Worldsheet Instantons and Torsion Curves, Part A: Direct Computation

TL;DR

This paper computes worldsheet instanton numbers in a specific Calabi-Yau threefold with torsion curves, developing new techniques to handle torsion and non-toric geometry, crucial for heterotic string compactifications.

Contribution

It introduces methods to distinguish torsion curve classes and directly computes instanton numbers, including torsion classes, in a non-toric Calabi-Yau threefold.

Findings

Identified all holomorphic rational curves in the Calabi-Yau threefold.

Developed techniques to analyze torsion curves and non-toric geometries.

Found homology classes with single instantons, preventing superpotential cancellations.

Abstract

As a first step towards studying vector bundle moduli in realistic heterotic compactifications, we identify all holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. Computing the homology, we find that H_2(X,Z)=Z^3+Z_3+Z_3. The torsion curves complicate our analysis, and we develop techniques to distinguish the torsion part of curve classes and to deal with the non-toric threefold X. In this paper, we use direct A-model computations to find the instanton numbers in each integral homology class, including torsion. One interesting result is that there are homology classes that contain only a single instanton, ensuring that there cannot be any unwanted cancellation in the non-perturbative superpotential.