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.

    Source link


    Leave A Reply