(Re)constructing Lie groupoids from their bisections and applications to prequantisation

TL;DR

This paper explores how to reconstruct Lie groupoids from their bisection groups using recent advances in infinite-dimensional Frobenius theorems, with applications to prequantisation of symplectic manifolds.

Contribution

It introduces a method to construct Lie groupoids from their bisection groups, leveraging recent progress in infinite-dimensional geometry, and applies this to prequantisation constraints.

Findings

Characterized integrability constraints via modified period discreteness.

Established a link between bisection groups and Lie groupoid reconstruction.

Applied the theory to prequantisation of symplectic manifolds.

Abstract

This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the bisections to a given Lie groupoid, where the second part is about the construction of Lie groupoids from candidates for their bisection Lie groups. The procedure of this second part becomes feasible due to some recent progress in the infinite-dimensional Frobenius theorem, which we heavily exploit. The main application to the prequantisation of (pre)symplectic manifolds comes from an integrability constraint of closed Lie subalgebras to closed Lie subgroups. We characterise this constraint in terms of a modified discreteness conditions on the periods of that manifold.