Symmetry breaking of monopoles

Below we list the main results and examples from Section 2.3 of my thesis: “Symmetry breaking”.

Summary

We identify the symmetry breaking of monopoles, part of their topological classification. In particular, for monopoles with structure group a Lie group with a classical, simple Lie algebra, we provide simple classifications based on eigenvalues of the Higgs field at infinity.

Introduction

Let \(G\) be a compact, connected, real Lie group and \(M\) a connected, geodesically complete three-manifold with infinite injectivity radius at some point. Let \((\Phi,A)\) be a monopole on \(M.\) That is, the pair satisfies the Bogomolny equations and a finite energy condition.

Let \(\mathfrak{g}\) be the Lie algebra of \(G\) and \(\mathfrak{z}(\mathfrak{g})\) be the centre of \(\mathfrak{g}.\)

When \(M=\mathbb{R}^3,\) it is conjectured that the finite energy condition for monopoles implies an extra boundary condition. That is, there exists some element \(\Phi_\infty\in\mathfrak{g}\setminus \mathfrak{z}(\mathfrak{g})\) such that along every oriented geodesic in \(M=\mathbb{R}^3,\) as you approach infinity, \(\Phi\) approaches \(\mathrm{Ad}_g(\Phi _\infty),\) for some \(g\in G.\) We assume that our monopoles satisfy this extra boundary condition for any \(M.\)

Given this boundary condition, the Higgs field is said to spontaneously break the symmetry group \(G\) to the isotropy group \(H\subseteq G\) given by \[H:=\lbrace g\in G\mid \mathrm{Ad}_g(\Phi _\infty)=\Phi _\infty\rbrace.\] The group \(H\) is a compact, connected, real Lie group. Let \(\mathfrak{h}\) be the Lie algebra of \(H\).

The adjoint orbit of \(\Phi_\infty\) is the orbit of \(\Phi_\infty\) under the adjoint action of \(G\) on \(\mathfrak{g}.\) This orbit is diffeomorphic to \(G/H.\) Associated to every monopole is a gauge-invariant class in \(\pi_2(G/H),\) the degree of the monopole. The group \(\pi_2(G/H)\) itself and the degree of a monopole comprise the topological classification of a monopole. We are only interested in the group \(\pi_2(G/H).\)

Classifying symmetry breaking

An adjoint orbit is also known as a generalized flag manifold. This viewpoint allows us to see the adjoint orbit as a complex manifold.

Proposition 2.3.10

For \(G\) compact and connected, the adjoint orbit \(G/H\) is a CW complex with cells only in even dimensions.

Using this viewpoint and the Hurewicz Theorem, we discover that \(\pi_2(G/H)\) is a free abelian group. Moreover, we know the rank of this group.

Theorem 2.3.13

For \(G\) a compact, connected Lie group, \[\pi_2(G/H)\simeq \mathbb{Z}^{\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{h}))-\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{g}))}.\]

Given this result, we only need to understand \(\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{h})).\)

Proposition 2.3.16 (reworded)

We have \[\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{g}))< \mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{h}))\leq \mathrm{rank}(\mathfrak{g}).\]

These bounds motivate the following definitions.

Definition 2.3.17

We say that \(\Phi_\infty\) has generalized minimal symmetry breaking if \(\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{h}))=\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{g}))+1.\) Additionally, we say that \(\Phi_\infty\) has maximal symmetry breaking if \(\mathrm{dim}_\mathbb{R}(\mathfrak{z}(\mathfrak{h}))=\mathrm{rank}(\mathfrak{g}).\)

There are many equivalent ways to view generalized minimal and maximal symmetry breaking. These are outlined in Theorem 2.3.27 in my thesis. Now we focus on classifying the symmetry breaking for simple groups.

Symmetry breaking for simple groups

The symmetry breaking of a monopole with a semi-simple structure group depends solely on the symmetry breaking of each component of the decomposition into simple groups. Thus, we need only look at such groups. We start with the most familiar: \(\mathrm{SU}(n)\) monopoles.

Theorem 2.3.31

Let \(\Phi_\infty\in\mathfrak{su}(n).\) Let \(N\) be the number of distinct eigenvalues of \(\Phi_\infty.\) Then \(\pi_2(G/H)\simeq \mathbb{Z}^{N-1}.\) In particular, \(\Phi_\infty\) has maximal symmetry breaking if and only if all eigenvalues of \(\Phi_\infty\) are distinct. Additionally, \(\Phi_\infty\) has generalized minimal symmetry breaking if and only if \(\Phi_\infty\) has two distinct eigenvalues.

Note that the notion of maximal symmetry breaking here agrees with the typical notion for \(\mathrm{SU}(n)\) monopoles. However, \(\Phi_\infty\) only has the usual minimal symmetry breaking if it has generalized minimal symmetry breaking and the multiplicity of one of the eigenvalues is one.

