Goldman form, flat connections and stable vector bundles

2021/05/08 by Leon A. Takhtajan, Takhtajan, Leon A.
Mathematics · #14D20 #32G13 #53D30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic and geometric function theory #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2105.03745

openalex publication_date 2021/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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

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