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Polyakov-Wiegmann Formula and Multiplicative Gerbes · Around

2009

Polyakov-Wiegmann Formula and Multiplicative Gerbes

Gawedzki, Krzysztof, Waldorf, Konrad

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An unambiguous definition of Feynman amplitudes in the Wess-Zumino-Witten sigma model and the Chern-Simon gauge theory with a general Lie group is determined by a certain geometric structure on the group.

For the WZW amplitudes, this is a (bundle) gerbe with connection of an appropriate curvature whereas for the CS amplitudes, the gerbe has to be additionally equipped with a multiplicative structure assuring its compatibility with the group multiplication.
We show that for simple compact Lie groups the obstruction to the existence of a multiplicative structure is provided by a 2-cocycle of phases that appears in the Polyakov-Wiegmann formula relating the Wess-Zumino action functional of the product of group-valued fields to the sum of the individual contributions.
These phases were computed long time ago for all compact simple Lie groups.

Built on

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1975
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1982
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L. D. Faddeev and S. L. Shatashvili
1984
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A. M. Polyakov and P. B. Wiegmann
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E. Witten
1984
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D. Gepner and E. Witten
1986
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G. Felder, K. Gawȩdzki and A. Kupiainen
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K. Gawȩdzki
1988
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G. Moore and N. Seiberg
1989
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