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Covariant theory of asymptotic symmetries, conservation laws and central charges

Glenn Barnich, Friedemann Brandt

TL;DR

This paper develops a covariant framework for asymptotic symmetries and conservation laws in Lagrangian gauge theories. By introducing asymptotic reducibility parameters and asymptotically conserved n-2 forms, it establishes a bijective correspondence between these two objects, and provides universal formulas for constructing the associated charges. The charges can form a covariant Poisson algebra that may admit central extensions, and the central charges are shown to be 2-cocycles in the corresponding Lie algebra; the results are demonstrated in electrodynamics, Yang-Mills theory, and Einstein gravity, including AdS3 gravity where Brown–Henneaux-type central charges arise. The formalism is further recast in BRST– antifield language, clarifying descent equations, cohomology groups, and the induced algebraic structures, and it connects naturally with Hamiltonian and covariant phase space approaches. Together, these results offer a unified, manifestly covariant account of asymptotic charges and their symmetry algebras with precise conditions for finiteness and central extensions, applicable to a broad class of gauge theories.

Abstract

Under suitable assumptions on the boundary conditions, it is shown that there is a bijective correspondence between equivalence classes of asymptotic reducibility parameters and asymptotically conserved n-2 forms in the context of Lagrangian gauge theories. The asymptotic reducibility parameters can be interpreted as asymptotic Killing vector fields of the background, with asymptotic behaviour determined by a new dynamical condition. A universal formula for asymptotically conserved n-2 forms in terms of the reducibility parameters is derived. Sufficient conditions for finiteness of the charges built out of the asymptotically conserved n-2 forms and for the existence of a Lie algebra g among equivalence classes of asymptotic reducibility parameters are given. The representation of g in terms of the charges may be centrally extended. An explicit and covariant formula for the central charges is constructed. They are shown to be 2-cocycles on the Lie algebra g. The general considerations and formulas are applied to electrodynamics, Yang-Mills theory and Einstein gravity.

Covariant theory of asymptotic symmetries, conservation laws and central charges

TL;DR

This paper develops a covariant framework for asymptotic symmetries and conservation laws in Lagrangian gauge theories. By introducing asymptotic reducibility parameters and asymptotically conserved n-2 forms, it establishes a bijective correspondence between these two objects, and provides universal formulas for constructing the associated charges. The charges can form a covariant Poisson algebra that may admit central extensions, and the central charges are shown to be 2-cocycles in the corresponding Lie algebra; the results are demonstrated in electrodynamics, Yang-Mills theory, and Einstein gravity, including AdS3 gravity where Brown–Henneaux-type central charges arise. The formalism is further recast in BRST– antifield language, clarifying descent equations, cohomology groups, and the induced algebraic structures, and it connects naturally with Hamiltonian and covariant phase space approaches. Together, these results offer a unified, manifestly covariant account of asymptotic charges and their symmetry algebras with precise conditions for finiteness and central extensions, applicable to a broad class of gauge theories.

Abstract

Under suitable assumptions on the boundary conditions, it is shown that there is a bijective correspondence between equivalence classes of asymptotic reducibility parameters and asymptotically conserved n-2 forms in the context of Lagrangian gauge theories. The asymptotic reducibility parameters can be interpreted as asymptotic Killing vector fields of the background, with asymptotic behaviour determined by a new dynamical condition. A universal formula for asymptotically conserved n-2 forms in terms of the reducibility parameters is derived. Sufficient conditions for finiteness of the charges built out of the asymptotically conserved n-2 forms and for the existence of a Lie algebra g among equivalence classes of asymptotic reducibility parameters are given. The representation of g in terms of the charges may be centrally extended. An explicit and covariant formula for the central charges is constructed. They are shown to be 2-cocycles on the Lie algebra g. The general considerations and formulas are applied to electrodynamics, Yang-Mills theory and Einstein gravity.

Paper Structure

This paper contains 107 sections, 5 theorems, 373 equations.

Table of Contents

  1. Introduction and summary
  2. The Noether-charge puzzle for gauge symmetries
  3. Various approaches
  4. Summary of results
  5. Asymptotic reducibility parameters and asymptotic symmetries
  6. Asymptotically conserved n--2 forms
  7. Bijective correspondence
  8. Charges
  9. Algebra of asymptotic reducibility parameters
  10. Induced global symmetries
  11. Algebra of asymptotically conserved n--2 forms
  12. Algebra of charges and central extensions
  13. Organization of the paper
  14. Section 2
  15. Section 3
  16. ...and 92 more sections

Key Result

Theorem 1

There is a bijective correspondence between the quotient space of asymptotic reducibility parameters factored by pure gauge parameters on the one hand, and equivalence classes of asymptotically conserved $n-2$ forms on the other hand.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1: Trivial linear characteristic cohomology