Liouville theory and special coadjoint Virasoro orbits

TL;DR

This paper explores the Hamiltonian reduction linking coadjoint Kac-Moody and Virasoro orbits, focusing on special orbits with double zeros, and connects these structures to Liouville theory, quantum mechanics, and sinh-Gordon integrable models.

Contribution

It explicitly describes the reduction process for special Virasoro coadjoint orbits and interprets their parameters in terms of quantum mechanics and Liouville insertions, revealing new links to integrable theories.

Findings

Special coadjoint Virasoro orbits correspond to quantum mechanics of particles on a circle.

Zeros of stabilizer vector fields relate to accessory parameters in uniformization.

Summing contributions yields a sinh-Gordon type integrable theory.

Abstract

We describe the Hamiltonian reduction of the coajoint Kac-Moody orbits to the Virasoro coajoint orbits explicitly in terms of the Lagrangian approach for the Wess-Zumino-Novikov-Witten theory. While a relation of the coajoint Virasoro orbit $Diff \; S^1 /SL(2,R)$ to the Liouville theory has been already studied we analyse the role of special coajoint Virasoro orbits $Diff \; S^1/\tilde{T}_{\pm ,n}$ corresponding to stabilizers generated by the vector fields with double zeros. The orbits with stabilizers with single zeros do not appear in the model. We find an interpretation of zeros $x_i$ of the vector field of stabilizer $\tilde{T}_{\pm ,n}$ and additional parameters $q_i$, $i = 1,...,n$, in terms of quantum mechanics for $n$ point particles on the circle. We argue that the special orbits are generated by insertions of "wrong sign" Liouville exponential into the path integral. The additional parmeters $q_i$ are naturally interpreted as accessory parameters for the uniformization map. Summing up the contributions of the special Virasoro orbits we get the integrable sinh-Gordon type theory.