Instantons with continuous conformal symmetries

Below we list the main results and examples from my paper “Instantons with continuous conformal symmetries: Hyperbolic and singular monopoles and more, oh my!” Other than Proposition 3.4, this work originally appeared as Chapter 3 of my thesis. I discussed this work during my talk at the Geometric Models of Matter Conference in 2024.

Summary

We classify the various continuous symmetries that an instanton can possess. We then linearize the symmetry equation for each continuous symmetry. By using representation theory, we can solve these linear equations given any representation. This makes the process of finding novel examples of instantons with higher rank structure groups much easier. In fact, for some symmetries, we know every example of instantons with that symmetry. Additionally, instantons with a variety of continuous symmetries are related to other gauge theoretic objects, so by investigating these symmetric instantons, we can better understand these other objects.

Introduction

We consider $\mathrm{Sp}(n)$ instantons: solutions to the self-dual equations with finite action that live over ${\mathbb{R}}^4$. Instantons correspond to ADHM data, using the ADHM transform.

As the self-dual equations are conformally invariant, we look at instantons with conformal symmetries. Based on the behaviour of the basic instanton and a similar result for isometries (proven in Appendix B of my thesis, though previously part of the folklore), I make the following conjecture about non-flat instantons—those with non-zero curvature.

Conjecture 2.14

If $\mathbb{A}$ is a non-flat instanton, then the group of conformal symmetries of $\mathbb{A}$ is conjugate to a subgroup of $\mathrm{Sp}(2).$

If the preceeding conjecture is true, then there are several continuous conformal symmetries, which I classify below. If the conjecture is false, then the only change is that there are additional kinds of continuous conformal symmetries.

Via the ADHM transform, instantons correspond with ADHM data. Up to gauge-equivalence, we can focus on ADHM data of a particular form.

Definition 2.16

Let ${\mathcal{M}}_{n,k}$ be the set of $(n+k)\times k$ quaternionic matrices $\hat{M}=\left[\begin{array}{c}L\\ M\end{array}\right]$ such that

• $M\in \mathrm{Mat}(k,k,\mathbb{H})$ is symmetric;
• $L\in \mathrm{Mat}(n,k,\mathbb{H})$ is such that $L{L}^{†}$ is a positive definite matrix;
• $R:=L^{†}L+M^{†}M$ is real and non-singular;
• for $x\in \mathbb{H}\simeq {\mathbb{R}}^4$, let $\mathrm{\Delta }(x):=\left[\begin{array}{c}L\\ M-I_{k}x\end{array}\right]$, then $\mathrm{\Delta }(x{)}^{†}\mathrm{\Delta }(x)$ is non-singular for all $x\in \mathbb{H}.$

ADHM data generally consist of pairs of matrices. In this case, if $\hat{M}\in {\mathcal{M}}_{n,k}$, then it is paired with $U:=\left[\begin{array}{c}{0}_{n,k}\\ I_k\end{array}\right].$ All $\hat{M}\in {\mathcal{M}}_{n,k}$ correspond to a $\mathrm{Sp}(n)$ instanton with instanton number $k.$ We use this form of ADHM data to discuss instantons with different continuous symmetries.

Circular $t$ symmetry

We start by studying instantons with circular $t$-symmetry, for $t\in [0,1].$

Theorem 3.1

Let $\hat{M}\in {\mathcal{M}}_{n,k}$ and $t\in [0,1].$ Then $\hat{M}$ has circular $t$-symmetry if and only if there exists $\rho \in \mathfrak{so}(k)$ such that $$tMi-iM+[\rho ,M]=0\text{and}[\rho ,R]=0.$$

Instantons with circular $t$-symmetry are interesting as hyperbolic monopoles with integral mass and singular monopoles with Dirac type singularities correspond to instantons with circular symmetry. In particular, the former correspond to instantons with circular $1$-symmetry and the latter instantons with circular $0$-symmetry. Thus, by better understanding instantons with circular $t$-symmetry, we better understand these other objects.

It turns out that we need only look at circular $t$-symmetry for $t\in [0,1]\cap \mathbb{Q}.$

Proposition 3.4

Let $\hat{M}\in {\mathcal{M}}_{n,k}$ and $t\in [0,1]\setminus \mathbb{Q}.$ Then $\hat{M}$ has circular $t$-symmetry if and only if it has toral symmetry.

Toral symmetry

Next, we study instantons with toral symmetry.

