Integrable Systems

6Griffiths Linearization Flows on Jacobians

The aim of this chapter is to present the Griffiths linearization method of studying integrable systems, summarizing the situations discussed in Chapter 4. Griffiths has found necessary and sufficient conditions on the matrix B, without reference to the Kac–Moody algebras, so that the flow of the Lax form [4.1] can be linearized on the Jacobian variety Jac(𝒞) for the spectral curve 𝒞 defined by [4.2]. These conditions are cohomological and we will see that the Lax equations turn out to have a very natural cohomological interpretation. These results are exemplified by the Toda lattice, the Lagrange top, Nahm’s equations and the n-dimensional rigid body.

6.1. Spectral curves

Suppose that for every p(h, z) belonging to the curve of affine equation [4.1], that is,

[6.1]

with dim ker(A − zI) = 1 (i.e. the corresponding eigenspace of A is one-dimensional) and generated by a vector v(t, p) ∈ V where V ≃ ℂⁿ is an n-dimensional vector space. There is then a family of holomorphic mappings that send (h, z) ∈ 𝒞 to ker(A − zI):

[6.2]

(We call this the eigenvector map associated with the Lax equation). We set:

[6.3]

where d = deg f_t(𝒞); 𝒪_ℙV(1) is the ...