Affine Kac-Moody Algebras and Tau-Functions for the Drinfeld-Sokolov Hierarchies: the Matrix-Resolvent Method

For each affine Kac-Moody algebra X(r) n of rank l, r = 1, 2, or 3, and for ev-ery choice of a vertex cm, m = 0, ... , l, of the corresponding Dynkin diagram, by using the matrix-resolvent method we define a gauge-invariant tau-structure for the associated Drinfeld-Sokolov hierarchy and give explicit formulas for generating series of logarithmic derivatives of the tau-function in terms of matrix resolvents, extending the results of [Mosc. Math. J. 21 (2021), 233-270, arXiv:1610.07534] with r = 1 and m = 0. For the case r = 1 and m = 0, we verify that the above-defined tau-structure agrees with the axioms of Hamil-tonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382-435, arXiv:1409.4616] and [arXiv:math.DG/0108160].