BMS Modules in Three Dimensions

TL;DR

The paper develops a framework for unitary representations of the $BMS_3$ algebra and its higher-spin extensions in three dimensions through induced modules, showing these arise naturally as ultrarelativistic limits of highest-weight Virasoro and ${\cal W}$-algebra modules. It contrasts this with non-relativistic (Galilean) contractions that typically yield non-unitary representations, highlighting how non-linear terms in higher-spin algebras influence the quantum structure. The authors construct explicit induced modules for $\mathfrak{bms}_3$ and its spin-3 extension, discuss vacuum and boosted states, and demonstrate the consistency of these representations with flat-space holographic expectations. They also compare ultrarelativistic and Galilean contractions at the level of the algebras, showing a genuine quantum higher-spin effect in the ordering and normal ordering of nonlinear terms. The results provide a concrete, unitary representation theory toolkit for flat-space holography and higher-spin theories in 3D.

Abstract

We build unitary representations of the BMS algebra and its higher-spin extensions in three dimensions, using induced representations as a guide. Our prescription naturally emerges from an ultrarelativistic limit of highest-weight representations of Virasoro and W algebras, which is to be contrasted with non-relativistic limits that typically give non-unitary representations. To support this dichotomy, we also point out that the ultrarelativistic and non-relativistic limits of generic W algebras differ in the structure of their non-linear terms.