Goldman form, flat connections and stable vector bundles

Abstract

We consider the moduli space N of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle A → N of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of of the quasi-Fuchsian projective connections -local holomorphic sections of A -that allow to pull back the Liouville symplectic form on T * N to A . We prove that the pullback of the Goldman form to 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 N by the Narasimhan-Seshadri connection is the natural symplectic form on N , introduced by Narasimhan and Atiyah & Bott.