The Geometry and Dynamics of Magnetic Monopoles [Course Book ed.] 9781400859306

Table of contents :
CONTENTS
PREFACE
INTRODUCTION
CHAPTER 1. The Monopole Equations
CHAPTER 2. Geometry of the Monopole Spaces
CHAPTER 3. Metric of Monopole Spaces
CHAPTER 4. Hyperkahler Property of the Metric
CHAPTER 5. The Twistor Description
CHAPTER 6. Particles and Symmetric Products
CHAPTER 7. The 2-monopole Space
CHAPTER 8. Spectral Radii and the Conformal Structure
CHAPTER 9. The Anti-self-dual Einstein Equations
CHAPTER 10. Some Inequalities
CHAPTER 11. The Metric on M02
CHAPTER 12. Detailed Properties of the Metric
CHAPTER 13. Geodesics on M02
CHAPTER 14. Particle Scattering
CHAPTER 15. Comparisons with KdV Solitons
CHAPTER 16. Background Material
BIBLIOGRAPHY
INDEX

Citation preview

The Geometry and Dynamics of Magnetic Monopoles

Μ. Β. PORTER LECTURES RICE UNIVERSITY, DEPARTMENT OF MATHEMATICS SALOMON BOCHNER, FOUNDING EDITOR

The Geometry And Dynamics of Magnetic Monopoles

MICHAEL ATIYAH AND NIGEL HITCHIN

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Copyright © 1988 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Guildford, Surrey All Rights Reserved Library of Congress Cataloging in Publication Data will be found on the last printed page of this book ISBN 0-691-08480-7 This book has been composed in Times Roman Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Paperbacks, although satisfactory for personal collections, are not usually suitable for library rebinding Printed in the United States of America by Princeton University Press, Princeton, New Jersey

CONTENTS

VB

PREFACE INTRODUCTION CHAPTER

1.

2. 3. CHAPTER 4. CHAPTER 5. CHAPTER 6. CHAPTER 7. CHAPTER 8. CHAPTER 9. CHAPTER 10. CHAPTER 11. CHAPTER 12. CHAPTER 13. CHAPTER 14. CHAPTER 15. CHAPTER 16. CHAPTER

CHAPTER

INDEX

The Monopole Equations Geometry of the Monopole Spaces Metric of Monopole Spaces Hyperkahler Property of the Metric The Twistor Description Particles and Symmetric Products The 2-monopole Space Spectral Radii and the Conformal Structure The Anti-self-dual Einstein Equations Some Inequalities The Metric on M~ Detailed Properties of the Metric Geodesics on M~ Particle Scattering Comparisons with KdV Solitons Background Material Bibliography

3 9 14

21 28 38 51 58 64 70 79 90 96 102 109 116 119 129 132

PREFACE

In January 1987 I gave the Milton Brockett Porter Lectures at Rice University. This provided me with the opportunity of presenting, at some length, the results on magnetic monopoles which Nigel Hitchin and I have been investigating over the past few years. This book, written jointly, is an expanded version of the lectures and it contains a full and detailed treatment of the essentially new results. Although dependent on earlier work by many authors we have endeavoured to make it more self-contained by adding some introductory and background material. Michael Atiyah

The Geometry and Dynamics of Magnetic Monopoles

INTRODUCTION

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 which behaves in certain respects like a particle: it should be approximately localized in space and should be "conserved" in collisions. There is now a very extensive theory of solitons for one-dimensional space, of which the prototype and best known example is the KdV equation [14]. These solitons have very remarkable properties and their evolution equations form an infinite-dimensional integrable Hamiltonian system. Some of the key features of the theory are: (1) there are explicit formulae describing the interaction of k solitons (for any integer k), (2) these formulae involve the theory of Riemann surfaces (or algebraic curves), (3) the inverse scattering method (associated to linear operators) is used to construct the solutions, (4) the scattering of solitons after a collision is essentially trivial (i.e. velocities are unaffected). Although the theory of the KdV equation is "exact" it should be recalled that the KdV equation arises naturally as an approximation for shallow waves in a channel. Thus the soliton features listed in (1) through (4) are only an approximate description of the real physical situation. The equations we shall be studying, governing the evolution of monopoles, will share some of the essential features (1), (2), (3) of the KdV theory. On the other hand the scattering of monopoles after a collision will be non-trivial in the sense that the velocities alter (i.e. there is "momentum transfer"). The monopole equations moreover take place in fully 3-dimensional space, and are correspondingly more complicated. The analogy with KdV will be examined in detail in chapter 15.

4

INTRODUCTION

Our results will not give exact solutions but only an approximation for small relative velocities. However, as we have pointed out, the KdV theory is itself only an approximation to a more realistic model. From a physical point of view there is no essential difference between an approxi­ mate solution to an exact equation and an exact solution to an approxi­ mate equation. The concept of a magnetic monopole, as an isolated point-source of magnetic charge, was introduced some 50 years ago by Dirac in a very influential paper [13]. In Maxwell's equations, electricity and magnetism appear on an equal footing but whereas electric charges occur naturally, magnetic charges or monopoles do not appear to do so. Nevertheless by postulating their existence Dirac was able to produce the only convincing argument leading to the quantization of electric charge, namely the fact that electric charges always appear in integer multiples of a fixed charge (that of the electron). With the advent of non-abelian gauge theories, in which the U(l) of Maxwell theory is enlarged to a non-abelian group such as SU(2), it was soon realized by 't Hooft and Polyakov that one could have smooth field configurations which behaved at large distances like a Dirac monopole. The essential point is that the non-linear equations, which generalize the linear Maxwell theory, admit "soliton" solutions in which the singular point-particle of Dirac is replaced by a smooth field approximately localized at the position of the "particle." The 't Hooft-Polyakov monopole is only known numerically but there is a simplified model introduced by Prasad and Sommerfield (in which the coefficient λ of the Higgs potential (1 — \φ\2)2 is put equal to 0) which has an explicit monopole solution. These are known as BPS (or self-dual) monopoles and are the ones we shall be concerned with. The BPS-monopoles are solutions of the Bogomolny equations which will be introduced in chapter 1. They describe static monopoles in R 3 and they have been extensively studied in recent years by many authors. They have remarkable properties which are best understood as a special case of the self-duality equations in 4-space (for solutions independent of one of the variables). In particular the Penrose twistor theory applies to these equations and this is the basis for all the methods of solution. It provides the link with complex-variable theory and ultimately explains the relevance of Riemann surfaces for the construction of monopoles. This material is explained and summarized in chapter 2 with further back­ ground given as an appendix in chapter 16. As we explain in chapter 16 there are several approaches to the study of monopoles, each with its own advantages. We will use all these at var­ ious places in our treatment. Although it would be logically simpler to stick to one point of view there are at present some technical problems which remain to be resolved. To avoid these we have adopted a hybrid approach, essentially taking the line of least resistance.

INTRODUCTION

5

The Bogomolny equations have solutions for all integer values k of the magnetic charge. Thesefc-monopoleswhen "well-separated" are approxi­ mately a superposition of k simple monopoles located at different points. The fact that such a configuration is static depends on the fact that the magnetic repulsion between the monopoles is balanced by an attractive force due to the Higgs field. The parameter space of all /c-monopoles is a 4fc-dimensional manifold Mk which is described in detail in chapter 2. For k = 1, we have Mx = R 3 χ S 1 indicating that a 1-monopole is deter­ mined by a point in R 3 (its "position") and a phase angle. For k > 1 there is a region (near co) in Mk which is approximately an (unordered) product of k copies of Mlt and this represents well-separated monopoles. However, this description breaks down in the interior of Mk, and this has profound implications for the interactions of monopoles which is the main thrust of this book. Manton [35] has argued that the geodesic flow on Mk, with respect to its natural metric (induced from L2-functions on R3), is the low-energy approximation to the true evolution of dynamic monopoles. His argument (reviewed in chapter 1) is based on the analogy of a particle in R" moving in a potential field V. The equilibrium positions are given by the subspace Μ c R" giving minima of V. For motion with energy near this mini­ mum the trajectory of a particle whose initial velocity is tangential to Μ is close to the corresponding geodesic on Μ (with some small oscillations in the transverse directions). The dynamics of monopoles can be put into this framework with R" replaced by an infinite-dimensional manifold, and Μ = Mk. The impor­ tant point is that the Bogomolny equations give the absolute minimum of the potential energy. Because we are now in an infinite-dimensional situation Manton's argument, although physically plausible, requires de­ tailed analytic justification. We shall not undertake this analysis, but we note that a similar situation has been studied in detail by Ebin [17]. To carry out Manton's programme we therefore have to investigate the Riemannian metric on Mk. In particular, we have to show that it is finite and complete. Lack of finiteness would mean the metric was infinite (or not defined) in certain directions, so that monopole motion was con­ strained, while incompleteness would mean that monopoles (and so magnetic charges) could disappear infinitetime. Finiteness and complete­ ness are deduced in chapter 3 from basic analytical results of Taubes. These results also yield the asymptotic behaviour of Mk when a fc-monopole separates into pieces. The direct definition of the metric on Mk involves computing the L 2 norms of the zero-modes (solutions of the linearized equations), and this is too complicated a procedure to be useful. Fortunately, however, there are some general symmetry principles which govern the metric on Mk and these can be effectively exploited. The basic result is that each Mk is a

