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

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