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A geometrical invitation to BMS group theory

Xavier Bekaert, Yannick Herfray, Lea Mele, Noémie Parrini

TL;DR

This work offers a self-contained, geometrical exposition of BMS group theory rooted in Carrollian boundary structures. It develops the intrinsic boundary viewpoint—via principal $\mathbb{R}$-bundles, Carrollian and conformal Carrollian geometries—to describe BMS transformations and their semidirect product composition with supertranslations. The text connects gravitational vacua, good cuts, and holographic reconstruction of Minkowski spacetime to canonical and non-canonical subgroups of BMS, and culminates with a group-theoretical classification of unitary irreducible representations, including hard/soft decompositions and their relation to infrared gravity data. The framework highlights how boundary geometry encodes bulk physics, clarifies the role of vacuum choices in decomposing states into Poincaré subgroups, and provides a path toward understanding BMS representations in any dimension, with explicit links to ambient realizations and the GJMS operator. Overall, the notes illuminate how asymptotic symmetries organize gravitational scattering data and illuminate the deeper structure tying infrared physics to boundary symmetries and vacua.

Abstract

In these lecture notes, a group-theoretical introduction to BMS symmetries is provided in a self-contained manner. More precisely, all definitions and structures are purely based on geometrical and group-theoretical notions defined at null infinity and valid in any dimension, in a way that circumvents its traditional bulk realisation as asymptotic symmetries. The topics which are reviewed are: the definition of BMS transformations as conformal Carrollian isometries of null infinity, the semidirect structure of the BMS group, the holographic reconstruction of Minkowski spacetime in terms of good cuts, the one-to-one correspondence between good cut subspaces and Poincaré subgroups (aka vacua), as well as a basic introduction to unitary representations of the BMS group.

A geometrical invitation to BMS group theory

TL;DR

This work offers a self-contained, geometrical exposition of BMS group theory rooted in Carrollian boundary structures. It develops the intrinsic boundary viewpoint—via principal -bundles, Carrollian and conformal Carrollian geometries—to describe BMS transformations and their semidirect product composition with supertranslations. The text connects gravitational vacua, good cuts, and holographic reconstruction of Minkowski spacetime to canonical and non-canonical subgroups of BMS, and culminates with a group-theoretical classification of unitary irreducible representations, including hard/soft decompositions and their relation to infrared gravity data. The framework highlights how boundary geometry encodes bulk physics, clarifies the role of vacuum choices in decomposing states into Poincaré subgroups, and provides a path toward understanding BMS representations in any dimension, with explicit links to ambient realizations and the GJMS operator. Overall, the notes illuminate how asymptotic symmetries organize gravitational scattering data and illuminate the deeper structure tying infrared physics to boundary symmetries and vacua.

Abstract

In these lecture notes, a group-theoretical introduction to BMS symmetries is provided in a self-contained manner. More precisely, all definitions and structures are purely based on geometrical and group-theoretical notions defined at null infinity and valid in any dimension, in a way that circumvents its traditional bulk realisation as asymptotic symmetries. The topics which are reviewed are: the definition of BMS transformations as conformal Carrollian isometries of null infinity, the semidirect structure of the BMS group, the holographic reconstruction of Minkowski spacetime in terms of good cuts, the one-to-one correspondence between good cut subspaces and Poincaré subgroups (aka vacua), as well as a basic introduction to unitary representations of the BMS group.
Paper Structure (43 sections, 5 theorems, 71 equations, 10 figures, 1 table)

This paper contains 43 sections, 5 theorems, 71 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. BMS group and Carrollian geometries
  3. Principal bundle geometry
  4. Carrollian geometry
  5. Temporal structure of Carrollian geometry
  6. Carrollian relativity
  7. Spatial structure of Carrollian geometry
  8. Weak Carrollian structure
  9. Strong Carrollian structure
  10. Conformal Carrollian geometry
  11. Weak conformal Carrollian structure
  12. Strong conformal Carrollian structure
  13. Semidirect product structure of BMS group
  14. Canonical subgroups
  15. Canonical subgroups of the Poincaré group
  16. ...and 28 more sections

Key Result

Proposition 3.1

The translation group is the maximal normal subgroup $\mathbb{R}^{d+1,1}$ of $ISO(d+1,1)$.

Figures (10)

  • Figure 1: Principal $\mathbb R$-bundle $\mathscr{M}$ over $\bar{\mathscr{M}}$ with its fundamental vector field $n$
  • Figure 2: Compactified Minkowski spacetime $\overline{\mathbb{R}^{d+1,1}}$ with its conformal boundary $\mathscr{I}^\pm_{d+1}$
  • Figure 3: A cut $s$ of $\mathscr{I}_{d+1}$
  • Figure 4: Values of a projectable vector field $X$ on $\mathscr{M}$ at various points $m_i\in\pi^{-1}(p)$ on the fibre above a base point $p\in\bar{\mathscr{M}}$
  • Figure 5: The cuts $s$ and $\hat{s}$ are related by a supertranslation.
  • ...and 5 more figures

Theorems & Definitions (54)

  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 44 more