Connections on Lie groupoids and Chern-Weil theory

Abstract

Let X = [X 1 ⇒ X 0 ] be a Lie groupoid equipped with a connection, given by a smooth distribution H ⊂ T X 1 transversal to the fibers of the source map. Under the assumption that the distribution H is integrable, we define an analog of de Rham cohomology for the pair (X, H) and study connections on principal G-bundles over (X, H) in terms of the associated Atiyah sequence of vector bundles. Finally, we develop the corresponding Chern-Weil theory and study characteristic classes. Contents 1. Introduction 1 2. Principal bundles over Lie groupoids 3 3. Integrable connections and de Rham cohomology 6 4. Connections on principal bundles over Lie groupoids 12 5. Differential forms associated to a connection 14 6. Chern-Weil map and characteristic classes for principal bundles over Lie groupoids with connections 20 Acknowledgements 25 References 25