Physical Representations of Corner Symmetries

TL;DR

This paper develops the full representation theory of gravitational corner symmetries in two dimensions, including projective representations linked to quantum symmetries, revealing their structure as conformal fields with harmonic oscillator indices.

Contribution

It provides a comprehensive analysis of the corner symmetry group's representations, including projective and quantum aspects, using Mackey's theory and the little group method.

Findings

Classified all irreducible unitary representations of the corner symmetry group.

Connected projective representations to one-dimensional conformal fields.

Applied the theory to describe local subsystems in gravitational contexts.

Abstract

We give the full representation theory of the gravitational extended corner symmetry group in two-dimensions. This includes projective representations, which correspond to representations of the quantum corner symmetry group. We find that they are described by one-dimensional conformal fields with an additional index in the Fock space of the harmonic oscillator. We begin with a review of Mackey's theory of induced representations and then proceed to its application to the corner symmetries. The field representations, induced from the irreducible representations of the special linear group are worked out first. The little group method is then applied to the extended corner symmetry group to obtain the irreducible unitary representations. Finally, we focus on projective representations and their application to the description of local subsystems.