Holomorphic SCFTs with small index

Abstract

We observe that every self-dual ternary code determines a holomorphic N=1 superconformal field theory. This provides ternary constructions of some well-known holomorphic N=1 SCFTs, including Duncan's "supermoonshine" model and the fermionic "beauty and the beast" model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by 24; we are able to prove this conjecture except in the case when the code has length 12 mod 24. Lastly, we discuss a conjecture of Stolz and Teichner relating N=1 SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic N=1 SCFT of central charge 12k and index 24/gcd(k,24). We give ternary code constructions of SCFTs realizing this suggestion for k \leq 5.