A short introduction to boundary symmetries

TL;DR

The work surveys how boundary conditions in gauge theories and gravity reveal physical boundary symmetries, epitomized by the BMS group at future null infinity. It develops the covariant phase space framework, clarifies ambiguities from boundary and corner terms, and introduces the anomaly operator to test background-independence, guiding a covariant, stationary choice of charges. By comparing Ashtekar–Streubel, Wald–Zoupas, improved Noether charges, and Barnich–Brandt approaches, the text provides a unified, practical route to well-defined BMS fluxes and charges, including a pedagogical Minkowski-based derivation of the BMS group. These tools underpin flat holography, memory effects, and soft-theorem connections, offering concrete prescriptions for conserved quantities in radiative spacetimes.

Abstract

Support material for lectures at the Mai '25 Galileo Galilei Institute school on asymptotic symmetries and flat holography. Contains an introduction to Noether theorem for gauge theories and gravity, covariant phase space formalism, boundary and asymptotic symmetries, future null infinity in Bondi-Sachs coordinates and in Penrose conformal compactification, BMS symmetries and their charges and fluxes. Includes an original and pedagogical derivation of the BMS group using only Minkowski, and an original derivation of an integral Hamiltonian generator for a scalar field on a null hypersurface.

Figures (4)

• Figure 1: Two space-like hypersurfaces $\Sigma_1$ and $\Sigma_2$ joined by a time-like boundary (left panel, $\cal T$) or a null boundary (right panel, $\cal N$).
• Figure 2: Different applications of the co-dimension 2 flux-balance laws in electromagnetism. Left panel: On a single space-like hypersurface $\Sigma$ with two boundaries, the surface charge difference is determine by the matter content in between. Right panel: Between different times, with dissipative boundary conditions -- allowing residual gauge transformations at the boundary, to which $\lambda$ must belong -- the surface charge difference is determined by the flux.
• Figure 3: Different cuts of ${\mathscr{I}}$, from bottom up: an initial good cut, a translated and super-translated. The translated one is still a good cut, its ray tracing identifies a point translated from the origin. The super-translated one is now a bad cut, its ray tracing forms caustics and does not identify an point in the bulk of flat spacetime.
• Figure 4: Two space-like hypersurfaces $\Sigma$ and $\Sigma'$ intersecting ${\mathscr{I}}$ and delimiting a portion ${\cal N}$ of it. By the conservation law $d\omega\,\hat{=}\, 0$, the canonical generator on the phase space defined on the portion of ${\mathscr{I}}$ is equal to the difference of the two canonical generators on the space-like hypersurfaces.