This article is cited in 1 paper

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

Boris Dubrovin^a, Daniele Valeri^bc, Di Yang<sup>d

a</sup> 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

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].

Keywords: Kac–Moody algebra, tau-function, Drinfeld–Sokolov hierarchy, matrix resolvent.

MSC: 37K10, 17B80, 17B67, 37K30

Received: April 7, 2022; in final form September 26, 2022; Published online October 14, 2022

Language: English

DOI: 10.3842/SIGMA.2022.077

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ArXiv: 2110.06655