Explicit abelian instantons on S1-invariant Kähler Einstein 6-manifolds

Abstract

We consider the dimensional reduction of the (deformed) Hermitian Yang–Mills condition on $S^1$-invariant Kähler Einstein 6-manifolds. This allows us to reformulate the (deformed) Hermitian Yang–Mills equations in terms of data on the quotient Kähler 4-manifold. In particular, when the gauge group is $U\left(1\right)$ we apply this construction to the canonical bundles of $C{P}^2$ and $C{P}^1\times C{P}^1$ endowed with the Calabi ansatz metric to find abelian instantons. We show that these are determined by a suitable subset of the spectrum of the zero section and are explicitly given in terms of certain hypergeometric functions. As a by-product of our investigation we find a coordinate expression for the holomorphic volume form on $O_{C{P}^2}(-3)$ and use it to construct a new special Lagrangian foliation. We also find 1-parameter families of explicit deformed Hermitian Yang–Mills connections on certain non-compact $S^1$-invariant Kähler Einstein 6-manifolds.