Mirror Symmetry for Two Parameter Models -- II

TL;DR

This paper uses mirror symmetry to analyze the moduli space of a specific Calabi-Yau manifold, computing instanton numbers and exploring symmetries in the complex structure and Kähler parameters.

Contribution

It provides a detailed description of the two-parameter moduli space, including monodromy, period basis, and instanton expansions, with new insights into symmetries and enumerative invariants.

Findings

Determined the monodromy and symplectic basis of periods.

Computed instanton expansion of Yukawa couplings.

Identified symmetries acting on the moduli space.

Abstract

We describe in detail the space of the two K\"ahler parameters of the Calabi--Yau manifold $\P_4^{(1,1,1,6,9)}[18]$ by exploiting mirror symmetry. The large complex structure limit of the mirror, which corresponds to the classical large radius limit, is found by studying the monodromy of the periods about the discriminant locus, the boundary of the moduli space corresponding to singular Calabi--Yau manifolds. A symplectic basis of periods is found and the action of the $Sp(6,\Z)$ generators of the modular group is determined. From the mirror map we compute the instanton expansion of the Yukawa couplings and the generalized $N=2$ index, arriving at the numbers of instantons of genus zero and genus one of each degree. We also investigate an $SL(2,\Z)$ symmetry that acts on a boundary of the moduli space.