Virasoro constraints in Drinfeld-Sokolov hierarchies

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.