Duality in Liouville Theory as a Reduced Symmetry

TL;DR

This paper explores the origin of duality symmetry in quantum Liouville theory by analyzing its classical and quantum reductions from a WZW-like theory, revealing that dual exponential potentials underpin the duality.

Contribution

It demonstrates that classical conformal invariance allows arbitrary potentials, but quantum invariance constrains them to dual exponentials, explaining Liouville duality as a consequence.

Findings

Classical conformal invariance does not fix the potential form.

Quantum theory requires a sum of two exponentials for the potential.

Duality arises from the quantum superposition of exponential potentials.

Abstract

The origin of the rather mysterious duality symmetry found in quantum Liouville theory is investigated by considering the Liouville theory as the reduction of a WZW-like theory in which the form of the potential for the Cartan field is not fixed a priori. It is shown that in the classical theory conformal invariance places no condition on the form of the potential, but the conformal invariance of the classical reduction requires that it be an exponential. In contrast, the quantum theory requires that, even before reduction, the potential be a sum of two exponentials. The duality of these two exponentials is the fore-runner of the Liouville duality. An interpretation for the reflection symmetry found in quantum Liouville theory is also obtained along similar lines.