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

arXiv comment23 pages

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• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• The Topological CP^1 Model and the Large-N Matrix Integral
• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• Simple Lie algebras, Drinfeld--Sokolov hierarchies, and multi-point correlation functions
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