# Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry

## TL;DR

This paper uses mirror symmetry to compute instanton numbers in a Calabi-Yau threefold with torsion homology, revealing novel torsion-dependent instanton effects and providing a non-toric example of torsion exchange in mirror symmetry.

## Contribution

It presents the first calculation of instanton numbers depending on torsion classes and demonstrates a non-toric example of torsion subgroup exchange in mirror symmetry.

## Key findings

- Instanton numbers depend on torsion homology classes.
- X is self-mirror at the quantum level.
- Provides a non-toric example of torsion exchange in mirror symmetry.

## Abstract

We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. We observe that X is self-mirror even at the quantum level. Using the self-mirror property, we derive the complete prepotential on X, going beyond the results of Part A. In particular, this yields the first example where the instanton number depends on the torsion part of its homology class. Another consequence is that the threefold X provides a non-toric example for the conjectured exchange of torsion subgroups in mirror manifolds.

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Source: https://tomesphere.com/paper/0704.0449