Extended Phase Space in General Gauge Theories

TL;DR

The work develops a universal geometric framework—the configuration algebroid—for extended phase spaces in gauge theories. By unifying diffeomorphisms and internal gauge transformations as Lie algebroid morphisms, it proves that the action of local symmetries on the covariant phase space is integrable and non-centrally extended, via a moment morphism, without relying on specific equations of motion. The approach recovers the known charge algebra in Chern-Simons theory and clarifies the role of edge modes and subregion data within a covariant, algebroid-based setting. This provides a robust, theory-agnostic foundation for understanding charges, corner symmetries, and potential quantum aspects of gauge theories.

Abstract

In a recent paper, it was shown that in diffeomorphism-invariant theories, Noether charges associated with a given codimension-2 surface become integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs $(\varphi, Φ)$, where $\varphi$ is a Lie algebroid morphism and $Φ$ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.