6

INTRODUCTION

hyperkahler manifold, which means that its holonomy group lies in the symplectic group Sp(fc) — oo: it is unique up to a constant. Writing this as a linear combination of s 0 and s l 5 s'0 = as 0 + bsj we can consider the ratio a/b which is then independent of the choice of s'0. This is the scattering S which we associate to the line u, that is, we put S(u) = % e C P 1 b It depends on the basis (e0, ej of Ε at the point at oo on the line u. We now fix an isomorphism (2.8)

(2.9)

R 3 = R χ C,

that is, we take coordinates (x 1 ; x 2 , x 3 ) and we combine x 2 and x 3 as the complex variable ζ = x 2 + ix 3 . We take for u = u(z) the line parallel to the x t -axis through the point (0, z), and we use t = x1 as a parameter on the line. For a monopole trivialized at the point * = (oo, 0, 0) we then have a well-defined point S(z) = S{u(z)) in C P 1 given by (2.8). Moreover this depends only on the point m e Mk(*) representing the monopole. Then the main result of Donaldson [15] can be put in the following form THEOREM (2.10) [Donaldson]. For any m e Mk{*) the scattering func­ tion Sm(z) is a rational function of degree k with Sm(oo) = 0, and m -» Sm gives a dijfeomorphism Mk(*) -> Rk onto the space of all such rational functions. The first part of (2.10) is essentially a simple consequence of the Bogomolny equations. The main content of Donaldson's theorem is the second part. In fact Donaldson does not describe his result in terms of the scat­ tering map. This is a refinement due to Hurtubise (see also [2] for the analogous hyperbolic case). The poles of the rational function Sm(z) associated to the monopole m are of particular interest. They arise when b = 0 in (2.8) so that the solution s 0 decays at both t -» + oo and ί -> — oo. A line u(z) with this property is called in [22] a spectral line. Thus the poles of Sm(z) represent the k spectral lines parallel to the χ x -axis. In chapter 5, when we describe the twistor approach to monopoles, the spectral lines in all directions will play a fundamental role. The space of all oriented straight lines in R 3 may be

CHAPTER 2

18

identified with the 2-dimensional complex manifold TP1, the tangent bundle of the complex projective line. The subspace consisting of all spectral lines is called the spectral curve (see also chapter 16). We will now identify Mk with Rk but, as we remarked earlier, we have to be careful about the SO(2)-actions. We will therefore spell out the way the group SO(2) χ R χ U(l) χ C has to act on Rk in order to be com­ patible with its natural action on Mk. Here R denotes translation in the xl -direction, C gives translation in the orthogonal plane, and U(l) repre­ sents the phase change, i.e. the action on the asymptotic isomorphism a. So let (λ, μ, ν) e SO(2) χ C* χ C s SO(2) χ R χ U(l) χ C where λ, μ, ν are complex numbers, |A| = 1, μ # 0 and log|ju| e R, μ/\μ\ e U(l). Then the scattering function S(z) is transformed by (λ, μ) into (2.11)

μ-^-^/ΤΗζ-ν)).

Note that the poles of S(z) are transformed in the natural way, which is as it should be since they have a natural interpretation as parametrizing the spectral lines parallel to the χt -axis. For k = 1 the function S takes the form (2.12)

S(z) =

^-r. ζ — b

The standard monopole centred at the origin is given by b = 0, a = 1. It follows from (2.11) that in general the monopole is located at the point (—|log|a|, b), while the argument of a describes its phase. Thus M x has the complex structure (2.13)

M , ^ C x C*.

Suppose now we consider the case of general k and assume the function S(z) has simple poles so that it can be written in the form:

(2.14) S(z)=£-A-· i = i ζ — bi

If the bi are far apart we shall see in chapter 3 (Proposition 3.12) that the monopole associated to S(z) approximates a combination of k simple monopoles corresponding to the summands in (2.14), that is, having centres at the points (—•jlog|aj|, fr;) and phase angles given by the arguments of the at. One noteworthy aspect of Donaldson's result (2.10) is that it demon­ strates that Mk and M° have a 2-parameter family of complex structures, 2 3 one for each point of S , representing the preferred axis in R . Note that the natural action of SO(3) does not preserve these complex structures: it

MONOPOLE SPACES

19

rotates them. Only the SO(2) subgroupfixingthe axis preserves the corres­ ponding complex structure. In chapter 3 we shall explain the natural origin of these complex structures and the way in which they relate to the natural metric on Mk. A rational function in Rk has the normal form k-l

(2.15)

S(z) =

Σ «k_ j = 0 modulo the equivalence S ~ XS for any non-zero complex number λ. Thus Mk can be identified with the subspace of C P * - 1 χ CPk~1 complementary to the variety with the equation bkA(a, b) = 0. Since this equation is homoge­ nous of bi-degree (k, k) we see that Mk is also the complement of an anticanonical divisor. Note that when k = 2, M° is an algebraic surface and the metric will be an anti-self-dual Einstein metric.

20

CHAPTER 2

The topology of the spaces of rational functions was studied in [42]. In particular, the fundamental group (2.16) is independent of k. More precisely there is, up to homotopy, a well-defined inclusion given on compact subsets by

where K is a large real number, and this induces an isomorphism on For k = 1 we have which implies (2.16). Up to homotopy we actually have a well-defined addition map given by In particular we have a map which on n^ induces the homomorphism (2.17) taking each generator to 1. Consider now the action of given by multiplying S(z) by a constant X . For k varying with this gives an Clearly on element of this gives the element (1, 1 , . . . , 1). Hence applying (2.17) we see that in we get k times the generator. Since is obtained from by factoring out by the multiplicative action of C* (and an additive action of C) it follows that (2.18)

Let us represent, as before, as the complement of where If we can normalize the a t so that Since A is homogeneous of degree k in the at this normalization is unique up to a fcth root of unity. Hence the variety defined by is a /c-fold covering of Since is conis an irreducible polynomial, nected. Hence from (2.18) it is the universal covering of For k = 2 our normalized rational functions are

and Hence (changing variables) in C 3 with equation (2.19) The manifold

is the quotient by the involution

is the algebraic surface

CHAPTER 3

Metric of Monopole Spaces

In the previous chapter we saw that the parameter space of based kmonopoles is a manifold Mk of 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 of Mk. This is given by the L2-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 R 3 this is not trivial, and it requires analytical justification. Fortunately, Taubes [44] has investigated such analytical questions carefully and all we have to do is to apply his results appropriately. It is perhaps worth pointing out that in a somewhat similar situation for σ-models studied by Ward [49], not all zero-modes are square-integrable which means that certain directions of motion on the parameter space are "forbidden", since they would require infinite energy. Let c = (Α, φ) be afe-monopole,i.e. a solution of the Bogomolny equa­ tions of charge k, and let Tc be the space of (α, φ) which are squareintegrable and satisfy the equations (3.1)

*DAa - ΩΑφ + [φ, α] = 0

(3.2)

*DA*a + [φ, φ] = 0.

Here α and φ are a Lie-algebra valued 1-form and function respectively. Equation (3.1) is the linearization of the Bogomolny equations, while (3.2) expresses the fact that (α, φ) is orthogonal to gauge directions—more precisely to directions arising from infinitesimal gauge transformations with compact support. In [44] Taubes establishes the following (3.3)

dim Tc = 4k.

The Higgs field φ is itself an infinitesimal gauge transformation (not van­ ishing at oo) and it gives rise to the special vector (ϋΑφ, 0) e Tc. Let T'c a Tc denote the orthogonal space to this vector, so that dim T'c = 4k — 1. It also follows from the results of [44] that T'c can be identified with the tangent space to the moduli space Nk at the point [c]. The essential point

22

CHAPTER 3

is that all directions in Nk can be represented by square integrable variations. The moduli space Mk of based /c-monopoles has dimension 4k and l fibres over Nk with S as fibre. This fibre direction corresponds to the infinitesimal gauge transformation given by the Higgs field. Thus the tangent space to Mk at [c] can be identified with T c . As pointed out in [44] the space of all pairs (α, φ) can be identified with the space of functions taking values in su(2) (χ) Η where Η denotes the quaternions. If a = α dx + β dy + γ dz, then (α, φ) corresponds to the function φ + α/ + βJ + yK, where I, J, Κ are the usual basis of H. More­ over the equations (3.1) and (3.2) are Η-invariant. It follows that (3.4)

Tc is a vector space over H.

