Construction of holomorphic SCFTs with minimal index

Abstract

It has been an open question whether there exist holomorphic VOAs with N=1 supersymmetry and Witten index 1 for any positive central charge c. The existence of such a theory with c=288 is closely related to the 576-periodicity of the TMF spectrum via the Segal-Stolz-Teichner conjecture. Here we report the construction of thousands of such theories by extending the approach initiated by Gaiotto and Johnson-Freyd via ternary codes. In addition, for every positive integer k, we also construct holomorphic N=1 VOAs of central charge c=12k with index 24/gcd(24,k) -- conjectured to be the minimum -- in a computer-aided search carried out by AI coding agents. We overcome the exponential-time barrier by using chains of smaller cells to build ternary codes, whose indices can then be evaluated via a transfer-matrix technique, and we use a symmetric orbifold construction to reduce the problem of k=12 and 24 to that of k=6. Along the way, we also establish several results on ternary codes and their indices. Using an observation relating codes and Landau-Ginzburg orbifold models, we argue that the non-chiral doubles of the ternary code VOAs can always be gapped supersymmetrically.