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Bundle gerbes
 J. London Math. Soc
, 1996
"... Abstract. An introduction to the theory of bundle gerbes and their relationship to HitchinChatterjee gerbes is presented. Topics covered are connective ..."
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Cited by 140 (9 self)
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Abstract. An introduction to the theory of bundle gerbes and their relationship to HitchinChatterjee gerbes is presented. Topics covered are connective
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 50 (3 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
From loop groups to 2groups
 HHA
"... We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2gr ..."
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We describe an interesting relation between Lie 2algebras, the Kac– Moody central extensions of loop groups, and the group String(n). A Lie 2algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the ‘Jacobiator’. Similarly, a Lie 2group is a categorified version of a Lie group. If G is a simplyconnected compact simple Lie group, there is a 1parameter family of Lie 2algebras gk each having g as its Lie algebra of objects, but with a Jacobiator built from the canonical 3form on G. There appears to be no Lie 2group having gk as its Lie 2algebra, except when k = 0. Here, however, we construct for integral k an infinitedimensional Lie 2group PkG whose Lie 2algebra is equivalent to gk. The objects of PkG are based paths in G, while the automorphisms of any object form the levelk Kac– Moody central extension of the loop group ΩG. This 2group is closely related to the kth power of the canonical gerbe over G. Its nerve gives a topological group PkG  that is an extension of G by K(Z, 2). When k = ±1, PkG  can also be obtained by killing the third homotopy group of G. Thus, when G = Spin(n), PkG  is none other than String(n). 1 1
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
CATEGORIFIED SYMPLECTIC GEOMETRY AND THE STRING LIE 2ALGEBRA
"... Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for ndimensional field theories, in which the nondegenerate 2form of symplectic geometry is replaced by a nondegenerate (n + 1)form. The case n = 2 is relevant to string theory: we call this ‘2plectic geometry ..."
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Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for ndimensional field theories, in which the nondegenerate 2form of symplectic geometry is replaced by a nondegenerate (n + 1)form. The case n = 2 is relevant to string theory: we call this ‘2plectic geometry’. Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2plectic manifold form a ‘Lie 2algebra’, which is a categorified version of a Lie algebra. Any compact simple Lie group G has a canonical 2plectic structure, so it is natural to wonder what Lie 2algebra this example yields. This Lie 2algebra is infinitedimensional, but we show here that the subLie2algebra of leftinvariant observables is finitedimensional, and isomorphic to the already known ‘string Lie 2algebra ’ associated to G. So, categorified symplectic geometry gives a geometric construction of the string Lie 2algebra. 1.
L∞algebra connections and applications to String and ChernSimons ntransport
, 2008
"... We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspect ..."
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We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a Dbrane the KalbRamond background field of the string restricts to a 2bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)bundle on the Dbrane to a U(H)bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the Stringextension BU(1) → String(G) → G. Here the obstruction to the lift is a 3bundle with connection (a 2gerbe): the ChernSimons 3bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a Stringstructure. We discuss how to describe this obstruction problem in terms of Lie nalgebras and their corresponding categorified CartanEhresmann connections. Generalizations even beyond Stringextensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized ChernSimons 7bundles classified by the second Pontrjagin class.
Central extensions of smooth 2groups and a finitedimensional string 2group, Geom. Topol. 15 (2011) 609–676
 Department of Mathematics, Massachusetts Institute of Technology
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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).