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

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.

38 pages, 1 figure, v2: corrected several errors

References in corpus (3)

• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• A Mirror Theorem for Toric Stacks
• A construction of Frobenius manifolds with logarithmic poles and applications

Cited by in corpus (4)

• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• Gromov-Witten Invariants of Local P^2 and Modular Forms
• Frobenius Manifolds near the Discriminant and Relations in the Tautological Ring
• A Fock Sheaf For Givental Quantization