Haploid Algebras in $$C^*$$-Tensor Categories and the Schellekens List

Abstract

AbstractWe prove that a haploid associative algebra in a$$C^*$$C∗-tensor category$$\mathcal {C}$$Cis equivalent to a Q-system (a special$$C^*$$C∗-Frobenius algebra) in$$\mathcal {C}$$Cif and only if it is rigid. This allows us to prove the unitarity of all the 70 strongly rational holomorphic vertex operator algebras with central charge$$c=24$$c=24and non-zero weight-one subspace, corresponding to entries 1–70 of the so called Schellekens list. Furthermore, using the recent generalized deep hole construction of these vertex operator algebras, we prove that they are also strongly local in the sense of Carpi, Kawahigashi, Longo and Weiner and consequently we obtain some new holomorphic conformal nets associated to the entries of the list. Finally, we completely classify the simple CFT type vertex operator superalgebra extensions of the unitary$$N=1$$N=1and$$N=2$$N=2super-Virasoro vertex operator superalgebras with central charge$$c<\frac{3}{2}$$c<32and$$c<3$$c<3respectively, relying on the known classification results for the corresponding superconformal nets.