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Sphere packing and quantum gravity

Journal of High Energy Physics volume 2019, Article number: 48 (2019) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We establish a precise relation between the modular bootstrap, used to con- strain the spectrum of 2D CFTs, and the sphere packing problem in Euclidean geometry. The modular bootstrap bound for chiral algebra U(1)c maps exactly to the Cohn-Elkies linear programming bound on the sphere packing density in d = 2c dimensions. We also show that the analytic functionals developed earlier for the correlator conformal bootstrap can be adapted to this context. For c = 4 and c = 12, these functionals exactly repro- duce the “magic functions” used recently by Viazovska [1] and Cohn et al. [2] to solve the sphere packing problem in dimensions 8 and 24. The same functionals are also applied to general 2D CFTs, with only Virasoro symmetry. In the limit of large central charge, we relate sphere packing to bounds on the black hole spectrum in 3D quantum gravity, and prove analytically that any such theory must have a nontrivial primary state of dimension \( {\Delta}_0\underset{\sim }{<}c/\mathrm{8.503.} \)

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Affiliations

  1. Department of Physics, Cornell University, Ithaca, New York, USA

    Thomas Hartman

  2. C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, USA

    Dalimil Mazáč & Leonardo Rastelli

  3. Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY, 11794, USA

    Dalimil Mazáč

  4. CERN, Theoretical Physics Department, 1211, Geneva 23, Switzerland

    Leonardo Rastelli

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  1. Thomas Hartman
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  2. Dalimil Mazáč
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Correspondence to Dalimil Mazáč.

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Hartman, T., Mazáč, D. & Rastelli, L. Sphere packing and quantum gravity. J. High Energ. Phys. 2019, 48 (2019). https://doi.org/10.1007/JHEP12(2019)048

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Keywords

  • AdS-CFT Correspondence
  • Conformal Field Theory
  • Black Holes
  • Conformal and W Symmetry