On the Convergence of Gromov-Witten Potentials and Givental's Formula

TL;DR

This paper investigates the convergence properties of Gromov-Witten potentials for smooth projective varieties, establishing conditions under which these generating functions converge, especially for varieties with analytic and semisimple quantum cohomology.

Contribution

It provides a detailed analysis of convergence notions for Gromov-Witten potentials and proves convergence for all-genus potentials under specific geometric conditions.

Findings

Convergence of Gromov-Witten potentials is established for varieties with analytic and semisimple quantum cohomology.

Results include convergence proofs for compact toric, complete flag, and certain non-compact toric varieties.

The paper links Gromov-Witten theory with analytic and algebraic geometry through convergence analysis.

Abstract

Let X be a smooth projective variety. The Gromov-Witten potentials of X are generating functions for the Gromov-Witten invariants of X: they are formal power series, sometimes in infinitely many variables, with Taylor coefficients given by Gromov-Witten invariants of X. It is natural to ask whether these formal power series converge. In this paper we describe and analyze various notions of convergence for Gromov-Witten potentials. Using results of Givental and Teleman, we show that if the quantum cohomology of X is analytic and generically semisimple then the genus-g Gromov-Witten potential of X converges for all g. We deduce convergence results for the all-genus Gromov-Witten potentials of compact toric varieties, complete flag varieties, and certain non-compact toric varieties.