Abstract
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 2c 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.
Highlights
Proving upper bounds for the packing density is difficult, and in most cases the best bounds currently known are obtained via the linear programming bound of Cohn and Elkies [8], which relies on harmonic analysis
While these problems sound completely unrelated, Hartman, Mazáč, and Rastelli [9] discovered a surprising connection between them: the spinless modular bootstrap for twodimensional CFTs is very nearly the same as the linear programming bound for sphere packing
We carry out the first large-scale numerical study of the U(1)c spinless modular bootstrap with c large, or equivalently the linear programming bound on sphere packing in high dimensions, by adapting the numerical techniques introduced by AfkhamiJeddi, Hartman, and Tajdini for the Virasoro case [12]
Summary
For sphere packing in high dimensions, the central question is the asymptotic behavior of the packing density. One of our primary results in this paper is a numerical estimate of the fully optimized U(1)c spinless modular bootstrap bound for the spectral gap (Conjecture 3.1). What our computations indicate is that the Kabatyanskii-Levenshtein upper bound can be decreased by an exponential factor through optimizing the linear programming bound. Much like the case of sphere packing, the asymptotic rate in the MRRW bound has not been beaten by any method, and it is an open problem whether it optimizes the linear programming bound. Barg and Jaffe [24] examined this issue numerically, and they conjectured that it is the optimal rate in the linear programming bound. Their conjecture is widely believed, but the evidence is not conclusive. The linear portion of the spectrum matches the spectrum of the generalized free fermion in one dimension, which was used to construct analytic functionals for CFT in [26] and adapted to sphere packing in [9]
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