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 ( 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).
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:
There is a natural equivalence between
The Nahm equations can be written in the Lax form as follows. Set
then the system of Nahm equations is equivalent to the Lax equation
As an immediate corollary, we obtain that the spectrum of the matrix does not depend on . Therefore, the characteristic equation
which determines the so-called spectral curve in the twistor space is invariant under the flow in .
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 ( 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).
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:
There is a natural equivalence between
The Nahm equations can be written in the Lax form as follows. Set
then the system of Nahm equations is equivalent to the Lax equation
As an immediate corollary, we obtain that the spectrum of the matrix does not depend on . Therefore, the characteristic equation
which determines the so-called spectral curve in the twistor space is invariant under the flow in .