Liouville and Toda field theories on Riemann surfaces

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 cohomolog ical 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 explic itly 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