Matrix quantization of gravitational edge modes

Abstract

Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra mathfrak{sdiff} (S2) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra mathfrak{su} (N), we derive novel finite-N deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra mathfrak{sdiff}left({S}^2right){oplus}_{mathcal{L}}{mathbb{R}}^{S^2} arises as the large-N limit of mathfrak{sl}left(N,mathbb{C}right)oplus mathbb{R} and that the full area-preserving corner symmetry algebra mathfrak{sdiff}left({S}^2right){oplus}_{mathcal{L}}mathfrak{sl}{left(2,mathbb{R}right)}^{S^2} is the large-N limit of the pseudo-unitary group mathfrak{su} (N, N). We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter N as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the mathfrak{su} (N) structure constants, as well as an explicit computation of the full mathfrak{diff} (S2) structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite N.