Coadjoint Orbits, Cocycles and Gravitational Wess-Zumino

TL;DR

This paper explores the geometric actions on coadjoint orbits, revealing their cocycle properties, constructing Wess-Zumino terms for central extensions, and deriving a Polyakov-Wiegmann formula that explains bulk-boundary decoupling phenomena in related models.

Contribution

It demonstrates that geometric actions are 1-cocycles for loop groups, constructs Wess-Zumino terms for central extensions, and derives a Polyakov-Wiegmann formula for gravitational WZ actions.

Findings

Geometric actions are 1-cocycles for loop groups.

Wess-Zumino terms are constructed for central extensions.

Bulk-boundary decoupling is explained via cocycle properties.

Abstract

About 30 years ago, in a joint work with L. Faddeev we introduced a geometric action on coadjoint orbits. This action, in particular, gives rise to a path integral formula for characters of the corresponding group $G$. In this paper, we revisit this topic and observe that the geometric action is a 1-cocycle for the loop group $LG$. In the case of $G$ being a central extension, we construct Wess-Zumino (WZ) type terms and show that the cocycle property of the geometric action gives rise to a Polyakov-Wiegmann (PW) formula. In particular, we obtain a PW type formula for the Polyakov's gravitational WZ action. After quantization, this formula leads to an interesting bulk-boundary decoupling phenomenon previously observed in the WZW model. We explain that this decoupling is a general feature of the Wess-Zumino terms obtained from geometric actions, and that in this case the path integral is expressed in terms of the 2-cocycle which defines the central extension. In memory of our teacher Ludwig Faddeev.