Nahm equations

Nahm equations

The Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators.

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 structure on coadjoint orbits of complex semisimple Lie groups, proved by Peter Kronheimer, Olivier Biquard, and A.G. Kovalev.



Let T1(z),T2(z), T3(z) be three matrix-valued meromorphic functions of a complex variable z. The Nahm equations are a system of matrix differential equations


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

\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.

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

Additional conditions

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};\,
  3. Ti 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 (T1,T2, T3) form an irreducible representation of the group SU(2).

Nahm–Hitchin description of monopoles

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 T1,T2, T3 by the group O(k,R).

Lax representation

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

& 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, 

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


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

det(λI + A(ζ,z)) = 0,

which determines the so-called spectral curve in the twistor space TP1, is invariant under the flow in z.

See also

  • Bogomolny equation
  • Yang–Mills–Higgs equations


External links

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia

  • Monopole (mathematics) — In mathematics, a monopole is a connection over a principal bundle G with a section (the Higgs field) of the associated adjoint bundle. The connection and Higgs field should satisfy the Bogomolnyi equation and be of finite action. See also Nahm… …   Wikipedia

  • Emmanuel Rudnitzky — Man Ray, 16. Juni 1934 in Paris Fotograf: Carl van Vechten Signatur Man Ray Man Ray [ …   Deutsch Wikipedia

  • Man Ray — Man Ray, 16. Juni 1934 in Paris, fotografiert von Carl van Vechten Man Ray [mæn reɪ] (* …   Deutsch Wikipedia

  • Michael Atiyah — Sir Michael Atiyah Born 22 April 1929 (1929 04 22) (age 82) …   Wikipedia

  • Augustin Louis Cauchy — [ogysˈtɛ̃ lwi koˈʃi] (* 21. August 1789 in Paris; † 23. Mai 1857 in Sceaux) war ein französischer Ma …   Deutsch Wikipedia

  • Jürgen Ehlers (Physiker) — Jürgen Ehlers anlässlich der Verleihung der Medaille der Karls Universität Prag Jürgen Ehlers (* 29. Dezember 1929 in Hamburg; † 20. Mai 2008) war ein deutscher Physiker, der wichtige Beiträge zur einsteinschen Allgemeinen Relativitätstheorie… …   Deutsch Wikipedia

  • Mick Harris — (eigentlich Michael John Harris, * 1967 in Birmingham) ist ein britischer Musiker und Produzent. Bekanntheit erlangte er Mitte der 1980er als Schlagzeuger der britischen Grindcore Band Napalm Death und gilt als einer der Begründer des Blastbeat.… …   Deutsch Wikipedia

  • Alexander Markowich Ostrowski — Alexander Markowitsch Ostrowski Alexander Markowitsch Ostrowski (russisch Александр Маркович Островский, wiss. Transliteration Aleksandr Markovič Ostrovskij; * 25. September 1893 in Kiew; † 20. November 1986 in …   Deutsch Wikipedia

  • Alexander Markowitsch Ostrowski — Ostrowski in Washington (1964) …   Deutsch Wikipedia