Simple Lie algebras, Drinfeld--Sokolov hierarchies, and multi-point correlation functions

Abstract

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 $\text{topological ODE}$.

minor corrections in Definition 1.1.1, Lemma 2.2.2 and Appendix; corrected typos; added a footnote on page 13; 36 pages