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| The '''Nahm equations''' are a system of [[ordinary differential equation]]s introduced by [[Werner Nahm]] in the context of the [[Nahm transform]] – an alternative to [[Richard S. Ward|Ward]]'s [[twistor]] construction of [[monopole (mathematics)|monopoles]]. The Nahm equations are formally analogous to the algebraic equations in the [[ADHM construction]] of [[instanton]]s, where finite order matrices are replaced by differential operators.
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| Deep study of the Nahm equations was carried out by [[Nigel Hitchin]] and [[Simon Donaldson]]. Conceptually, the equations arise in the process of infinite-dimensional [[hyperkähler reduction]]. Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the [[moduli space]] of monopoles; and the existence of [[hyperkähler manifold|hyperkähler structure]] on [[coadjoint orbit]]s of complex semisimple Lie groups, proved by [[Peter Kronheimer]], Olivier Biquard, and A.G. Kovalev.
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| == Equations ==
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| Let ''T''<sub>1</sub>(''z''),''T''<sub>2</sub>(''z''), ''T''<sub>3</sub>(''z'') be three matrix-valued meromorphic functions of a complex variable ''z''. The Nahm equations are a system of matrix differential equations
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| :<math>
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| \begin{align}
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| \frac{dT_1}{dz}&=[T_2,T_3]\\[3pt]
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| \frac{dT_2}{dz}&=[T_3,T_1]\\[3pt]
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| \frac{dT_3}{dz}&=[T_1,T_2],
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| \end{align}
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| </math>
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| together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the [[Levi-Civita symbol]], in the form
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| :<math>\frac{dT_i}{dz}=\frac{1}{2}\sum_{j,k}\epsilon_{ijk}[T_j,T_k]=\sum_{j,k}\epsilon_{ijk}T_j T_k. </math>
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| More generally, instead of considering ''N'' by ''N'' matrices, one can consider Nahm's equations with values in a Lie algebra '''g'''.
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| === Additional conditions ===
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| The variable ''z'' is restricted to the open interval (0,2), and the following conditions are imposed:
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| # <math>T^*_i = -T_i;</math>
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| # <math>T_i(2-z)=T_i(z)^{T};\,</math>
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| # ''T''<sub>''i''</sub> can be continued to a meromorphic function of ''z'' in a neighborhood of the closed interval [0,2], analytic outside of 0 and 2, and with simple poles at ''z'' = 0 and ''z'' = 2; and
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| # At the poles, the residues of (''T''<sub>1</sub>,''T''<sub>2</sub>, ''T''<sub>3</sub>) form an irreducible representation of the group [[SU(2)]].
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| == Nahm–Hitchin description of monopoles ==
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| There is a natural equivalence between
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| # the monopoles of charge ''k'' for the group SU(2), modulo gauge transformations, and
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| # the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of ''T''<sub>1</sub>,''T''<sub>2</sub>, ''T''<sub>3</sub> by the group O(k,'''R''').
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| == Lax representation ==
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| The Nahm equations can be written in the [[Lax form]] as follows. Set
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| :<math>
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| \begin{align}
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| & A_0=T_1+iT_2, \quad A_1=-2i T_3, \quad A_2=T_1-iT_2 \\[3 pt]
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| & A(\zeta)=A_0+\zeta A_1+\zeta^2 A_2, \quad B(\zeta)=\frac{1}{2}\frac{dA}{d\zeta}=\frac{1}{2}A_1+\zeta A_2,
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| \end{align}
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| </math> | |
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| then the system of Nahm equations is equivalent to the Lax equation
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| :<math> \frac{dA}{dz}=[A,B]. </math>
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| As an immediate corollary, we obtain that the spectrum of the matrix ''A'' does not depend on ''z''. Therefore, the characteristic equation
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| :<math> \det(\lambda I+A(\zeta,z))=0, </math>
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| which determines the so-called '''spectral curve''' in the twistor space ''TP''<sup>1</sup>, is invariant under the flow in ''z''.
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| == See also ==
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| *[[Bogomolny equation]]
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| *[[Yang–Mills–Higgs equations]]
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| == References ==
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| *{{cite paper |last=Nahm |first=W. |title=All self-dual multimonopoles for arbitrary gauge groups |work=CERN, preprint TH. 3172 |year=1981 |url=http://cdsweb.cern.ch/record/131817 }}
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| *{{cite journal |authorlink=Nigel Hitchin |first=Nigel |last=Hitchin |title=On the construction of monopoles |journal=Communications in Mathematical Physics |volume=89 |issue=2 |year=1983 |pages=145–190 |doi=10.1007/BF01211826 }}
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| *{{cite journal |authorlink=Simon Donaldson |first=Simon |last=Donaldson |title=Nahm's equations and the classification of monopoles |journal=Communications in Mathematical Physics |volume=96 |issue=3 |year=1984 |pages=387–407 |doi=10.1007/BF01214583 }}
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| *{{cite book |authorlink=Michael Atiyah |first=Michael |last=Atiyah |last2=Hitchin |first2=N. J. |title=The geometry and dynamics of magnetic monopoles |series=M. B. Porter Lectures |publisher=Princeton University Press |location=Princeton, NJ |year=1988 |isbn=0-691-08480-7 }}
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| *{{cite journal |last=Kovalev |first=A. G. |title=Nahm's equations and complex adjoint orbits |journal=[[Quarterly Journal of Mathematics|Quart. J. Math. Oxford]] |volume=47 |year=1996 |issue=185 |pages=41–58 |doi=10.1093/qmath/47.1.41 }}
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| *{{cite journal |last=Biquard |first=Olivier |title={{lang|fr|Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes}} |trans_title=Nahm equations and Poisson structure of complex semisimple Lie algebras |journal=[[Mathematische Annalen|Math. Ann.]] |volume=304 |year=1996 |issue=2 |pages=253–276 |doi=10.1007/BF01446293 }}
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| == External links ==
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| *[http://www.maths.tcd.ie/~islands/index.php?title=Main_Page Islands project] – a wiki about the Nahm equations and related topics
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| [[Category:Differential equations]]
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| [[Category:Mathematical physics]]
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| [[Category:Integrable systems]]
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| [[Category:Equations of physics]]
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