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.
Categories: math.AG math-ph math.MP