Twisted K-Theory and K-Theory of Bundle Gerbes

Bouwknegt, Peter; Carey, Alan L.; Mathai, Varghese; Murray, Michael K.; Stevenson, Danny

doi:10.1007/s002200200646

Cited by 191 publications

(409 citation statements)

References 32 publications

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“…which is a 2-morphism in the bi-category BGrb X [4] , between the induced stable isomorphisms of bundle gerbes over X [4] making the following diagram commute up to an associator φ (represented by a 2-arrow in the diagram): Here we write π * 1 m as a stable isomorphism Q 34 ⊗ Q 23 → Q 24 over X [4] , similarly for π * 2 m, π * 3 m and π * 4 m. Hence, the associator φ as a 2-morphism in BGrb X [4] defines a line bundle L φ over X [4] , which is required to have a trivialisation section.…”

Section: Bundle 2-gerbesmentioning

confidence: 99%

“…Let us write Q ijk = Q jk ⊗ Q ij , Q ijkl = Q kl ⊗ Q jk ⊗ Q ij and so on. So for example, Q 123 = Q 23 ⊗ Q 12 , Q 1234 = Q 34 ⊗ Q 23 ⊗ Q 12 and the diagram (4.2) in BGrb X [4] can be written as…”

Section: Bundle 2-gerbesmentioning

confidence: 99%

“…G) for a bundle gerbe G over X together with compatible conditions on bundle 2-gerbe products in BGrb X [3] and associator natural transformations in BGrb X [4] (see [44] for more details). A bundle 2-gerbe (Q 1 , Y 1 ; X, M ) is called stably isomorphic to a bundle 2-gerbe (Q 2 , Y 2 ; X, M ) if and only if Q 1 is isomorphic to Q 2 ⊗ δ(G) for a bundle gerbe G over X together with extra conditions involving the associator natural isomorphisms for Q 1 and Q 2 .…”

Section: Bundle 2-gerbesmentioning

confidence: 99%

“…Recall that there is a bundle called the associator bundle on X [4] which is the obstruction to the bundle 2-gerbe product being associative. It can be defined by considering the product [3] are the face maps in the simplicial complex.…”

Section: Bundle 2-gerbesmentioning

confidence: 99%

“…(2) A stable isomorphism m: π * 1 Q⊗π * 3 Q → π * 2 Q in BGrb X [3] defining the bundle 2-gerbe product which is associative up to a 2-morphism φ in BGrb X [4] . (3) The 2-morphism φ satisfies a natural coherency condition in BGrb X [5] .…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

Carey

Johnson

Murray

et al. 2005

Commun. Math. Phys.

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146

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Abstract. We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal G-bundle with connection and a class in H 4 (BG, Z) for a compact semi-simple Lie group G. The ChernSimons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-ZuminoWitten models associated to the group G. We do this by introducing a lifting to the level of bundle gerbes of the natural map from H 4 (BG, Z) to H 3 (G, Z). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.

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Section: Bundle 2-gerbesmentioning

confidence: 99%

Section: Bundle 2-gerbesmentioning

confidence: 99%

Section: Bundle 2-gerbesmentioning

confidence: 99%

Section: Bundle 2-gerbesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

Carey

Johnson

Murray

et al. 2005

Commun. Math. Phys.

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146

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Applying geometricK-cycles to fractional indices

Deeley

Goffeng

2017

Math. Nachr.

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A geometric model for twisted K‐homology is introduced. It is modeled after the Mathai–Melrose–Singer fractional analytic index theorem in the same way as the Baum–Douglas model of K‐homology was modeled after the Atiyah–Singer index theorem. A natural transformation from twisted geometric K‐homology to the new geometric model is constructed. The analytic assembly mapping to analytic twisted K‐homology in this model is an isomorphism for torsion twists on a finite CW‐complex. For a general twist on a smooth manifold the analytic assembly mapping is a surjection. Beyond the aforementioned fractional invariants, we study T‐duality for geometric cycles.

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The BV-algebra structure of W 3 cohomology

Bouwknegt

Pilch

Strings and Symmetries

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We summarize some recent results obtained in collaboration with J. McCarthy on the spectrum of physical states in W 3 gravity coupled to c = 2 matter. We show that the space of physical states, defined as a semi-infinite (or BRST) cohomology of the W 3 algebra, carries the structure of a BV-algebra. This BV-algebra has a quotient which is isomorphic to the BV-algebra of polyvector fields on the base affine space of SL(3, C). Details will appear elsewhere.Throughout this paper we will use the notation h for the Cartan subalgebra, h * Z for the set of integral weights, P + for the set of dominant integral weights, P ++ for the set of strictly dominant integral weights, ∆ + for the positive roots and W for the Weyl group of some Lie algebra g. L(Λ) will denote the finite dimensional irreducible representation of g with highest weight Λ ∈ P + and ℓ(w) the length of w ∈ W . In the following g will always refer to sl 3 . The W 3 algebra and its modulesThe W 3 algebra with central charge c ∈ C (denoted simply by W in the sequel) is defined as the quotient of the free Lie algebra generated by L m , W m , m ∈ Z, by the ideal generated by the following commutation

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Twisted K-Theory and K-Theory of Bundle Gerbes

Abstract: Abstract. In this note we introduce the notion of bundle gerbe K-theory and investigate the relation to twisted K-theory. We provide some examples. Possible applications of bundle gerbe K-theory to the classification of D-brane charges in nontrivial backgrounds are briefly discussed.

Cited by 191 publications

References 32 publications

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

Applying geometricK-cycles to fractional indices

The BV-algebra structure of W 3 cohomology

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