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~~Systems governed by non-linear differential equations are of fundamental importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics… See more details below

~~Systems governed by non-linear differential equations are of fundamental importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics for slowly moving monopoles. This picture clarifies the important notion of solitons, which has attracted much attention in recent years. The soliton idea bridges the gap between the concepts of "fields" and "particles," and is here explored in a fully three-dimensional context. While the background and motivation for the work comes from physics, the presentation is mathematical.This book is interdisciplinary and addresses concerns of theoretical physicists interested in elementary particles or general relativity and mathematicians working in analysis or geometry. The interaction between geometry and physics through non-linear partial differential equations is now at a very exciting stage, and the book is a contribution to this activity.

- ISBN-13:
- 9780691604114
- Publisher:
- Princeton University Press
- Publication date:
- 07/14/2014
- Series:
- Porter Lectures Series
- Pages:
- 142
- Product dimensions:
- 6.80(w) x 9.90(h) x 0.30(d)

All rights reserved.

ISBN: 978-0-691-08480-0

CHAPTER 1

The Monopole Equations

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 field ? in Maxwell theory is considered as a 1-form on **R**3 and the magnetic field *B* as a 2-form. The Maxwell field tensor *F = B + c dt [conjunction] E* is then, as a consequence of the Maxwell equations, a *closed* 2-form in Minkowski space **R**4 and can therefore be expressed as *F = dA* for a 2-form *A,* the electromagnetic potential. To *A* is associated a vector particle — the photon.

There is an ambiguity *A* ->*A + d*λ in the choice of *A.* From the point of view of gauge theory this has an interpretation as the ambiguity ψ ->*e*iλ ψ in the choice of phase of a wave function. Geometrically this requires us to consider *A* as a connection form on a (trivial) principal bundle with structure group U(l) and *F* as its curvature. The circle group U(l) measures the difference in phase. Even at this stage there is a noticeable difference in the role of the electric and magnetic fields, for at a fixed time *t*= t0 , the *magnetic field B* is the curvature of a connection on **R**3, whereas *E* has no such geometrical interpretation.

The equation of motion of the field *A* is determined by an action density which is required to be *gauge-invariant.* For electromagnetism the density is given by (*F, F*), using the Minkowski space inner product. This gives rise to the source-free Maxwell equations

dF = 0 = d * F.

Pure Yang-Mills theory is a gauge theory directly modelled on electromagnetism, but using a non-abelian Lie group *G* instead of the abelian group U(l). The vector potential *A,* whose geometrical interpretation is that of a connection on a principal G-bundle, has the physical interpretation of giving rise to a vector particle for each generator of the Lie algebra of G. The action density (*F, F*), analogous to that of electromagnetism, satisfies the gauge-invariance condition and gives rise to the Yang-Mills equations

DAF = 0 = DA * F

where *DA* = *d + A* is the covariant exterior derivative, depending on the connection *A.*

One of the restrictions of Yang-Mills theory, which was recognized early on, is the fact that like a photon the Yang-Mills particle is constrained to have zero mass. To incorporate mass, for physical reasons one requires a term like *m2Q(A)* in the action, where *Q* is quadratic in the vector potential. It is impossible, however to construct such terms in a gauge-invariant manner. One way around this problem is to incorporate a new field — the **Higgs field** φ — into the gauge theory.

Mathematically speaking, φ is a section of a vector bundle associated to the principal G-bundle, on which *A* is defined, by a representation. Physically, φ is a scalar field transforming under some representation of *G,* often the adjoint representation. The action density is taken to be of the form

a = (F, F) + (Dφ, Dφ) + V(φ)

where *V* is a gauge-invariant potential function and *D*φ the covariant derivative of φ. An action of the above form is called a **Yang-Mills-Higgs** action.

The way in which the mass of *A* enters may be seen for example by considering the Higgs field to be in a ground state — that is at a minimum of the potential function *V.* Since *V* is gauge-invariant, the minimum is not attained at a unique field φ, but in general at an *orbit* of φ under the group of gauge transformations. Any φ in a ground state can then be gauge transformed to a constant Higgs field φ0. In this gauge, since φ0 is constant,

Dφ0 = dφ0 + Aφ0 = Aφ0.

