All-Genus Large-Degree Asymptotics for Gromov--Witten Invariants of the Projective Plane

Abstract

This is the first part of a series of papers on the large-degree asymptotics of Gromov--Witten invariants. In this paper, we prove complete large-degree asymptotic expansions, at every fixed genus, for the primary Gromov--Witten invariants of the complex projective plane. The proof uses singularity analysis to transfer the local expansions of generating functions at their dominant singularities to asymptotic expansions of Gromov--Witten invariants. The genus-zero asymptotic expansion is obtained from an analysis of the Witten--Dijkgraaf--Verlinde--Verlinde (WDVV) equation. The higher-genus cases are obtained from the Givental--Teleman reconstruction theorem for semisimple cohomological field theories and from the graph-sum formula for the $R$-matrix action.