Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies

TL;DR

This paper introduces a general method to compute topological tau functions for Drinfeld-Sokolov hierarchies using Fredholm determinants and the Borodin-Okounkov formula, applicable to any untwisted affine Kac-Moody algebra.

Contribution

It provides a novel, systematic approach to evaluate topological tau functions for a broad class of integrable hierarchies, expanding the computational toolkit in the field.

Findings

Explicit formulas for tau functions in several examples

Connection established between Fredholm determinants and string equations

Method applicable to all untwisted affine Kac-Moody algebras

Abstract

We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. A number of explicit examples are given.