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 [11]. 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 ɛ → 0asymptotics of the analytic k-point function, and also compute three more asymptotics of the analytic function for ɛ →∞, q → 0, q →∞, thus defining new invariants for the Riemann sphere.