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# Communications in Analysis and Geometry

## Volume 26 (2018)

### Number 4

### Hitchin’s equations on a nonorientable manifold

Pages: 857 – 886

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n4.a6

#### Authors

#### Abstract

We define Hitchin’s moduli space $\mathcal{M}^{\textrm{Hitchin}} (P)$ 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 $K^{\mathbb{C}}$-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 $\tilde{M}$ of $M$ is a Kähler manifold with odd complex dimension and if the Kähler form is odd under the non-trivial deck transformation $\tau$ on $\tilde{M}$, Hitchin’s moduli space $\mathcal{M}^{\textrm{Hitchin}} (\tilde{P})$ of the pull-back bundle $\tilde{P} \to \tilde{M}$ has a hyper-Kähler structure and admits an involution induced by $\tau$. The fixed-point set $\mathcal{M}^{\textrm{Hitchin}} (\tilde{P})^{\tau}$ is symplectic or Lagrangian with respect to various symplectic structures on $\mathcal{M}^{\textrm{Hitchin}} (\tilde{P})$. We show that there is a local diffeomorphism from $\mathcal{M}^{\textrm{Hitchin}} (P)$ to $\mathcal{M}^{\textrm{Hitchin}} (\tilde{P})^{\tau}$. We compare the gauge theoretical constructions with the algebraic approach using representation varieties.

#### Keywords

moduli spaces, non-orientable manifolds, symplectic and hyper-Kähler geometry, representation varieties

#### 2010 Mathematics Subject Classification

53D30, 58D27

The research of N.H. was supported by grant number 99-2115-M-007-008-MY3 from the National Science Council of Taiwan.

The research of G.W. was supported by grant number R-146-000-200-112 from the National University of Singapore.

The research of S.W. was partially supported by RGC grant HKU705612P (Hong Kong) and by MOST grant 105-2115-M-007-001-MY2 (Taiwan).

Received 7 August 2015

Published 6 September 2018