Thus

(Dφ0, Dφ0) = (Aφ0, Aφ0)

giving the required quadratic term in *A.* The actual value of the mass is determined by the constant φ0. When φ transforms under the adjoint representation, the most common form for the potential is the quartic expression *V* = λ(1 – [absolute value of φ]2)2. This corresponds to the action density

(1.1) a = (F, F) + (Dφ, Dφ) + λ(1 – [absolute value of φ]2)2

and the corresponding variational equations are the **Yang-Mills-Higgs equations:**

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If magnetic monopoles exist, then the classical limit of their quantum theory should be describable as a soliton-like solution to equation (1.2).

Although the equations (1.2) were motivated for the physical requirements of gauge in variance, mass and simplicity, they have mathematically a cohesion and applicability which gives further cause for their study. We may consider some special cases of the Yang-Mills-Higgs system:

(1) Quite apart from the gauge theory of particles, Ginzburg and Landau introduced in 1950 the action (1.1) to mathematically describe superconductivity. In their situation the gauge group was U(1) and the Higgs field was complex, but it can in any case be considered as the off-diagonal component of an SU(2) Higgs field, with an SU(2) connection reduced to U(1). The quartic potential was introduced for largely phenomenological reasons. One particularly interesting situation is the 2-dimensional version where static pole-like solutions — "vortices" — have been shown to exist [30].

(2) In considering the pure Yang-Mills equations in Euclidean space **R**4, instead of Minkowski space, the solutions which are invariant under the action of various groups of Euclidean motions may be reinterpreted as field equations on lower-dimensional spaces. Witten showed that the SO(3)-invariant solutions to the Yang-Mills equations with group SU(2) correspond to the full abelian Yang-Mills-Higgs system (with a particular value of λ) on a 2-dimensional space of constant negative curvature. As shown in [30], the 2-dimensional system on **R**2 — the Ginzburg-Landau equations — arise from pure Yang-Mills theory on S2 × **R**2 with the product metric by an analogous dimensional reduction.

(3) If we consider the pure Yang-Mills equations in Euclidean **R**4 which are invariant under x4-translation, then we may write the connection form as

A = A1dx1 + A2 dx2 + A3 dx3 + φ dx4

where *Ai* and φ are Lie algebra-valued functions on **R**3. The Euclidean Yang-Mills action can then be written as

a = (F, F) + (Dφ, Dφ)

for the curvature and covariant derivative of the connection *A1 dx1* + *A2 dx2 + A3 dx3,* on R3. It follows that the Yang-Mills-Higgs theory on **R**3 with λ = 0 is equivalent to the dimensional reduction of pure Yang-Mills in **R**4, with its Euclidean metric. As a special case of solutions to the Yang-Mills equations in **R**4 we have the **self-dual connections** which satisfy *F = *F.* These will generate special solutions to the Yang-Mills-Higgs equations in **R**3 and will be the ones we shall use to describe static monopoles.

There is finally a formal mechanical interpretation of the equations (1.2) with λ = 0 which suggests the procedure for approximating the dynamics of monopoles which we have adopted in this book, due to Manton [35]. We begin by considering the space A of all pairs (*A*, φ) on **R**3 of vector potentials and Higgs fields with appropriate decay at infinity. The group G of gauge transformations acts on A to produce the quotient C = A/G. A tangent vector *c* of C may be represented by a vector ([??],[??]) in A which is orthogonal to the gauge group orbit through (*A*, φ), which means that *d*A]*??] + [φ, [??]] = 0. With this choice then

(1.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

defines (formally) a Riemannian metric on C. There is also a potential function *U* on C defined by:

(1.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We consider the motion of a particle on the infinite-dimensional manifold C with potential *U.* Formulating this in Hamiltonian terms one finds that the particle evolves according to a representative of a solution to the Yang-Mills-Higgs equations with λ = 0.

