Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons and Mirror Symmetry

TL;DR

This paper explores the geometric and gauge theory foundations of mirror symmetry in Calabi-Yau manifolds, linking Hodge duality, spinor representations, and Hermitian Yang-Mills instantons to explain mirror pairs.

Contribution

It presents a novel gauge theory perspective on mirror symmetry, connecting the doubling of form spaces to Hermitian Yang-Mills instantons in Calabi-Yau manifolds.

Findings

Mirror symmetry corresponds to pairs of Hermitian Yang-Mills instantons.

Doubling of two-form spaces relates to mirror pairs.

Gauge theory formulation explains the existence of mirror Calabi-Yau manifolds.

Abstract

We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.