Worldsheet Instantons, Torsion Curves, and Non-Perturbative Superpotentials

TL;DR

This paper computes Gromov-Witten invariants for a specific Calabi-Yau threefold with torsion in homology, enabling the calculation of instanton contributions to superpotentials in heterotic models.

Contribution

It provides the first explicit calculation of Gromov-Witten invariants for homology classes with torsion, advancing understanding of non-perturbative effects in string theory.

Findings

Complete genus-0 prepotential computed

Some curve classes contain only a single instanton

First explicit calculation of Gromov-Witten invariants with torsion

Abstract

As a first step towards computing instanton-generated superpotentials in heterotic standard model vacua, we determine the Gromov-Witten invariants for a Calabi-Yau threefold with fundamental group pi_1(X)=Z_3 x Z_3. We find that the curves fall into homology classes in H_2(X,Z)=Z^3+(Z_3+Z_3). The unexpected appearance of the finite torsion subgroup in the homology group complicates our analysis. However, we succeed in computing the complete genus-0 prepotential. Expanding it as a power series, the number of instantons in any integral homology class can be read off. This is the first explicit calculation of the Gromov-Witten invariants of homology classes with torsion. We find that some curve classes contain only a single instanton. This ensures that the contribution to the superpotential from each such instanton cannot cancel.