Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 1

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

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n1.a4


Boris Dubrovin (Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy)

Di Yang (Max-Planck-Institut für Mathematik, Bonn, Germany; and School of Mathematical Sciences, University of Science and Technology of China, Hefei, China)

Don Zagier (Max-Planck-Institut für Mathematik, Bonn, Germany)


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 $\epsilon \to 0$ 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