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72
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 44 (4 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.
Thom isomorphism and Pushforward map in twisted Ktheory
"... Abstract. We establish the Thom isomorphism in twisted Ktheory for any real vector bundle and develop the pushforward map in twisted Ktheory for any differentiable proper map f: X → Y (not necessarily Koriented). The pushforward map generalizes the pushforward map in ordinary Ktheory for any K ..."
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Cited by 38 (5 self)
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Abstract. We establish the Thom isomorphism in twisted Ktheory for any real vector bundle and develop the pushforward map in twisted Ktheory for any differentiable proper map f: X → Y (not necessarily Koriented). The pushforward map generalizes the pushforward map in ordinary Ktheory for any Koriented differentiable proper map and the AtiyahSinger index theorem of Dirac operators on Clifford modules. For Dbranes satisfying FreedWitten’s anomaly cancellation condition in a manifold with a nontrivial Bfield, we associate a canonical element in the twisted Kgroup to get the socalled Dbrane charges. Contents
Topological and smooth stacks
"... Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are ..."
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Cited by 19 (0 self)
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Abstract. We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a “stack over a stack ” and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces). 1.
GROUPOID COHOMOLOGY AND EXTENSIONS
, 2006
"... Abstract. We show that Haefliger’s cohomology for étale groupoids, Moore’s cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Čech) cohomology for topological simplicial spaces. 1. ..."
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Cited by 18 (4 self)
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Abstract. We show that Haefliger’s cohomology for étale groupoids, Moore’s cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Čech) cohomology for topological simplicial spaces. 1.
Momentum Maps and Morita Equivalence
, 2003
"... We introduce quasisymplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including the ordinary Hamiltonian Gspaces, Lu’s momentum maps of Poisson group actions, and group valued momentum ..."
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Cited by 14 (3 self)
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We introduce quasisymplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including the ordinary Hamiltonian Gspaces, Lu’s momentum maps of Poisson group actions, and group valued momentum maps of Alekseev–Malkin–Meinrenken. More precisely, we carry out the following program: (1) Define and study properties of quasisymplectic groupoids; (2) Study the momentum map theory defined by a quasisymplectic groupoid Γ ⇒ P. In particular, we study the reduction theory and prove that J −1 (O)/Γ is a symplectic manifold (even though ωX ∈ Ω 2 (X) may be degenerate), where O ⊂ P is a groupoid orbit. More generally, we prove that the intertwiner space (X1 ×P X2)/Γ between two Hamiltonian Γspaces X1 and X2 is a symplectic manifold (whenever it is a smooth manifold); (3) Study Morita equivalence of quasisymplectic groupoids. In particular, we prove that Morita equivalent quasisymplectic groupoids give rise to equivalent momentum map theories. Moreover the intertwiner space (X1 ×P X2)/Γ is independent of the Morita equivalence. As a result, we recover various wellknown results concerning equivalence of momentum maps including the AlekseevGinzburgWeinstein linearization theorem and the Alekseev–Malkin–Meinrenken equivalence theorem between quasiHamiltonian spaces and Hamiltonian loop group spaces.
A Stringy product on twisted orbifold Ktheory
"... Abstract. In this paper we define an associative stringy product for the twisted orbifold K–theory of a compact, almost complex orbifold X. This product is defined on the twisted K–theory τ Korb(∧X) of the inertia orbifold ∧X, where the twisting gerbe τ is assumed to be in the image of the inverse t ..."
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Cited by 13 (0 self)
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Abstract. In this paper we define an associative stringy product for the twisted orbifold K–theory of a compact, almost complex orbifold X. This product is defined on the twisted K–theory τ Korb(∧X) of the inertia orbifold ∧X, where the twisting gerbe τ is assumed to be in the image of the inverse transgression H 4 (BX, Z) → H 3 (B ∧ X, Z). 1.
The ring structure for equivariant twisted Ktheory
, 2006
"... We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1: H ∗ (Γ•; A) → H∗−1 ((N ⋊ Γ) •; A) for any crossed module N ..."
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Cited by 13 (0 self)
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We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1: H ∗ (Γ•; A) → H∗−1 ((N ⋊ Γ) •; A) for any crossed module N → Γ and prove that any element in the image is ∞multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module N → Γ and any e ∈ ˇ Z3 (Γ•; S1), that (N) admits a ring structure. As an applithe equivariant twisted Ktheory group K ∗ e,Γ cation, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted Ktheory group K ∗ [c],G(G) is endowed with a canonical ring structure K i+d [c],G(G)⊗Kj+d [c],G(G) → Ki+j+d [c],G (G), where d = dim G and [c] ∈ H2 ((G⋊G) •; S1).
Continuous representations of groupoids
, 2007
"... Abstract. We introduce unitary representations of continuous groupoids on continuous fields of Hilbert spaces. We investigate some properties of these objects, using several examples. We present a palette of results, including, among others: a comparison of the different notions of continuity for r ..."
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Cited by 11 (0 self)
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Abstract. We introduce unitary representations of continuous groupoids on continuous fields of Hilbert spaces. We investigate some properties of these objects, using several examples. We present a palette of results, including, among others: a comparison of the different notions of continuity for representations, a description of the representations of families of groups, and a version of the PeterWeyl theorem for groupoids. 1.
Geometric quantization of Hamiltonian actions of Lie algebroids and Lie groupoids
 Int. J. Geom. Methods Mod. Phys
"... We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a p ..."
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Cited by 9 (1 self)
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We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a MarsdenWeinstein quotient for our setting and prove a “quantization commutes with reduction ” theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Finally, we investigate the functorial behaviour of these constructions under generalized morphisms of groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of families of Lie groups, and foliations, as well as some general constructions from differential geometry.