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

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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 H4(BG, ℤ) 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 H4(BG, ℤ) to H3(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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Authors and Affiliations

  1. Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia

    Alan L. Carey

  2. Department of Pure Mathematics, University of Adelaide, Adelaide, SA, 5005, Australia

    Stuart Johnson & Michael K. Murray

  3. Department of Mathematics, University of California at Riverside, 202 Surge Building, Riverside, CA, 92521-0135, USA

    Danny Stevenson

  4. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Bai-Ling Wang

Authors
  1. Alan L. Carey
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  2. Stuart Johnson
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  3. Michael K. Murray
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  4. Danny Stevenson
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  5. Bai-Ling Wang
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Correspondence to Alan L. Carey.

Additional information

Communicated by A. Connes

The authors acknowledge the support of the Australian Research Council. ALC thanks MPI für Mathematik in Bonn and ESI in Vienna and BLW thanks CMA of Australian National University for their hospitality during part of the writing of this paper.

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Carey, A., Johnson, S., Murray, M. et al. Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories. Commun. Math. Phys. 259, 577–613 (2005). https://doi.org/10.1007/s00220-005-1376-8

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  • DOI: https://doi.org/10.1007/s00220-005-1376-8

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