Pure and Applied Mathematics Quarterly
Volume 16 (2020)
Special Issue in Honor of Yuri Manin: Part 3 of 3
Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau
Gromov–Witten invariants of the Riemann sphere
Pages: 153 – 190
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 . 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.
2010 Mathematics Subject Classification
Primary 14N35, 37K10. Secondary 05A15, 53D45.
Boris Dubrovin passed away on March 19, 2019.
Received 2 February 2018
Published 6 February 2020