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 \( \mathfrak{sdiff}\left({S}^2\right){\oplus}_{\mathcal{L}}{\mathbb{R}}^{S^2} \) 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 \( \mathfrak{sdiff}\left({S}^2\right){\oplus}_{\mathcal{L}}\mathfrak{sl}{\left(2,\mathbb{R}\right)}^{S^2} \) 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.
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Acknowledgments
We thank Rob Leigh and Lee Smolin for helpful discussions. We are grateful to the organizers of the conference “Quantum Gravity Around the Corner” held at Perimeter Institute. The work of SFM is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) and also in part by the Alfred P. Sloan Foundation, grant FG-2020-13768. AJS is supported by the Air Force Office of Scientific Research under award number FA9550-19-1-036. Research at Perimeter Institute is partly supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and the Province of Ontario through the Ministry of Colleges and Universities.
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Donnelly, W., Freidel, L., Moosavian, S.F. et al. Matrix quantization of gravitational edge modes. J. High Energ. Phys. 2023, 163 (2023). https://doi.org/10.1007/JHEP05(2023)163
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DOI: https://doi.org/10.1007/JHEP05(2023)163