Fixed points on moduli spaces

Below we list the main results and examples from my paper “Fixed points of Lie group actions on moduli spaces: A tale of two actions”. Theorem 1.1 originally appeared in my thesis as Theorem 1.1.1. I discussed this result during my talk at the Geometric Models of Matter Conference in 2024.

The results in this paper allow us to study Euclidean monopoles, hyperbolic monopoles, and instantons with continuous symmetries. However, it is a general result that can also be used outside of gauge theory to easily find fixed points of Lie group actions on moduli spaces. These applications comprise the majority of my thesis.

Summary

We examine Lie group actions on moduli spaces and their fixed points. We show that when the Lie group is compact and connected, we obtain a linear constraint and the problem is one of representation theory. This greatly simplifies the search for fixed 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.

Introduction

Gauge theory involves two ingredients: objects and a gauge group acting on the objects. Physically, we only care about objects up to the gauge action. Denoting the space of objects by \(\mathcal{X}\) and the gauge group by \(\mathcal{G},\) we only care about the moduli space \(\mathcal{X}/\mathcal{G}.\) For instance, in electromagnetism, the objects of study are one-forms \(A\) representing the electromagnetic potential. We are only interested in the electric and magnetic fields generated by \(A\): the components of the curvature two-form \(F_A=dA+A\wedge A.\) The space of one-forms is acted on by the group of smooth maps \(g\colon \mathbb{R}^4\rightarrow \mathrm{U}(1)\) via \(g.A:=gAg^{-1}-(dg)g^{-1}.\) As these smooth maps are \(\mathrm{U}(1)\)-valued, we see that \(F_{g.A}=F_A.\) As such, physically, \(A\) and \(g.A\) are equivalent.

Gauge theoretic objects are difficult to find, they often involve solving non-linear constraints. We want to use symmetry to find examples. Let \(\mathcal{S}\) be a group of symmetries acting on \(\mathcal{X}\). Generally, when discussing symmetry, we mean points fixed by an action. However, looking for points on \(\mathcal{X}\) fixed by \(\mathcal{S}\) is too restrictive. Indeed, physically, two objects in the same gauge equivalence class are identical. As such, we want them to have the same symmetry, so we require the symmetry and gauge actions to commute. This means that the symmetry action descends to an action on the moduli space.

We are interested in fixed points on the moduli space. If \(A\in\mathcal{X}\) is such that \([A]\) is fixed by the symmetry action, then for all \(s\in \mathcal{S},\) there is some \(g\in\mathcal{G}\) such that \(s.A=g.A.\) That is, physically, the object is unaffected by the symmetry action.

Let \(M\) be a smooth manifold and \(G\) a Lie group acting smoothly on \(M\). The smoothness of the Lie group actions grants Lie algebra actions. If we have a right (left) group action, we obtain a Lie algebra (anti-)homomorphism taking value in \(\mathfrak{X}(M).\) Given \(x\in\mathrm{Lie}(G)\) and \(p\in M,\) the Lie algebra action is given by \(x.p:=\frac{d}{dt}\Big|_{t=0}\mathrm{exp}(tx).p\in T_pM.\)

Theorem 1.1

Let \(\mathcal{X}\) be a smooth manifold, \(\mathcal{G}\) a compact Lie group, and \(\mathcal{S}\) a compact, connected Lie group. Suppose that \(\mathcal{G}\) and \(\mathcal{S}\) act smoothly on \(\mathcal{X}\) on the left and the two actions commute. We have that \([A]\in\mathcal{X}/\mathcal{G}\) is fixed by \(\mathcal{S}\) if and only if there is a Lie algebra homomorphism \(\rho\colon\mathrm{Lie}(\mathcal{S})\rightarrow\mathrm{Lie}(\mathcal{G})\) such that, for all \(x\in\mathrm{Lie}(\mathcal{S}),\) \[x.A+\rho(x).A=0.\]

This result tells us that if we satisfy the proper technical conditions, then we can differentiate the equations of symmetry, obtaining linear equations. Thus, we can use linear algebra to find every example of our symmetric objects. As compact Lie groups can be realized as matrix Lie groups, the Lie algebra homomorphisms give us representations of Lie algebras, so we reduce the problem to the realm of representation theory.