Example 2.3.34

In my joint paper “Construction of Nahm data and BPS monopoles with continuous symmetries”, we identify several novel spherically symmetric \(\mathrm{SU}(n)\) Euclidean monopoles. Up to gauge, the Higgs fields have the following values at infinity: \[\begin{gather} i\cdot\mathrm{diag}(1,1,-1,-1)\in\mathfrak{su}(4),\\ i\cdot\mathrm{diag}(1,1,1,-1,-1,-1)\in\mathfrak{su}(6),\\ \frac{i}{5}\cdot\mathrm{diag}(4,4,4,-6,-6)\in\mathfrak{su}(5). \end{gather}\] These all have generalized minimal symmetry breaking, but neither maximal nor minimal symmetry breaking in the usual sense.

Now we move to \(\mathrm{SO}(2m+1)\) monopoles.

Theorem 2.3.35

Let \(\Phi_\infty\in\mathfrak{so}(2m+1),\) with \(m\geq 1.\) Let \(\pm \lambda_1 i,\ldots,\pm \lambda_m i,\) and \(\lambda_{m+1}:=0\) be the eigenvalues of \(\Phi_\infty.\) Let \(N\) be the number of distinct values of \(\lambda_i^2.\) Then \(\pi_2(G/H)\simeq \mathbb{Z}^{N-1}.\) In particular, \(\Phi_\infty\) has maximal symmetry breaking if and only if all \(\lambda_i^2\) are distinct. Additionally, \(\Phi_\infty\) has generalized minimal symmetry breaking if and only if there is some \(\lambda\neq 0\) such that at least one \(\lambda_i^2=\lambda^2\) and the rest of the \(\lambda_j^2\) are either \(\lambda^2\) or zero.

Next is \(\mathrm{Sp}(n)\) monopoles.

Theorem 2.3.35

Let \(\Phi_\infty\in\mathfrak{sp}(n).\) Let \(\alpha_1,\ldots,\alpha_n\) be the modulus of the right eigenvalues of \(\Phi_\infty.\) Let \(N\) be the number of distinct, non-zero values of \(\alpha_i.\) Then \(\pi_2(G/H)\simeq \mathbb{Z}^N.\) In particular, \(\Phi_\infty\) has maximal symmetry breaking if and only if all \(\alpha_i\) are non-zero and distinct. Additionally, \(\Phi_\infty\) has generalized minimal symmetry breaking if and only if there is some \(\alpha>0\) such that at least one \(\alpha_i=\alpha\) and the rest of the \(\alpha_j\) are either \(\alpha\) or zero.

Example 2.3.39

In my paper “Hyperbolic monopoles with continuous symmetries”, I identify several novel spherically symmetric \(\mathrm{Sp}(n)\) hyperbolic monopoles. Up to gauge, the Higgs fields have the following values at infinity: \[\begin{gather} \frac{i}{2}I_k\in\mathfrak{sp}(k), \\ \frac{i}{2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\in\mathfrak{sp}(2),\\ \begin{bmatrix} \frac{17i}{50} & \frac{12}{175}\sqrt{14} & \frac{8}{3325}\sqrt{3990} & \frac{6i}{475}\sqrt{285} \\ -\frac{12}{175}\sqrt{14} & \frac{31i}{350} & -\frac{48i}{3325}\sqrt{285} & \frac{18}{3325}\sqrt{3990} \\ -\frac{8}{3325}\sqrt{3990} & -\frac{48i}{3325}\sqrt{285} & \frac{473i}{1330} & \frac{36}{665}\sqrt{14} \\ \frac{6i}{475}\sqrt{285} & -\frac{18}{3325}\sqrt{3990} & -\frac{36}{665}\sqrt{14} & \frac{41i}{190} \end{bmatrix}\in\mathfrak{sp}(4). \end{gather}\] These all have right eigenvalues in \(\lbrace\pm i/2\rbrace.\) Thus, they all have generalized minimal symmetry breaking.

Finally, we have \(\mathrm{SO}(2m)\) monopoles. Note that we ignore \(\mathrm{SO}(2)\) monopoles, as these are abelian. We do however, consider \(\mathrm{SO}(4)\) monopoles, even though this group is not simple. This is for the sake of completeness.

Theorem 2.3.50

Let \(\Phi_\infty\in\mathfrak{so}(2m),\) with \(m\geq 2.\) Let \(\pm \lambda_1 i,\ldots,\pm \lambda_m i\) be the eigenvalues of \(\Phi_\infty.\) Let \(N\) be the number of distinct values of \(\lambda_i^2\) and let \(\mu\) be one if zero is an eigenvalue of \(\Phi_\infty\) with multiplicity greater than two and zero otherwise. Then \(\pi_2(G/H)\simeq \mathbb{Z}^{N-\mu}.\) In particular, \(\Phi_\infty\) has maximal symmetry breaking if and only if all \(\lambda_i^2\) are distinct. Additionally, \(\Phi_\infty\) has generalized minimal symmetry breaking if and only if there is some \(\lambda\neq 0\) such that at least one \(\lambda_i^2=\lambda^2\) and the rest of the \(\lambda_j^2=\lambda^2\) or at least two \(\lambda_j^2=0\) and the rest are \(\lambda^2.\)

For methods classifying the symmetry breaking when dealing with the exceptional simple Lie algebras, see Section 2.3.3.2 of my thesis.