Magnetic Monopole - Appendix

Appendix

In physics the phrase "magnetic monopole" usually denoted a Yang–Mills potential A and Higgs field ϕ whose equations of motion are determined by the Yang–Mills action

In mathematics, the phrase customarily refers to a static solution to these equations in the Bogomolny–Parasad–Sommerfeld limit λ → ϕ which realizes, within topological class, the absolutes minimum of the functional

This means that it in a connection A on a principal G-bundle over R3 (c.f. also Connections on a manifold; principal G-object) and a section ϕ of the associated adjoint bundle of Lie algebras such that the curvature F_A and covariant derivative D_A ϕ satisfy the Bogomolny equations

and the boundary conditions.

Pure mathematical advances in the theory of monopoles from the 1980's onwards have often proceeded on the basis of physically motived questions.

The equations themselves are invariant under gauge transformation and orientation-preserving symmetries. When γ is large, ϕ/||ϕ|| defines a mapping from a 2-sphere of radius γ in R3 to an adjoint orbit G/k and the homotopy class of this mapping is called the magnetic charge. Most work has been done in the case G = SU(2), where the charge is a positive integer k. The absolute minimum value of the functional is then 8πk and the coefficient m in the asymptotic expansion of ϕ/||ϕ|| is k/2.

The first SU(2) solution was found by E. B. Bogomolny, J. K. Parasad and C. M. Sommerfield in 1975. It is spherically symmetric of charge 1 and has the form

In 1980, C.H.Taubes showed by a gluing construction that there exist solutions for all large k and soon after explicit axially-symmetric solutions were found. The first exact solution in the general case was given in 1981 by R.S.Ward for in terms of elliptic function.

There are two ways of solving the Bogomolny equations. The first is by twistor methods. In the formulation of N.J.Hitchin, an arbitrary solution corresponds to a holomorphic vector bundle over the complex surface TP1, the tangent bundle of the projective line. This is naturally isomorphic to the space of oriented straight lines in R3.

The boundary condition show that the holomorphic bundle is an extension of line bundles determined by a compact algebraic curve of genus (k − 1)2 (the spectral curve) in TP1, satisfying certain constraints.

The second method, due to W.Nahm, involves solving an eigen value problem for the coupled Dirac operator and transforming the equations with their boundary conditions into a system of ordinary differential equations, the Nahm equations.

where T_i(s) is a k×k -matrix valued function on (0,2).

Both constructions are based on analogous procedures for instantons, the key observation due to N.S.Manton being of the self-dual Yang–Mills equations (c.f. also Yang–Mills field) in R4.

The equivalence of the two methods for SU(2) and their general applicability was established in (see also). Explicit formulas for A and are difficult to obtain by either method, despite some exact solutions of Nahm's equations in symmetric situations.

The case of a more general Lie group G, where the stabilizer of ϕ at infinity is a maximal torus, was treated by M.K.Murray from the twistor point of view, where the single spectral curve of an SU(2)-monopole is replaced by a collection of curves indexed by the vortices of the Dynkin diagram of G. The corresponding Nahm construction was designed by J.Hustubise and Murray.

The moduli space (c.f. also Moduli theory) of all SU(2) monopoles of charge k up to gauge equivalence was shown by Taubes to be a smooth non-compact manifold fo dimension 4k − 1. Restricting to gauge transformations that preserve the connection at infinity gives a 4k-dimensional manifold M_k, which is a circle bundle over the true moduli space and carries a natural complete hyperKähler metric (c.f. also Kähler–Einstein manifold). With suspected to any of the complex structures of the hyper-Kähler family, this manifold is holomorphically equivalent to the space of based rational mapping of degree k from P₁ to itself.

The metric is known in twistor terms, and its Kähler potential can be written using the Riemann theta functions of the spectral curve, but only the case k = 2 is known in a more conventional and usable form (as of 2000). This Atiyah–Hitchin manifold, the Einstein Taub-NUT metric and R4 are the only 4-dimensional complete hyperKähler manifolds with a non-triholomorphic SU(2) action. Its geodesics have been studied and a programme of Manton concerning monopole dynamics put into effect. Further dynamical features have been elucidated by numerical and analytical techniques.

A cyclic k-fold conering of M_k splits isometrically us a product, where is the space of strongly centred monopoles. This space features in an application of S-duality in theoretical physics, and in G.B.Segal and A.Selby studied its topology and the L2 harmonic forms defined on it, partially confirming the physical prediction.

Magnetic monopole on hyperbolic three-space were investigated from the twistor point of view b M.F.Atiyah (replacing the complex surface TP1 by the comoplement of the anti-diagonal in P1 × P1) and in terms of discrete Nahm equations by Murray and M.A.Singer.