Euclidean monopoles with continuous symmetries

Below we list the main results and examples from my joint paper “Construction of Nahm data and BPS monopoles with continuous symmetries”, including improvements mentioned in my paper “Hyperbolic monopoles with continuous symmetries”.

Summary

By linearizing the symmetry equations and using representation theory, we identify many novel examples of Euclidean monopoles with higher rank structure groups and non-maximal symmetry breaking. In the paper, we identify many infinite families of spherically symmetric, Euclidean monopoles with varying rank structure groups.

Introduction

In this paper, we consider \(\mathrm{SU}(N)\) Euclidean monopoles. Using the Nahm transform, such monopoles correspond to Nahm data, solutions to the Nahm equations.

Definition

A triple \(\mathcal{T}=(T_1,T_2,T_3)\) of \(\mathfrak{su}(n)^{\oplus 3}\)-valued functions on an open subset of the real line is called Nahm data if it satisfies the Nahm equations: \[\frac{dT_1}{dt}=[T_2,T_3], \quad \frac{dT_2}{dt}=[T_3,T_1], \quad\textrm{and}\quad \frac{dT_3}{dt}=[T_1,T_2].\]

We use Nahm data to study Euclidean monopoles with continuous symmetries. We start with axial symmetry.

Theorem 3.1

Let \(\mathcal{T}\) be \(\mathfrak{su}(n)^{\oplus 3}\)-valued Nahm data. Then \(\mathcal{T}\) is axially symmetric about the \(z\)-axis if and only if there exists \(Y\in\mathfrak{su}(n)\) such that \[T_1=\left[T_2,Y\right], \quad T_2=\left[Y,T_1\right], \quad\textrm{and}\quad 0=[T_3,Y].\]

We then look at spherical symmetry. Let \(\epsilon\) be the Levi-Civita symbol and \(\lbrace\upsilon_1,\upsilon_2,\upsilon_3\rbrace\subseteq\mathfrak{so}(3)\) be the standard basis given by \((\upsilon_i)_ {jk}=-\epsilon_{ijk}.\)

Theorem 4.1

Let \(\mathcal{T}\) be \(\mathfrak{su}(n)^{\oplus 3}\)-valued Nahm data. Then \(\mathcal{T}\) is spherically symmetric if and only if there exists complex representation \((\mathbb{C}^n,\rho)\) with \(\rho\colon\mathfrak{so}(3)\rightarrow\mathfrak{su}(n),\) such that for all \(i,j\in\lbrace 1,2,3\rbrace,\) \[[\rho(\upsilon_i),T_j]=\sum_{k=1}^3\epsilon_{ijk}T_k.\] The induced representation is said to generate the spherical symmetry of \(\mathcal{T}.\)

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 Euclidean monopoles very easy. We list some examples below, though more are found in the paper.

Example 1 (from Theorem 4.11)

The following \(\mathcal{T}\) corresponds with a spherically symmetric \(\mathrm{SU}(4)\) monopole. Let \(f(t):=-\frac{t}{t^2-1}\) and \(g(t):=\frac{i}{t^2-1}.\) Then let \[T_1:=\begin{bmatrix}0 & 0 & 0 & g \\ 0 & 0 & f & 0 \\ 0 & -f & 0 & 0 \\ g & 0 & 0 & 0\end{bmatrix}, \quad T_2:=\begin{bmatrix}0 & 0 & f & 0 \\ 0 & 0 & 0 & -g \\ -f & 0 & 0 & 0 \\ 0 & -g & 0 & 0\end{bmatrix}, \quad\textrm{and}\quad T_3:=\begin{bmatrix}0 & f & 0 & 0 \\ -f & 0 & 0 & 0 \\ 0 & 0 & 0 & g \\ 0 & 0 & g & 0\end{bmatrix}.\] This monopole was generated from the representation \((\mathbb{C}^3,\mathrm{ad})\oplus (\mathbb{C},0)\), which is the \(\underline{3}\oplus\underline{1}\) representation.

Along any ray emanating from the origin, up to gauge, the asymptotic expansion of the Higgs field of the corresponding monopole \(\Phi\) is given by \[\Phi(r):=i\cdot\mathrm{diag}(1,1,-1,-1)-\frac{i}{2r}\mathrm{diag}(2,2,-2,-2)+\mathcal{O}\left(r^{-2}\right).\] Thus, the monopole topologically decomposes the trivial \(\mathrm{SU}(4)\) bundle over the “sphere at infinity” into two, nontrivial \(\mathrm{U}(2)\) bundles, both of which are further decomposed holomorphically into line bundles with Chern numbers \(\pm 2.\) We see that this monopole has neither maximal nor minimal symmetry breaking. Specifically, it 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 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.

Example 2 (from Theorem 4.12)

In Theorem 4.12, we identify Nahm data corresponding to a spherically symmetric \(\mathrm{SU}(6)\) monopole. This monopole was generated from the \(\underline{5}\oplus\underline{3}\oplus\underline{1}\) representation. The explicit Nahm data is too large to display here, see the paper for more information.

