Abstract
We define Hitchin’s moduli space M Hitchin (P) 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 K C -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 M Hitchin (P) of the pullback bundle P → M has a hyper-Kähler structure and admits an involution induced by τ. The fixed-point set M Hitchin (P) τ is symplectic or Lagrangian with respect to various symplectic structures on M Hitchin (P). We show that there is a local diffeomorphism from M Hitchin (P) to M Hitchin (P) τ . We compare the gauge theoretical constructions with the algebraic approach using representation varieties.