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

TL;DR

The paper establishes a connection between haploid algebras and Q-systems in $C^*$-tensor categories, proving unitarity for certain vertex operator algebras and classifying specific superalgebra extensions.

Contribution

It demonstrates that haploid associative algebras are equivalent to Q-systems when rigid, proving unitarity of 70 holomorphic VOAs on the Schellekens list and classifying certain superalgebra extensions.

Findings

Proved unitarity of 70 holomorphic VOAs with $c=24$ and non-zero weight-one subspace.

Established the equivalence of haploid algebras and Q-systems under rigidity.

Classified simple CFT type vertex operator superalgebra extensions for $c<rac{3}{2}$ and $c<3$.

Abstract

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.