# On tau-functions for the Toda lattice hierarchy

## TL;DR

This paper extends tau-function techniques from the KdV hierarchy to the Toda lattice hierarchy, providing explicit formulas for correlation functions and applications to GUE and Gromov--Witten invariants.

## Contribution

It introduces a method to define wave functions for the Toda hierarchy and derives explicit generating series for correlation functions, expanding the scope of integrable systems analysis.

## Key findings

- Explicit formulas for k-point correlation functions
- Application to GUE correlators
- Potential use in Gromov--Witten invariants

## Abstract

We extend a recent result of [13] for the KdV hierarchy to the Toda lattice hierarchy. Namely, for an arbitrary solution to the Toda lattice hierarchy, we define a pair of wave functions, and use them to give explicit formulae for the generating series of $k$-point correlation functions of the solution. Applications to computing GUE correlators and Gromov--Witten invariants of the Riemann sphere are under consideration.