Hitchin's equations on a nonorientable manifold

We define Hitchin’s moduli space for a principal bundle P, whose structure group is a compact semisimple Lie group K, over a compact non-orientable Riemannian manifold M. We use the Donaldson–Corlette correspondence, which identifies Hitchin’s moduli space with the moduli space of flat connections, which remains valid when M is non-orientable. This enables us to study Hitchin’s moduli space both by gauge theoretical methods and algebraically by using representation varieties. If the orientable double cover M~ of M is a Kähler manifold with odd complex dimension and if the Kähler form is odd under the non-trivial deck transformation τ on M~, Hitchin’s moduli space of the pull-back bundle has a hyper-Kähler structure and admits an involution induced by τ. The fixed-point set is symplectic or Lagrangian with respect to various symplectic structures on the moduli space. We compare the gauge theoretical constructions with the algebraic approach using representation varieties.