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

Abstract

For each affine Kac-Moody algebra 𝛸⁽ʳ⁾ₙ of rank ℓ, 𝑟 = 1,2, or 3, and for every 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 Hamiltonian tau-symmetry in the sense of [Adv. Math. 293 (2016), 382-435, arXiv:1409.4616] and [arXiv:math.DG/0108160].