Instantons on Cylindrical Manifolds

TL;DR

This paper proves that instantons with finite energy on cylindrical manifolds over certain special geometries must be flat connections, extending understanding of gauge theory on these manifolds.

Contribution

It establishes a rigidity result for instantons on cylindrical manifolds with special geometric structures, showing they must be flat.

Findings

Instantons with finite energy are necessarily flat on these manifolds.

The result applies to manifolds with nearly Kähler and nearly parallel G2 structures.

Provides new insights into gauge theory on special holonomy manifolds.

Abstract

We consider an instanton,$\textbf{A}$,with $L^{2}$-curvature $F_{\textbf{A}}$ on the cylindrical manifold $Z=\mathbf{R}\times M$,where $M$ is a closed Riemannian $n$-manifold, $n\geq 4$.We assume $M$ admits a $3$-form $P$ and a $4$-form $Q$ satisfy $dP=4Q$ and $d\ast{Q}=(n-3)\ast P$.Manifolds with these forms include nearly K\"{a}hler 6-manifolds and nearly parallel $G_{2}$-manifolds in dimension 7.Then we can prove that the instanton must be a flat connection.