Goldman form, flat connections and stable vector bundles

TL;DR

This paper explores the geometric structures of moduli spaces of stable vector bundles and flat connections on Riemann surfaces, establishing new links between Goldman forms, Liouville forms, and the Riemann-Hilbert correspondence.

Contribution

It introduces the analogue of quasi-Fuchsian projective connections for vector bundles and proves the equivalence of Goldman and Liouville forms via the Riemann-Hilbert correspondence.

Findings

Goldman form pullback matches Liouville form on the affine bundle

Riemann-Hilbert correspondence preserves symplectic structures

Classic symplectic form on moduli space derived from Narasimhan-Seshadri connection

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 Teichm\"{u}ller spaces and the moduli of bundles, we introduce the analogue of 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.