We are unable to perform the Nahm transform on this data due to computational limitations. However, from general theory, we know that along any ray from the origin, up to gauge, the asymptotic expansion of the Higgs field of the corresponding monopole \(\Phi\) is given by \[\Phi(r):=i\cdot\mathrm{diag}\left(1,1,1,-1,-1,-1\right)-\frac{i}{2r}\mathrm{diag}(3,3,3,-3,-3,-3)+\mathcal{O}\left(r^{-2}\right).\] Thus, the monopole topologically decomposes the trivial \(\mathrm{SU}(6)\) bundle over the “sphere at infinity” into two, nontrivial \(\mathrm{U}(3)\) bundles, both of which are further decomposed holomorphically into line bundles with Chern numbers \(\pm 3.\) We see that this monopole has neither maximal nor minimal symmetry breaking. Specifically, it has generalized minimal symmetry breaking.

Example 3 (from Theorem 4.13)

The following \(\mathcal{T}\) corresponds with a spherically symmetric \(\mathrm{SU}(5)\) monopole. Let \(f^+(t):=-\frac{3t+1}{3(t^2-1)},\) \(f^-(t):=-\frac{3t+5}{3(t^2-1)},\) and \(g(t):=\frac{2}{\sqrt{3}(t^2-1)}.\) Then let \[\begin{aligned}T_1&:=\begin{bmatrix} \frac{3i}{2}f^+ & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{i}{2}f^+ & 0 & 0 & \sqrt{\frac{2}{3}}g & 0 \\ 0 & 0 & -\frac{i}{2}f^+ & 0 & 0 & \sqrt{\frac{2}{3}}g \\ 0 & 0 & 0 & -\frac{3i}{2}f^+ & 0 & 0 \\ 0 & -\sqrt{\frac{2}{3}}g & 0 & 0 & \frac{i}{2}f^- & 0 \\ 0 & 0 & -\sqrt{\frac{2}{3}}g & 0 & 0 & -\frac{i}{2}f^- \end{bmatrix}, \\\ T_2&:=\begin{bmatrix} 0 & \frac{\sqrt{3}}{2}f^+ & 0 & 0 & \frac{i}{\sqrt{2}}g & 0 \\ -\frac{\sqrt{3}}{2}f^+ & 0 & f^+ & 0 & 0 & \frac{i}{\sqrt{6}}g \\ 0 & -f^+ & 0 & \frac{\sqrt{3}}{2}f^+ & \frac{i}{\sqrt{6}}g & 0 \\ 0 & 0 & -\frac{\sqrt{3}}{2}f^+ & 0 & 0 & \frac{i}{\sqrt{2}}g \\ \frac{i}{\sqrt{2}}g & 0 & \frac{i}{\sqrt{6}}g & 0 & 0 & \frac{1}{2}f^- \\ 0 & \frac{i}{\sqrt{6}}g & 0 & \frac{i}{\sqrt{2}}g & -\frac{1}{2}f^- & 0 \end{bmatrix}, \quad\textrm{and}\\\ T_3&:=\begin{bmatrix} 0 & \frac{i\sqrt{3}}{2}f^+ & 0 & 0 & -\frac{1}{\sqrt{2}}g & 0 \\ \frac{i\sqrt{3}}{2}f^+ & 0 & if^+ & 0 & 0 & -\frac{1}{\sqrt{6}}g \\ 0 & if^+ & 0 & \frac{i\sqrt{3}}{2}f^+ & \frac{1}{\sqrt{6}}g & 0 \\ 0 & 0 & \frac{i\sqrt{3}}{2}f^+ & 0 & 0 & \frac{1}{\sqrt{2}}g \\ \frac{1}{\sqrt{2}}g & 0 & -\frac{1}{\sqrt{6}}g & 0 & 0 & \frac{i}{2}f^- \\ 0 & \frac{1}{\sqrt{6}}g & 0 & -\frac{1}{\sqrt{2}}g & \frac{i}{2}f^- & 0 \end{bmatrix}. \end{aligned}\] This monopole was generated from the \(\underline{4}\oplus\underline{2}\) representation.

Along any ray emanating from the origin, up to gauge, the asymptotic expansion of the Higgs field of the corresponding monopole \(\Phi\) is given by \[\Phi(r):=i\cdot\mathrm{diag}\left(\frac{4}{5},\frac{4}{5},\frac{4}{5},-\frac{6}{5},-\frac{6}{5}\right)-\frac{i}{2r}\mathrm{diag}(2,2,2,-3,-3)+\mathcal{O}\left(r^{-2}\right).\] Thus, the monopole topologically decomposes the trivial \(\mathrm{SU}(5)\) bundle over the “sphere at infinity” into the direct sum of a \(\mathrm{U}(3)\) and a \(\mathrm{U}(2)\) bundle, both of which are further decomposed holomorphically into line bundles with Chern numbers \(2\) and \(-3\), respectively. We see that this monopole has neither maximal nor minimal symmetry breaking. Specifically, it 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 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.