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

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arxiv math/0410013 v2 pith:WSO2Z3UJ submitted 2004-10-01 math.DG math-phmath.MP

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

Alan L. Carey , Stuart Johnson , Michael K. Murray , Danny Stevenson , Bai-Ling Wang This is my paper
classification math.DG math-phmath.MP
keywords bundlechern-simonsgerbeswess-zumino-wittenassociatedcorrespondencegeometricallygerbe
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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.

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