  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.
