Coadjoint orbits of the quantum corner symmetry group factorize into SL(2,R) and Heisenberg orbits, and their geometric quantization reproduces the known unitary representations, apart from the complementary series and light-like orbits.
New edge modes and corner charges for first-order symmetries of 4D gravity
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abstract
We present a set of noncommuting frame-translation symmetries in 4D gravity in tetrad-connection variables, which allow expressing diffeomorphisms as composite transformations. Working on the phase space level for finite regions, we pay close attention to the corner piece of the generators, discuss various possible charge brackets, relative definitions of the charges, coupling to spinors and relations to other charges. What emerges is a picture of the symmetries and edge modes of gravity that bears local resemblance to a Poincare group $SO(1,3)\ltimes \mathbb{R}^{1,3}$, but possesses structure functions. In particular, we argue that the symmetries and charges presented here are more amenable to discretisation, and sketch a strategy for this charge algebra, geared toward quantum gravity applications.
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Orbit method for Quantum Corner Symmetries
Coadjoint orbits of the quantum corner symmetry group factorize into SL(2,R) and Heisenberg orbits, and their geometric quantization reproduces the known unitary representations, apart from the complementary series and light-like orbits.