The Many Faces of the Quantum Liouville Exponentials

TL;DR

This paper proves the equivalence of three main approaches to quantum Liouville exponentials, expresses them using quantum group structures, and connects them to operator tau-functions and quantum integrable systems.

Contribution

It demonstrates the equivalence of different operator approaches, provides q-binomial sum representations, and links Liouville exponentials to quantum tau-functions and integrable equations.

Findings

Proves equivalence of three main approaches to quantum Liouville exponentials.

Expresses Liouville exponentials as q-binomial sums over chiral fields.

Connects Liouville exponentials to quantum tau-functions and Hirota equations.

Abstract

First, it is proven that the three main operator-approaches to the quantum Liouville exponentials --- that is the one of Gervais-Neveu (more recently developed further by Gervais), Braaten-Curtright-Ghandour-Thorn, and Otto-Weigt --- are equivalent since they are related by simple basis transformations in the Fock space of the free field depending upon the zero-mode only. Second, the GN-G expressions for quantum Liouville exponentials, where the $U_q(sl(2))$ quantum group structure is manifest, are shown to be given by q-binomial sums over powers of the chiral fields in the $J=1/2$ representation. Third, the Liouville exponentials are expressed as operator tau-functions, whose chiral expansion exhibits a q Gauss decomposition, which is the direct quantum analogue of the classical solution of Leznov and Saveliev. It involves q exponentials of quantum group generators with group "parameters" equal to chiral components of the quantum metric. Fourth, we point out that the OPE of the $J=1/2$ Liouville exponential provides the quantum version of the Hirota bilinear equation.