Tau functions and Virasoro symmetries for Drinfeld–Sokolov hierarchies

Abstract

For each Drinfeld–Sokolov integrable hierarchy associated to affine Kac–Moody algebra, we obtain a uniform construction of tau function by using tau-symmetric Hamiltonian densities, moreover, we represent its Virasoro symmetries as linear/nonlinear actions on the tau function. The relations between the tau function constructed in this paper and those defined for particular cases of Drinfeld–Sokolov hierarchies in the literature are clarified. We also show that, whenever the affine Kac–Moody algebra is simply-laced or twisted, the tau function of the Drinfeld–Sokolov hierarchy coincides with the solution of the corresponding Kac–Wakimoto hierarchy constructed from the principal vertex operator realization of the affine algebra.