Free fermions, W-algebras and isomonodromic deformations

TL;DR

This paper constructs representations of W-algebras using free fermions and links vertex operators to isomonodromy problems, providing new insights into tau-functions of multicomponent Toda hierarchies.

Contribution

It introduces a novel approach to represent W-algebras via free fermions and connects vertex operators with isomonodromic deformations, enhancing understanding of tau-functions.

Findings

New representation of W-algebras at integer central charges

Vertex operators expressed through isomonodromy solutions

Enhanced understanding of tau-functions in Toda hierarchies

Abstract

We consider the theory of multicomponent free massless fermions in two dimensions and use it for construction of representations of W-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to get some new insights on tau-functions of the multicomponent Toda type hierarchies for the class of solutions, given by the isomonodromy vertex operators and get useful representation for the tau-function of isomonodromic deformations.