Hyperbolic monopoles with continuous symmetries

Below we list the main results and examples from my paper “Hyperbolic monopoles with continuous symmetries”.

Summary

By linearizing the symmetry equations and using representation theory, we identify many novel examples of hyperbolic monopoles with higher rank structure groups. In fact, in the paper, we identify many infinite families of spherically symmetric, hyperbolic monopoles with varying rank structure groups. These monopoles have interesting properties; in Example 2 below, we identify a spherically symmetric \(\mathrm{Sp}(4)\) hyperbolic monopole that vanishes nowhere.

Introduction

In this paper, we consider \(\mathrm{Sp}(n)\) hyperbolic monopoles with integral mass using the ball model \(H^3=\lbrace X\in\mathfrak{sp}(1)\mid r^2:=|X|^2<1\rbrace.\) Such hyperbolic monopoles correspond to instantons with circular symmetry (specifically circular \(1\)-symmetry, using the notation from my thesis). Instantons in turn correspond to ADHM data, using the ADHM transform. In this paper, we look at a specific subset of ADHM data.

Definition 1

Let \(\mathcal{M}_{n,k}\) be the set of \((n+k)\times k\) quaternionic matrices \(\hat{M}=\begin{bmatrix} L \\ M\end{bmatrix}\) such that

\(M=M_1i+M_2j+M_3k\in\mathrm{Mat}(k,k,\mathbb{H})\) is symmetric and \(M_1,M_2,M_3\) are real;
\(L\in\mathrm{Mat}(n,k,\mathbb{H})\) is such that \(LL^\dagger\) is a positive definite matrix;
\(L^\dagger L-M^2=I_k;\)
for \(x\in\mathbb{H}\simeq \mathbb{R}^4,\) let \(\Delta(x):=\begin{bmatrix} L \\ M-I_kx \end{bmatrix}\), then \(\Delta(x)^\dagger \Delta(x)\) is non-singular for all \(x\in\overline{H^3}.\)

All \(\hat{M}\in\mathcal{M}_{n,k}\) correspond to \(\mathrm{Sp}(n)\) hyperbolic monopoles with instanton number \(k\).

Symmetric hyperbolic monopoles

We use the above ADHM data to study hyperbolic monopoles with continuous symmetries. We start with axial symmetry.

Theorem 1

Let \(\hat{M}\in\mathcal{M}_{n,k}\) and \(\upsilon:=\frac{k}{2}\in\mathfrak{sp}(1).\) Then \(\hat{M}\) is axially symmetric about the \(z\)-axis if and only if there exists \(Y\in\mathfrak{so}(k)\) such that \[[M,Y]=\left[\upsilon,M\right].\]

We then look at spherical symmetry.

Theorem 2

Let \(\hat{M}\in\mathcal{M}_{n,k}.\) Then \(\hat{M}\) is spherically symmetric if and only if there exists real representation \((\mathbb{R}^k,\rho)\) with \(\rho\colon\mathfrak{sp}(1)\rightarrow\mathfrak{so}(k),\) such that for all \(\upsilon\in\mathfrak{sp}(1),\) \[[M,\rho(\upsilon)]=[\upsilon,M].\] The induced representation is said to generate the spherical symmetry of \(\hat{M}.\)

In the paper, using representation theory, we prove the Structure Theorem, which solves the above symmetry equation for any given representation. This result makes finding spherically symmetric hyperbolic monopoles very easy. We list some examples below, though more are found in the paper.

Example 1 (from Proposition 3)

Let \(\hat{M}:=\begin{bmatrix} I_k \\ 0_k\end{bmatrix}\in\mathcal{M}_{k,k}.\) This ADHM data corresponds with a spherically symmetric \(\mathrm{Sp}(k)\) hyperbolic monopole with instanton number \(k\). This monopole corresponds to the basic instanton.

