Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories

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 $H^4(BG, \ZZ)$ for a compact semi-simple Lie group $G$. The Chern-Simons 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-Zumino-Witten models associated to the group $G$. We do this by introducing a lifting to the level of bundle gerbes of the natural map from $H^4(BG, \ZZ)$ to $H^3(G, \ZZ)$. 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.

35 pages, xy-pic diagrams, published version

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Cited by in corpus (51)

• Higher Gauge Theory: 2-Connections on 2-Bundles
• Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group
• A higher stacky perspective on Chern-Simons theory
• Extended higher cup-product Chern-Simons theories
• L-infinity algebra connections and applications to String- and Chern-Simons n-transport
• Notes on topological insulators
• Twenty-five years of two-dimensional rational conformal field theory
• Twisted Cohomotopy implies twisted String structure on M5-branes
• From Loop Groups to 2-Groups
• A Construction of String 2-Group Models using a Transgression-Regression Technique
• Fusion of symmetric $D$-branes and Verlinde rings
• Spin structures on loop spaces that characterize string manifolds
• From Loop Space Mechanics to Nonabelian Strings
• Higher U(1)-gerbe connections in geometric prequantization
• Generalized Kahler geometry and gerbes
• Polyakov-Wiegmann Formula and Multiplicative Gerbes
• String geometry vs. spin geometry on loop spaces
• Abelian BF theory and Turaev-Viro invariant
• Hamiltonian anomalies from extended field theories
• Transgressive loop group extensions
• Massey products in differential cohomology via stacks
• Ninebrane structures
• Global gauge anomalies in two-dimensional bosonic sigma models
• The caloron correspondence and higher string classes for loop groups
• String Connections and Chern-Simons Theory
• Categorified Symplectic Geometry and the String Lie 2-Algebra
• The gauging of two-dimensional bosonic sigma models on world-sheets with defects
• Differential Twisted K-theory and Applications
• Defects, dualities and the geometry of strings via gerbes. I. Dualities and state fusion through defects
• Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation
• Thom isomorphism and Push-forward map in twisted K-theory
• Bicommutant categories from conformal nets
• Loop Groups, Higgs Fields and Generalised String Classes
• Courant Algebroids in Parabolic Geometry
• Instanton Density Operator in Lattice QCD from Higher Category Theory
• Affine su(2) fusion rules from gerbe 2-isomorphisms
• Categorical Tori
• Sign choices for orientifolds
• Geometric cycles, index theory and twisted K-homology
• The ring structure for equivariant twisted K-theory
• Chern correspondence for higher principal bundles
• Bigerbes
• Explicit Non-Abelian Gerbes with Connections
• Bundle gerbes and the Weyl map
• Abelian Topological Order on Lattice Enriched with Electromagnetic Background
• Twisted Weil Algebras for the String Lie 2-Algebra
• Equivariant Cartan-Eilenberg supergerbes II. Equivariance in the super-Minkowskian setting
• Many finite-dimensional lifting bundle gerbes are torsion
• Gerbes in Geometry, Field Theory, and Quantisation
• Higher Structures, Self-Dual Strings and 6d Superconformal Field Theories
• Geometry of infinite dimensional Grassmannians and the Mickelsson-Rajeev cocycle