We now endow Mk with the natural L 2 -metric, that is, we take the L2norm on each tangent space Tc. The actions of I, J, Κ on Tc in (3.4) are then isometries. Since I2 = — 1, we get an almost complex structure on Mk defined by I. The same applies for any imaginary quaternion of norm one. On the other hand, as explained in chapter 2, Donaldson's theorem identifies Mk with the space Rk of rational functions of a complex variable, and this exhibits Mk as a complex manifold. Moreover this identification depends on a choice of direction in R 3 , so that these complex structures on Mk are naturally parametrized by the same S2 as the almost complex structures introduced above. In fact it is fairly straightforward to see that the al­ most complex structure defined by I coincides with that coming from Donaldson's theorem. In particular this shows that I (and similarly all other unit imaginary quaternions) actually defines a complex structure, that is, it satisfies the integrability condition. An alternative and more direct proof of this will be given in chapter 5, where it will be shown that I is actually covariant constant. The compatibility between the almost complex structure I and the complex structure arising from Donaldson's theorem is best understood as follows. Having distinguished the first axis in R 3 we can identify R 3 χ S1 ^ C χ C* by (x 1 } x2, x3, Θ) -y [x 2 + s — ε we have 1 ^ ( 0 - - ^ ( 0 II2 = 0(ε). This energy estimate, together with the basic results of Uhlenbeck [46], then shows that, after gauge transformations we get a convergent subse­ quence as required. Having shown that Mk is complete the next question is to investigate its asymptotic behaviour. The main picture is described in the following proposition: PROPOSITION (3.8).

Given an infinite sequence of points ofMk there exists ι a subsequence mv, a partition k = £ /c; with kt > 0, a sequence of points >=i

x\ e R 3 for i= 1,2,. ..,1, such that (0 the sequence m\ = T'v(mv), where T\ is the translation y -* y — x\, con­ verges weakly (i.e. on compact subsets of R 3 ) to a k{-monopole m' with centre at the origin, (ii) as ν -* oo the distances between any pair of points x'v, x{ tend to oo and the direction of the line x'vx{ converges to a fixed direction. Roughly speaking we can envisage the situation as an "expanding uni­ verse" in which I different "galaxies" recede from one another in definite directions. The points x\ represent the "centres" of the galaxies. The main step in proving Proposition (3.8) is the following lemma: LEMMA (3.9). Let (Av, φν) be a sequence of k-monopoles with φν(0) = 0. Then there is a subsequence v, and gauge transformations gt so that the sequence g^AVi, φν) converges weakly to a k^-monopole with 1 < k1 < k. Proof: The main results of Uhlenbeck [46] assert the existence of a subsequence and gt so that gt{AVi, φν) converges weakly to a solution (Α, φ) of the Bogomolny equations. The total energy of (Α, φ) is still finite and so, as in [30], \φ\ tends to some constant c as |x| -» oo. In fact we must have c = 1, since the L6 norm of \φν\ — 1 is uniformly bounded [30]. Thus (Α, φ) is a /cj-monopole for some kx < k. Finally φ(0) = 0 shows that [Α, φ) is not the 0-monopole, so k1 > 1. To prove (3.8) we start by picking a zero x v of each φν and then trans­ lating back to the origin. Applying (3.8) we then get a subsequence con­ verging weakly to a fcj-monopole. Finally, centering this monopole, gives (on retranslation) our first subsequence xj. If k1 = k the process ends here. If ki < k then on topological grounds there must (for large v) exist further zeros of φν a long way from the x\. Repeating the construction, and using the compactness of S2 (the sphere of directions in R 3 ), we end up eventually with a subsequence satisfying (i) and (ii) of Proposition (3.8).

25

MONOPOLE METRICS

Next we investigate the behaviour of spectral curves: PROPOSITION (3.10). Let mv e Mk be a sequence as constructed in (3.8) with the translated weak limits m\ Let Γ ν and Ρ be the associated spectral curves. Then the translated spectral curves P„ = 7^(Γν) converge to the degenerate curve Ρ + £ (-^y + LJtX where Ltj is the line in TP1 correi spending to the straight lines in the limiting direction lim (x'vx{). V

Proof: Proposition (3.9) implies that, for large r, the monopole m„ is approximately made up of a number of smaller monopoles very far apart. This is an example of the cluster decomposition of Taubes [45]. It is shown in [45] (see Lemmas C.3.1, C.4.1 and D.2.3) that in the regions "between" the monopoles we have strong decay estimates for \φ\ — 1, and ΌΑφ. These are sufficient to give continuity for spectral lines which do not get close to more than one monopole. This proves that outside a neigh­ bourhood of the lines Ltj and L}i the curve T\ converges to P . The same argument applied with i and j interchanged show that Ltj and Ljt are limit components of P„. Hence T\ degenerates in the limit as stated. REMARKS: (1) The degeneration in (3.10) is simply a set-theoretical one. If one takes into account multiplicities then Ltj should have multi­ plicity kj. (2) For large v, Proposition (3.10) implies that Γ ν approximates the union of spectral curves of degrees kt, which are "far apart". If we translate so that one of these say Ρ has fixed centre then the other curves Ρ are "far away" and approximate the line pair Ltj + Ln. The spectral lines asso­ ciated with Ρ are nearly all parallel to this direction (or its opposite). Consider next the rational function description of monopoles given by Donaldson's theorem. If we pick the preferred direction in R 3 to be differ­ ent from the special directions LtJ associated with the sequence mv of (3.9), then the decay estimates in [45] show that the scattering functions Sv converge weakly (i.e. on compact sets in the z-plane) to the scattering function S ; associated to the limiting monopole m;. On the other hand if we take the special direction to be one of the Lfj· then the scattering functions Sv do not converge. This can be seen as follows. Suppose that (Α, φ) is approximately a combination of [Au φ J and the λ-translate (A2, φ2)λ of (A2, φ2). By this we mean (with λ large) that (Α, φ) looks like (Au φγ) near 0 and like (A2, φ2)λ near (0, 0, λ) and decays appropriately in between. Let s'0 be a solution to (WAl — ΐφ^ = 0 along a line parallel to the x3-axis which decays as t -» — oo. If the line is not a spectral line, then s'0(t) is close to the — i eigenspace of φ1 at t = λ/2, if λ is large. Because (Au φ^ and (A2, φ2)λ are close in this region, s'0 is therefore also close to

CHAPTER 3

26

a solution t'0 of (V^2 — ίφ2)λ$ = 0 which decays as t -> — oo. Consequently, the total scattering of (Α, φ) approaches that of (A2, φ2)λ. However by (2.11) the effect of a translation by λ is to multiply the rational function of (Α2,φ2) by e2\ Applying this to our sequence of monopoles mv, relative to one of the special lines Ltj, then shows that the scattering function S(z) does not converge as ν -* oo. In fact S(z) grows exponentially with the separation of the centres x'v, x{. From this discussion we can now deduce the following convergence property of monopoles in Donaldson's rational function parametrization. PROPOSITION (3.11). Let Sv(z) be a sequence of rational functions of degree k, normalized so that Sv(co) = 0, and assume that Sv(z) converges weakly to a rational function S(z) of degree I < k. Then the corresponding monopoles mv converge weakly to the monopole m determined by S. Proof: By (3.8) and (3.10) there is a subsequence m„ of the mv con­ verging weakly to a monopole m and the corresponding spectral curves Γ μ converge to the spectral curve Γ of m, together with a finite number of lines Ly Since the rational functions Sv converge it follows from our discussion above that none of the lines Lj is parallel to the χ t -axis (used in the Donaldson parametrization) and it then follows that S is the ra­ tional function of the monopole m. Thus m is uniquely determined. Now start again, replacing S v by any infinite subsequence. We again get a sub­ sequence of monopoles converging to m. This implies that the whole se­ quence mv in fact converges weakly to m. In particular consider the sequence. Sv(z) = S(z) + -Ϊ—, 2 — vb

b^O.

By (3.14) the monopoles mv converge weakly to the monopole m associated to S. Repeating the argument and using the description of 1-monopoles given in chapter 2 shows the following: PROPOSITION (3.12). Given au ..., ak e C*, there exists a constant R{au . . . ,ak) such that, for \bt — b}\ > R the rational function k

a-

i=l

Ζ — ft;

represents a k-monopole which is an approximate superposition of 1monopoles located at the points (—•£ log|a ; |, bt) with phases arg at. The rational functions occurring in (3.11) with widely separated poles describe an asymptotic part of the monopole space Mk where we can use

MONOPOLE METRICS

27

a "particle" description. For k = 2, once we have fixed the centre and chosen our axes suitably, this accounts for all asymptotic directions on M2. The situation will be studied in greater detail in the later chapters. Return now to the metric on Mk. Note first that the natural physical normalization arising from the interpretation as kinetic energy is one-half the usual L2-norm. In particular the length squared of the U(Indirection generated by the Higgs field φ (viewed as an infinitesimal gauge trans­ formation) is \ I DA4> 1 2 = \ IFA || 2 = Ink.

by the Bogomolny equations

Since the almost complex structures, I, J, K, as applied to this U(l)generator give the Regenerators of translation, we see that each of these also has length squared Ink. In particular for k = 1 this shows that a 1-monopole, viewed as a classical particle, has mass Απ. The metric on Mi = S1 χ R 3 is the standard flat metric, normalized on R 3 to be In ds2 where ds2 is the standard Euclidean metric. Since S1 = U(l)/Z2 it has radius ^Λ/2ΤΓ. In the asymptotic region given by (3.12), where Mk can be described as a product of k copies of Mj it follows from the results of Taubes [45] that the metric is asymptotically a product and therefore asymptotically flat. More precise information on the asymptotic behaviour will be given in the later chapters.

