Fixed points of Lie group actions on moduli spaces: A tale of two actions

TL;DR

The paper addresses fixed points of commuting Lie group actions on moduli spaces $\mathcal{X}/\mathcal{G}$ by deriving a linear Lie-algebra constraint that characterizes symmetric points. The core method reduces the fixed-point problem to representation theory: when the symmetry group $\mathcal{S}$ is compact and connected, a fixed class $[A]$ is governed by a Lie algebra homomorphism $\rho: \mathrm{Lie}(\mathcal{S}) \rightarrow \mathrm{Lie}(\mathcal{G})$ satisfying $x.A+\rho(x).A=0$ for all $x$, with a one-dimensional analogue available without compactness. For compact but possibly disconnected $\mathcal{S}$, the fixed-point test reduces to checking $\mathcal{S}_0$ and finitely many cosets; in all cases the stabilizer $S_A$ is a compact Lie subgroup. The results offer a general, representation-theoretic framework for locating fixed points of Lie-group actions on moduli spaces, with broad applicability beyond gauge theory, including explicit symmetric examples via linear constraints.

Abstract

In this paper, we examine Lie group actions on moduli spaces (sets themselves built as quotients by group actions) and their fixed points. We show that when the Lie group is compact and connected, we obtain a linear constraint. This constraint makes the problem of finding fixed points one of representation theory, greatly simplifying the search for such points. We obtain a similar result when the Lie group is one-dimensional. For compact and disconnected Lie groups, we show that we need only additionally check a finite number of points. Finally, we show that the subgroup fixing an equivalence class in the moduli space is a compact Lie subgroup.

Theorems & Definitions (10)

• Theorem 1.1
• Proposition 1.2
• Proposition 2.1
• proof
• proof : Proof of Theorem \ref{['thm:mainthm']}
• proof : Proof of Proposition \ref{['prop:R']}
• Example 2.3
• proof
• Proposition 2.4
• proof