Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list

Abstract

We prove that a haploid associative algebra in a $C^{\ast }$-tensor category $\mathcal{C}$ is equivalent to a Q-system (a special $C^{\ast }$-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.

44 pages, revised version

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Cited by in corpus (10)

• Subfactors and Mathematical Physics
• Wightman fields for two-dimensional conformal field theories with pointed representation category
• Energy bounds for vertex operator algebra extensions
• From vertex operator superalgebras to graded-local conformal nets and back
• Geometric Positivity of the Fusion Products of Unitary Vertex Operator Algebra Modules
• Unitarity and strong graded locality of holomorphic vertex operator superalgebras with central charge at most 24
• Separable algebras in multitensor C${}^{\ast }$-categories are unitarizable
• Interpolation categories for Conformal Embeddings
• Conformal nets from minimal W-algebras
• On a Connes Fusion Approach to Finite Index Extensions of Conformal Nets