Holomorphic SCFTs with small index

<jats:title>Abstract</jats:title><jats:p>We observe that every self-dual ternary code determines a holomorphic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline1.png" /><jats:tex-math> $\mathcal N=1$ </jats:tex-math></jats:alternatives></jats:inline-formula> superconformal field theory. This provides ternary constructions of some well-known holomorphic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline2.png" /><jats:tex-math> $\mathcal N=1$ </jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline3.png" /><jats:tex-math> $24$ </jats:tex-math></jats:alternatives></jats:inline-formula>; we are able to prove this conjecture except in the case when the code has length <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline4.png" /><jats:tex-math> $12$ </jats:tex-math></jats:alternatives></jats:inline-formula> mod <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline5.png" /><jats:tex-math> $24$ </jats:tex-math></jats:alternatives></jats:inline-formula>. Lastly, we discuss a conjecture of Stolz and Teichner relating <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline6.png" /><jats:tex-math> $\mathcal N=1$ </jats:tex-math></jats:alternatives></jats:inline-formula> 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 <jats:italic>k</jats:italic>, a holomorphic <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline7.png" /><jats:tex-math> $\mathcal N=1$ </jats:tex-math></jats:alternatives></jats:inline-formula> SCFT of central charge <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline8.png" /><jats:tex-math> $12k$ </jats:tex-math></jats:alternatives></jats:inline-formula> and index <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline9.png" /><jats:tex-math> $24/\gcd (k,24)$ </jats:tex-math></jats:alternatives></jats:inline-formula>. We give ternary code constructions of SCFTs realizing this suggestion for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0008414X2100002X_inline10.png" /><jats:tex-math> $k\leq 5$ </jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>