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In physics, the phrase "magnetic monopole" usually denotes a Yang–Mills potential $A$ and Higgs field $\phi$ whose equations of motion are determined by the Yang–Mills–Higgs action

\begin{equation*} \int ( F _ { A } , F _ { A } ) + ( D _ { A } \phi , D _ { A } \phi ) - \lambda ( 1 - \| \phi \| ^ { 2 } ) ^ { 2 }. \end{equation*}

In mathematics, the phrase customarily refers to a static solution to these equations in the Bogomolny–Prasad–Sommerfield limit $\lambda \rightarrow 0$ which realizes, within its topological class, the absolute minimum of the functional

\begin{equation*} \int _ { \mathbf{R} ^ { 3 } } ( F _ { A } , F _ { A } ) + ( D _ { A } \phi , D _ { A } \phi ). \end{equation*}

This means that it is a connection $A$ on a principal $G$-bundle over $\mathbf{R} ^ { 3 }$ (cf. also [[Connections on a manifold|Connections on a manifold]]; [[Principal G-object|Principal $G$-object]]) and a section $\phi$ of the associated adjoint bundle of Lie algebras such that the [[Curvature|curvature]] $F _ { A }$ and [[Covariant derivative|covariant derivative]] $D _ { A } \phi$ satisfy the Bogomolny equations

\begin{equation*} F _ { A } = * D _ { A } \phi \end{equation*}

and the boundary conditions

\begin{equation*} \| \phi \| = 1 - \frac { m } { r } + O ( r ^ { - 2 } ) , \| D _ { A } \phi \| = O ( r ^ { - 2 } ). \end{equation*}

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

The equations themselves are invariant under gauge transformations and orientation-preserving isometries. When $r$ is large, $\phi / \| \phi \|$ defines a mapping from a $2$-sphere of radius $r$ in $\mathbf{R} ^ { 3 }$ 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 $G = \operatorname{SU} ( 2 )$, where the charge is a positive integer $k$. The absolute minimum value of the functional is then $8 \pi k$ and the coefficient $m$ in the asymptotic expansion of $\| \phi \|$ is $k / 2$.

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

\begin{equation*} A = \left( \frac { 1 } { \operatorname { sinh } r } - \frac { 1 } { r } \right) \epsilon _ { i j k } \frac { x _ { j } } { r } \sigma _ { k } d x _ { i }, \end{equation*}

\begin{equation*} \phi = ( \frac { 1 } { \operatorname { tanh } r } - \frac { 1 } { r } ) \frac { x _ { i } } { r } \sigma _ { i }. \end{equation*}

In 1980, C.H. Taubes [[#References|[a10]]] showed by a gluing construction that there exist solutions for all larger $k$ 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 $k = 2$ in terms of elliptic functions.

There are two ways of solving the Bogomolny equations. The first is by twistor methods. In the formulation of N.J. Hitchin [[#References|[a6]]], an arbitrary solution corresponds to a holomorphic vector bundle over the complex surface $T P ^ { 1 }$, the tangent bundle of the projective line. This is naturally isomorphic to the space of oriented straight lines in $\mathbf{R} ^ { 3 }$. The boundary conditions show that the holomorphic bundle is an extension of line bundles determined by a compact [[Algebraic curve|algebraic curve]] of genus $( k - 1 ) ^ { 2 }$ (the spectral curve) in $T P ^ { 1 }$, satisfying certain constraints. The second method, due to W. Nahm [[#References|[a12]]], involves solving an eigenvalue problem for the coupled Dirac operator and transforming the equations with their boundary conditions into a system of ordinary differential equations, the Nahm equations,

\begin{equation*} \frac { d T _ { 1 } } { d s } = [ T _ { 2 } , T _ { 3 } ] , \frac { d T _ { 2 } } { d s } = [ T _ { 3 } , T _ { 1 } ] , \frac { d T _ { 3 } } { d s } = [ T _ { 1 } , T _ { 2 } ], \end{equation*}

where $T _ { i } ( s )$ is a $( k \times k )$-matrix-valued function on $( 0,2 )$. Both constructions are based on analogous procedures for instantons, the key observation due to N.S. Manton being that the Bogomolny equations are dimensional reductions of the self-dual Yang–Mills equations (cf. also [[Yang–Mills field|Yang–Mills field]]) in $\mathbf{R} ^ { 4 }$. The equivalence of the two methods for $\operatorname{SU} ( 2 )$ and their general applicability was established in [[#References|[a7]]] (see also [[#References|[a8]]]). Explicit formulas for $A$ and $\phi$ are difficult to obtain by either method, despite some exact solutions of Nahm's equations in symmetric situations [[#References|[a5]]].

The case of a more general [[Lie group|Lie group]] $G$, where the stabilizer of $\phi$ at infinity is a maximal torus, was treated by M.K. Murray [[#References|[a11]]] from the twistor point of view, where the single spectral curve of an $\operatorname{SU} ( 2 )$-monopole is replaced by a collection of curves indexed by the vertices of the [[Dynkin diagram|Dynkin diagram]] of $G$. The corresponding Nahm construction was described by J. Hurtubise and Murray [[#References|[a9]]].

