Line bundles on moduli and related spaces

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, on a smooth open stratum when the moduli space is not smooth. The construction also yields an alternative geometric object, distinct from the familiar gerbe, representing the fundamental class in the third integral cohomology group of K or, equivalently, the first Pontrjagin class of the classifying space of K.