Surface charge algebra in gauge theories and thermodynamic integrability

TL;DR

This work develops a covariant Lagrangian framework for surface charges and their algebras in gauge theories, deriving charges from the linearized theory via the variational bicomplex. It shows that for exact solutions and symmetries the charges form a Pfaff system with integrability governed by $Frobenius$, and, crucially, that charges associated with the derived symmetry algebra vanish, reducing to the abelian quotient of exact symmetries. In the asymptotic regime, the charges furnish centrally extended representations of the asymptotic symmetry algebra, encoded by a 2-cocycle $K_{g_1,g_2}$ and compatible with Hamiltonian and covariant phase-space formalisms. The results unify exact and asymptotic analyses, clarify ambiguities inherent in covariant phase-space methods, and connect with prior Hamiltonian approaches, including gravity examples, through detailed appendices and constructions.

Abstract

Surface charges and their algebra in interacting Lagrangian gauge field theories are investigated by using techniques from the variational calculus. In the case of exact solutions and symmetries, the surface charges are interpreted as a Pfaff system. Integrability is governed by Frobenius' theorem and the charges associated with the derived symmetry algebra are shown to vanish. In the asymptotic context, we provide a generalized covariant derivation of the result that the representation of the asymptotic symmetry algebra through charges may be centrally extended. Finally, we make contact with Hamiltonian and with covariant phase space methods.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2
• Corollary 3
• Proposition 4
• Corollary 5
• Corollary 6
• Corollary 7
• Proposition 8
• Corollary 9
• Corollary 10