Worldsheet Instantons and Torsion Curves, Part B: Mirror Symmetry

TL;DR

This work computes worldsheet instanton counts on a non-toric Calabi–Yau $X$ with $\mathbb{Z}_3\times\mathbb{Z}_3$ torsion by leveraging mirror symmetry and covering-space techniques. It demonstrates that $X$ is self-mirror at the quantum level and derives the complete nonperturbative genus-zero prepotential, revealing the first explicit dependence of instanton numbers on torsion in homology. The study provides evidence for a torsion-exchange mirror relation in a non-toric setting and proposes a closed modular-form–based ansatz for the prepotential, with implications for heterotic model building and moduli stabilization. Altogether, the results extend mirror symmetry to Calabi–Yau manifolds with torsion curves and suggest a rich modular structure governing torsion-sensitive instanton data.

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.

Figures (2)

• Figure 1: Secondary fan of the non-simplicial facets. Chambers are indicated by dashed lines.
• Figure 2: Secondary fan of the codimension two face $[a_ia_jb_ib_jc_ld_l]$.

Theorems & Definitions (2)

• Lemma 1
• Lemma 2