# The Geometry and Dynamics of Magnetic Monopoles

MICHAEL FRANCIS ATIYAH
NIGEL HITCHIN
Series: Porter Lectures
Pages: 142
https://www.jstor.org/stable/j.ctt7zv206

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. PREFACE
(pp. vii-2)
Michael Atiyah
4. INTRODUCTION
(pp. 3-8)

The purpose of this book is to apply geometrical methods to investigate solutions of the non-linear system of hyperbolic equations which describe the time evolution of non-abelian magnetic monopoles. The problem we study is, in various respects, a somewhat simplified model but it retains sufficient features to be physically interesting. It gives information about the low-energy scattering of monopoles and it exhibits some new and significant phenomena.

From a purely mathematical point of view our investigation should be seen as a contribution to the area of “soliton” theory. In general a soliton is a solution of some non-linear differential equation...

5. CHAPTER 1 The Monopole Equations
(pp. 9-13)

We give here an outline of the physical background out of which the monopoles we analyse, described by solutions of the Bogomolny equations, arise. The reader is directed to [30], [12], and [11] for further information on the links between the mathematical and physical theory.

We are concerned here with a gauge theory, the prototype of which is electromagnetic theory. In differential geometric terms the electric fieldEin Maxwell theory is considered as a 1-form on R3and the magnetic fieldBas a 2-form. The Maxwell field tensor F = B + c dt ˄ E is then,...

6. CHAPTER 2 Geometry of the Monopole Spaces
(pp. 14-20)

We begin by reviewing rapidly the definitions of monopoles and their parameter or moduli spaces. For further details we refer to [22] or [30]. We shall throughout take the gauge groupGto be SU(2) although our methods can in principle be extended to allG.

The data for a monopole on R3consist of a gauge field or connectionAμ(x),μ= 1, 2, 3, and a Higgs fieldϕ(x). All these are smooth functions of x ∈ R3and take their values in the Lie algebra of SU(2). As usual one defines the covariant derivativeDμϕby${{D}_{\mu }}\phi ={{\partial }_{\mu }}\phi +\left[ {{A}_{\mu }},\phi \right]$...

7. CHAPTER 3 Metric of Monopole Spaces
(pp. 21-27)

In the previous chapter we saw that the parameter space of basedk-monopoles is a manifoldMkof dimension 4k, and Donaldson’s theorem gives us a simple explicit model of this manifold. We shall now go on to introduce and investigate the natural Riemannian metric ofMk. This is given by theL2-norm of the “zero-modes”, i.e. the solutions of the linearized equations, and the first thing is to show that this is finite, that is, that the zero-modes are square-integrable. Because of the non-compactness of R3this is not trivial, and it requires analytical justification. Fortunately, Taubes [44] has...

8. CHAPTER 4 Hyperkähler Property of the Metric
(pp. 28-37)

The Riemannian metric defined on the moduli spaceMkin the previous chapter has, as noted in (3.4), the property of being hermitian with respect to the almost complex structuresI,J,K, given by an action of the quaternions on the tangent bundle ofMk. Moreover, as a consequence of Donaldson’s theorem, these complex structures are integrable. We shall show here that the metric is Kähler with respect to these three complex structures. Such a metric is called hyperkähler. Its holonomy is a subgroup of Sp(k), and in particular it has vanishing Ricci tensor and so may be regarded...

9. CHAPTER 5 The Twistor Description
(pp. 38-50)

One of the features of a hyperkähler manifold which makes its metric relatively amenable is the existence of a twistor description, which generalizes the Penrose non-linear graviton construction [37]. This falls within the general theory of quaternionic manifolds developed by Salamon [40,41], but the specific case of hyperkähler metrics is dealt with in [26].

The starting point is the fact, noted earlier, that ifM4nis a hyperkähler manifold with covanant constant complex structuresI,JandK, then (aI+bJ+cK) is also a covariant constant complex structure ifa2+b2+c2= 1. The...

10. CHAPTER 6 Particles and Symmetric Products
(pp. 51-57)

In the previous chapters we have emphasized the role of the space of rational functionsRkas a model of the moduli space ofk-monopole configurations. In chapter 2 it was defined as the complement of a subvariety inCP2kand is therefore clearly a smooth variety. However, when it came to defining the holomorphicsymplecticstructure in (5.5) we broke the symmetry of this description and began to consider the roots of polynomials rather than their coefficients. Although this point of view was adopted for the convenience of describing the symplectic form, it does in fact suggest another description...

