Asymptotic Symmetries in the Gauge Fixing Approach and the BMS Group
Romain Ruzziconi
TL;DR
The notes provide a coherent framework for studying asymptotic symmetries in gauge theories using the gauge-fixing approach, with a detailed treatment of how to fix gauges, enforce boundary conditions, and determine the residual symmetry algebra and its action on the solution space. They develop the Barnich–Brandt formalism to construct meaningful surface charges for asymptotic symmetries and relate it to the covariant Iyer–Wald approach, illustrating with four-dimensional GR in asymptotically (A)dS4 and flat spacetimes. The work highlights the connections between symmetry algebras, charges, and dynamics, including issues of integrability, finiteness, and conservation, and discusses important applications to holography and infrared physics, such as soft theorems and memory effects. Overall, it provides a pedagogical, self-contained account linking boundary conditions, residual gauge transformations, and conserved charges in GR, with explicit examples of AdS/CFT-relevant setups and the Λ-BMS4 structure in asymptotically flat spacetimes. The discussion emphasizes how these structures inform holography, infrared structure, and fundamental questions like black hole information, while outlining practical computational tools for charges and algebras.
Abstract
These notes are an introduction to asymptotic symmetries in gauge theories, with a focus on general relativity in four dimensions. We explain how to impose consistent sets of boundary conditions in the gauge fixing approach and how to derive the asymptotic symmetry parameters. The different procedures to obtain the associated charges are presented. As an illustration of these general concepts, the examples of four-dimensional general relativity in asymptotically (locally) (A)dS$_4$ and asymptotically flat spacetimes are covered. This enables us to discuss the different extensions of the Bondi-Metzner-Sachs-van der Burg (BMS) group and their relevance for holography, soft gravitons theorems, memory effects, and black hole information paradox. These notes are based on lectures given at the XV Modave Summer School in Mathematical Physics.