On tau-functions for the KdV hierarchy

TL;DR

This paper derives new formulas for tau-functions of the KdV hierarchy using matrix resolvent and wave functions, with applications to important tau-functions like Witten--Kontsevich and BGW.

Contribution

Introduces two novel formulas for generating series of tau-functions using wave functions, expanding analytical tools for the KdV hierarchy.

Findings

New formulas for tau-function generating series derived

Applications demonstrated for Witten--Kontsevich and BGW tau-functions

Extension to Lamé tau-function with modular deformation

Abstract

For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten--Kontsevich tau-function, to the generalized Br\'ezin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lam\'e tau-function are also given.