A quantum group structure in integrable conformal field theories

Abstract

We discuss a quantization prescription of the conformal algebras of a class ofd=2 conformal field theories which are integrable. We first give a geometrical construction of certain extensions of the classical Virasoro algebra, known as classicalW algebras, in which these algebras arise as the Lie algebra of the second Hamiltonian structure of a generalized Korteweg-de Vries hierarchy. This fact implies that theW algebras, obtained as a reduction with respect to the nilpotent subalgebras of the Kac-Moody algebra, describe the integrability of a Toda field theory. Subsequently we determine the coadjoint operators of theW algebras, and relate these to classical Yang-Baxter matrices. The quantization of these algebras can be carried out using the concept of a so-called quantum group. We derive the condition under which the representations of these quantum groups admit a Hilbert space completion by exploring the relation with the braid group. Then we consider a modification of the Miura transformation which we use to define a quantumW algebra. This leads to an alternative interpretation of the coset construction for Kac-Moody algebras in terms of nonlinear integrable hierarchies. Subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of Riemann surfaces to identify an action of theW algebras on the moduli space of stable curves, and we give the invariants of this action. This provides a generalization of the situation for the Virasoro algebra, where such as invariant is given by the so-called Mumford form which describes the partition function of the bosonic string.