11. CHAPTER 7 The 2-Monopole Space
(pp. 58-63)

From now on we shall concentrate on the case of magnetic chargek= 2 and investigate in detail the geometry of the moduli spacesM2and$M_{2}^{0}$. The twistor approach fork= 2 has been carefully explored by Hurtubise [27] and we begin by reviewing his results, which we reformulate in slightly different terms.

A 2-monopole has a centre, which we shall take as our origin in R3, and through the centre there are just two (unoriented) spectral linesαandβsay. Ifα=βthe monopole is axially symmetric withαas axis, and...

12. CHAPTER 8 Spectral Radii and the Conformal Structure
(pp. 64-69)

In the previous chapter we saw that a 2-monopole with given centre is determined, up to rotation, by a single angular parameterθwith 0 ≤θ<π/2. Moreover in any given (unoriented) directionχthere are just two spectral lines, and by symmetry they are equidistant from the centre. Letρ(χ) denote this distance: we shall call it the spectral radius in the directionχ. In particular takingχto be one of the three axeseiof the monopole we obtain the three principal spectral radiiρi=ρ(ei). We shall now compute theρias functionsθand...

13. CHAPTER 9 The Anti-self-dual Einstein Equations
(pp. 70-78)

We know, from the general considerations explained in chapter 4, that$M_{k}^{0}$is a hyperkähler manifold for allk. Fork= 2,$\dim\text{ }M_{k}^{0}=4$and in four dimensions a manifold is hyperkähler if and only if it is an anti-self-dual Einstein manifold, since the holonomy group is now SU(2) ⊂ SO(4). Since$M_{2}^{0}$admits SO(3) as a group of isometries we can start by looking at all SO(3)-invariant anti-self-dual Einstein metrics. The differential equations for such metrics reduce, because of the SO(3)-symmetry, to a system of ordinary differential equations (at least in the region where orbits are 3-dimensional). This problem...

14. CHAPTER 10 Some Inequalities
(pp. 79-89)

In chapter 9 we saw that there was a unique trajectory F of our differential equations (9.3) which, in the projective (a,b,c)-plane went from Pʹ to Q. In this section we shall derive a number of inequalities which effectively give further information about F. As before we shall use affine coordinates (x,y) centred atC. First we prove

Lemma (10.1).The curveFlies entirely in the region$-1+\frac{x}{2}\le y\le -1+x.$

Proof: It will be sufficient to prove that the gradient$\frac{dy}{dx}$of the differential equation (9.6) satisfies

(a)$\frac{dy}{dx}\textless 1\text{ on }y=-1+x,0\textless x\textless 1$,

(b)$\frac{dy}{dx}>\frac{1}{2}\text{ on }y=-1+\frac{x}{2},0\textless x\textless 1$, and that

(c) near Pʹ,...

15. CHAPTER 11 The Metric on $M_{2}^{0}$.
(pp. 90-95)

We shall now put together the results of chapters 8 and 9 to derive the explicit form of the metric on the monopole space$M_{2}^{0}$. We recall that in chapter 8 we obtained in formula (8.18) the explicit form of the conformal structure on$M_{2}^{0}$, so that only an overall scalar function remains to be determined. On the other hand in chapter 9, from an entirely different viewpoint, we analysed the differential equations (9.3) for an SO(3)-invariant hyperkähler metric. Comparing these two results (including the correct signs) we see that the curve F of chapter 9, describing a...

16. CHAPTER 12 Detailed Properties of the Metric
(pp. 96-101)

Having found the explicit form of the metric on$M_{2}^{0}$we shall now investigate some of its properties in further detail. In particular we shall investigate its behaviour neark= 0 andk= 1. Recall thatk= 0 corresponds to the RP2representing axially-symmetric monopoles whilek→ 1 is the region “at ∞”.