CHAPTER 4

Hyperkahler Property of the Metric

The Riemannian metric denned on the moduli space Mk in the previous chapter has, as noted in (3.4), the property of being hermitian with respect to the almost complex structures I, J, K, given by an action of the qua­ ternions on the tangent bundle of Mk. Moreover, as a consequence of Donaldson's theorem, these complex structures are integrable. We shall show here that the metric is Kahler with respect to these three complex structures. Such a metric is called hyperkahler. Its holonomy is a subgroup of Sp(fc), and in particular it has vanishing Ricci tensor and so may be re­ garded as a (positive definite) vacuum solution to Einstein's equations. The existence of this special metric structure on the moduli space is a consequence of the structure of the Bogomolny equations themselves. The moduli space of solutions may formally be regarded as a hyperkahler quotient in the sense of [26], automatically possessing a hyperkahler metric. Similar remarks apply to the moduli space of instantons, or to the 2-dimensional reduction of the self-duality equations [25]. We shall describe this quotient point of view later on in this section. To prove that the metric is Kahler with respect to I, J and K, we must show that the 2-forms ω1, ω2, ω 3 corresponding to the three complex structures are closed. Actually, as the following lemma shows, this will also prove, independently of Donaldson's theorem, the integrability of the complex structures. LEMMA (4.1). Let g be a metric which is hermitian with respect to almost complex structures I, J, Κ arising from a quaternionic structure on the tan­ gent bundle. Then the metric is hyperkahler if and only if the corresponding 2-forms are closed. Proof: Note first that ω2(Χ, Υ) = g(JX, Υ) = g(KIX, Υ) = ω3(ΙΧ, Υ) and so (4.2)

ι(Χ)ω2 = ι{ΙΧ)ω3.

HYPERKAHLER METRICS

29

It follows that a complex vector field X is of type (1,0) with respect to I, that is IX = iX, if and only if (4.3)

I(X)W2

= i(I(X)W3).

By the Newlander-Nirenberg theorem the complex structure I is integrable if the Lie bracket of two vector fields X and Y of type (1,0) is again of type (1,0), so we must show that if X and Y satisfy (4.3), so does [X, Y]. However,

Now .2XW2 = d(I(X)W 2)

= d(i(I(X)W3)) = i.2xW3

since dW2 = 0 from (4.3) since dW3 = 0,

and from (4.3), I(Y)W2 = i(I(Y)W3).

Thus 1([X, Y])W2 = i(.2x(I(Y)W 3) - I(Y).2XW3) = i(I[X, Y]W3).

Thus I is in fact integrable and since dW l = 0, the metric is Kahler with respect to I. Repeating with J and K shows it to be hyperkahler. In order to consider more explicitly the 2-forms Wt, w 2 and W3 on the moduli space of monopoles, we review here in slightly more detail Taubes' analysis of the tangent space of the moduli space mentioned in chapter 3. This is based on the study of the elliptic complex associated to the deformation problem: QO(g) ~ Ql(g) Ei1 QO(g) ~ Q2(g)

where and The equation dia, 1/1) = 0 is the linearization of the Bogomolny equations, and solutions in the image of d 1 are those arising from infinitesimal gauge transformations. The standard way to analyse such a complex on a compact manifold is to consider the single elliptic operator d 2 + d! = g A operating on

30

CHAPTER 4

Q*(g) © Q°(g) where d^ is the formal adjoint of dt. This operator is essen­ tially the Dirac operator coupled to the connection on S ® g (where S is the spinor bundle with Levi-Civita connection) and Higgs field φ. Over R3, trivializing S with covariant constant sections, it can be identified with the space of quaternions which makes Q)A an operator which commutes with the quaternions. Its null-space is given by equations (3.1) and (3.2) and is a quaternionic vector space. The analysis of the complex over the non-compact space R 3 was given by Taubes [44] who used the norm (3.7)

\\(a, ψ)\\ϊ = \\VAa\\> + | D > | 2 + \\ίΦ, αψ + \\[φ, φψ ate = (Α, φ) and proved that 9)A and 3)\ are Fredholm operators of index 4/c and — 4k respectively operating from Q*(g) © n°(g) with || ||c norm to the same space with L 2 norm. He showed moreover that the kernels of ΟιA and Q)*A consist of the square integrable solutions. A Weitzenbock formula gives the vanishing of Ker $>% and it follows, as noted in chapter 3, that the space Tc of square-integrable solutions of 2Α(α, φ) = 0 is a vector space over the quaternions of real dimension 4k. Using this Fredholm theory, one shows that if (α, φ) is a smooth squareintegrable solution to d2(a, φ) = 0 and (α', φ') is its orthogonal projection onto the finite-dimensional kernel of S>A, then (4.4)

(α, φ) = (α', φ') + ά,θ

for some θ e Q°(g). This is the basis of the relationship between Tc and the tangent space to the moduli space: we associate to each square integrable solution to d2(a, φ) = 0 its harmonic representative. Now suppose c = (Α, φ) is a solution of the Bogomolny equations satisfying the boundary conditions (2.4). As shown in [22] there is an asymptotic gauge in which the Higgs field is diagonal and the connection matrix has the form

2

2

where ||B| = 0(r~ ), A ( — \ = 0, and \A - A\ = 0{r~ ) where A is the connection form for a homogeneous connection on a line bundle of degree 3 k over the 2-sphere of directions in R . Every monopole is therefore gauge-equivalent to one which differs from a standard homogeneous con­ nection and Higgs field on R 3 — {0} by an expression (α, φ) of order r~2, and hence in particular square-integrable. Let us denote by Ψ" the infinite-dimensional vector space of all such pairs, not necessarily satisfying the Bogomolny equations.

HYPERKAHLER METRICS

31

The space "Y has a quaternionic structure given as in chapter 3 by writing (a, 1/1) = (ex dx + /3 dy + y dz, 1/1) as

(4.6)

(a, 1/1)

= 1/1 + exl + /3J + yK.

(This is the same structure we considered on the kernel of !!fi A)' Furthermore, since the coefficients are square integrable we may define 2-forms (1)1> (1)2' (1)3 corresponding to I, J and K:

= fR3 (/(a1> 1/11)' (a 2, 1/12))

w 1((a 1, 1/11)' (a2' 1/12»

(4.7)

=

fR3 (1/11' O(2) -

(OC 1, 1/12)

+ (/31' Y2) -

(Yl' /32)'

These forms are constant, and therefore closed. Suppose instead of a single solution to the Bogomolny equations we have a smooth family c(x) parametrized by an open set of Rn. The gauging procedure which transforms an individual configuration to lie in "Y can be applied uniformly over a compact set in R n to yield a map f: U -+ "Y of some neighbourhood in Rn. Pulling back the three 2-forms w 1 , W2' W3 gives three closed forms on U. A tangent vector X on U defines a tangent vector in the moduli space according to (4.4): setting (a, 1/1) = df(X) we take (a', 1/1') E Tc where (a', 1/1') is the projection of (a, 1/1) onto the kernel of!!fi A' Now from [30, VI] the square integrable solutions of !!fi A(a, 1/1) = 0 are actually of order r- 2, so we have (4.8)

(a,

1/1) = (a', 1/1') + d/J

where d 10 = (D AO, [. We can construct a complex manifold fibering over with a twisted symplectic form by finding a holomorphic map from to the group of holomorphic symplectic diffeomorphisms of Substituting for z in the result will give the necessary twist. Now a symplectic diffeomorphism can be obtained by exponentiating a vector field which preserves the symplectic structure, and such a vector field can be generated from a single function. Suppose, then, we take the function

CHAPTER 5

50

If ι(Χ)ω = dH, then

But this is just the vector field defined by dq(z) = 0 dt dp(z)

2z

dt

πζ

p(z)

modulo q(z)

and exponentiating to t = π gives the identification of Theorem (5.5). Thus the whole twistor space is generated by the single function

Constructions of this form were produced in four dimensions by Penrose [37]. The Legendre transform construction described in [26] is another case.

