Lie Algebroids and the Geometry of Off-shell BRST

TL;DR

This paper reformulates gauge theories using Lie algebroids, simplifying the geometric structure and naturally incorporating BRST, which may enhance understanding of quantization, anomalies, and entanglement in gauge theories.

Contribution

It introduces a Lie algebroid-based formulation of gauge theories that inherently includes BRST, providing a fully off-shell geometric framework.

Findings

Simplifies gauge theory geometry to vector bundles over space-time.

Integrates BRST into the geometric structure without adjuncts.

Applicable to all gauge theories, including Yang-Mills and gravity.

Abstract

It is well-known that principal bundles and associated bundles underlie the geometric structure of classical gauge field theories. In this paper, we explore the reformulation of gauge theories in terms of Lie algebroids and their associated bundles. This turns out to be a simple but elegant change, mathematically involving a quotient that removes spurious structure. The payoff is that the entire geometric structure involves only vector bundles over space-time, and we emphasize that familiar concepts such as BRST are built into the geometry, rather than appearing as adjunct structure. Thus the formulation of gauge theories in terms of Lie algebroids provides a fully off-shell account of the BRST complex. We expect that this formulation will have appealing impacts on the geometric understanding of quantization and anomalies, as well as entanglement in gauge theories. The formalism covers all gauge theories, and we discuss Yang-Mills theories with matter as well as gravitational theories explicitly.