High-dimensional sphere packing and the modular bootstrap

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

doi:10.1007/JHEP12(2020)066

paper

High-dimensional sphere packing and the modular bootstrap

arXiv:2006.02560 · doi:10.1007/JHEP12(2020)066

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 $2 c$ 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.

48 pages, 19 figures