On the Narasimhan–Seshadri correspondence for real and quaternionic vector bundles

Abstract

Let $(M,\sigma)$ be a compact Klein surface of genus $g \geq 2$ and let $E$ be a smooth Hermitian vector bundle on $M$. Let $\tau$ be a Real or Quaternionic structure on $E$ and denote respectively by $\mathcal{G}^{\tau}_{\mathbb{C}}$ and $\mathcal{G}^{\tau}_{E}$ the groups of complex linear and unitary automorphisms of $E$ that commute to $\tau$. In this paper, we study the action of $\mathcal{G}^{\tau}_{\mathbb{C}}$ on the space $\mathcal{A}^{\tau}_{E}$ of $\tau$-compatible unitary connections on $E$ and show that the closure of a semi-stable $\mathcal{G}^{\tau}_{\mathbb{C}}$-orbit contains a unique $\mathcal{G}^{\tau}_{E}$-orbit of projectively flat connections. We then use this invariant-theoretic perspective to prove a version of the Narasimhan–Seshadri correspondence in this context: $S$-equivalence classes of semi-stable Real and Quaternionic vector bundes are in bijective correspondence with equivalence classes of certain appropriate representations of orbifold fundamental groups of Real Seifert manifolds over the Klein surface $(M,\sigma)$.