Equivariant gluing theory on regular instanton moduli spaces

TL;DR

This work extends classical instanton gluing to a finite-group-equivariant setting on 4-manifolds with isolated fixed points, aiming to produce $\Gamma$-invariant ASD connections on the connected sum $X_1\# X_2$. It first constructs a Γ-invariant approximate solution via Γ-equivariant neck gluing and flatting near fixed points, then obtains a genuine ASD connection by solving a Γ-invariant equation using a right inverse and a contraction mapping. A key contribution is the explicit description of the equivariant gluing parameter space $Gl^{\Gamma}$ as the centraliser of the isotropy representation in $G$, together with conditions such as equivalent isotropy representations and $(H^2_{A_i})^{\Gamma}=0$ that guarantee existence and uniqueness up to gauge, all while respecting orbit equivalence under $\Gamma_{A_1}\times \Gamma_{A_2}$. The results enable compactifications of equivariant instanton moduli spaces and provide a framework for symmetry-respecting gauge theories on orbifolds with isolated fixed points.

Abstract

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

Figures (2)

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Theorems & Definitions (12)

• Lemma 1
• Lemma 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• Lemma 6
• proof