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

Abstract

We consider the pull-back of a natural sequence of cohomology classes ${\Theta }_{g,n}\in H^{2(2g-2+n)}({\overline{\mathcal{M}}}_{g,n})$ to the moduli space of stable maps ${\mathcal{M}}_n^g({\mathbb{P}}^1,d)$. These classes are related to the Brézin-Gross-Witten tau function of the KdV hierarchy via $Z^{BGW}(\hslash ,t_0,t_1,...)=\exp \sum \frac{{\hslash }^{2g-2}}{n!}{\int }_{{\overline{\mathcal{M}}}_{g,n}}{\Theta }_{g,n}\cdot {\prod }_{j=1}^n{\psi }_j^{k_j}\prod t_{k_j}$. 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.

27 pages

Cited by in corpus (3)

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