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The BGW KdV tau function coupled to Gromov-Witten invariants of $\mathbb{P}^1$ Norbury, Paul


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 $\overline{\mathcal{M}}^g_n(\mathbb{P}^1,d)$. These classes are related to the Brezin-Gross-Witten tau function of the KdV hierarchy via $$Z^{BGW}(\hbar,t_0,t_1,...)=\exp\sum\frac{1}{n!}\int_{\overline{\mathcal{M}}_{g,n}}\Theta_{g,n}\cdot\prod_{j=1}^n\psi_j^{k_j}\prod t_{k_j}.$$ Insertions of the pull-backs of the classes $\Theta_{g,n}$ into the integrals defining Gromov-Witten invariants define new invariants. In the case of target $\mathbb{P}^1$ we show that these are computable and satisfy the Toda equation.

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