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.
They can also be viewed as a dimensional reduction of the anti-self-dual Yang-Mills equations (Donaldson 1984).
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 (Kronheimer 1990), (Biquard 1996), and (Kovalev 1996).
The three equations can be written concisely using the Levi-Civita symbol, in the form More generally, instead of considering
matrices, one can consider Nahm's equations with values in a Lie algebra
, and the following conditions are imposed: There is a natural equivalence between The Nahm equations can be written in the Lax form as follows.
Therefore, the characteristic equation which determines the so-called spectral curve in the twistor space