid induces a Poisson structure on its base manifold, and that to every Poisson manifold there is an associated symplectic Weinstein groupoid. 1
Orbivariant Ktheory
, 2006
"... Orbispaces are spaces with extra structure. The main examples come from topological group actions X G and are denoted [X/G], their underlying space, or coarse moduli space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. We shall work with orbispace ..."
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Orbispaces are spaces with extra structure. The main examples come from topological group actions X G and are denoted [X/G], their underlying space, or coarse moduli space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. We shall work with orbispaces whose coarse moduli spaces are CWcomplexes, and whose stabilizer groups are compact Lie groups. We also require the stratification of the coarse moduli space by the type of stabilizer group to be compactible with the CW structure. A convenient model for an orbispace is then given by a topological groupoid (see [2], [3]). An orbispace always comes with a map to its coarse moduli space. By a suborbispace X ′ ⊂ X, we shall mean an orbispace obtained by pulling back along a subspace of the coarse moduli space. If X is an orbispace modeled by a topological groupoid G, then a vector bundle over X is a vector bundle over the space of objects of G equipped with an action of the arrows of G. It is tempting to define Ktheory as the Grothendieck
FOUNDATIONS OF TOPOLOGICAL STACKS I
, 2005
"... Abstract. This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as far as introducing the homotopy groups and establish ..."
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Abstract. This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as far as introducing the homotopy groups and establishing their basic properties. We also develop a Galois theory of covering spaces for a (locally connected semilocally 1connected) topological stack. Built into the Galois theory is a method for determining the stacky structure (i.e., inertia groups) of covering stacks. As a consequence, we get for free a characterization of topological stacks that are quotients of topological spaces by discrete group actions. For example, this give a handy characterization of good orbifolds. Orbifolds, graphs of groups, and complexes of groups are examples of topological (DeligneMumford) stacks. We also show that any algebraic stack (of finite type over C) gives rise to a topological stack. We also prove a Riemann Existence Theorem for stacks. In particular, the algebraic fundamental group
Contents
, 704
"... Abstract. Topological Tduality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dualtorus bundle. We give a new geometric construction of Tdualization, which allows the duality to be extended to the following situations: bundles of groups other than tori, e ..."
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Abstract. Topological Tduality is a transformation taking a gerbe on a principal torus bundle to a gerbe on a principal dualtorus bundle. We give a new geometric construction of Tdualization, which allows the duality to be extended to the following situations: bundles of groups other than tori, even bundles of some nonabelian groups, can be dualized; bundles whose duals are families of noncommutative groups (in the sense of noncommutative geometry) can be treated; and the base manifold parameterizing the bundles may be replaced by a topological stack. Some methods developed for the construction may be of independent interest: these are a Pontryagin type duality between commutative principal bundles and gerbes, nonabelian Takai duality for groupoids, and the computation of certain equivariant Brauer groups.
On state sums, internalisation and unification
, 2004
"... In this mostly expository article, elements of higher category theory essential to the construction of a class of four dimensional quantum geometric models are reviewed. These models improve current state sum models for Quantum Gravity, such as the BarrettCrane model [1], in that they appear, for i ..."
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In this mostly expository article, elements of higher category theory essential to the construction of a class of four dimensional quantum geometric models are reviewed. These models improve current state sum models for Quantum Gravity, such as the BarrettCrane model [1], in that they appear, for instance, to remove degeneracies which swamp the partition function. Much work remains to be done before a complete construction is reached, but the crucial categorical notion of internalisation already illuminates the idea that a full unified model may result from few, albeit as yet poorly understood, additional principles. In particular, a spacetime and matter duality principle is employed through an understanding of the role of pseudomonoidal objects
On state sums, internalisation and unification
, 2004
"... In this mostly expository article, elements of higher category theory essential to the construction of a class of four dimensional quantum geometric models are reviewed. These models improve current state sum models for Quantum Gravity, such as the BarrettCrane model [3], in that they appear, for i ..."
Abstract
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In this mostly expository article, elements of higher category theory essential to the construction of a class of four dimensional quantum geometric models are reviewed. These models improve current state sum models for Quantum Gravity, such as the BarrettCrane model [3], in that they appear, for instance to remove degeneracies which swamp the partition function. Much work remains to be done before a complete construction is reached, but the crucial categorical notion of internalisation already illuminates the idea that a full unified model may result from few, albeit as yet poorly understood, additional principles. In particular, a spacetime and matter duality principle is employed through an understanding of the role of pseudomonoidal objects in categorified
Vector bundles on Orbispaces
, 2005
"... Orbispaces are spaces with extra structure. The main examples come from compact Lie group actions X G and are denoted [X/G], their underlying space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. To be more precise, an orbispace is a topological st ..."
Abstract
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Orbispaces are spaces with extra structure. The main examples come from compact Lie group actions X G and are denoted [X/G], their underlying space being X/G. By definition, every orbispace is locally of the form [X/G], but the group G might vary. To be more precise, an orbispace is a topological stack which is locally equivalent to [X/G] for G a Lie group and X a GCWcomplex [2] [4] [5]. However, if all the stabilizer groups are finite, there exists a more concrete alternative definition [1]. Definition 1 An orbispace is a map p: E → Y satisfying the following condition: there exists an open cover {Ui} of Y and finite group actions Xi Gi such that p −1 (Ui) → Ui is fiberwise homotopy equivalent to (Xi ×EGi)/Gi → Xi/Gi. The space Y is called the coarse moduli space, and the space E is called the total space. A morphism of orbispaces (E, Y) → (E ′ , Y ′ ) is a commutative diagram: E Y