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Concept
Nahm equations
Summary
In differential geometry and gauge theory, 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. They can also be viewed as a dimensional reduction of the anti-self-dual Yang-Mills equations . 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 , , and .
Let be three matrix-valued meromorphic functions of a complex variable . 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 by matrices, one can consider Nahm's equations with values in a Lie algebra .
The variable is restricted to the open interval , and the following conditions are imposed:
can be continued to a meromorphic function of in a neighborhood of the closed interval , analytic outside of and , and with simple poles at and ; and
At the poles, the residues of form an irreducible representation of the group SU(2).
There is a natural equivalence between
the monopoles of charge for the group , modulo gauge transformations, and
the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of by the group .
Official source
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