High-dimensional sphere packing and the modular bootstrap

TL;DR

This paper explores high-dimensional sphere packing bounds using the modular bootstrap approach, revealing exponential improvements in density bounds as dimension increases and identifying cases where bounds can be tight.

Contribution

It provides a detailed numerical analysis of the modular bootstrap for sphere packing in high dimensions and extends understanding of when bounds are tight.

Findings

Exponential improvement in sphere packing density bounds with increasing dimension.

Numerical evidence ruling out sharp bounds for most dimensions below 90.

Identification of specific dimensions where bounds can be tight, including known and conjectured cases.

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.