Topological String Theory on Compact Calabi-Yau: Modularity and Boundary Conditions

TL;DR

This paper computes the topological string partition function on compact Calabi-Yau manifolds, revealing its modular properties and boundary conditions, enabling high-genus calculations up to genus 51 for the quintic.

Contribution

It introduces a method to determine the topological string partition function using modularity and boundary conditions, extending calculations to very high genus.

Findings

Partition function Z transforms as a wave function on H^3(M,Z)

Boundary data fixed using gap condition, orbifold regularity, and Castelnuovo bounds

Achieved genus 51 calculations for the quintic

Abstract

The topological string partition function Z=exp(lambda^{2g-2} F_g) is calculated on a compact Calabi-Yau M. The F_g fulfill the holomorphic anomaly equations, which imply that Z transforms as a wave function on the symplectic space H^3(M,Z). This defines it everywhere in the moduli space of M along with preferred local coordinates. Modular properties of the sections F_g as well as local constraints from the 4d effective action allow us to fix Z to a large extend. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovos theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.