  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.