The functionK(k) is analytic fork< 1 and has a power series expansion [32](12.1) $K(k)=\frac{\pi }{2}\left\{ 1+\frac{{{k}^{2}}}{4}+\frac{9{{k}^{4}}}{64}+\cdots \right\}.$Ask→ 1, i.e.kʹ → 0, it has an asymptotic expansion [43], [32].(12.2) $K(k)\sim -\log {{k}^{\prime }}\left\{ 1+\frac{{{({{k}^{\prime }})}^{2}}}{4} \right\}+\cdots .$Substituting (12.1) into the formulas of (11.15) gives$ab=-2k(1-{{k}^{2}})\frac{{{\pi }^{2}}}{4}\left\{ 1+\frac{{{k}^{2}}}{4}+\cdots \right\}\left\{ \frac{k}{2}+\frac{9{{k}^{3}}}{16}+\cdots \right\}=-\frac{{{\pi }^{2}}{{k}^{2}}}{4}\left( 1+\frac{3{{k}^{2}}}{8}+\cdots \right),$$bc=\frac{{{\pi }^{2}}{{k}^{2}}}{4}\left( 1+\frac{3{{k}^{2}}}{8}+\cdots \right)-2(1-{{k}^{2}})\frac{{{\pi }^{2}}}{4}\left( 1+\frac{{{k}^{2}}}{2}+\frac{11{{k}^{4}}}{32}+\cdots \right)=-\frac{{{\pi }^{2}}}{2}\left( 1+\frac{{{k}^{4}}}{32}+\cdots \right),$$ca=-\frac{{{\pi }^{2}}{{k}^{2}}}{4}\left( 1+\frac{3{{k}^{2}}}{8}+\cdots \right)+2{{k}^{2}}\frac{{{\pi }^{2}}}{4}\left( 1+\frac{{{k}^{2}}}{2}+\cdots \right)=\frac{{{\pi }^{2}}{{k}^{2}}}{4}\left( 1+\frac{5{{k}^{2}}}{8}+\cdots \right).$...

17. CHAPTER 13 Geodesies on $M_{2}^{0}$
(pp. 102-108)

Now that we have explicit and detailed information concerning the metric on the monopole parameter space$M_{2}^{0}$we shall investigate its geodesies. Since the metric is asymptotically flat, geodesies near ∞ are asymptotically straight lines. We shall in particular be interested in following a geodesic in from ∞ and seeing how it emerges at the other end. In other words we shall study the scattering of geodesies. This scattering is produced by the non-trivial curvature (and topology) in the “finite part” of$M_{2}^{0}$, i.e. the part near RP2.

The study of all geodesies in full generality is a complicated...

18. CHAPTER 14 Particle Scattering
(pp. 109-115)

We shall now interpret the geodesic scattering of chapter 13 in terms of the scattering of monopoles, regarded as point-particles. We recall that a point on$M_{2}^{0}$withξlarge gives a 2-monopole configuration which approximately represents two single monopoles a long way apart. More precisely (12.7) shows that the Euclidean distance between these two monopoles is$\xi /\sqrt{2}$.

As we saw in chapter 7 a 2-monopole has three principal axese1,e2,e3. Of these the “Higgs axis”e2is the line joining the particles in the asymptotic region. Rotation about the Higgs axis therefore leaves the locations of...

19. CHAPTER 15 Comparison with KdV Solitons
(pp. 116-118)

In the introduction we indicated that BPS-monopoles in R3share many of the essential features of the usual one-dimensional solitons, as in the KdV equation. Having seen how BPS-monopoles interact (at low energy) we shall now review the similarities and differences in detail.

Let us recall that in the KdV theory one considers a functionu(x) on the line as a potential for the operator${{L}_{u}}=\frac{{{d}^{2}}}{d{{x}^{2}}}+u.$WithLuone considers the scattering matrix which contains transmission and reflection coefficients. If the reflection coefficient is zero,uis said to be a reflectionless potential. The solution of the KdV equation with...

20. CHAPTER 16 Background Material
(pp. 119-128)

Our analysis of the geometry of the moduli space of monopoles has depended on a variety of approaches to solving the Bogomolny equations. Each one has its strengths and weaknesses, hence the use of the most appropriate one when needed. Here we try to give a survey, part historical, of these different points of view in order to impart some perspective on the previous chapters and aid the reader in understanding the proofs. We give in particular an extended description of Donaldson’s theorem on monopoles and rational maps.

The starting point for the whole subject of non-abelian magnetic monopoles was...

21. BIBLIOGRAPHY
(pp. 129-131)
22. INDEX
(pp. 132-133)
23. Back Matter
(pp. 134-134)