Nahm Equations - Lax Representation

Lax Representation

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

$\begin{align} & A_0=T_1+iT_2, \quad A_1=-2i T_3, \quad A_2=T_1-iT_2 \\ & A(\zeta)=A_0+\zeta A_1+\zeta^2 A_2, \quad B(\zeta)=\frac{1}{2}\frac{dA}{d\zeta}=\frac{1}{2}A_1+\zeta A_2, \end{align}$

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

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

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