When \(\mathcal{S}\) is one-dimensional, using a similar method, we obtain the same result even when we no longer have compactness.

Proposition 1.2

Let \(\mathcal{X}\) be a smooth manifold, \(\mathcal{G}\) a Lie group, and \(\mathcal{S}\) a connected, one-dimensional Lie group (isomorphic to either \(S^1\) or \(\mathbb{R}\)). Suppose that \(\mathcal{G}\) and \(\mathcal{S}\) act smoothly on \(\mathcal{X}\) on the left and the two actions commute. We have that \([A]\in\mathcal{X}/\mathcal{G}\) is fixed by \(\mathcal{S}\) if and only if there is some \(\rho\in\mathrm{Lie}(\mathcal{G})\) such that, for all \(t\in\mathbb{R}\), \[t.A+t\rho.A=0.\]

Note that in both of the above results, the two terms in the equations belong to \(T_A\mathcal{X}\) and thus can be added together to obtain zero. Nothing changes in these equations if we have two right group actions. However, if we have a mix of left and right group actions, then the equations become \(x.A-\rho(x).A=0\) and \(t.A-t\rho.A=0\), respectively.

Now we consider the case where we do not have connectedness.

Proposition 2.1

Let \(\mathcal{X}\) be a smooth manifold and let \(\mathcal{G}\) as well as \(\mathcal{S}\) be Lie groups. Denote the connected component of \(\mathcal{S}\) containing the identity by \(\mathcal{S}_0\subseteq\mathcal{S}\). Suppose that \(\mathcal{G}\) and \(\mathcal{S}\) act smoothly on \(\mathcal{X}\) and the two actions commute. We have that \([A]\in\mathcal{X}/\mathcal{G}\) is fixed by \(\mathcal{S}\) if and only if \([A]\) is fixed by some element in each connected component of \(\mathcal{S}\) and \([A]\) is fixed by \(\mathcal{S}_0\).

If the symmetry group \(\mathcal{S}\) is compact, then it has a finite number of connected components. As such, if \(\mathcal{S}\) is compact, then we need only check that a point is fixed by \(\mathcal{S}_0\) and a finite number of additional points. In this case, \(\mathcal{S}_0\) is a compact and connected Lie group, falling under the umbrella of Theorem 1.1.

Armed with Theorem 1.1, we can examine objects that are fixed by different symmetries. It turns out that we can ignore certain symmetries, as we can investigate their closure instead.

Proposition 2.4

Let \(\mathcal{X}\) be a smooth manifold and let both \(\mathcal{G}\) and \(\mathcal{S}\) be Lie groups. Suppose that \(\mathcal{G}\) and \(\mathcal{S}\) act on \(\mathcal{X}\) and their actions commute. Additionally, suppose that \(\mathcal{G}\) acts on \(\mathcal{X}\) properly. For \(A\in\mathcal{X}\), let \(H_A:=\{s\in \mathcal{S}\mid s\cdot [A]=[A]\}\). Then \(H_A\) is a closed Lie subgroup of \(\mathcal{S}\). Moreover, if \(\mathcal{S}\) is compact, then \(H_A\) is as well.

As an example of the importance of Proposition 2.4, suppose we look at objects symmetric under different connected Lie subgroups of \(S^1\times S^1\). Excluding the trivial subgroup, we must consider the Lie subgroups \(R_t:=\{(e^{i\theta},e^{it\theta})\mid \theta\in\mathbb{R}\}\), for all \(t\in\mathbb{R}\), \(R_\infty:=\{1\}\times S^1\), and \(S^1\times S^1\).

When \(t\in\mathbb{Q}\), \(R_t\simeq S^1\). Otherwise, \(R_t\simeq \mathbb{R}\), which is a problem as \(\mathbb{R}\) is not compact. Proposition 1.2 allows us to study objects symmetric under \(R_t\), for any \(t\), as \(\mathbb{R}\) is one-dimensional. However, Proposition 2.4 tells us that, in our case, we can ignore those symmetries that are not compact. Indeed, when \(t\notin\mathbb{Q}\), \(R_t\) is dense in \(S^1\times S^1\), so an element in the moduli space is fixed by \(R_t\) if and only if it is fixed by \(S^1\times S^1\). Thus, we are only left with compact and connected Lie groups for symmetry groups, exactly those we can study using Theorem 1.1.