# Moscow Mathematical Journal

Volume 21, Issue 2, April–June 2021  pp. 233–270.

Simple Lie Algebras, Drinfeld–Sokolov Hierarchies, and Multi-Point Correlation Functions

Authors:  Marco Bertola (1), Boris Dubrovin (2) and Di Yang (3)
Author institution:(1) SISSA, via Bonomea 265, Trieste 34136, Italy,
Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, H3G 1M8, Canada
Centre de recherches mathématiques, Université de Montréal, C. P. 6128, succ. centre ville, Montréal, Québec, H3C 3J7, Canada
(2) SISSA, via Bonomea 265, Trieste 34136, Italy,
N. N. Bogolyubov Laboratory for Geometrical Methods in Mathematical Physics, Moscow State University, Moscow 119899, Russia
(3) Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P.R. China

Summary:

For a simple Lie algebra $\mathfrak{g}$, we derive a simple algorithm for computing logarithmic derivatives of tau-functions of Drinfeld–Sokolov hierarchy of $\mathfrak{g}$-type in terms of $\mathfrak{g}$-valued resolvents. We show, for the topological solution to the lowest-weight-gauge Drinfeld–Sokolov hierarchy of $\mathfrak{g}$-type, the resolvents evaluated at zero satisfy the topological ODE.

2020 Math. Subj. Class. 37K10; 53D45, 17B80, 14N35.

Keywords: Simple Lie algebra, tau-function, Drinfeld–Sokolov hierarchy, matrix resolvent, topological ODE.