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

Let M ( (8, C) ) be the moduli space of (8, C) -Higgs bundles over a compact Riemann surface X of genus g 2. The triality automorphism of (8, C) acts on M ( (8, C) ) and those Higgs bundles that admit a reduction of structure group to G₂ are fixed points of this action. This defines a map of moduli spaces of Higgs bundles M (G₂) ( (8, C) ). In this work, the action of the triality automorphism is extended to an action on the Hitchin integrable system associated to M ( (8, C) ). In particular, it is checked that the map M (G₂) ( (8, C) ) restricts to a map at the level of Prym varieties. Necessary and sufficient conditions are also provided for the Prym varieties associated with the moduli spaces of G₂ and (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.