CHAPTER 6

Particles and Symmetric Products

In the previous chapters we have emphasized the role of the space of rational functions Rk as a model of the moduli space of /c-monopole configurations. In chapter 2 it was defined as the complement of a subvariety in CP2k and is therefore clearly a smooth variety. However, when it came to defining the holomorphic symplectic structure 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 of the moduli space which is more in keeping with the physical origins of the problem—that of describing particle-like behaviour. Recall that in the previous chapter we took a rational function S{Z) =

W)

and chose the roots βχ,..., βκ of q(z) and the values p(J?i),..., p(/y of p(z) at those roots to parametrize the moduli space. These were only good coordinates when βί φ β} (i φ j). Since p{z) and q(z) have no common factors, p(/?;) is a non-zero complex number so this parametrization is by an unordered set of k points in C χ C*. Now the moduli space of a single monopole consists of its position in R 3 and phase in S1, represented holomorphically by the rational function a/(z + b), or a single point in C χ C* ^ R 3 χ S1. If the objects we were describing were pure particles, then the moduli space would be simply the set of all unorderedfc-tuplesof points in C χ C*—thefe-foldsymmetric product S*(C χ C*). This space has singularities, however, where the po­ sition and phase of two of the particles coincide and is not therefore our moduli space. There is nevertheless a map Rk -+ Sk(C χ C*) obtaining by associating the pairs (fiu ρ(βύ) to the rational function p{z)/q(z), which is biholomorphic on the open set β( φ βΓ We may thus

128

CHAPTER 16

consider as a desingularization of the configuration space of k points in particles with a phase and position. There is in fact an abstract way of defining such desingularizations in algebraic geometry—the theory of Hilbert schemes. These Have been used by Beauville [6] to construct compact complex manifolds with symplectic forms, but the theory may be adapted also to deal with our situation as follows. For a complex manifold Y we have the fc-fold product the symmetric product and the Hilbert scheme A point of is by definition either (i) an ideal sheaf (ii) a sheaf S of cyclic.

with dim or equivalently. with finite support, dim

The correspondence is is a module of finite length

and then

and

We have a birational map

which for dim Y = 1 is an isomorphism. For dim is smooth and is a desingularization of If Y is symplectic with 2-form then there is a naturally induced symplectic structure, with form on the regular points of for dim is therefore a symplectic manifold [6], Let be a complex fibration with dim We define an open set

to consist of sheaves S (as in (ii)) with a cyclic This is equivalent to either of the following: (a) there exist local sections with for some cyclic sheaf T on X. (b) the map Supp is bijective and if are local coordinates centred around a point of Supp S (with x a local coordinate on X) the local ideal / (kernel of is generated by is the multiplicity and the are polynomials of degree less than m. To see these equivalences note first that is cyclic. Also in local coordinates, the section will be given by equations with convergent power series. Since every cyclic sheaf on X is locally we see that Finally cyclic implies the first part of (b) and also that powers of x (the local coordinate on X) give a

SYMMETRIC PRODUCTS

53

basis. This means that each y\ = p^x) mod I for some polynomial pt and so the second part of (b) follows. Put more geometrically the ideals I of Y ^ are (locally over X) just the complete intersections of multiple fibres and local sections. For k = 2, Y [2] S2Y is the desingularization defined by 2 blowing up the diagonal A a Y and then dividing by Z 2 , Y2 -> Y [2] is a double cover branched along A (the projectivized normal bundle of A). Consider the divisor C in S2 Y mapping to the diagonal in S2X, and let C be the proper transform of C in Yl2] (i.e. the complete inverse image consists of C and the exceptional divisor A). Our open set Yl2] is in this case just the complement of C. Note that in this example C is just the singular set of the map Y [2] -> S2X, so that Yl2] is the regular set where the differential has maximal rank. EXAMPLE:

From (b) we can describe explicit local coordinates for Y^1 which in particular show that it is a manifold and that Y™ -+ X[k] is a fibration. Assume first that m = k, i.e. that Supp S consists of just one point. Then the ideal

with deg pi = k — 1 describes canonically a neighbourhood on Y™, i.e. local coordinates are the coefficients au ... ,ak and all the r(k — 1) coefficients of the pt. In general if Supp S consists of several points, lying over the point e Xlk] where = k, then Yjf1 is locally a product of Y^'1 and inherits local coordinates from the factors (which reduces us to the previous case when Supp S is one point). Consider now the special case when X = C and Y = C x C with projection onto the last factor. We can now use global coordinates ..., yr, x) and the ideals of Y^1 can be globally described as the complete intersections

where q(x) is a monic polynomial of degree k and all pt(x) have degree less than k. The previous local product description comes from factorizing q(x) and localizing the pt accordingly. Thus Y[kl = C k CP" of degree k with /(oo) = (1, 0 , 0 , . . . , 0).

Specializing further to r = 1 we see that the space Rk of rational functions / of degree k (with /(oo) = 0) is identified with for 7t:C* x C -> C.

54

CHAPTER 6

Now recall that the total space of L2 over TP1 (or equivalently (CP 3 - CP^/Z) is a complex 3-manifold Ζ fibered over C P 1 with fibre C* χ C. This is the twistor space for S1 χ R 3 . Applying our constructions above fibre-wise we get a new complex manifold Zk fibered over C P 1 with fibre Rk. This is the twistor space for k-monopoles. This is just a reinterpretation of our previous descriptions: the spectral curve in TP1 having L2 trivial lifts to the principal bundle, giving a curve on Ζ meeting each fibre in k points, or better inducing an ideal I of colength k on each fibre. This gives a section of Zk and represents the twistor description of the /c-monopole. dy Since C* χ C has the natural 2-form — Λ dx we get a symplectic strucy ture naturally induced on the fibres of Zk. those determine in principle the hyperkahler metric. NOTE:

The fact that Zk is a fibre-wise symmetric product corresponds

to the fact that the Hamiltonian function Η = —- Υ β2 used to twist it is just the sum of the basic function for the k factors I and that — β2 is the twist function of Ζ itself I. The generality of the above construction suggests other applications to monopole systems which possess both a particle-like description and selfdual solutions. For example, one may consider monopoles with gauge group SU(n) where the Higgs field at infinity has (n — 1) equal eigenvalues. The analogue of Donaldson's theorem on rational maps is the statement that the moduli space of such monopoles is equivalent to the space of 1 -1 rational maps from C P to C P " of degree k. Such a theorem is indeed true in the hyperbolic case [2]. We would then expect a twistor space construction based on the general case r = η — 1 given above. We return now to the special case which occupies most of our attention in this book—the moduli space M°2 of centred charge 2 monopoles. Its double covering M% is the space of rational functions of the form

where Δ(α, b) = 1, i.e. a\ + a2b0 = 1. Set χ = a0, y = a1 and ζ = — b0, then the equation is that of the affine surface x2 - zy2 = 1.

55

SYMMETRIC PRODUCTS

Using the "particle" description,

-Jz

Pl = Jz P(Pl) = x

P2 = P(P2) = x - Jzy.

+ Jzy

Thus the symplectic form is

dz Ady x

(since 2x dx = y2 dz

=7T,---

+ 2yz dy).

This is the standard holomorphic 2-form on an affine surface. The spectral curve r can be put in the standard form: (6.1)

1]2

= 4ki(C -

3k 2 C2

-

().

Following Hurtubise [27], the constraint that L2 should be trivial is expressed by OJ

= 4kl

where OJ is the real period of the elliptic curve. If ((u) is the Weierstrass (-function for the normal form of the curve (6.1) then the trivialization of L 2 can be written as b(1],O = ex p ( -4k 1(((U)- : ) )

where C(u) has real quasi period 1], (no relation with 1] and ( of (6.1». The section of the twistor space Z~ corresponding to some 2-monopole is obtained by transforming the standard spectral curve r and the trivialization of L2 by an element of PSU(2) = SO(3) acting projectively on Cpl and with the induced action on L2 over Tpl. Writing an element of SU(2) as Z~

(_~ ~). and relating the section of

to the coordinates of R~ by evaluation of the section at ( = 0, we obtain jj20J

Jz = P= - 4 (6.2)

1

210g(x

f.J'(u)

+ Jzy) = 2 log p(f3) =

OJ

(((U) -

1]U jjbOJ ~) + -2- f.J'(u)

56

CHAPTER 6

where u is determined by (6.3)

- = p{u) + k2. a

This expresses, in an implicit but analytical form, the relationship be­ tween the parametrization of the moduli space by the modulus of an ellip­ tic curve and an element of SO(3) and that of a rational map. In particular it describes the SO(3) action on the complex manifold R2 a n d , knowing the holomorphic symplectic form, this gives as discussed above all the information for finding the metric. Now right multiplication of I

-

-b

_ J by I

aj

_ .„) takes a to ema

and b to e~'eb so from (6.3) keeps u fixed. It also fixes ab and so generates the holomorphic circle action ζ-»e~AWz y -

e2iey

o f x 2 - - zy2 = 1. ,

Thus right action b y

/

COS f

„ ) gives a non-holomorphic circle cos θ J action preserving ω2 . This action is generated by the vector field X given by

Uine

da dt db dt dp Since the Lie derivative of ω 2 + ia>3 — In — Adfiby this vector field gives Ρ ί'ωΐ5 the required Kahler form, it can be determined by differentiating (6.2) and (6.3) with respect to X. We shall not follow this course here, preferring to give a more elemen­ tary and convenient derivation of the metric later on, though we may remark that our original analytical solution of the metric equation was arrived at by twistor methods. 2

2

REMARK: The equation x — zy = 1 can be thought of as a degenerate 2 2 form of the series of surfaces with equation x — zy = z\ For k > 2 these surfaces have a rational double point at the origin which is a quotient singularity of C 2 by the binary dihedral group Dk. For k = 1 it has two ordinary double points. The resolution of these singularities admits hyperkahler metrics of different types. For k > 2, there are asymptotically locally

SYMMETRIC PRODUCTS

57

Euclidean solutions, given by Kronheimer [31]. A brief discussion of the case k = 1 as an "approximation" to the metric on a K3 surface is given in [24]. In all cases, the twistor approach to constructing this metric involves elliptic curves in TPX. The second homology of the resolution for k > 1 is generated by (k + 1) 2-spheres which intersect according to the Dynkin diagram of the group SO(2k + 2), and in particular have self-intersection — 2. The homology for k = 0 is generated by a single 2-sphere of self-intersection — 4, the double covering of the space of axi-symmetric monopoles.

