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

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.