Local Mirror Symmetry at Higher Genus

TL;DR

This paper explores local mirror symmetry for higher-genus curves in Calabi-Yau spaces, computes the topological string partition function for local P^2, and investigates the integrality and growth of Gromov-Witten invariants.

Contribution

It provides a method to compute higher-genus partition functions using Kodaira-Spencer theory and verifies Gromov-Witten predictions through localization, highlighting the mysterious integrality of invariants.

Findings

Partition function expressed in integer coefficients.

Verification of Gromov-Witten predictions via localization.

Analysis of asymptotic growth of invariants.

Abstract

We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa and the local mirror map, the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa.