Exact Operator Solution for Liouville Theory with $q$ A Root of Unity

TL;DR

This paper extends the exact operator solution of quantum Liouville theory to cases where the quantum deformation parameter q is a root of unity, revealing a nilpotent screening charge and finite polynomial expressions for Liouville exponentials.

Contribution

It provides a novel extension of Liouville theory solutions to root of unity cases, highlighting the nilpotency of the screening charge operator.

Findings

Screening charge operator becomes nilpotent at roots of unity.

Liouville exponentials can be expressed as finite polynomials.

Extension of the operator solution to new quantum deformation regimes.

Abstract

The exact operator solution for quantum Liouville theory constructed for the generic quantum deformation parameter $q$ is extended to the case with $q$ being a root of unity. The screening charge operator becomes nilpotent in such cases and arbitrary Liouville exponentials can be obtained in finite polynomials of the screening charge.