Gromov--Witten invariants of the Riemann sphere

Abstract

A conjectural formula for the $k$-point generating function of Gromov--Witten invariants of the Riemann sphere for all genera and all degrees was proposed in \cite{DY2}. In this paper, we give a proof of this formula together with an explicit analytic (as opposed to formal) expression for the corresponding matrix resolvent. We also give a formula for the $k$-point function as a sum of $(k-1)!$ products of hypergeometric functions of one variable. We show that the $k$-point generating function coincides with the $\hat{I}\mu \to 0$ asymptotics of the analytic $k$-point function, and also compute three more asymptotics of the analytic function for $\hat{I}\mu \to \infty$, $q\to 0$, $q\to \infty$, thus defining new invariants for the Riemann sphere.

23 pages