Nahm equations

Let $T\_1(z),\; T\_2(z),\; T\_3(z)$ be three matrix-valued meromorphic functions of a complex variable $z$. The Nahm equations are a system of matrix differential equations

:$$

\begin{align}

\frac{dT_1}{dz}&=[T_2,T_3]\\[3pt]

\frac{dT_2}{dz}&=[T_3,T_1]\\[3pt]

\frac{dT_3}{dz}&=[T_1,T_2],

\end{align}

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

:$\backslash frac\{dT\_i\}\{dz\}=\backslash frac\{1\}\{2\}\backslash sum\_\{j,k\}\backslash epsilon\_\{ijk\}[T\_j,T\_k]=\backslash sum\_\{j,k\}\backslash epsilon\_\{ijk\}T\_j\; T\_k.$

More generally, instead of considering $N$ by $N$ matrices, one can consider Nahm's equations with values in a Lie algebra $g$.

The variable $z$ is restricted to the open interval $(0,2)$, and the following conditions are imposed:

1. $T^*\_i\; =\; -T\_i;$
2. $T\_i(2-z)=T\_i(z)^\{T\};\backslash ,$
3. $T\_iN$ 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
4. At the poles, the residues of $T\_1,\; T\_2,\; T\_3$ form an irreducible representation of the group SU(2).

There is a natural equivalence between

1. the monopoles of charge $K$ for the group $SU(2)$, modulo gauge transformations, and
2. the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of $T\_1,\; T\_2,\; T\_3$ by the group $O(k,R)$.

The Nahm equations can be written in the Lax form as follows. Set

:$$

\begin{align}

& A_0=T_1+iT_2, \quad A_1=-2i T_3, \quad A_2=T_1-iT_2 \\[3 pt]

& 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,

\end{align}

then the system of Nahm equations is equivalent to the Lax equation

:$\backslash frac\{dA\}\{dz\}=[A,B].$

As an immediate corollary, we obtain that the spectrum of the matrix $A$ does not depend on $z$. Therefore, the characteristic equation

:$\backslash det(\backslash lambda\; I+A(\backslash zeta,z))=0,$

which determines the so-called spectral curve in the twistor space $TP^1$ is invariant under the flow in $z$.

• Bogomolny equation
• Yang–Mills–Higgs equations

• {{cite journal |last=Nahm |first=W. |title=All self-dual multimonopoles for arbitrary gauge groups |journal=CERN, Preprint TH. 3172 |year=1981 |url=http://cdsweb.cern.ch/record/131817 }}
• {{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 |bibcode = 1983CMaPh..89..145H |s2cid=120823242 }}
• {{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 |bibcode = 1984CMaPh..96..387D |s2cid=119959346 |url=http://projecteuclid.org/euclid.cmp/1103941858 }}
• {{cite book |authorlink=Michael Atiyah |first1=Michael |last1=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 }}
• {{cite journal |last=Kronheimer |first=Peter B. |authorlink=Peter B. Kronheimer |title=A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group |journal=Journal of the London Mathematical Society |volume=42 |year=1990 |number=2 |pages=193–208 |doi=10.1112/jlms/s2-42.2.193 |url=https://doi.org/10.1112/jlms/s2-42.2.193}}
• {{cite journal |last=Kovalev |first=A. G. |title=Nahm's equations and complex adjoint orbits |journal=Quart. J. Math. Oxford |volume=47 |year=1996 |issue=185 |pages=41–58 |doi=10.1093/qmath/47.1.41 }}
• {{cite journal |last=Biquard |first=Olivier |title=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=Math. Ann. |volume=304 |year=1996 |issue=2 |pages=253–276 |doi=10.1007/BF01446293 |s2cid=73680531 }}

• [https://web.archive.org/web/20130419100240/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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