Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 077, 32 pages      arXiv:2110.06655      https://doi.org/10.3842/SIGMA.2022.077

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

Boris Dubrovin a, Daniele Valeri bc and Di Yang d
a) Deceased
b) Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy
c) INFN, Section of Rome, Italy
d) School of Mathematical Sciences, USTC, Hefei 230026, P.R. China

Received April 07, 2022, in final form September 26, 2022; Published online October 14, 2022

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].

Key words: Kac-Moody algebra; tau-function; Drinfeld-Sokolov hierarchy; matrix resolvent.

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