Holomorphic SCFTs with small index

Gaiotto, Davide ; Johnson-Freyd, Theo

Canadian journal of mathematics, Tome 74 (2022) no. 2, pp. 573-601

Résumé

We observe that every self-dual ternary code determines a holomorphic $\mathcal N=1$ superconformal field theory. This provides ternary constructions of some well-known holomorphic $\mathcal N=1$ superconformal field theories (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 $\mathcal N=1$ 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 $\mathcal N=1$ SCFT of central charge $12k$ and index $24/\gcd (k,24)$. We give ternary code constructions of SCFTs realizing this suggestion for $k\leq 5$.

Détail

DOI : 10.4153/S0008414X2100002X

Mots-clés : Supersymmetry, vertex operator algebras, linear codes, moonshine, TMF