Liouville and Toda field theories on Riemann surfaces

TL;DR

This paper investigates classical and quantum Liouville and Toda field theories on Riemann surfaces, analyzing their monodromy properties, solution spaces, and operator structures, with explicit constructions and extensions to higher rank theories.

Contribution

It provides a detailed analysis of Liouville and Toda theories on Riemann surfaces, including solution representations and quantum operator structures, extending to $sl_n$ Toda theories.

Findings

Explicit solutions in terms of Krichever--Novikov oscillators

Determination of quantum exchange algebras

Conditions for univalence and locality in quantum theory

Abstract

We study the Liouville theory on a Riemann surface of genus g by means of their associated Drinfeld--Sokolov linear systems. We discuss the cohomological properties of the monodromies of these systems. We identify the space of solutions of the equations of motion which are single--valued and local and explicitly represent them in terms of Krichever--Novikov oscillators. Then we discuss the operator structure of the quantum theory, in particular we determine the quantum exchange algebras and find the quantum conditions for univalence and locality. We show that we can extend the above discussion to $sl_n$ Toda theories.