Virasoro constraints in Drinfeld-Sokolov hierarchies

TL;DR

This paper develops a geometric framework for Virasoro constraints within Drinfeld-Sokolov hierarchies, linking solutions to principal bundles with Higgs fields and deriving differential equations akin to those in 2D quantum gravity.

Contribution

It introduces a geometric description of solutions and Virasoro constraints in Drinfeld-Sokolov hierarchies, extending the understanding of string solutions and tau-functions.

Findings

Characterization of string solutions via compatible connections.

Tau-functions satisfy generalized Virasoro constraints.

Connection between geometric data and differential equations in quantum gravity.

Abstract

We describe a geometric theory of Virasoro constraints in generalized Drinfeld-Sokolov hierarchies. Solutions of Drinfeld-Sokolov hierarchies are succinctly described by giving a principal bundle on a complex curve together with the data of a Higgs field near infinity. String solutions for these hierarchies are defined as points having a big stabilizer under a certain Lie algebra action. We characterize principal bundles coming from string solutions as those possessing connections compatible with the Higgs field near infinity. We show that tau-functions of string solutions satisfy second-order differential equations generalizing the Virasoro constraints of 2d quantum gravity.