Goldman form, flat connections and stable vector bundles

Abstract

We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle $\mathscr{A}\to \mathscr{N}$ of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections – local holomorphic sections of $\mathscr{A}$ – that allow to pull back the Liouville symplectic form on $T\*\mathscr{N}$ to $\mathscr{A}$. We prove that the pullback of the Goldman form to $\mathscr{A}$ by the Riemann–Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to $\mathscr{N}$ by the Narasimhan–Seshadri connection is the natural symplectic form on $\mathscr{N}$, introduced by Narasimhan and Atiyah & Bott.