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High-dimensional sphere packing and the modular bootstrap

Nima Afkhami-Jeddi, Henry Cohn, Thomas Hartman, David de Laat, Amirhossein Tajdini

TL;DR

A numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)c× U( 1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions, giving a more detailed picture of the behavior for finite c than was previously available.

Abstract

We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra $U(1)^c \times U(1)^c$, or equivalently the linear programming bound for sphere packing in $2c$ dimensions. We give a more detailed picture of the behavior for finite $c$ than was previously available, and we extrapolate as $c \to \infty$. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimensions. Furthermore, we study when these bounds can be tight. Besides the known cases $c=1/2$, $4$, and $12$ and the conjectured case $c=1$, our calculations numerically rule out sharp bounds for all other $c<90$, by combining the modular bootstrap with linear programming bounds for spherical codes.

High-dimensional sphere packing and the modular bootstrap

TL;DR

A numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)c× U( 1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions, giving a more detailed picture of the behavior for finite c than was previously available.

Abstract

We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra , or equivalently the linear programming bound for sphere packing in dimensions. We give a more detailed picture of the behavior for finite than was previously available, and we extrapolate as . Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimensions. Furthermore, we study when these bounds can be tight. Besides the known cases , , and and the conjectured case , our calculations numerically rule out sharp bounds for all other , by combining the modular bootstrap with linear programming bounds for spherical codes.

Paper Structure

This paper contains 26 sections, 4 theorems, 80 equations, 19 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Results from the spinless modular bootstrap for large $c$
  3. New constraints on tight sphere packing bounds
  4. The spinless modular bootstrap
  5. Setting up the bootstrap
  6. Uncertainty principle
  7. Sphere packing
  8. Numerics
  9. Numerical results
  10. Data and plots
  11. Extrapolation
  12. Properties of the spectrum and degeneracies
  13. Convergence to the free fermion spectrum
  14. Deviations from the free fermion spectrum
  15. Growth rate of the degeneracies
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $h \colon \mathbb{R}^d \to \mathbb{R}$ be an integrable, continuous, radial function such that $\widehat{h}$ is integrable, and let $r$ be a positive real number. If $h(0)=\widehat{h}(0)=1$, $h(x) \le 0$ whenever $|x| \ge r$, and $\widehat{h}(y) \ge 0$ for all $y$, then every sphere packing in $

Figures (19)

  • Figure 1: The linear programming bound for the sphere packing density.
  • Figure 2: The ratio $\Delta^{\textrm{LP}}_1(c)/c$, together with lower bounds from sphere packings.
  • Figure 3: A comparison of the linear programming bound with other bounds. The annotations on the right show the limits of the colored curves in high dimensions, including our conjectured limit for the linear programming bound from Conjecture \ref{['conjecture:LP']}.
  • Figure 4: The spectra $\Delta_n^\textup{LP}(c)$ for $2 \le c \le 64$ and $c \in \{1/2,1\}$, drawn in black. The dots highlight the spectra for $c \in \{1/2,1,4,12\}$, and the green lines show the 1d generalized free fermion spectra.
  • Figure 5: The linear programming bound and the hypothetical bound based on the 1d generalized free fermion spectrum $\Delta_n(c) = n + (c-4)/8$.
  • ...and 14 more figures

Theorems & Definitions (10)

  • Theorem 2.1: Cohn and Elkies CE
  • Conjecture 3.1
  • Conjecture 3.2
  • Conjecture 4.1
  • Conjecture 4.2
  • Conjecture 4.3
  • Theorem 5.1: DGS77
  • Theorem 5.2: LOV12
  • Theorem 5.3
  • proof