The moduli space (cf. also [[Moduli theory|Moduli theory]]) of all $\operatorname{SU} ( 2 )$ monopoles of charge $k$ up to gauge equivalence was shown by Taubes [[#References|[a14]]] to be a smooth non-compact manifold of dimension $4 k - 1$. Restricting to gauge transformations that preserve the connection at infinity gives a $4 k$-dimensional manifold $M _ { k }$, which is a circle bundle over the true moduli space and carries a natural complete hyper-Kähler metric [[#References|[a1]]] (cf. also [[Kähler–Einstein manifold|Kähler–Einstein manifold]]). With respect to any of the complex structures of the hyper-Kähler family, this manifold is holomorphically equivalent to the space of based rational mappings of degree $k$ from $P^{1}$ to itself [[#References|[a3]]]. The metric is known in twistor terms [[#References|[a1]]], and its Kähler potential can be written using the [[Riemann theta-function|Riemann theta-function]] of the spectral curve [[#References|[a8]]], but only the case $k = 2$ is known in a more conventional and usable form [[#References|[a1]]] (as of 2000). This Atiyah–Hitchin manifold, the Euclidean Taub–NUT metric and $\mathbf{R} ^ { 4 }$ are the only $4$-dimensional complete hyper-Kähler manifolds with a non-triholomorphic $\operatorname{SU} ( 2 )$ action. Its geodesics have been studied and a programme of Manton concerning monopole dynamics put into effect. Further dynamical features have been elucidated by P.M. Sutcliffe and C.J. Houghton [[#References|[a15]]] using a mixture of numerical and analytical techniques.

A cyclic $k$-fold covering of $M _ { k }$ splits isometrically as a product $\widetilde{ M } _ { k } \times S ^ { 1 } \times \mathbf{R} ^ { 3 }$, where $\tilde { M } _ { k }$ is the space of strongly centred monopoles. This space features in an application of [[S-duality|$S$-duality]] in theoretical physics, and in [[#References|[a13]]] G.B. Segal and A. Selby studied its topology and the $L^{2}$ harmonic forms defined on it, partially confirming the physical predictions.

Magnetic monopoles on hyperbolic three-space were investigated from the twistor point of view by M.F. Atiyah [[#References|[a2]]] (replacing the complex surface $T P ^ { 1 }$ by the complement of the anti-diagonal in $P ^ { 1 } \times P ^ { 1 }$) and in terms of discrete Nahm equations by Murray and M.A. Singer, [[#References|[a16]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> M.F. Atiyah, N.J. Hitchin, "The geometry and dynamics of magnetic monopoles" , Princeton Univ. Press (1988) {{MR|0934202}} {{ZBL|0671.53001}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> M.F. Atiyah, "Magnetic monopoles in hyperbolic space" , ''Vector bundles on algebraic varieties'' , Oxford Univ. Press (1987) pp. 1–34 {{MR|0893593}} {{ZBL|}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> S.K. Donaldson, "Nahm's equations and the classification of monopoles" ''Commun. Math. Phys.'' , '''96''' (1984) pp. 397–407 {{MR|769355}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> N.J. Hitchin, M.K. Murray, "Spectral curves and the ADHM method" ''Commun. Math. Phys.'' , '''114''' (1988) pp. 463–474 {{MR|0929140}} {{ZBL|0646.14021}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> N.J. Hitchin, N.S. Manton, M.K. Murray, "Symmetric monopoles" ''Nonlinearity'' , '''8''' (1995) pp. 661–692 {{MR|1355037}} {{ZBL|0846.53016}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> N.J. Hitchin, "Monopoles and geodesics" ''Commun. Math. Phys.'' , '''83''' (1982) pp. 579–602 {{MR|0649818}} {{ZBL|0502.58017}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> N.J. Hitchin, "On the construction of monopoles" ''Commun. Math. Phys.'' , '''89''' (1983) pp. 145–190 {{MR|0709461}} {{ZBL|0517.58014}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> N.J. Hitchin, "Integrable systems in Riemannian geometry" C.-L. Terng (ed.) K. Uhlenbeck (ed.) , ''Surveys in Differential Geometry'' , '''4''' , Internat. Press, Cambridge, Mass. (1999) pp. 21–80 {{MR|1726926}} {{ZBL|0939.37039}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> J. Hurtubise, M.K. Murray, "On the construction of monopoles for the classical groups" ''Commun. Math. Phys.'' , '''122''' (1989) pp. 35–89 {{MR|0994495}} {{ZBL|0682.32026}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> A. Jaffe, C.H. Taubes, "Vortices and monopoles" , ''Progress in Physics'' , '''2''' , Birkhäuser (1980) {{MR|0614447}} {{ZBL|0457.53034}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> M.K. Murray, "Monopoles and spectral curves for arbitrary Lie groups" ''Commun. Math. Phys.'' , '''90''' (1983) pp. 263–271 {{MR|0714438}} {{ZBL|0517.58015}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> W. Nahm, "The construction of all self-dual monopoles by the ADHM method" N.S. Craigie (ed.) P. Goddard (ed.) W. Nahm (ed.) , ''Monopoles in Quantum Field Theory'' , World Sci. (1982)</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> G.B. Segal, A. Selby, "The cohomology of the space of magnetic monopoles" ''Commun. Math. Phys.'' , '''177''' (1996) pp. 775–787 {{MR|1385085}} {{ZBL|0854.57029}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> C.H. Taubes, "Stability in Yang–Mills theories" ''Commun. Math. Phys.'' , '''91''' (1983) pp. 235–263 {{MR|0723549}} {{ZBL|0524.58020}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> P.M. Sutcliffe, "BPS monopoles" ''Internat. J. Modern Phys. A'' , '''12''' (1997) pp. 4663–4705 {{MR|1474144}} {{ZBL|0907.58081}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> M.K. Murray, "On the complete integrability of the discrete Nahm equations" ''Commun. Math. Phys.'' , '''210''' (2000) pp. 497–519 {{MR|1776842}} {{ZBL|1002.53016}} </td></tr></table>