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

We consider the pull-back of a natural sequence of cohomology classes ${\mathrm{\Theta }}_{g,n}\in H^{2(2g-2+n)}({\bar{\mathcal{M}}}_{g,n})$ to the moduli space of stable maps ${\mathcal{M}}_n^g({\mathbb{P}}^1,d)$. 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 ${\mathrm{\Theta }}_{g,n}$ 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.