Worldsheet Instantons and Torsion Curves Part B: Mirror Symmetry

We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z3 ⊕ Z3 Wilson lines. As we found in Part A [1], the integral homology group H2(X,Z) = Z 3 ⊕ Z3 ⊕ Z3 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 selfmirror 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. Email: vbraun, [email protected], [email protected], [email protected]