Line bundles on moduli and related spaces
Johannes Huebschmann (Universite de Lille 1)

TL;DR
This paper constructs principal circle bundles with connections on fiber spaces related to moduli spaces, providing a geometric perspective on line bundles and their relation to cohomology classes in the context of Lie groups and Riemann surfaces.
Contribution
It introduces a new geometric construction of line bundles on moduli spaces using relative 2-forms, extending the understanding of their topological and geometric properties.
Findings
Constructed principal G-circle bundles with prescribed curvature.
Applied the construction to moduli spaces of representations and vector bundles.
Provided an alternative geometric object representing the fundamental class in cohomology.
Abstract
Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the principal G-circle bundles with connection on P having the given relative 2-form as curvature. Given a compact Lie group K, a biinvariant Riemannian metric on K, and a closed Riemann surface S of genus s, when we apply the construction to the particular case where f is the familiar relator map from a product of 2s copies of K to K we obtain the principal K-circle bundles on the associated extended moduli spaces which, via reduction, then yield the corresponding line bundles on possibly twisted moduli spaces of representations of the fundamental group of S in K, in particular, on moduli spaces of semistable holomorphic vector bundles or, more precisely,…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Advanced Algebra and Geometry · Geometry and complex manifolds