CHAPTER 7

The 2-Monopole Space

From now on we shall concentrate on the case of magnetic charge k = 2 and investigate in detail the geometry of the moduli spaces M 2 and M 2 . The twistor approach for k = 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 lines α and β say. If α = β the monopole is axially symmetric with α as axis, and this axis uniquely defines the corresponding axially symmetric monopole. If α φ β there are precisely two monopoles for a given (unordered) pair (α, β): they are distinguished by the choice of a bisector ex of the angle between α and β (in the plane (α, β)). We shall call the (unoriented) line ex the first or main axis of the monopole. When α = β the main axis is the axis of symmetry. If α φ β we shall denote by e 2 the second bisector of the angle between α and β and call it (for reasons which will become clear later) the Higgs axis. The perpendicular (through the origin) to ex and e2, i.e. to the plane (α, β), will be denoted by e3 and called the third axis. Clearly, having fixed the axes eu e2, e3, the monopole will be deter­ mined by an angle θ with 0 < θ < π/2 as in Diagram 1. Note that when θ = 0 the monopole is axially symmetric about eu and e2 becomes in­ determinate. The rotation group SO(3) acts naturally on the moduli space M° of 2monopoles with fixed centre. It follows from what we have said that the orbits are parametrized by Θ. For θ = 0 the orbit is a real projective plane 2 RP , while for θφΟ the orbit is SO(3)/D where D s Z 2 χ Z 2 is the subgroup of diagonal matrices. This shows in particular that M 2 retracts 2 onto RP and so has fundamental group of order 2 as we have already seen in chapter 2. Suppose, having fixed our axes, we look for thefixedpoint set C of the group D acting on M 2 : it is necessarily a sub-manifold. From our de­ scription above of M 2 it is easy to see that C has three one-dimensional components Cu C2, C 3 , all points me C; having et as their main axis. The component Cx contains a unique point mi on RP 2 which divides Ογ in

2-MONOPOLE SPACE

59

β

e

2

Diagram 1

two pieces. On one piece (i.e. a component of C t — m^ the Higgs axis of the monopole will be e2 while on the other piece it will be e3. In other words as we move along C l 5 crossing RP 2 , the second and third axes of the monopole interchange their roles. This phenomenon will be very im­ portant in our description later of 2-monopole scattering. If we consider the Z 2 subgroup Dy of D given by reflection in the first axis its fixed point set Σ in M° will also be a sub-manifold, this time of dimension 2. It has two components which we shall denote by Σι and Σ 2 3 . The component Σ ΐ 5 which contains Cu consists of all monopoles 2 whose main axis is ex. It intersects RP in the unique point mt e Cv The component Σ 2 3 contains both C 2 and C 3 and consists of all monopoles whose main axis is perpendicular to eu i.e. lies in the plane e2e3. It inter­ 2 1 sects RP in the RP (or circle) joining the points m2 and m3. The surfaces Σ1 and Σ 2 3 will both be investigated in detail later when we describe the scattering of monopoles. It is instructive to describe the surfaces Σι and Σ 2 3 in terms of Donaldson's rational function description of M2. Recall from chapter 2 that M2 can be represented by rational functions S(z) = z\2 ^+ -b Z 0

60

CHAPTER 7

modulo the identification S ~ AS for λ e C*. It follows that the fixed points of the reflection ζ ^ —z correspond either to functions of the form: (I)

S(z) = -5 ζ

,

ueC

— u

or to the functions of the form (II)

S(z) = - ^ - , ζ

ν e C*

— V

Note that in (II) we cannot allow ν = 0 since the rational function would then reduce to degree one. If, in our decomposition R 3 = R χ C, we take R as giving the first axis ex then the two surfaces (I) and (II) must correspond to the two components Σ1 and Σ 2 3 . Since (I) contains for u = 0 the axiallysymmetric τηγ we see that (I) is Σ1 while (II) is Σ 2 3 . Rotation about the e^axis through an angle φ corresponds to multi­ plying u or ν in (I) or (II) by β2ίφ. Since Σ 2 3 intersects R P 2 in a circle, invariant under this rotation, it follows that there is some value σ of \v\ in (II) which represents axially-symmetric monopoles. The identification of the axes of the monopoles given by (I) and (II) will be determined a little later, while in chapter 8 we shall show that σ = π/4. Hurtubise [27] gives the following normal form for the spectral curve Γ c Γ Ρ 1 of a monopole: (7.1)

η2 = Γ ι ζ 3 - r2C2 -

Γι

ζ,

r,eR,r,>0.

Here ζ is an inhomogeneous coordinate on C P 1 and η is the corresponding fibre coordinate. The branch points of the elliptic curve are the four points - 1 1 0, oo, — a, a where —a and a' are the two roots of the quadratic (7.2)

ζ2 - % - 1 = 0. »"i

These two anti-podal pairs give α, β, the spectral lines through the centre of the monopole (the centre corresponds to η = 0). If we define θ by (7.3)

tan 2Θ =

^ r2

then the roots of (7.2) are — tan θ and cot Θ. The following diagram, 2 1 representing stereographic projection of S = C P onto C (and hence of 1 S onto R), then shows that 2Θ is the angle between the lines α and β. The axially symmetric monopole in (7.1) occurs when r1 = 0, i.e. when 9 = 0. This shows that the main axis e1 for general θ is the bisector shown in the diagram, corresponding to ζ = — tan Θ/2. Thus our definition of θ

2-MONOPOLE SPACE

61

Diagram 2

in (7.3) is consistent with our previous notation. In fact this is the way to deduce our previous description from that of [27], The parameters in (7.1) are not of course independent. In fact (as noted in chapter 6) Hurtubise shows that is the real period of the holomorphic differential

If we introduce the standard notation for elliptic integrals, namely (7.4)

Then putting

we find that

(7.5) Thus we can take k instead of as the orbit parameter in which case the spectral curve (7.1) takes the form (7.6) Note that

with k = 0 giving the axially symmetric monopoles.

Consider now the asymptotic situation as Then [32] (7.7)

i.e. as

62

CHAPTER 7

and (7.6) gives (7.8)

η2 ~ Κ2ζ2.

Thus the spectral curve is asymptotic to the pair of lines (7.9)

η=±Κζ

which correspond to a pair of points + Ρ in R 3 . Recall now [22] that in general the point (xu x2, x 3 ) of R 3 corresponds to the line (7.10)

η = (xj + ix2) - 2χ3ζ - {χ, - ϊχ2)ζ2.

Hence the points + P are the points (0, 0, + K/2). Moreover the x3-axis in these coordinates corresponds to the direction ζ = 0, which when θ -» π/2, tends to the Higgs axis e2. More precisely we have the diagram

p/

V Θ

^

2

Diagram 3 Hence the point Ρ on ζ = 0 at distance K/2 has ex and e 2 components K/2 cos θ and K/2 sin θ respectively. In view of (7.7), and recalling that k' = cos Θ, it follows that as θ -» π/2 the e^ -component tends to zero while the e 2 -component is asymptotic to K/2. Thus as θ -» π/2 the spectral lines approximate the two stars of lines through +K/2 on the Higgs axis. This is consistent with the general asymptotic picture explained in chapter 3. With this preliminary picture of the asymptotic behaviour of the 2-monopole let us return to the two surfaces I and II given in Donaldson's

2-MONOPOLE SPACE

63

description by the rational functions

ζ —u

ζ —υ

Consider first the case I. The poles, i.e. ζ = + V" represent the two spectral lines parallel to the main axis. The points (0, +V") in R 3 where these lines intersect the (e 2 e 3 )-plane are invariantly associated to the monopole. The line joining them must therefore be invariant under the symmetry group D of the monopole. This can only happen if the line is either e2 or e3. Moreover, by continuity the choice between e2 and e3 must be independent of \u\ or of Θ. But we have just seen that when θ -> π/2 all spectral lines approximately pass through the points ±K/2 on the Higgs axis e2. This shows that the poles ±~Ju must always lie on the Higgs axis and that asymptotically they represent the "locations" of the two "particles" in agreement with the general results of chapter 3. We turn next to the case II. Now the main axis lies in the plane (0, C) and we have two asymptotic regions, namely |i;| -* oo and |i;| -»· 0. When θ -* π/2 the spectral lines parallel to the e3-axis are at distance K/2, which tends to oo. Thus the case \v\ -» oo corresponds to e 3 = (R, 0) and the poles ±y/v lie on the Higgs axis e2. On the other hand, when θ -» π/2 the spectral lines parallel to the Higgs axis must be close to the Higgs axis so \v\ -> 0 corresponds to e2 = (R, 0). Finally the line joining the poles must this time be the e3-axis, i.e. it must be orthogonal to the e^axis; this follows by continuity from the region \v\ ->• oo. The point is that at the critical value |nj = σ, representing axially symmetric monopoles, e2 and e 3 become indeterminate (changing over as we cross the critical circle), but e t is always well defined. The discussion for the type I surface, shows that a monopole is deter­ mined by its main axis et together with a pair of opposite points on the Higgs axis e2. This means that the moduli space M2 can be naturally 3 2 identified with pairs of vectors (±x, ± y) in R , where \y\ = 1 and χ • y = 0: the notation is meant to indicate that all four signs give the same point oiM°2.

