Classical W-algebras, Drinfeld-Sokolov hierarchies and tau functions

Abstract:  Classical W-algebras W(g,O) are a family of Poisson vertex algebras associated to a simple Lie algebra g and a nilpotent orbit O. For (almost) every W(g,O) it is possible to construct an integrable hierarchy of PDEs which generalizes the Drinfeld-Sokolov hierarchy (which is recovered for the principal nilpotent orbit). For example, when g=sl_2, one gets the Korteweg-de Vries (KdV) hierarchy. More generally, for g=sl_n, these hierarchies are suitable reductions of the Kadomtsev-Petviashvili (KP) hierarchy and using the theory of vertex operators is then possible to construct tau function solutions. In the talk I will review these results and recent progress towards the construction of tau functions beyond the sl_n case.