Spin(8,C)-Higgs Bundles and the Hitchin Integrable System

Let M(Spin(8,C)) be the moduli space of Spin(8,C)-Higgs bundles over a compact Riemann surface X of genus g≥2. This admits a system called the Hitchin integrable system, induced by the Hitchin map, the fibers of which are Prym varieties. Moreover, the triality automorphism of Spin(8,C) acts on M(Spin(8,C)), and those Higgs bundles that admit a reduction in the structure group to G2 are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M(G2)→M(Spin(8,C)). In this work, the action of triality automorphism is extended to an action on the Hitchin integrable system associated with M(Spin(8,C)). In particular, it is checked that the map M(G2)→M(Spin(8,C)) is restricted to a map at the level of the Prym varieties induced by the Hitchin map. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G2 and Spin(8,C)-Higgs bundles to be disconnected. Finally, some consequences are drawn from the above results in relation to the geometry of the Prym varieties involved.