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Matrix Quantization of Gravitational Edge Modes

William Donnelly, Laurent Freidel, Seyed Faroogh Moosavian, Antony J. Speranza

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 $N$.

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 of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra , we derive novel finite- deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra arises as the large- limit of and that the full area-preserving corner symmetry algebra is the large- limit of the pseudo-unitary group . 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 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 structure constants, as well as an explicit computation of the full 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 .
Paper Structure (47 sections, 2 theorems, 366 equations, 1 figure)

This paper contains 47 sections, 2 theorems, 366 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results of the paper
  3. Corner symmetries and their Poisson brackets
  4. Corner symmetry algebra and its subalgebras
  5. Poisson bracket representations
  6. Mode expansion of Hamiltonian generators
  7. Matrix regularizations of classical symmetry algebras
  8. From functions on phase space to linear operators on Hilbert space
  9. Matrix regularization of $\mathfrak{sdiff}(S)$
  10. Matrix regularization of $\mathfrak{c}_{\mathbb{R}}(S)$
  11. Matrix regularization of $\mathfrak{c}_{\mathfrak{sl}(2,\mathbb{R})}(S)$
  12. The large-$N$ correspondence of Casimirs
  13. $\mathfrak{sdiff}(S)$ and $\mathfrak{su}(N)$
  14. Matching Casimirs
  15. $\mathfrak{c}_{\mathbb{R}}(S)$ and $\mathfrak{sl}(N,\mathbb{C})\oplus \mathbb{R}$
  16. ...and 32 more sections

Key Result

Lemma B.1

Multiplication by a scalar function $\lambda$ acts on the vectors $B_\phi$, $E_\phi$ according to

Figures (1)

  • Figure 1: The corner symmetry algebra $\mathfrak{g}_{\mathfrak{sl}(2,\mathbb{R})}(S)$, its subalgebras and their regularizations. The regularized algebra whose large-$N$ limit is $\mathfrak{g}_{\mathfrak{sl}(2,\mathbb{R})}(S)$ is missing in our analysis.

Theorems & Definitions (4)

  • Lemma B.1
  • proof
  • Lemma B.2
  • proof