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

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

2012-03-19

arxiv.org/abs/1203.4193v1

Mathematics

Algebraic Geometry

37 pages, 1 figure

Scientific paper

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.

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