Gromov-Witten invariants of $\mathbb{P}^1$ coupled to a KdV tau function

TL;DR

This paper introduces new Gromov-Witten invariants for al P^1 coupled with a KdV tau function, showing they can be computed via a matrix integral and satisfy the Toda equation.

Contribution

It establishes a novel connection between Gromov-Witten invariants, the Bre9zin-Gross-Witten tau function, and matrix models for the case of al P^1.

Findings

New invariants are given by a matrix integral

They satisfy the Toda equation

Linked to the Bre9zin-Gross-Witten tau function

Abstract

We consider the pull-back of a natural sequence of cohomology classes $\Theta_{g,n}\in H^{2(2g-2+n)}(\overline{\cal M}_{g,n})$ to the moduli space of stable maps ${\cal M}^g_n(\mathbb{P}^1,d)$. These classes are related to the Br\'ezin-Gross-Witten tau function of the KdV hierarchy via $Z^{BGW}(\hbar,t_0,t_1,...)=\exp\sum\frac{\hbar^{2g-2}}{n!}\int_{\overline{\cal M}_{g,n}}\Theta_{g,n}\cdot\prod_{j=1}^n\psi_j^{k_j}\prod t_{k_j}$. Insertions of the pull-backs of the classes $\Theta_{g,n}$ into the integrals defining Gromov-Witten invariants define new invariants which we show in the case of target $\mathbb{P}^1$ are given by a random matrix integral and satisfy the Toda equation.