Multipoint Series of Gromov–Witten Invariants of CP¹

Abstract

We derive explicit formulas for the multipoint series of $\mathbb{C}\mathbb{P}^{\text{1}}$ in degree 0 from the Toda hierarchy, using the recursions of the Toda hierarchy. The Toda equation then yields inductive formulas for the higher degree multipoint series of $\mathbb{C}\mathbb{P}^{\text{1}}$. We also obtain explicit formulas for the Hodge integrals $\int {_{\overline {\mathcal{M}} _{g,n} } \psi } _1^{k_1 } ...\psi _n^{k_n } \lambda _{g - i}$, in the cases i=0 and 1.