Separation of variables for type Hitchin systems on a hyperelliptic curve

Abstract

For Hitchin systems Darboux variables were until recently known only in the case of genus 2 and rank 2 (see [2]). For arbitrary simple Lie algebras a description of the class of spectral curves for Hitchin systems on hyperelliptic curves of any genus was given in [5]. For Lie algebras of types An, Bn, Cn Darboux variables were found in an explicit form using separation of variables. The goal of this paper is to find Darboux coordinates for Hitchin systems on a hyperelliptic curve for simple Lie algebras of type Dn on the basis of the explicit description of the spectral curve given in [5]. Nevertheless, we were able to find Darboux coordinates explicitly only in the simplest case of the Lie algebra so(4). Note that the isomorphism so(4) ∼= sl(2)⊕sl(2) does not help because it is an outer isomorphism, which does not preserve the spectral curve. Consider the Hitchin system on the hyperelliptic curve Σg of genus g given by y = P2g+1(x), for a classical simple Lie algebra g. This system is defined by the Lax operator L ([4], [5]). The spectral curve is given by the equation det(λ−L) = 0, and for the Lie algebra g of type Dn it has the form