Matrix quantization of gravitational edge modes

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 𝔰𝔡𝔦𝔣𝔣 ( S 2 ) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra 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 𝔰𝔡𝔦𝔣𝔣(S^2)⊕_ℒℝ^S^2 arises as the large- N limit of 𝔰𝔩(N,ℂ)⊕ℝ and that the full area-preserving corner symmetry algebra 𝔰𝔡𝔦𝔣𝔣(S^2)⊕_ℒ𝔰𝔩(2,ℝ)^S^2 is the large- N limit of the pseudo-unitary group 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 𝔰𝔲 ( N ) structure constants, as well as an explicit computation of the full 𝔡𝔦𝔣𝔣 ( S 2 ) 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 .