Surface Charges Toolkit for Gravity

TL;DR

This work presents a comprehensive toolkit for computing quasi-local surface charges in gravity and gauge theories, unifying metric and differential-form formalisms through the Iyer–Wald/Barnich–Brandt symplectic approach. It derives explicit charge densities for a wide class of theories (EH, EH–Maxwell, EH–scalar, Skyrme, LL, EC, CS, BF) and demonstrates how exact symmetries yield conserved charges on closed $(D-2)$-surfaces, with integrability and quasilocal interpretations clarified. The paper validates the framework through concrete examples, including BTZ black holes and self-dual Taub–NUT solutions, revealing structural patterns such as torsion insensitivity in several cases and compact asymptotic expressions. Overall, the toolkit provides practical, theory-rich formulas for charges across diverse gravity models, with implications for black hole thermodynamics, asymptotic symmetries, and potential quantum extensions.

Abstract

These notes provide a detailed catalog of surface charge formulas for different classes of gravity theories. The present catalog reviews and extends the existing literature on the topic. Part of the focus is on reviewing the method to compute quasi-local surface charges for gauge theories in order to clarify conceptual issues and their range of applicability. Many surface charge formulas for gravity theories are expressed in metric, tetrads-connection, Chern-Simons connection, and even BF variables. For most of them, the language of differential forms is exploited and contrasted with the more popular metric components language. The gravity theory is coupled with matter fields as scalar, Maxwell, Skyrme, Yang-Mills, and spinors. Furthermore, three examples with ready-to-download notebook codes, show the method in full action. Several new results are highlighted through the notes.