Nahm transform

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

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:

There is a natural equivalence between

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