High-dimensional sphere packing and the modular bootstrap

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 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. Contents 1 Introduction 2 The spinless modular bootstrap 3 Numerical results 4 Properties of the spectrum and degeneracies 5 Spherical codes and implied kissing numbers A The limiting case of the spinless modular bootstrap B Convergence of the spinless bootstrap

Dispersive CFT Sum Rules

“…along any (non-real) direction in the complex plane. 15 The statement in (2.35) that the schannel OPE of P s|u ∆,J contains a single physical block, plus double-twist and their derivatives, means that its only poles are at s = ∆ − J + 2m with residue given in (2.45) and no poles in t. (Double-twist blocks and their derivatives are generated by the gamma factors in (2.42).) Since t = 4∆ φ − s − u, all singularities of P s|u ∆,J (s, t) (at fixed u) are the poles in s. There is a unique function with these properties:…”

“…See [2][3][4] for reviews and e.g. [5][6][7][8][9][10][11][12][13][14][15][16] for a partial list of recent results.…”

“…The idea to define d > 1 functionals as double contour integrals appeared independently in[45] and examples of genuine extremal functionals were also constructed there. Another intersting example of an extremal functional in the two-variable situation appeared in the context of the modular bootstrap[15].…”

Comments on the holographic description of Narain theories

“…This is the analog of the Cardy formula for Narain theories, as can be deduced directly from modular invariance [16]. It applies universally for ∆ c but as we will see below, in certain cases its validity extends to much smaller values of ∆.…”

“…where by (∆) we understand the density of U (1) c × U (1) c primary states. Provided this condition is satisfied, at leading order in 1/c the density of states is then given by the analog of the Cardy formula [16]…”