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In physics, the phrase  "magnetic monopole"  usually denotes a Yang–Mills potential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300201.png" /> and Higgs field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300202.png" /> whose equations of motion are determined by the Yang–Mills–Higgs action
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In mathematics, the phrase customarily refers to a static solution to these equations in the Bogomolny–Prasad–Sommerfield limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300204.png" /> which realizes, within its topological class, the absolute minimum of the functional
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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
  
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\begin{equation*} \int ( F _ { A } , F _ { A } ) + ( D _ { A } \phi , D _ { A } \phi ) - \lambda ( 1 - \| \phi \| ^ { 2 } ) ^ { 2 }. \end{equation*}
  
This means that it is a connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300206.png" /> on a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300207.png" />-bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300208.png" /> (cf. also [[Connections on a manifold|Connections on a manifold]]; [[Principal G-object|Principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m1300209.png" />-object]]) and a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002010.png" /> of the associated adjoint bundle of Lie algebras such that the [[Curvature|curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002011.png" /> and [[Covariant derivative|covariant derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002012.png" /> satisfy the Bogomolny equations
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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
  
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\begin{equation*} \int _ { \mathbf{R} ^ { 3 } } ( F _ { A } , F _ { A } ) + ( D _ { A } \phi , D _ { A } \phi ). \end{equation*}
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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
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\begin{equation*} F _ { A } = * D _ { A } \phi \end{equation*}
  
 
and the boundary conditions
 
and the boundary conditions
  
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\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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002015.png" /> is large, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002016.png" /> defines a mapping from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002017.png" />-sphere of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002019.png" /> to an adjoint orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002020.png" /> and the homotopy class of this mapping is called the magnetic charge. Most work has been done in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002021.png" />, where the charge is a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002022.png" />. The absolute minimum value of the functional is then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002023.png" /> and the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002024.png" /> in the asymptotic expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002025.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002026.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002027.png" /> solution was found by E.B. Bogomolny, M.K. Prasad and C.M. Sommerfield in 1975. It is spherically symmetric of charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002028.png" /> and has the form
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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
  
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\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*}
  
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002031.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002032.png" /> in terms of elliptic functions.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002033.png" />, the tangent bundle of the projective line. This is naturally isomorphic to the space of oriented straight lines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002034.png" />. The boundary conditions show that the holomorphic bundle is an extension of line bundles determined by a compact [[Algebraic curve|algebraic curve]] of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002035.png" /> (the spectral curve) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002036.png" />, 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,
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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,
  
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002038.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002039.png" />-matrix-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002040.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002041.png" />. The equivalence of the two methods for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002042.png" /> and their general applicability was established in [[#References|[a7]]] (see also [[#References|[a8]]]). Explicit formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002044.png" /> are difficult to obtain by either method, despite some exact solutions of Nahm's equations in symmetric situations [[#References|[a5]]].
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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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002045.png" />, where the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002046.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002048.png" />-monopole is replaced by a collection of curves indexed by the vertices of the [[Dynkin diagram|Dynkin diagram]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002049.png" />. The corresponding Nahm construction was described by J. Hurtubise and Murray [[#References|[a9]]].
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002050.png" /> monopoles of charge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002051.png" /> up to gauge equivalence was shown by Taubes [[#References|[a14]]] to be a smooth non-compact manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002052.png" />. Restricting to gauge transformations that preserve the connection at infinity gives a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002053.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002054.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002055.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002056.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002057.png" /> is known in a more conventional and usable form [[#References|[a1]]] (as of 2000). This Atiyah–Hitchin manifold, the Euclidean Taub–NUT metric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002058.png" /> are the only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002059.png" />-dimensional complete hyper-Kähler manifolds with a non-triholomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002060.png" /> 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.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002061.png" />-fold covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002062.png" /> splits isometrically as a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002063.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002064.png" /> is the space of strongly centred monopoles. This space features in an application of [[S-duality|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002065.png" />-duality]] in theoretical physics, and in [[#References|[a13]]] G.B. Segal and A. Selby studied its topology and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002067.png" /> harmonic forms defined on it, partially confirming the physical predictions.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002068.png" /> by the complement of the anti-diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130020/m13002069.png" />) and in terms of discrete Nahm equations by Murray and M.A. Singer, [[#References|[a16]]].
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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====
 