Theorem 4.1

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has toral symmetry if and only if there exists ${\rho }_1,{\rho }_2\in \mathfrak{so}(k)$ such that $[{\rho }_1,{\rho }_2]=0$ and we have $$iM=[{\rho }_1,M],-Mi=[{\rho }_2,M],[{\rho }_1,R]=0,\text{and}[{\rho }_2,R]=0.$$

Instantons with toral symmetry correspond to axially symmetric hyperbolic monopoles with integral mass and axially symmetric singular monopoles with Dirac type singularities.

Simple spherical symmetry

Now we move onto simple spherical symmetry.

Theorem 5.3

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has simple spherical symmetry if and only if there exists a real representation $\rho :\mathfrak{sp}(1)\to \mathfrak{so}(k)$ such that for all $\upsilon \in \mathfrak{sp}(1),$ $$[\upsilon ,M]+[\rho (\upsilon ),M]=0\text{and}[\rho (\upsilon ),R]=0.$$

In my paper, using representation theory, we prove the Structure Theorem, which solves the above symmetry equation for any given representation. This result makes finding instantons with simple spherical symmetry very easy. For examples of instantons with this kind of symmetry, we can look at spherically symmetric hyperbolic monopoles that I previously found.

Much like our previous symmetries, instantons with simple spherical symmetry correspond to another kind of object: a hyperbolic analogue to Higgs bundles. This is an analogue to Higgs bundles as it is a dimensional reduction of the self-dual equations, but invariant on a two-sphere instead of two axes. Additionally, if an instanton has only simple spherical symmetry, then it corresponds to a hyperbolic monopole with integral mass that possesses no continuous symmetries.

Isoclinic spherical symmetry

Next we examine isoclinic spherical symmetry.

Theorem 5.23

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has isoclinic spherical symmetry if and only if there exists a real representation $\rho :\mathfrak{sp}(1)\to \mathfrak{so}(k)$ such that for all $\upsilon \in \mathfrak{sp}(1),$ $$\upsilon M+[\rho (\upsilon ),M]=0\text{and}[\rho (\upsilon ),R]=0.$$

In my paper, using representation theory, we prove the Structure Theorem, which solves the above symmetry equation for any given representation. This result makes finding instantons with isoclinic spherical symmetry very easy. We list an example below, though we can find another when we examine rotational symmetry.

Example 1 (from Proposition 5.36)

Let $\lambda >0.$ The following $\hat{M}:=\left[\begin{array}{c}L\\ M\end{array}\right]\in {\mathcal{M}}_{3,7}$ corresponds with a $\mathrm{Sp}(3)$ instanton with isoclinic spherical symmetry and instanton number $7.$ $$M:=\lambda \left[\begin{array}{ccccccc}0& 0& 0& 0& 1& -i& j\\ 0& 0& 0& 0& i& 1& -k\\ 0& 0& 0& 0& j& -k& -1\\ 0& 0& 0& 0& k& j& i\\ 1& i& j& k& 0& 0& 0\\ -i& 1& -k& j& 0& 0& 0\\ j& -k& -1& i& 0& 0& 0\end{array}\right]\text{and}L:=\lambda \left[\begin{array}{ccccccc}3& -i& -j& -k& 0& 0& 0\\ 0& 2\sqrt{2}& k\sqrt{2}& -j\sqrt{2}& 0& 0& 0\\ 0& 0& \sqrt{6}& i\sqrt{6}& 0& 0& 0\end{array}\right].$$ This instanton was generated from the real representation whose complexification is the $\underset{―}{3}\oplus \underset{―}{2}\oplus \underset{―}{2}$ representation.

Instantons with isoclinic spherical symmetry correspond to another kind of object: a hyperbolic analogue to Nahm data. This is an analogue to Nahm data as it is a dimensional reduction of the self-dual equations, but along a three-sphere instead of three axes. Additionally, if an instanton has only isoclinic spherical symmetry, then it corresponds to a singular monopole with Dirac type singularities that possesses no continuous symmetries.

Isoclinic superspherical symmetry

We postpone our glance at conformal spherical symmetry until later and instead focus on isoclinic superspherical symmetry.

Theorem 6.2

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has isoclinic superspherical symmetry if and only if there exists a real representation $\rho :\mathfrak{sp}(1)\oplus \mathbb{R}\to \mathfrak{so}(k)$ such that for all $\upsilon \in \mathfrak{sp}(1)$ and $t\in \mathbb{R},$ $$\upsilon M+[\rho (\upsilon ,t),M]-2tMi=0\text{and}[\rho (\upsilon ,t),R]=0.$$

Instantons with isoclinic superspherical symmetry are interesting as they correspond with spherically symmetric singular monopoles with Dirac type singularities.

