BMS Particles in Three Dimensions

TL;DR

This thesis develops a comprehensive group-theoretic framework for three-dimensional quantum gravity on AdS3 and Minkowski backgrounds, focusing on how asymptotic symmetries (BMS3) and Virasoro structures organize gravitational perturbations and particle-like excitations. It constructs and classifies unitary representations of semi-direct products via induced representations, coadjoint orbits, and central extensions, tying these to physical notions of BMS particles and soft gravitons. A detailed analysis of Poincaré and Galilean (Bargmann) cases in three dimensions provides explicit characters, partition functions, and Wigner-rotation effects, offering a concrete laboratory for flat-space holography and higher-spin extensions. The work culminates in a rigorous framework for BMS3, its universal central extension, and the interpretation of one-loop partition functions as BMS3 characters, with broad implications for flat-space holography and the role of soft degrees of freedom in quantum gravity.

Abstract

This thesis is devoted to the group-theoretic aspects of three-dimensional quantum gravity on Anti-de Sitter and Minkowskian backgrounds. In particular we describe the relation between unitary representations of asymptotic symmetry groups and gravitational perturbations around a space-time metric. In the asymptotically flat case this leads to "BMS particles", representing standard relativistic particles dressed with gravitational degrees of freedom accounted for by coadjoint orbits of the Virasoro group. Their thermodynamics are described by BMS characters, which coincide with gravitational one-loop partition functions. We also extend these considerations to higher-spin theories and supergravity.

Figures (30)

• Figure 1: The coordinates $u$ and $r$ in space-time. The time coordinate $x^0$ points upwards. The wavy red line represents an outgoing radial massless particle emitted at $r=0$ and moving to some non-zero distance $r$ away from the observer at $r=0$; the particle moves along one of the generators of the light cone given by $u=\text{const}$. The drawing is three-dimensional, so the circle of radius $r$ in this picture would actually be a sphere (spanned by the coordinate $z$) in a four-dimensional space-time.
• Figure 2: A representation of celestial spheres on the Penrose diagram of Minkowski space. As in fig. \ref{['BondiC']}, the wavy red line represents an outgoing radial light ray. The drawing is three-dimensional, so the red circle at the top of the picture would really be a sphere --- a celestial sphere --- in a four-dimensional space-time. Future null infinity is the cone $\mathbb{R}\times S^2$ on the upper half of the image, spanned by $u$ and $z$.
• Figure 3: The group $\text{U}(1)$ is diffeomorphic to a circle $S^1$, whose universal cover is the real line $\mathbb{R}$. The projection $\mathbb{R}\rightarrow S^1\cong\mathbb{R}/\mathbb{Z}$ is obtained by identifying points of $\mathbb{R}$ that differ by some periodicity, typically $\theta\sim\theta+2\pi$. In particular, paths in $\mathbb{R}$ which are not closed may be projected on closed paths in $S^1$. As an application we can picture topological projective representations: if ${\mathcal{T}}$ is projective and if $\gamma$ is a closed path in the circle, the sequence ${\mathcal{T}}[\gamma(t)]$ may not be a closed path in the space of operators.
• Figure 4: A wavefunction $\Psi$ on ${\mathcal{M}}=\mathbb{R}$ centred around some point $k$ is acted upon by a unitary operator ${\mathcal{T}}[f]$ that implements the transformation $k\mapsto f\cdot k$. The resulting transformed wavefunction ${\mathcal{T}}[f]\cdot\Psi$ is the old one, translated by $f$.
• Figure 5: A manifold ${\mathcal{M}}$ acted upon by a rotation around some axis. The points that belong to the axis are the only ones left fixed by the rotation, and are therefore the only ones that contribute to the integral of formula (\ref{['frob']}).

Theorems & Definitions (40)

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