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 show that the Donaldson-Corlette correspondence, which identifies Hitchin's moduli space with the moduli space of flat K^C-connections, 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 K"ahler manifold with odd complex dimension and if the K"ahler form is odd under the non-trivial deck transformation on M~, Hitchin's moduli space of the pull-back bundle P~ over M~ has a hyper-K"ahler structure and admits an involution induced by the deck transformation. The fixed-point set is symplectic or Lagrangian with respect to various symplectic structures on Hitchin's moduli space over M~. We show that there is a local diffeomorphism from Hitchin's moduli space over (the nonorientable manifold) M to the fixed point set of the Hitchin's moduli space over (its orientable double cover) M~. We compare the gauge theoretical constructions with the algebraic approach using representation varieties.