Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of $BF$ theory

TL;DR

This paper interprets the holographic relationship between 2D $BF$ theory with gauge group PSL(2,R) and Schwarzian quantum mechanics through a Drinfeld-Sokolov reduction, revealing a connection to edge states and Virasoro orbits.

Contribution

It establishes a novel interpretation of the holographic duality as a Drinfeld-Sokolov reduction, linking $BF$ theory to Schwarzian quantum mechanics via edge states and coadjoint orbits.

Findings

Partition function as a sum over topological sectors.

Localization on exceptional Virasoro coadjoint orbits.

Reduced theory governed by Schwarzian action.

Abstract

We give an interpretation of the holographic correspondence between two-dimensional $BF$ theory on the punctured disk with gauge group ${\rm PSL}(2,\mathbb R)$ and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian $S^1$-action on the orbits. The partition function is given by a sum over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.