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 $\mathfrak{sdiff}(S^2)$ 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 $\mathfrak{sdiff}(S^2){\oplus }_{\mathcal{L}}{\mathbb{R}}^{S^2}$ arises as the large-$N$ limit of $\mathfrak{sl}(N,\mathbb{C})\oplus \mathbb{R}$ and that the full area-preserving corner symmetry algebra $\mathfrak{sdiff}(S^2){\oplus }_{\mathcal{L}}\mathfrak{sl}(2,\mathbb{R}{)}^{S^2}$ 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 $\mathfrak{diff}(S^2)$ 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 $N$.

v3: 57 pages + Appendices + References = 99 pages; Typos fixed; Published in JHEP

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• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Asymptotic symmetries and subleading soft graviton theorem
• New symmetries for the Gravitational S-matrix
• Notes on the BMS group in three dimensions: I. Induced representations
• Large N algebras and generalized entropy
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• Quantum Mechanics of a Spherically Symmetric Causal Diamond in Minkowski Spacetime
• Minimal Areas from Entangled Matrices
• Physical Representations of Corner Symmetries
• Dynamical edge modes in Maxwell theory from a BRST perspective, with an application to the Casimir energy
• On the covariant formulation of gauge theories with boundaries
• Dynamical Edge Modes in Yang-Mills Theory
• De Sitter Horizon Edge Partition Functions
• Gravitons on Nariai Edges