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

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.

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References in corpus (3)

• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• The equivariant Gromov-Witten theory of P^1
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra

Cited by in corpus (7)

• Drinfeld-Sokolov hierarchies and diagram automorphisms of affine Kac-Moody algebras
• Simple Lie algebras, Drinfeld--Sokolov hierarchies, and multi-point correlation functions
• Matrix model for the total descendant potential of a simple singularity of type $D$
• Affine Kac-Moody Algebras and Tau-Functions for the Drinfeld-Sokolov Hierarchies: the Matrix-Resolvent Method
• Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials
• Fermionic Computations for Integrable Hierarchies
• Fredholm Pfaffian $\tau$-functions for orthogonal isospectral and isomonodromic systems