Polyakov-Wiegmann Formula and Multiplicative Gerbes

TL;DR

This paper explores the geometric structures underlying Feynman amplitudes in Wess-Zumino-Witten and Chern-Simons theories, focusing on gerbes and their multiplicative properties related to the Polyakov-Wiegmann formula.

Contribution

It identifies the obstruction to multiplicative gerbes for simple compact Lie groups as a 2-cocycle, linking it to the Polyakov-Wiegmann phases, and establishes conditions for their existence and uniqueness.

Findings

Obstruction to multiplicative gerbes is given by a 2-cocycle of phases.

Polyakov-Wiegmann phases are computed for all compact simple Lie groups.

Existence and uniqueness of multiplicative gerbes depend on triviality of these phases.

Abstract

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. If they are trivial, then the multiplicative structure exists and is unique up to isomorphism.