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

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.