Line Bundles and Central Extensions

Abstract

We discuss line bundles, connection and curvature, and the group of isomorphism classes of line bundles L equipped with a connection ∇ (§2.1 and §2.2). This group turns out to be isomorphic to a Deligne cohomology group (§2.2). In fact Deligne cohomology provides a tool for constructing line bundles with connections. If the infinitesimal action of some Lie algebra on the underlying manifold preserves the isomorphism class of (L, ∇), then a central extension of this Lie algebra acts on sections of the line bundle L. The action is written down explicitly using hamiltonian vector fields (prequantization à la Kostant-Souriau) (§2.3). Similarly, if a Lie group action preserves (L, ∇) up to isomorphism, a central extension of the Lie group acts on sections of the line bundle (Kostant). A similar central extension exists in a holomorphic context (Mumford). In §2.5, we discuss results of Weinstein, which give a similar central extension when there is no line bundle (because the given 2-form is not integral).