Abstract
We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.
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Acknowledgements
The authors acknowledge the support of the contract ANR-05-BLAN-0029-03 in the early stage of this collaboration. The work of K.W. was supported by a Feodor-Lynen scholarship granted by the Alexander von Humboldt Foundation. That of R.R.S. was partially funded by the Collaborative Research Centre 676 “Particles, Strings and the Early Universe - the Structure of Matter and Space-Time”.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Gawȩdzki, K., Suszek, R.R. & Waldorf, K. Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models. Commun. Math. Phys. 302, 513–580 (2011). https://doi.org/10.1007/s00220-010-1162-0
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DOI: https://doi.org/10.1007/s00220-010-1162-0