Complex Anti-Self-Dual Connections on a Product of Calabi–Yau Surfaces and Triholomorphic Curves

Abstract:

We study the adiabatic limit of a sequence of Ω-anti-self-dual connections on unitary bundles over a product of two compact Calabi–Yau surfaces M×N by scaling metrics to shrink N to a point. We show that after fixing gauge transformations, a subsequence of the N-components of these connections converges to a triholomorphic curve from M away from a Cayley cycle in M×N to the moduli space ${\cal M}_N$ of instantons on M×N modulo gauge equivalence in the Hausdorff topology, and converges on the blow-up locus to a family, which is parameterized by the Cayley cycle, of triholomorphic curves from C ² to ${\cal M}_N$.