# Sphere Packing and Quantum Gravity

## TL;DR

This paper reveals a deep connection between 2D conformal field theory constraints, sphere packing problems, and quantum gravity, providing exact bounds and functional methods that unify these areas.

## Contribution

It establishes a precise mathematical link between the modular bootstrap in 2D CFTs and sphere packing bounds, including reproducing known optimal solutions in specific dimensions.

## Key findings

- Modular bootstrap bounds map to sphere packing bounds in 2c dimensions.
- Functional methods reproduce optimal sphere packings in dimensions 8 and 24.
- Large central charge limits relate sphere packing to black hole spectrum bounds.

## Abstract

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 $\Delta_0 \lesssim c/8.503$.