Goldman form, flat connections and stable vector bundles

  • Leon A. Takhtajan

    Stony Brook University, USA; Euler International Mathematical Institute, Saint Petersburg, Russia
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Abstract

We consider the moduli space of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle 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 – that allow to pull back the Liouville symplectic form on to . We prove that the pullback of the Goldman form to 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 by the Narasimhan–Seshadri connection is the natural symplectic form on , introduced by Narasimhan and Atiyah & Bott.

Cite this article

Leon A. Takhtajan, Goldman form, flat connections and stable vector bundles. Enseign. Math. 68 (2022), no. 3/4, pp. 409–440

DOI 10.4171/LEM/1036