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Sphere Packing and Quantum Gravity

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arxiv 1905.01319 v2 pith:QGKERAP7 submitted 2019-05-03 hep-th math.MGmath.NT

Sphere Packing and Quantum Gravity

Thomas Hartman , Dalimil Maz\'a\v{c} , Leonardo Rastelli This is my paper
classification hep-th math.MGmath.NT
keywords packingspherebootstrapfunctionalsboundcftsdimensionsexactly
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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We establish a precise relation between the modular bootstrap, used to constrain 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 reproduce 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 $\Delta_0 \lesssim c/8.503$.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Cusp Form Dimensions, Lattice Uniqueness, and LP Sharpness for Sphere Packing in Dimensions 8 and 24

    math.CO 2026-04 unverdicted novelty 7.0

    Three independent conditions on cusp forms, dual LP obstructions, and extremal CFTs are conjectured to be equivalent for d ≡ 0 mod 8 and together explain LP sharpness exclusively in dimensions 8 and 24.

  2. Descending into the Modular Bootstrap

    hep-th 2026-04 unverdicted novelty 7.0

    Machine-learning optimization produces candidate truncated modular-invariant partition functions for 2d CFTs in the central-charge window 1 to 8/7, indicating a continuous solution space and a stricter spectral-gap bo...

  3. Neural Spectral Bias and Conformal Correlators II: Modular and Annulus Bootstrap

    hep-th 2026-07 conditional novelty 6.0

    Lightweight neural nets with spectral bias reconstruct full 2d CFT torus and annulus partition functions from crossing, a gap, and a single interior anchor to sub-percent accuracy on known theories.

  4. A dual linear programming bound for sphere packing in dimension 36

    math.MG 2026-07 conditional novelty 6.0

    An exact dual Cohn–Elkies certificate in dimension 36 proves the LP bound exceeds the Kschischang–Pasupathy packing density by ≥32.91, so that packing cannot be certified optimal by any Cohn–Elkies auxiliary function.

  5. A dual linear programming bound for sphere packing in dimension 36

    math.MG 2026-07 accept novelty 6.0

    In dimension 36, a modular-form certificate proves the Cohn–Elkies LP bound exceeds the density of the best known packing by at least a factor of 32.9.

  6. Upgrading Extremal Flows in the Space of Derivatives

    hep-th 2026-04 unverdicted novelty 6.0

    A prototype successfully upgrades low-order extremal flow solutions to high numerical order for gap maximization in a simple spinning modular bootstrap test case.