Rotational symmetry

Now we look at rotational symmetry.

Theorem 7.1

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has rotational symmetry if and only if there exists a real representation $\rho :\mathfrak{sp}(1)\oplus \mathfrak{sp}(1)\to \mathfrak{so}(k)$ such that for all $\upsilon ,\omega \in \mathfrak{sp}(1),$ $$\upsilon M+[\rho (\upsilon ,\omega ),M]-M\omega =0\text{and}[\rho (\upsilon ,\omega ),R]=0.$$

In my paper, using representation theory, we prove the Structure Theorem, which solves the above symmetry equation for any given representation. This result makes finding instantons with rotational symmetry very easy. We list an example below.

Example 2 (from Proposition 7.17)

Let $\lambda >0.$ The following $\hat{M}:=\left[\begin{array}{c}L\\ M\end{array}\right]\in {\mathcal{M}}_{3,5}$ corresponds with a $\mathrm{Sp}(3)$ instanton with rotational symmetry and instanton number $5.$ $$M:=\lambda \left[\begin{array}{ccccc}0& 0& 0& 0& 1\\ 0& 0& 0& 0& i\\ 0& 0& 0& 0& j\\ 0& 0& 0& 0& k\\ 1& i& j& k& 0\end{array}\right]\text{and}L:=\lambda \left[\begin{array}{ccccc}\sqrt{3}& -\frac{i}{\sqrt{3}}& -\frac{j}{\sqrt{3}}& -\frac{k}{\sqrt{3}}& 0\\ 0& 2\sqrt{\frac{2}{3}}& k\sqrt{\frac{2}{3}}& -j\sqrt{\frac{2}{3}}& 0\\ 0& 0& \sqrt{2}& i\sqrt{2}& 0\end{array}\right].$$ This instanton was generated from the real representation whose complexification is the $(\underset{―}{2},\underset{―}{2})\oplus (\underset{―}{1},\underset{―}{1})$ representation.

For all the above symmetries, there is a constraint on the upper part of $\hat{M}$ as well. This constraint allows us to determine when there is no element of ${\mathcal{M}}_{n,k}$ for a given $n$ and $k$ and symmetry, among other things. See my paper for more information.

Conformal superspherical symmetry

Next we investigate conformal superspherical symmetry.

Corollary 6.16

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has conformal superspherical symmetry if and only if it has simple spherical symmetry and there exists $\rho \in \mathfrak{sp}(n+k)$ such that $U^T\rho U-U^T\hat{M}$ is real and we also have $$\rho \hat{M}+U-\hat{M}{U}^T\rho U+\hat{M}{U}^T\hat{M}=0\text{and}\rho U-\hat{M}-U{U}^T\rho U+U{U}^T\hat{M}=0.$$

Instantons with conformal superspherical symmetry correspond with hyperbolic analogues of Higgs bundles with axial symmetry and spherically symmetric hyperbolic monopoles with integral mass. As such, we can look at spherically symmetric hyperbolic monopoles that I previously found for examples of instantons with this kind of symmetry. However, there are examples outside of that set of ADHM data.

Example 3 (from Proposition 6.19)

Let $B\in (0,\frac{2}{3}\sqrt{6}).$ Define $$A:=\frac{\sqrt{12-15B^2+12\sqrt{B^4-B^2+1}}}{3}\text{and}a:=\frac{2B^2-\sqrt{B^4-B^2+1}-1}{\sqrt{-3B^2+6+6\sqrt{B^4-B^2+1}}}.$$ Note that $A>0$ and $a\in \mathbb{R}.$ The following $\hat{M}\in {\mathcal{M}}_{2,2}$ corresponds to a $\mathrm{Sp}(2)$ instanton with conformal superspherical symmetry and instanton number $2$: $$\hat{M}:=\left[\begin{array}{cc}A& 0\\ 0& B\\ 0& a\\ a& 0\end{array}\right].$$ Additionally, when $B\ne 1$, this is not in the set of ADHM data considered in my paper on hyperbolic monopoles with continuous symmetries.

Conformal spherical and full symmetry

It turns out that we know all of the instantons with the remaining conformal symmetries.

Theorem 5.43

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has conformal spherical symmetry if and only if $n=k$ and the instanton is the direct sum of $k$ copies of the basic instanton.

Corollary 5.44

Let $\hat{M}\in {\mathcal{M}}_{n,k}.$ Then $\hat{M}$ has full symmetry if and only if $n=k$ and the instanton is the direct sum of $k$ copies of the basic instanton.