====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)</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</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</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</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</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</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</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</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</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)</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</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</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</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</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</TD></TR></table>
+
<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>

Latest revision as of 16:59, 1 July 2020

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; Principal $G$-object) and a section $\phi$ of the associated adjoint bundle of Lie algebras such that the curvature $F _ { A }$ and 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 [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 [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 of genus $( k - 1 ) ^ { 2 }$ (the spectral curve) in $T P ^ { 1 }$, satisfying certain constraints. The second method, due to W. Nahm [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) in $\mathbf{R} ^ { 4 }$. The equivalence of the two methods for $\operatorname{SU} ( 2 )$ and their general applicability was established in [a7] (see also [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 [a5].

The case of a more general Lie group $G$, where the stabilizer of $\phi$ at infinity is a maximal torus, was treated by M.K. Murray [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 of $G$. The corresponding Nahm construction was described by J. Hurtubise and Murray [a9].

The moduli space (cf. also Moduli theory) of all $\operatorname{SU} ( 2 )$ monopoles of charge $k$ up to gauge equivalence was shown by Taubes [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 [a1] (cf. also 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 [a3]. The metric is known in twistor terms [a1], and its Kähler potential can be written using the Riemann theta-function of the spectral curve [a8], but only the case $k = 2$ is known in a more conventional and usable form [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 [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 in theoretical physics, and in [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 [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, [a16].

References

[a1] M.F. Atiyah, N.J. Hitchin, "The geometry and dynamics of magnetic monopoles" , Princeton Univ. Press (1988) MR0934202 Zbl 0671.53001
[a2] M.F. Atiyah, "Magnetic monopoles in hyperbolic space" , Vector bundles on algebraic varieties , Oxford Univ. Press (1987) pp. 1–34 MR0893593
[a3] S.K. Donaldson, "Nahm's equations and the classification of monopoles" Commun. Math. Phys. , 96 (1984) pp. 397–407 MR769355
[a4] N.J. Hitchin, M.K. Murray, "Spectral curves and the ADHM method" Commun. Math. Phys. , 114 (1988) pp. 463–474 MR0929140 Zbl 0646.14021
[a5] N.J. Hitchin, N.S. Manton, M.K. Murray, "Symmetric monopoles" Nonlinearity , 8 (1995) pp. 661–692 MR1355037 Zbl 0846.53016
[a6] N.J. Hitchin, "Monopoles and geodesics" Commun. Math. Phys. , 83 (1982) pp. 579–602 MR0649818 Zbl 0502.58017
[a7] N.J. Hitchin, "On the construction of monopoles" Commun. Math. Phys. , 89 (1983) pp. 145–190 MR0709461 Zbl 0517.58014
[a8] 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 MR1726926 Zbl 0939.37039
[a9] J. Hurtubise, M.K. Murray, "On the construction of monopoles for the classical groups" Commun. Math. Phys. , 122 (1989) pp. 35–89 MR0994495 Zbl 0682.32026
[a10] A. Jaffe, C.H. Taubes, "Vortices and monopoles" , Progress in Physics , 2 , Birkhäuser (1980) MR0614447 Zbl 0457.53034
[a11] M.K. Murray, "Monopoles and spectral curves for arbitrary Lie groups" Commun. Math. Phys. , 90 (1983) pp. 263–271 MR0714438 Zbl 0517.58015
[a12] 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)
[a13] G.B. Segal, A. Selby, "The cohomology of the space of magnetic monopoles" Commun. Math. Phys. , 177 (1996) pp. 775–787 MR1385085 Zbl 0854.57029
[a14] C.H. Taubes, "Stability in Yang–Mills theories" Commun. Math. Phys. , 91 (1983) pp. 235–263 MR0723549 Zbl 0524.58020
[a15] P.M. Sutcliffe, "BPS monopoles" Internat. J. Modern Phys. A , 12 (1997) pp. 4663–4705 MR1474144 Zbl 0907.58081
[a16] M.K. Murray, "On the complete integrability of the discrete Nahm equations" Commun. Math. Phys. , 210 (2000) pp. 497–519 MR1776842 Zbl 1002.53016
How to Cite This Entry:
Magnetic monopole. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Magnetic_monopole&oldid=17620
This article was adapted from an original article by N.J. Hitchin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article