This is a formal statement, but pushing the analogy further we consider the submanifold *M* of C on which *U* attains its minimum. If the motion of the particle is initially tangential to *M*, then we expect the variation of *U* in the subsequent motion to be small. It will thus be determined by the kinetic energy term (1.3), which yields the geodesic motion on *M* with respect to the induced metric.

To make this procedure work, we need then to know the absolute minimum of *U* on A, and to determine the induced metric.

We consider then the SU(2) connections *A* and Higgs fields φ (in the adjoint representation) such that [absolute value of F] = *O(r*-2) and [absolute value of *D*φ] = *O(r*-2) to ensure that *U* exists. The second condition implies that [absolute value of φ] tends to a constant value at infinity which we take to be 1. This is the last vestige of the potential term λ(1 – [absolute value of φ]2)2 which we have set to zero.

Integrating over a ball of radius *R,* we have

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the Bianchi identity *DF =* 0, so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The second term in the integral is therefore a surface integral of the 2-form (φ, F) over the sphere of radius *R:*

(1.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [absolute value of φ] -> 1 as *R* -> ∞ then considering the 2-dimensional complex vector bundle with connection associated to *A,* the eigenspaces of φ are complex line bundles over *SR* and as such have Chern classes + [+ or -]*k.* With the decay condition [absolute value of *D*φ] = *O(r*- 2) the curvature of these line bundles approaches the projection of the curvature *F* onto them. Thus, together with [absolute value of φ] -> 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In (1.5), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore if *k* ≥ 0, the absolute minimum of *U* is 4π*k* and occurs when (*A*, φ) satisfy the **Bogomolny equations:**

(1.7) F = *Dφ.

This is the dimensional reduction to **R**3 of the self-dual Yang-Mills equations in Euclidean **R**4. The integer *k* is the **charge** of the solution.

We see then finally how we are led to consider the metric on the moduli space of gauge-equivalence classes of the solutions to the Bogomolny equations: these provide the absolute minimum of the potential *U* on C and the mechanical analogue suggests we should obtain this way an approximation to the full Yang-Mills-Higgs time-dependent system.

In the case of the group U(1), the Bogomolny equations reduce to

B = grad φ

and this, with φ = *k/2r* is the origin of the first magnetic monopole proposed by Dirac in 1931 [13]. In modern terms the quantization condition that *k* should be an integer is precisely the gauge-theoretical statement that *B* is the curvature of a connection on a line bundle over **R**3 – {0}. Our monopoles behave at infinity like a Dirac monopole but because of their non-abelian nature, have finite energy.

Geometry of the Monopole Spaces

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 group *G* to be SU(2) although our methods can in principle be extended to all *G.*

The data for a monopole on **R**3 consist of a gauge field or connection *A*µ(*x*), µ = 1, 2, 3, and a Higgs field φ(*x*). All these are smooth functions of *x* [member of] **R**3 and take their values in the Lie algebra of SU(2). As usual one defines the covariant derivative *D*µφ by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [partial derivative]µ = [partial derivative]/[partial derivative]xµ, and the field or curvature *F* by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For a monopole these functions must satisfy the **Bogomolny equations**

(2.1) Dφ = *F

where * is the duality operator on **R**3. Moreover we require the energy, essentially the L2-norm of *F,* to be finite. Then as shown in [30] this implies that

(2.2) [absolute value of φ] -> α as [absolute value of x] -> ∞

for some constant α. Here [absolute value of φ] stands for an invariant norm on the Lie algebra which for definiteness we shall take as

(2.3) [absolute value of T]2 = –½ Trace T2.

Although α is a physically significant parameter we can by a mathematical rescaling of **R**3 assume (if α ≠ 0) that α = 1. Moreover we shall impose the somewhat stronger asymptotic assumptions used in [22], namely

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where *r* is the distance from an origin in **R**3 and [partial derivative]/[partial derivative]Ω refers to angular derivatives. It is shown in [30] that these are indeed all consequences of the finiteness of the energy and that they can be proved by a refinement of the proof of (2.2).

Excerpted fromThe Geometry and Dynamics of Magnetic MonopolesbyMichael Francis Atiyah, Nigel Hitchin. Copyright © 1988 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.

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