Finding the Mirror of the Beauville Manifold

TL;DR

This paper constructs the mirror of the Beauville manifold, a Calabi-Yau with non-abelian fundamental group, confirming a threshold bound state and analyzing D-brane symmetries at the conifold point.

Contribution

It provides the first explicit mirror construction for the Beauville manifold using Batyrev-Borisov conjecture, and explores D-brane actions consistent with mirror symmetry.

Findings

Confirmed the existence of a threshold bound state at the conifold point.

Identified the mirror of the Beauville manifold via monomial-divisor mirror map.

Analyzed the quantum symmetry group's action on D-branes at the conifold.

Abstract

We construct the mirror of the Beauville manifold. The Beauville manifold is a Calabi-Yau manifold with non-abelian fundamental group. We use the conjecture of Batyrev and Borisov to find the previously misidentified mirror of its universal covering space, $\mathbb{P}^7[2,2,2,2]$. The monomial-divisor mirror map is essential in identifying how the fundamental group of the Beauville manifold acts on the mirror of $\mathbb{P}^7[2,2,2,2]$. Once we find the mirror of the Beauville manifold, we confirm the existence of the threshold bound state around the conifold point, which was originally conjectured in hep-th/0106262. We also consider how the quantum symmetry group acts on the D-branes that become massless at the conifold point and show the action proposed in hep-th/0102018 is compatible with mirror symmetry.