High-dimensional sphere packing and the modular bootstrap

A bstract We carry out 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 2 c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. 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.