Discrete connections on principal bundles: the Discrete Atiyah Sequence

TL;DR

This paper develops a discrete analogue of Atiyah's exact sequence for principal bundles, establishing correspondences between splittings, discrete connections, and curvature in discrete and local Lie groupoid categories.

Contribution

It introduces a discrete Atiyah sequence framework, linking splittings, discrete connections, and curvature, extending classical geometric concepts to discrete settings.

Findings

Discrete Atiyah sequence corresponds to discrete connections.

Right splittings in FBS may not be splittings in lLgpdC, leading to a notion of discrete curvature.

Semidirect product extensions relate to splittings in local Lie groupoids.

Abstract

In this work we study discrete analogues of an exact sequence of vector bundles introduced by M. Atiyah in 1957, associated to any smooth principal $G$-bundle $\pi:Q\rightarrow Q/G$. In the original setting, the splittings of the exact sequence correspond to connections on the principal bundle $\pi$. The discrete analogues that we consider here can be studied in two different categories: the category of fiber bundles with a (chosen) section, FBS, and the category of local Lie groupoids, lLgpdC. In FBS we find a correspondence between a) (semi-local) splittings of the discrete Atiyah sequence (DAS) of $\pi$, b) discrete connections on the same bundle $\pi$, and c) isomorphisms of the DAS with certain fiber product extensions in FBS. We see that the right splittings of the DAS (in FBS) are not necessarily right splittings in lLgpdC: we use this obstruction to define the discrete curvature of a discrete connection. Then, there is a correspondence between the splittings of the DAS in lLgpdC and discrete connections with trivial discrete curvature. We also introduce a semidirect product between (some) local Lie groupoids and prove that there is a correspondence between semidirect product extensions and splittings of the DAS in lLgpdC.