Coadjoint Orbits, Cocycles and Gravitational Wess-Zumino

TL;DR

The paper revisits geometric actions on coadjoint orbits, showing that orbit actions constitute a 1-cocycle for the loop group $LG$ and give rise to a PW-type structure for Wess–Zumino terms in central extensions. It systematically develops PW formulas for both the WZW and gravitational (Virasoro–Bott) WZ actions, tying the boundary contributions to 2-cocycles that define the central extensions. A key outcome is a bulk–boundary decoupling phenomenon in the quantum theory: the path integral is governed by the central extension data, reducing to boundary information encoded in $\alpha_2$. The results unify orbit-method constructions with WZ terms and provide explicit decoupling formulas for both current-algebra and Virasoro/Weyl gravitational cases, with implications for open/closed string contexts and boundary states $|a\rangle$.

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.