# Haploid algebras in $C^*$-tensor categories and the Schellekens list

November 23, 2022

We prove that a haploid associative algebra in a $C^*$-tensor category
$\mathcal{C}$ is equivalent to a Q-system (a special $C^*$-Frobenius algebra)
in $\mathcal{C}$ if 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 $c=24$ and 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 $N=1$
and $N=2$ super-Virasoro vertex operator superalgebras with central charge
$c<\frac{3}{2}$ and $c<3$ respectively, relying on the known classification
results for the corresponding superconformal nets.

Keywords:

*none*