CHAPTER 8

Spectral Radii and the Conformal Structure

In the previous chapter we saw that a 2-monopole with given centre is determined, up to rotation, by a single angular parameter θ with 0 < θ < π/2. Moreover in any given (unoriented) direction χ there are just two spectral lines, and by symmetry they are equidistant from the centre. Let ρ(χ) denote this distance: we shall call it the spectral radius in the direction χ. In particular taking χ to be one of the three axes et of the monopole we obtain the three principal spectral radii pi = p(et). We shall now compute the pt as functions of θ and we shall also show that (8.1)

p3 is the largest of all the spectral radii, i.e. p3 = Max ρ(χ). X

(8.2)

For directions perpendicular to e3, the principal spectral radii p t and p2 are the two local maxima.

In connection with (8.2) recall that the (e^J-plane contains the two spec­ tra/ lines α and β through the centre so that ρ(χ) = 0 for χ = a and χ = β. Thus ρ(χ) has, in this plane, two (local) maxima and two minima. We shall use the normal form for the spectral curve given by (7.1) and (7.3), namely (8.3)

η2 = rxC(C2 - 2ζ cot 2Θ - 1).

Oriented directions are now parametrized by ζ e C u oo, and the two spectral lines in the ζ direction are parametrized by the two solutions for η given by (8.3). Moreover the distance to the origin is given by

as one can check from the formula (7.9) relating (η, ζ) to (xu x2, x 3 ). Using (8.3) we shall now find the maximum value of (8.4). First we have (8.5)

\η\2 = Γ1\ζ\2\ζ-ζ~1 -2 cot 2Θ\.

CONFORMAL STRUCTURE

65

Now put, = seit/>, then (8.6)

I' -

C 1 - 2 cot 2W = I(s - S-l) cos ¢ - 2 cot 20 + i(s + S-l) sin ¢12 = (s - S-1)2 cos 2 ¢ - 4(s - S-l) cos ¢ cot 20 + 4 cot 220 + (s + S-1)2 sin 2 ¢

= (s + s - 1 f

+ (s -

-

S-1)2

4 ( cos ¢ cot220

+

(s -

S-l)

2

(s

)2

+ 4 cot 2 20

= (s + S-1)2 cosec 2 20 - 4 ( cos ¢ + ~

cot 20

(s -

S-l)

2

cot 20

)2

+ S-1)2 cosec 2 20

with equality when (8.7)

cos ¢

+

(S-S-l)

2

cot 2B

=

O.

Hence, from (8.4), (8.5) and (8.6) we deduce 2

P :s; r 1 (1

S

2

+ s2f

(s

+s

-1

S

) cosec 20 = r 1 1 + S2 cosec 20


oo and so ds would not remain finite. Thus trajectories through the vertices can be excluded. This only leaves the two trajectories QI and F, together with the point trajectory I. For this last one we have a = b = c, all orbits are spherical and equations (9.3) give η = z~2, ξ = la. This corresponds to the flat metric on R 4 /Z 2 : this has a conical singularity at the origin which arises

CHAPTER 9

76

because we are looking at manifolds with SO(3) action not with SU(2) action. For the trajectory QI we have a = c, so that (9.3) gives b = η~* and - — = (f/_* — a)2 — a2 = η'1 ααη

— 2αη~^.

For this we deduce

which integrates immediately to give (9.10)

a - 1 f7* = η — γ

for some constant y.

Since we have chosen a to be positive, η* must be taken to be the positive square root. The point I corresponds to η -» oo while Q corresponds to η -* y. Note that η is always positive so that y > 0 (y φ 0, otherwise α = fr = c and we are back .to the fixed point I). Finally the geodesic distance ξ is given by — άξ = abed*] —

η'

^2 ^1

(n - i)

so that

«--/srb?*'· (i - if Since this integral converges as η -» oo while it diverges (to + oo) as η -» y, we must (to get a complete metric) add the degenerate orbit cor­ responding to η = oo (i.e. to the point I). Since a, b, c tend to 0 as η -» oo, the degenerate orbit is a single point. Moreover the above formulas show that a ~ ξ/2 so that the added point is a conical singularity. As for the flat case this means that we have a complete non-singular hyperkahler manifold acted on by SU(2), not SO(3). The solution we have just described is well known and belongs to the Taub-NUT family. Note that, because we have a = c, this metric has an additional U(l) symmetry. At oo, that is as η -> γ, we have a = c ~ ξ -> oo, b -> y~*. Finally we come to the solution represented by the curve F connecting P' to Q. This is of course the one we are really interested in. It must represent the metric on the 2-monopole space M°, since we already know that this is a complete hyperkahler 4-manifold acted on by SO(3) and we have just seen that there are no other possibilities. However, as an inde­ pendent check, we shall verify directly that F does indeed define a com­ plete non-singular manifold.

EINSTEIN EQUATIONS

77

Since, near Q, the curve F is exponentially close, to the line x = 1 it follows easily that the formulas for a, b, c are exponentially close to those of the Taub-NUT metric, except that b is now negative. The equality a = c is now replaced by a — c being exponentially small (as functions of £). Moreover, since rj now increases as we approach Q, the t e r m i s replaced by so that we have (9.11) where means that the difference is exponentially small. To examine the curve F near the saddle point we recall that so that (9.12)

Equations (9.3) then give asymptotically

(9.13)

which, on division, yield (9.14) for some constant

Since

at Q we must have

Also

(from (9.13)).

(9.15) Hence, from (9.13) again, (9.16) for some positive constant Since We now choose the origin in

so that

(9.17)

when

and we have by taking

From (9.14) and (9.15) it follows that

and

78

CHAPTER 9

Finally, from (9.12), we get (9.18)

b ~-δ

+

ξ

-.

The leading terms in equations (9.17) and (9.18) show that, by adding the degenerate 2-dimensional orbit at ξ = 0 we get a smooth manifold. This 2-dimensional orbit has a metric with a = \b\ — δ and so is a sphere of radius δ or else the associated projective plane. Also, in the plane normal to this orbit the metric is άξ2 + Αξ2σ\, which shows that the SO(2) in SO(3) denned by σχ winds twice round the normal circle. If ξ = 0 is a 2-sphere then the orbits ξ φ 0 must be SO(3)/Z2, where Z 2 c SO(2) is the subgroup of order 2. If ξ = 0 is a R P 2 then the orbits ξ φ 0 must be SO(3)/D where D is the subgroup of diagonal matrices. This checks with our description in chapter 6 of the 2-monopole space Μ°. In the analysis in [20] a 2-dimensional orbit of the type just described is termed a bolt, but the case a ~ 2ξ which leads, as we have seen, to 3dimensional orbits of the form SO(3)/Z2 = SU(2)/Z4 was believed not to occur. To sum up we see that there is (up to a double covering) a unique com­ plete hyperkahler 4-manifold Μ acted on by SO(3) so that (a) the generic orbit is 3-dimensional, (b) SO(3) rotates the complex structures of M. This must therefore be our 2-monopole space M° (or its covering M2)We have determined the behaviour of the metric near the "bolt" and at oo. In chapter 10 we shall continue the qualitative analysis by deriving various inequalities on a, b, c. Then in chapter 11 we shall give an explicit analytical form for the metric. Note that the flat metric and the Taub-NUT metric are acted on by SU(2) and not by SO(3). This distinguishes them from our manifold.

C H A P T E R 10

Some Inequalities

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 P' 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 at C. First we prove LEMMA (10.1).

The curve F lies entirely in the region < - 1 + x.