Up to gauge, we have that the Higgs field of the corresponding hyperbolic monopole is given by \(\Phi(X)=\frac{X}{1+r^2}I_k.\) As \(r=|X|\rightarrow 1\), we see that the modulus of the eigenvalues of \(\Phi(X)\) approach \(\frac{1}{2}.\) Thus, this monopole has generalized minimal symmetry breaking. Additionally, we have \[|\Phi(r)|^2=k\left(\frac{r}{1+r^2}\right)^2.\] The energy density of the monopole is \[\varepsilon=\frac{3k}{2}\left(\frac{1-r^2}{1+r^2}\right)^4.\] Note that the norm squared of the Higgs field and the energy density scale linearly with \(k\).

Below, we see the norm squared of the Higgs field as well as the energy density of the monopole when \(k=1\). From the figure, we see that the Higgs field only vanishes at the origin and the monopole looks like a point-particle, in that the energy density has a global maximum at the origin.

The norm squared of the Higgs field and the energy density for the spherically symmetric \mathrm{Sp}(1) hyperbolic monopole. The solid line is \varepsilon with the left vertical axis and the dashed line is |\Phi|^2 with the right vertical axis. — The norm squared of the Higgs field and the energy density for the spherically symmetric \(\mathrm{Sp}(1)\) hyperbolic monopole. The solid line is \(\varepsilon\) with the left vertical axis and the dashed line is \(|\Phi|^2\) with the right vertical axis.

Example 2 (from Proposition 6)

The following \(\hat{M}\in\mathcal{M}_{2,4}\) corresponds with a spherically symmetric \(\mathrm{Sp}(2)\) hyperbolic monopole with instanton number \(4\): \[\hat{M}:=\frac{1}{\sqrt{3}}\begin{bmatrix} \sqrt{2} & -\frac{i}{\sqrt{2}} & \frac{j}{\sqrt{2}} & 0 \\ 0 & \sqrt{\frac{3}{2}} & -k\sqrt{\frac{3}{2}} & 0 \\ 0 & 0 & 0 & k \\ 0 & 0 & 0 & j \\ 0 & 0 & 0 & i \\ k & j & i & 0 \end{bmatrix}.\] This monopole was generated from the real representation \((\mathbb{R}^3,\mathrm{ad})\oplus (\mathbb{R},0),\) whose complexification is the \(\underline{3}\oplus\underline{1}\) representation.

Along any ray emanating from the origin, up to gauge, the Higgs field of the corresponding monopole \(\Phi\) is given by \[\Phi(r):=\begin{bmatrix} \frac{ir}{r^2+1} & 0 \\ 0 & -\frac{ir(r^4+6r^2+1)}{(r^2+1)(3r^4+2r^2+3)}\end{bmatrix}.\] Note that the eigenvalues of \(\Phi(r)\) as \(r\rightarrow 1\) are \(\pm\frac{i}{2}\). Thus, this monopole has generalized minimal symmetry breaking. Then we have \[|\Phi(r)|^2=\frac{2(5r^8+12r^6+30r^4+12r^2+5)r^2}{(3r^4+2r^2+3)^2(r^2+1)^2}.\] The energy density of the monopole is \[\varepsilon=\frac{(1-r^2)^4}{(1+r^2)^4}\cdot\frac{135r^{16}+840r^{14}+5252r^{12}+13304r^{10}+18282r^8 +13304r^6+5252r^4+840r^2+135}{(3r^4+2r^2+3)^4}.\]

Below, we see the norm squared of the Higgs field as well as the energy density of the monopole. From the figure, we see that the Higgs field only vanishes at the origin and the monopole looks like a point-particle, in that the energy density has a global maximum at the origin.

The norm squared of the Higgs field and the energy density for the spherically symmetric \mathrm{Sp}(2) hyperbolic monopole. The solid line is \varepsilon with the left vertical axis and the dashed line is |\Phi|^2 with the right vertical axis. — The norm squared of the Higgs field and the energy density for the spherically symmetric \(\mathrm{Sp}(2)\) hyperbolic monopole. The solid line is \(\varepsilon\) with the left vertical axis and the dashed line is \(|\Phi|^2\) with the right vertical axis.

