Sphere Packing and Quantum Gravity

TL;DR

The paper builds a rigorous bridge between the modular bootstrap for 2D CFTs and the classical sphere-packing problem, identifying the spinless $U(1)^c$ bound with the Cohn–Elkies linear programming bound in $d=2c$ dimensions and showing that analytic functionals from the 1D crossing problem reproduce the sphere-packing magic functions at $c=4$ and $c=12$. It extends these functionals to the Virasoro modular bootstrap and to general 2D CFTs, including a large-$c$ analysis that yields the bound $\Delta_0 \lesssim c/8.503$ and a nontrivial primary below $c/8$ for AdS$_3$ gravity. The results unify constraints from modular invariance, sphere packing, and 1D bootstrap, and illuminate how black-hole spectra in 3D gravity relate to high-dimensional packing. They also provide a framework for complete functional bases and Fourier interpolation in any dimension, with explicit constructions in eight and twenty-four dimensions and conjectures towards higher dimensions.

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 $Δ_0 \lesssim c/8.503$.

Figures (7)

• Figure 1: Universal features of the spectrum of 3D gravity
• Figure 2: Density of sphere packings in $\mathbb{R}^d$. For the best known packings, see table I.1 of ConwaySloane.
• Figure 3: Upper bounds from linear programming on the gap for $U(1)^c$ and $\mathrm{Vir}_c$, denoted $\Delta_{U}(c)$ and $\Delta_V(c)$. We have subtracted $\frac{c+4}{8}$ from both bounds. $\frac{c+4}{8}$ is the optimal bound in the case of the four-point function bootstrap in 1D, translated to modular bootstrap variables.
• Figure 4: Left: The fundamental domain of $\Gamma(2)$ in the upper half-plane. Right: Its image in the space of the four-point cross-ratio $z$ under the mapping $\tau\mapsto\lambda(\tau)$. The image of the interior of the fundamental domain is the complex plane without the branch cuts $(-\infty,0]$ and $[1,\infty)$. The four boundary segments are mapped as shown by the different arrows.
• Figure 5: Left: Contour integral definition of the analytic extremal functional in the $z$-plane, see \ref{['eq:funDef']}, \ref{['eq:ourContours']}. Right: Viazovska's contour integral definition of the magic functions, see \ref{['eq:bViazovska']}. The two definitions are related by the transformation $z=\lambda(\tau)$. The contours labelled by the same Roman numerals get mapped to each other.