Dimension Reduction of Generalized ASD Instantons

TL;DR

This paper investigates generalized anti-self-dual instantons on Riemannian manifolds with special forms, providing criteria for dimension reduction and explicit descriptions of moduli spaces, including compactifications for certain cases.

Contribution

It introduces a topological criterion for dimension reduction of generalized ASD instantons on product manifolds and describes the structure of their moduli spaces, including special instantons.

Findings

Established a topological criterion for dimension reduction.

Derived explicit descriptions of moduli spaces for various instantons.

Constructed compactifications for moduli spaces when one factor is a 4-manifold.

Abstract

We study generalized anti-self-dual instantons defined over Riemannian manifolds equipped with a parallel codimension-$4$ differential form. In particular, for product Riemannian manifolds possessing such a form, we study dimension reduction phenomena, finding a topological criterion for bundles which, when satisfied, allows for a complete characterization of dimension reduction for the corresponding moduli space of generalized ASD instantons. By establishing an integrability result for families of connections, we then deduce explicit descriptions for these moduli spaces, including those of Hermitian Yang--Mills connections, $G_2$-, and $\Spin(7)$-instantons. When one factor in the product is a $4$-manifold, we establish well-behaved compactifications for these moduli spaces.