# Darboux transformations and Fay identities for the extended bigraded Toda hierarchy

## TL;DR

This paper explores the extended bigraded Toda hierarchy's tau-function, deriving Fay identities and showing Darboux transformations act via vertex operators, advancing understanding of integrable systems linked to Gromov-Witten theory.

## Contribution

It introduces generalized Fay identities and characterizes Darboux transformations as vertex operators within the EBTH framework.

## Key findings

- Derived bilinear equations for the tau-function.
- Established Fay identities for the EBTH.
- Linked Darboux transformations to vertex operators.

## Abstract

The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the Gromov-Witten total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We write a bilinear equation for the tau-function of the EBTH and derive Fay identities from it. We show that the action of Darboux transformations on the tau-function is given by vertex operators. As a consequence, we obtain generalized Fay identities.