Real Higgs pairs and non-abelian Hodge correspondence on a Klein surface

We introduce real structures on $L$-twisted Higgs pairs over a compact connected Riemann surface $X$ equipped with an antiholomorphic involution, where $L$ is a holomorphic line bundle on $X$ with a real structure, and prove a Hitchin–Kobayashi correspondence for the $L$-twisted Higgs pairs. Real $G^\mathbb{R}$-Higgs bundles, where $G^\mathbb{R}$ is a real form of a connected semisimple complex affine algebraic group $G$, constitute a particular class of examples of these pairs. In this case, the real structure of the moduli space of $G$-Higgs pairs is defined using a conjugation of $G$ that commutes with the one defining the real form $G^\mathbb{R}$ and a compact conjugation of $G$ preserving $G^\mathbb{R}$. We establish a homeomorphism between the moduli space of real $G^\mathbb{R}$-Higgs bundles and the moduli space of representations of the fundamental group of $X$ in $G^\mathbb{R}$ that can be extended to a representation of the orbifold fundamental group of $X$ into a certain enlargement of $G^\mathbb{R}$ with quotient $\mathbb{Z}/2 \mathbb{Z}$. Finally, we show how real $G^\mathbb{R}$-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of $G^\mathbb{R}$-Higgs bundles, constructed using the real structures on $G$ and $X$. A similar result is proved for the representations of the orbifold fundamental group.