Asymptotic behaviour of instantons on Cylinder Manifolds

TL;DR

This paper investigates the asymptotic behavior of instantons on cylinder manifolds, showing exponential decay for good manifolds and triviality of solutions on real Killing spinor manifolds under certain conditions.

Contribution

It provides new results on the decay and triviality of instantons on specific classes of cylinder manifolds with particular geometric structures.

Findings

Instantons decay exponentially on good manifolds.

Solutions are trivial on real Killing spinor manifolds under mild conditions.

Characterization of instanton behavior based on manifold properties.

Abstract

IIn this article, we study the instanton equation on the cylinder over a closed manifold $X$ which admits non-zero smooth $3$-form $P$ and $4$-form $Q$. Our results are (1) if $X$ is a \textbf{good} manifold, i.e., $P,Q$ satisfying $d\ast_{X}P=d\ast_{X}Q=0$, then the instanton with integrable curvature decays exponentially at the ends, and, (2) if $X$ is a real Killing spinor manifold, i.e., $P,Q$ satisfying $dP=4Q$ and $d\ast_{X}Q=(n-3)\ast_{X}P$, we prove that the solution of instanton equation is trivial under some mild conditions.