The Conformal Packing Problem

TL;DR

This paper introduces the conformal packing problem, deriving explicit bounds for the minimal dual conformal weight of certain vertex operator algebras, with exact bounds for central charges 8 and 24, inspired by sphere packing solutions.

Contribution

It formulates the conformal packing problem and provides explicit numerical bounds for specific central charges, connecting to sphere packing methods.

Findings

Bounds of 1 and 2 for c=8 and c=24

Exact bounds achieved by known vertex operator algebras

Methods similar to sphere packing solutions

Abstract

We formulate the conformal packing problem and dual packing problem in analogy to similar problems for binary codes and lattices. We obtain explicit numerical upper bounds for the minimal dual conformal weight of a unitary strongly-rational vertex operator algebra for several central charges c and we also discuss asymptotic bounds. As a main result, we find the bounds 1 and 2 for the minimal dual conformal weight for the central charges c=8 and c=24, respectively. These bounds are reached by the vertex operator algebra associated to the affine Kac-Moody algebra E_8 at level 1 of central charge c=8 and the moonshine module of central charge c=24. The optimal bounds for these two central charges are obtained by methods similar to the one used by Viazovska and Cohn et al. in solving the sphere packing problem in dimension 8 and 24, respectively.