Gromov--Witten invariants of the Riemann sphere

TL;DR

This paper proves a conjectural formula for Gromov--Witten invariants of the Riemann sphere, providing explicit analytic expressions and asymptotics, thereby advancing the understanding of these enumerative invariants across all genera and degrees.

Contribution

It offers a proof of the conjectural formula, explicit analytic expressions for the matrix resolvent, and new asymptotic invariants for the Riemann sphere.

Findings

Proof of the conjectural formula for all genera and degrees.

Explicit analytic expression for the matrix resolvent.

New invariants derived from asymptotics of the generating functions.

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 $\epsilon\rightarrow 0$ asymptotics of the analytic $k$-point function, and also compute three more asymptotics of the analytic function for $\epsilon\rightarrow \infty$, $q\rightarrow 0$, $q\rightarrow\infty$, thus defining new invariants for the Riemann sphere.