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Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions

Geoffrey Compère

TL;DR

The work develops a unified, cohomological framework for exact and asymptotic conservation laws in Lagrangian gauge theories. It constructs gauge-invariant surface charges as one-forms in field space, encoded by surface forms $k_f$, and links their algebra to reducibility parameters via a covariant presymplectic structure. The formalism recovers and unifies Hamiltonian and covariant phase space approaches, including Regge–Teitelboim and Abbott–Deser constructions, and naturally yields central extensions of the asymptotic symmetry algebra. It is then applied to gravity with matter, including p-forms and the Chern–Simons term, deriving the first law and Smarr relations for a broad class of black holes, Gödel-type geometries in three and five dimensions, Kerr–AdS spacetimes, black rings, and plane-wave backgrounds. These results advance robust, geometry-driven definitions of energy, angular momentum, and charges in diverse spacetimes, and illuminate the role of asymptotic symmetries and their central extensions in quantum gravity contexts, particularly in three dimensions and AdS/CFT-inspired settings.

Abstract

The treatment of exact conservation laws in Lagrangian gauge theories constitutes the main axis of the first part of the thesis. The formalism is developed as a self-consistent theory but is inspired by earlier works, mainly by cohomological results, covariant phase space methods and by the Hamiltonian formalism. The thermodynamical properties of black holes, especially the first law, are studied in a general geometrical setting and are worked out for several black objects: black holes, strings and rings. Also, the geometrical and thermodynamical properties of a new family of black holes with closed timelike curves in three dimensions are described. The second part of the thesis is the natural generalization of the first part to asymptotic analyses. We start with a general construction of covariant phase spaces admitting asymptotically conserved charges. The representation of the asymptotic symmetry algebra by a covariant Poisson bracket among the conserved charges is then defined and is shown to admit generically central extensions. The asymptotic structures of three three-dimensional spacetimes are then studied in detail and the consequences for quantum gravity in three dimensions are discussed.

Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions

TL;DR

The work develops a unified, cohomological framework for exact and asymptotic conservation laws in Lagrangian gauge theories. It constructs gauge-invariant surface charges as one-forms in field space, encoded by surface forms , and links their algebra to reducibility parameters via a covariant presymplectic structure. The formalism recovers and unifies Hamiltonian and covariant phase space approaches, including Regge–Teitelboim and Abbott–Deser constructions, and naturally yields central extensions of the asymptotic symmetry algebra. It is then applied to gravity with matter, including p-forms and the Chern–Simons term, deriving the first law and Smarr relations for a broad class of black holes, Gödel-type geometries in three and five dimensions, Kerr–AdS spacetimes, black rings, and plane-wave backgrounds. These results advance robust, geometry-driven definitions of energy, angular momentum, and charges in diverse spacetimes, and illuminate the role of asymptotic symmetries and their central extensions in quantum gravity contexts, particularly in three dimensions and AdS/CFT-inspired settings.

Abstract

The treatment of exact conservation laws in Lagrangian gauge theories constitutes the main axis of the first part of the thesis. The formalism is developed as a self-consistent theory but is inspired by earlier works, mainly by cohomological results, covariant phase space methods and by the Hamiltonian formalism. The thermodynamical properties of black holes, especially the first law, are studied in a general geometrical setting and are worked out for several black objects: black holes, strings and rings. Also, the geometrical and thermodynamical properties of a new family of black holes with closed timelike curves in three dimensions are described. The second part of the thesis is the natural generalization of the first part to asymptotic analyses. We start with a general construction of covariant phase spaces admitting asymptotically conserved charges. The representation of the asymptotic symmetry algebra by a covariant Poisson bracket among the conserved charges is then defined and is shown to admit generically central extensions. The asymptotic structures of three three-dimensional spacetimes are then studied in detail and the consequences for quantum gravity in three dimensions are discussed.

Paper Structure

This paper contains 119 sections, 23 theorems, 582 equations, 8 figures.

Table of Contents

  1. Conservation laws and symmetries
  2. Puzzles in gauge theories
  3. Results in local cohomology
  4. Windows on the literature
  5. The central idea: the linearized theory
  6. Classical theory of surface charges
  7. Global symmetries and Noether currents
  8. Gauge symmetries and vanishing Noether currents
  9. Surface charge one-forms and their algebra
  10. Hamiltonian formalism
  11. Exact solutions and symmetries
  12. Charges for gravity coupled to matter fields
  13. Diffeomorphic invariant theories
  14. Surface one-form
  15. Properties of the surface one-form
  16. ...and 104 more sections

Key Result

Lemma 1

The surface one-forms satisfy where $W_{{\delta\mathcal{L}}/{\delta\phi}}[\text{d}_V\phi,R_{f}]\equiv -i_{R_f} W_{{\delta\mathcal{L}}/{\delta\phi}}$.

Figures (8)

  • Figure 1: Two Cauchy surfaces $\Sigma_1$, $\Sigma_2$ and their intersection with the spatial boundary $\mathcal{T}^\infty$.
  • Figure 3.1: Penrose diagram of an asymptotically flat spacetime with spherically symmetric collapsing star. Each point is a $n-2$-dimensional sphere. Radial light rays propagate along $45^\circ$ diagonals. The star region is hatched and the black hole region is indicated in grey.
  • Figure 3.2: The null vector $n$ is defined with respect to $\xi$.
  • Figure 3.3: If two null generators of $\mathcal{H}$ cross, they may be deformed to a timelike curve.
  • Figure 4.1: Gödel cosmons
  • ...and 3 more figures

Theorems & Definitions (23)

  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 5
  • Corollary 6
  • Proposition 7
  • Theorem 8
  • Theorem 9: Area law
  • Theorem 10: Zero law
  • ...and 13 more