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

TL;DR

This paper develops a gauge-invariant tau-structure for Drinfeld-Sokolov hierarchies associated with affine Kac-Moody algebras using the matrix-resolvent method, extending previous results and verifying Hamiltonian tau-symmetry.

Contribution

It introduces a new matrix-resolvent-based tau-structure for these hierarchies, providing explicit formulas and extending prior work to more general cases.

Findings

Explicit formulas for tau-functions in terms of matrix resolvents

Extension of previous results to all affine Kac-Moody algebras of rank $ ext{ell}$

Verification of tau-structure's consistency with Hamiltonian tau-symmetry

Abstract

For each affine Kac-Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice of a vertex $c_m$, $m=0,\dots,\ell$, 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].