Cusp Form Dimensions, Lattice Uniqueness, and LP Sharpness for Sphere Packing in Dimensions 8 and 24

TL;DR

This paper explores why the Cohn-Elkies LP bound for sphere packing is sharp only in dimensions 8 and 24 by examining conditions from number theory, lattice theory, and conformal field theory.

Contribution

It identifies three necessary conditions for LP sharpness, linking them through number theory, modular forms, and quantum statistical systems, and proposes their equivalence in certain dimensions.

Findings

LP sharpness only in dimensions 8 and 24 explained by three conditions.

Conditions rule out LP sharpness in other dimensions like 16, 32, and above 48.

Connections established between LP bounds, modular forms, and quantum systems.

Abstract

The Cohn-Elkies linear programming (LP) bound for sphere packing is known to be sharp in dimensions 8 and 24 but in no other dimension above 2. We investigate why by examining three independent necessary conditions for LP sharpness, drawn from number theory, lattice theory, and conformal field theory. The first condition, dim S_{d/2}(SL_2(Z)) <= 1, bounds the freedom in theta series and rules out all d >= 48. The second, derived from Cohn and Triantafillou's dual LP obstruction via cusp forms for the congruence subgroup Gamma_0(2), explains why LP sharpness fails in dimensions 16 and 32 despite the first condition being satisfied. The third, via the Hartman-Mazac-Rastelli correspondence between LP bounds and the modular bootstrap for Narain conformal field theories, reinterprets LP sharpness as the existence of an extremal CFT. We formulate a conjecture that these three conditions are equivalent for d congruent to 0 mod 8, and observe that the Bost-Connes quantum statistical system provides a natural algebraic framework in which all three perspectives are connected through the Hecke algebra.