I +- - on y = - 1 + - , 0 < χ < 1, and that αχ 2 2 (c) near P', the curve F lies in the required region. The flow lines on the boundary of the region will then look like

Q

c /

P' Diagram 8

1

1

80

CHAPTER 10

which shows that, once inside the region, F cannot emerge from it before reaching Q. dy Now (a) is trivial because in fact — = 0 on y = — 1 + x. To prove (b) ax we have to show that dj_ dx

-x) y(ix(l -- x)(l + χ -y)

>

1 2

on y == - 1 - - , 0 < x < 1,

i.e. that

(-

- 3 ΉΧ τ)

1 > * *(i- x ) (

Since χ > 0 this is equivalent to

Η

( 2 - - x)(4 -- x) > 2(1 - x)(4 + x) 2 2 8 -6x 4• x > 8 -6x — 2x χ2 > 0 , so that (b) is established. Finally, (c) in fact follows from (b) and the fact that P' is a saddle-point. From (b) we see that no trajectory in the sector bounded by P'C and y = — 1 + x/2 can cross this latter line. This shows that F cannot start from P' below its tangent line y = — 1 + x/2. In other words F starts above its tangent and so lies in the required region. Lemma (10.1) implies in particular that x, \y\, 1 and hence also a, \b\, c satisfy the inequalities to be the sides of a triangle. It will be convenient to use this triangle and to introduce its angles A, B, C. Since b2 = a2 + c 2 — lac cos Β the inequality — 1 + x/2 < y, that is,

implies that a2 + c2 - b2 lac

1 1

3a 1 8c ~~ 2"

Moreover both inequalities used are equalities only when α = 0, i.e. at the "bolt". Thus we have established (10.2)

π Β < —

with equality only at the bolt.

SOME INEQUALITIES

81

Lemma (10.1) implies that \y\ < 1 and since χ < 1 we see that c is the longest side of our triangle, so that C is its largest angle. The next lemma gives an upper bound for C. LEMMA (10.3). Proof:

C < 2π/3 with equality only at the bolt

If s = \{a + \b\ + c) then

_ (s - a){s - |b|) _ 2 r n sin2C/2 = a\b\

{-a-b

+ c){a + b + c) Aab

(χ + y - l)(* + y + i) 4xy

=

Jx +

yf-l

4xv Now C < 2π/3 is equivalent to sin 2 C/2 < 3/4. Hence we have to show

fr + ^ o . xy Since xy < 0 on the curve F this inequality multiplies up to give (x + y)2 - 1 > 3xy or (10.4) x2 + y2 - xy - 1 > 0. Geometrically we have therefore to prove that our curve F lies below the ellipse (10.5)

x2 + y2 - xy - 1 = 0.

Note that this ellipse passes through both P' = (0, — 1) and Q = (1, 0). Moreover at P' it has the same tangent y = — 1 + x/2 as F. If we expand the equation of the ellipse to second order in χ we get x

Λ

χ

,

2

3x \* 3x

2

= 2 - 1 + Ί Γ + ··· with a positive coefficient of x 2 . On the other hand our flow in the projective (a, b, c)-plane given by (9.3) is symmetric under the interchange of b and c. Since this fixes the point P' and takes Q to R (see diagram 4 in chapter 9), it must take the trajectory F to a trajectory F' in region 3 joining P' to R. The symmetry of the configuration (locally a reflection in P') shows that

82

CHAPTER 10

F' lies below the tangent line y = — 1 + x/2. This implies that, in the Taylor expansion of y + 1 — x/2 at P' for the curve F, the first non-vanishing term is of odd degree. In particular there is no coefficient of x 2 . Thus the ellipse (10.5) starts above F near P'. To show that the ellipse remains above F it will be sufficient to show that, on the ellipse, the gradient of our flow is less than the gradient of the ellipse, i.e. that, for 0 < χ < 1, y < 0 2x — y χ — 2y

y(l - y)(l + y — x) x(l — x)(l + χ — y)

,

-

-

Now both denominators are positive in our range so that we can multiply up to get (2x - y)x(l - x)(l +x-y)>(x-

2y)y(\ - y)(l + y - x).

But on the ellipse we have (1 — x)(l +x — y)=l-x2

+ xy — y = 2

y2-y 2

(1 — y)(l + y — x) = 1 — y + xy — χ = χ — χ. Hence we have to show xy{2x - y)(y - 1) > xy(x - 2y)(x - 1) or, since xy < 0, (2x - y)(y - 1) < (x - 2y)(x - 1) 2xy — y2 — 2x + y < x2 — 2xy — χ + 2y x2 + y2 — Axy + χ + y > 0. Using the equation for the ellipse again this reduces to 1 — 3xy + χ + y > 0

H-0H)>* But y — 1/3 < 0, so the above expression is a minimum when χ = 0, y = — 1 and has minimum value 4 , 1 4 „ - - 3 - - - - = 0. Hence in our range, where χ > 0 and y > — 1, the expression is positive and the Lemma is proved. At the bolt we have π

A=0,

Β = -, 3

^ 2π C = — 3

SOME INEQUALITIES

83

while at oo, since \b\ ->· constant and a, c -* oo with an exponentially small difference (see chapter 9), it follows that Β -> 0,

and

A,C-*^,

i.e. A and C tend to π/2 from below. Since the angles are clearly continuous functions of ξ (which parametrizes the SO(3)-orbits) we see that C must take the value π/2 at least once. In fact we shall prove: LEMMA (10.6). Proof:

C = π/2 for exactly one value of ξ.

C = π/2 is equivalent to c2 = a2 + b2 or

x 2 + y2 = 1. Thus we must show that our curve F crosses this circle precisely once (other than at P' and Q). At P' the circle has tangent y = — 1 and so it starts below F, while at Q' it is above F (since F is exponentially close to χ = 1). It will be sufficient therefore to prove that on the circle in the quadrant χ > 0, y < 0, the gradient of our differential equation equals the gradient of the circle at just one point. Now the difference φ of these gradients is given by j;(l - y)(l +y-x)

χ

x(l - x)(l +x-y)

y

_ y(i - y)(i +y-x)

+ x{i-

χ)(ΐ

+x-y)

xy(\ — x)(l + χ — y) The numerator of this expression is A

-(x

4

2

2

2

+ j/ ) + (x + v ) - xy(x + y) + xy(x 2

2

+ y)

2

which on the circle x + y = 1 simplifies to xy(2xy — χ — y + 1). But the hyperbola 2xy — x — y + I =0 2

2

intersects the circle x + y = 1 in the four points

ll0M

tt

°' (r^(-^)

of which only the third lies strictly in the relevant quadrant. Thus φ changes sign just once as required, completing the proof of the Lemma. 2

2

Since the line y = — 1 + x/2 intersects the circle x + y = 1 where χ = 0 or χ = 4/5, Lemma (10.1) shows that the unique point, given by

CHAPTER 10

84

Lemma (10.6), where F crosses the circle must have χ > 4/5. Thus we deduce: (10.7)

When C = π/2, cos Β > 4/5.

We shall rewrite our basic equations (9.3) in trigonometric terms using our triangle with sides a, \b\, c and angles A, B, C. Also we shall use the geodesic distance ξ related as before to η by άξ = — abcdn. Then da άξ

(b- -c)2- -a2 2bc

1 da abc άη

=

2 2 \b\2 + 2\b\c + c - -a 2\b\c

5

(since b < 0 ) ,

-- cos A + 1. Similar calculations for b and c, making allowances for signs lead then to the system da άξ (10.8)

= cos A + 1

d\b\ - with equality only at P'. dx 2

86

CHAPTER 10

Proof: By definition the slope of F is given by the differential equation (9.6). Consider therefore the curve G in the region 0 < χ < 1, y 0, i.e. \b\

is a concave function of ξ. Moreover equality occurs only at ξ = 0, i.e. at the bolt. As in the proof of Lemma (10.10) it will be sufficient to prove

Proof: that

d -— (cos B) > 0. dx Now „ a2 + c2 - b2 x2 + 1 - y2 cos Β = = —, lac 2x and so x2 - 1 + y2 - 2xy - / dx - 2 .

" / -(cosU) =

This will be positive provided 2

2

x + y - 1 > 2xy

/ dx

or, since y < 0, dv dx

2

2

x + v — 1 . 2xy

(10.20) / > — ± f

But we have already proved that, on the curve F,

TA 2

2

x + y - 1 > xy

(10.3) (in the equivalent form (10.4))

which together (since y < 0) imply (10.20).

CHAPTER 10

88

We shall conclude this section by tabulating, for convenience, the range of values of all our variables. In doing so it is instructive to adjoin to our curve F its mirror image F', by the symmetry b c, since the two together form a complete smooth solution of our differential equation, preserved by the symmetry (a, b, c, ξ) -> ( - a , -c, -b,

-ξ)

Ά

ξ

a

b

R

y

— 00

— 00

— 00

F

— OO

0

0

- 0 and ξ < 0 respectively. We have shown in Lemmas (10.10) and (10.19) that, from P' to Q, A and Β are monotonic. However, C is not monotonic since (by (10.6)) it takes the value π/2 at some point strictly between P' and Q. If we consider a complete geodesic on our 4-manifold, orthogonal to the SO(3)-orbits, then using ξ as geodesic parameter (from — oo to + oo)

SOME INEQUALITIES

89

the table shows how a, b, c, A, B, C behave along the geodesic. Note the way the roles and values of b and c interchange as we cross ξ = 0. This is a reflection of the fact that ± ξ are on the same SO(3)-orbit. The constants in the table are only determined by F up to an overall factor. However, the ratio