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Adjoint L-Infinity Actions and Conserved Charges in GR

Changsun Choi, Ryan E. Grady

TL;DR

This work develops an L_inf-algebraic framework to derive conserved currents and charges associated with Killing symmetries in EinsteinCartanPalatini gravity, grounded in CostelloGwilliam Noether theory and the CGRS formulation of gravity. By introducing the infinity adjoint action, the authors build equivariant action functionals whose Noether currents recover familiar expressions; they demonstrate the method concretely by extracting the Schwarzschild entropy. The approach connects BV/perturbative gravity with factorization-algebraic perspectives, and provides a purely algebraic justification for a higher (∞) adjoint action. The results yield explicit currents and charges, and validate the formalism by reproducing the Wald-type black hole entropy in Schwarzschild spacetime from the L_inf structure. This work thus advances the unification of modern homotopy-algebra methods with classical gravitational observables.

Abstract

In this work we compute the conserved currents and charges associated to the action of an infinitesimal isometry (Killing field) in Einstein--Cartan--Palatini gravity. We offer a new approach to these quantities through the formalism of $L_\infty$-algebras and the work of Ćirić, Giotopoulos, Radovanović, and Szabo, and Costello and Gwilliam. We demonstrate our approach by computing the entropy of the Schwarzchild black hole. Along the way, we prove a purely algebraic result about the existence and utility of a higher (a full $\infty$) version of the adjoint action of an $L_\infty$-algebra.

Adjoint L-Infinity Actions and Conserved Charges in GR

TL;DR

This work develops an L_inf-algebraic framework to derive conserved currents and charges associated with Killing symmetries in EinsteinCartanPalatini gravity, grounded in CostelloGwilliam Noether theory and the CGRS formulation of gravity. By introducing the infinity adjoint action, the authors build equivariant action functionals whose Noether currents recover familiar expressions; they demonstrate the method concretely by extracting the Schwarzschild entropy. The approach connects BV/perturbative gravity with factorization-algebraic perspectives, and provides a purely algebraic justification for a higher (∞) adjoint action. The results yield explicit currents and charges, and validate the formalism by reproducing the Wald-type black hole entropy in Schwarzschild spacetime from the L_inf structure. This work thus advances the unification of modern homotopy-algebra methods with classical gravitational observables.

Abstract

In this work we compute the conserved currents and charges associated to the action of an infinitesimal isometry (Killing field) in Einstein--Cartan--Palatini gravity. We offer a new approach to these quantities through the formalism of -algebras and the work of Ćirić, Giotopoulos, Radovanović, and Szabo, and Costello and Gwilliam. We demonstrate our approach by computing the entropy of the Schwarzchild black hole. Along the way, we prove a purely algebraic result about the existence and utility of a higher (a full ) version of the adjoint action of an -algebra.
Paper Structure (17 sections, 17 theorems, 88 equations)

This paper contains 17 sections, 17 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries I: Local $L_\infty$-algebras
  4. Local $L_\infty$-algebras and Classical Field Theory
  5. Chains and Cochains of an $L_\infty$-algebra
  6. Preliminaries II: Actions of Local $L_\infty$-algebras
  7. Action Complexes
  8. Equivariant Action Functionals
  9. Quick Comparison to Others
  10. The Infinity Adjoint Action
  11. Definition of Infinity Adjoint Action
  12. Properties of the Infinity Adjoint
  13. Conserved Currents and Charges from $L_\infty$-actions
  14. The Ćirić--Giotopoulos--Radovanović--Szabo $L_\infty$-algebra
  15. Conserved Charges from the CGRS $L_\infty$-algebra: Black Hole Entropy
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathcal{M}_\text{ECP}$ be the local cyclic $L_\infty$-algebra presenting Einstein--Cartan--Palatini gravity on the four manifold $M$. Let $\xi \in \mathrm{Vect}(M)$ be a Killing vector field. If $\Lambda=0$, then on shell, the current associated to $\xi$ is the map where

Theorems & Definitions (46)

  • Theorem 1.1: Theorem \ref{['thm:main2']}
  • Theorem 1.2: Theorem \ref{['thm:main']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 36 more