Geometric structures on Lie groupoids and differentiable stacks

TL;DR

This thesis explores geometric structures on Lie groupoids and differentiable stacks, focusing on gerbes, connections, and extensions, advancing the understanding of their interrelations and applications in differential geometry.

Contribution

It introduces new notions of gerbes, connections, and topological groupoid extensions, establishing correspondences and frameworks for their study over stacks and groupoids.

Findings

Established a correspondence between topological groupoid extensions and gerbes over stacks.

Developed a Chern-Weil map for principal bundles over Lie groupoids with connections.

Discussed the notion of connections on principal bundles over Deligne Mumford stacks.

Abstract

In this PhD thesis, we have studied certain geometric structures over Lie groupoids and differentiable stacks. This thesis is based on the work [arXiv:2103.04560, arXiv:2012.08447, arXiv:2012.08442, arXiv:1907.00375]. In [arXiv:1907.00375], we have discussed the correspondence between two notions of a gerbe over a stack. In [arXiv:2012.08447], we have developed the notion of Chern-Weil map for principal bundles over a Lie groupoid when the Lie groupoid is equipped with a connection. In [arXiv:2012.08442], we discuss the notion of connection on the principal bundle over a Deligne Mumford stack using the notion of Atiyah sequence. The work in [arXiv:2103.04560] introduces the notion of a topological groupoid extension and discusses the correspondence between (Morita equivalent classes of) topological groupoid extensions and gerbes over topological stacks.