  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.
