Lie Algebroids and the Geometry of Off-shell BRST

TL;DR

This work presents a unified geometric reformulation of gauge theories using Atiyah Lie algebroids, showing that BRST symmetry is inherently encoded in the algebroid structure via the vertical part of the extended exterior derivative $\hat{d}$ and the connection reform $\omega$, with ghost fields arising as components of the algebroid data. By passing from the principal bundle $P$ to the quotient $TP/G$, the framework eliminates redundant right actions and yields a vector bundle over space-time $M$, enabling a fully off-shell BRST description that applies to Yang–Mills theories with matter and to gravitational theories via solder forms. The algebraic and differential-geometric machinery developed for transitive Lie algebroids, derivations on vector bundles, and the horizontal/vertical split of $\hat{d}$ is then concretely connected to standard local physics through Atiyah algebroids, trivializations, and BRST transformations of gauge fields and ghosts. The approach also clarifies the role of diffeomorphism ghosts in gravity, showing they do not arise as independent geometric degrees of freedom in this formalism and that solder forms provide a natural gravitational extension within the same geometric language. Overall, the paper lays a foundation for a geometry-driven variational calculus for gauge theories, with potential implications for quantization, anomalies, and entanglement in gauge/gravity systems.

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.