Example 3 (from Proposition 7)

The following \(\hat{M}\in\mathcal{M}_{4,6}\) corresponds with a spherically symmetric \(\mathrm{Sp}(4)\) hyperbolic monopole with instanton number \(6\): \[Y:=\begin{bmatrix} 0 & i & -j \\ -i & 0 & k \\ j & -k & 0 \end{bmatrix}, \quad l:= \begin{bmatrix} \sqrt{2} & -\frac{i}{\sqrt{2}} & \frac{j}{\sqrt{2}} \\ 0 & \sqrt{\frac{3}{2}} & -k\sqrt{\frac{3}{2}} \end{bmatrix}, \quad\textrm{and}\quad \hat{M}:=\frac{1}{2}\begin{bmatrix} l & 0 \\ 0 & l \\ 0 & Y \\ -Y & 0 \end{bmatrix}.\] This monopole was generated from the real representation \((\mathbb{R}^3,\mathrm{ad})\oplus (\mathbb{R}^3,\mathrm{ad}),\) whose complexification is the \(\underline{3}\oplus\underline{3}\) representation.

The Higgs field of the corresponding hyperbolic monopole \(\Phi\) is too large to output here. However, the norm squared of the Higgs field is sufficiently small: \[|\Phi(r)|^2=\frac{r^{12}+9r^{10}+33r^8+58r^6+33r^4+9r^2+1}{4(r^4+r^2+1)^2(r^2+1)^2}.\] The energy density of the monopole is \[\varepsilon=\frac{3(1-r^2)^4}{8(1+r^2)^4}\cdot\frac{5r^{16}+70r^{14}+381r^{12}+942r^{10}+1260r^8+942r^6+381r^4+70r^2+5}{(r^4+r^2+1)^4}.\] As \(r\rightarrow 1\), we have that \[\Phi\rightarrow \begin{bmatrix} \frac{17i}{50} & \frac{12}{175}\sqrt{14} & \frac{8}{3325}\sqrt{3990} & \frac{6i}{475}\sqrt{285} \\ -\frac{12}{175}\sqrt{14} & \frac{31i}{350} & -\frac{48i}{3325}\sqrt{285} & \frac{18}{3325}\sqrt{3990} \\ -\frac{8}{3325}\sqrt{3990} & -\frac{48i}{3325}\sqrt{285} & \frac{473i}{1330} & \frac{36}{665}\sqrt{14} \\ \frac{6i}{475}\sqrt{285} & -\frac{18}{3325}\sqrt{3990} & -\frac{36}{665}\sqrt{14} & \frac{41i}{190} \end{bmatrix}.\] This matrix has eigenvalues \(\frac{i}{2}\) and \(-\frac{i}{2}\) with multiplicity \(3\) and \(1\), respectively. Thus, this monopole has generalized minimal symmetry breaking

Below, we see the norm squared of the Higgs field as well as the energy density of the monopole. From the figure, we see that the Higgs field never vanishes and the monopole looks like a shell of charge, in that the energy density has a global maximum away from the origin.

The norm squared of the Higgs field and the energy density for the spherically symmetric \mathrm{Sp}(4) hyperbolic monopole. The solid line is \varepsilon with the left vertical axis and the dashed line is |\Phi|^2 with the right vertical axis. — The norm squared of the Higgs field and the energy density for the spherically symmetric \(\mathrm{Sp}(4)\) hyperbolic monopole. The solid line is \(\varepsilon\) with the left vertical axis and the dashed line is \(|\Phi|^2\) with the right vertical axis.

The remainder of the paper focuses on a constraint on the ADHM data of spherically symmetric hyperbolic monopoles obtained using representation theory. This constraint allows us to determine when there is no spherically symmetric element of \(\mathcal{M}_{n,k}\) for a given \(n\) and \(k\), among other things.