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

Yang, Hyun Seok  (Center for Quantum Spacetime, Sogang University, Seoul 121-741, Republic of Korea) ; Yun, Sangheon (Institute for the Early Universe, Ewha Womans University, Seoul 120-750, Republic of Korea)

09 November 2017

Abstract: We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group $\text{Spin}\left(\mathrm{6}\right)$ 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.

Published in: Advances in High Energy Physics 2017 (2017) 7962426