Equivariant gluing theory on regular instanton moduli spaces

TL;DR

This paper develops an equivariant gluing theory for instanton moduli spaces, focusing on how to glue invariant anti-self-dual connections over 4-manifolds with finite group actions to produce new invariant solutions.

Contribution

It introduces a method for gluing $ ext{Gamma}$-invariant ASD connections on 4-manifolds with group actions, extending classical gluing techniques to the equivariant setting.

Findings

Established a framework for equivariant gluing of instantons

Demonstrated the construction on connected sums with group actions

Provided conditions for the existence of invariant ASD connections

Abstract

We follow the idea of gluing theory in instanton moduli spaces and discuss the case when there is a finite group $\Gamma$ acting on the 4-manifolds $X_1, X_2$ with $x_1, x_2$ as isolated fixed points, how to glue two $\Gamma$-invariant ASD connections over $X_1, X_2$ together to get a $\Gamma$-invariant ASD connection on the connected sum $X_1\# X_2$.