Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order n such that $\Gamma _{0}(n)$ is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6, and 8 to establish that each of the following fifteen Lie algebras is the weight-one space $V_{1}$ of exactly one holomorphic, $C_{2}$-cofinite vertex operator algebra V of CFT type and central charge 24: $A_{5}C_{5}E_{6,2}$, $A_{3}A_{7,2}{C_{3}^{2}}$, $A_{8,2}F_{4,2}$, $B_{8}E_{8,2}$, ${A_{2}^{2}}A_{5,2}^{2}B_{2}$, $C_{8}{F_{4}^{2}}$, $A_{4,2}^{2}C_{4,2}$, $A_{2,2}^{4}D_{4,4}$, $B_{5}E_{7,2}F_{4}$, $B_{4}{C_{6}^{2}}$, $A_{4,5}^{2}$, $A_{4}A_{9,2}B_{3}$, $B_{6}C_{10}$, $A_{1}C_{5,3}G_{2,2}$, and $A_{1,2}A_{3,4}^{3}$.

A simple sufficient condition for triviality of obstructions in the orbifold construction for subfactors

We present a simple sufficient condition for triviality of obstructions in the orbifold construction. As an application, we can show the existence of subfactors with principal graph $D_{2n}$ without full use of Ocneanu's paragroup theory.

Generalized orbifold construction for conformal nets

Let [Formula: see text] be a conformal net. We give the notion of a proper action of a finite hypergroup [Formula: see text] acting by vacuum preserving unital completely positive (so-called stochastic) maps on [Formula: see text] which generalizes the proper action of a finite group [Formula: see text]. Taking the fixed point under such an action gives a finite index subnet [Formula: see text] of [Formula: see text], which generalizes the [Formula: see text]-orbifold net. Conversely, we show that if [Formula: see text] is a finite inclusion of conformal nets, then [Formula: see text] is a generalized orbifold [Formula: see text] of the conformal net [Formula: see text] by a unique finite hypergroup [Formula: see text]. There is a Galois correspondence between intermediate nets [Formula: see text] and subhypergroups [Formula: see text] given by [Formula: see text]. In this case, the fixed point of [Formula: see text] is the generalized orbifold by the hypergroup of double cosets [Formula: see text]. If [Formula: see text] is a finite index inclusion of completely rational nets, we show that the inclusion [Formula: see text] is conjugate to an intermediate subfactor of a Longo–Rehren inclusion. This implies that if [Formula: see text] is a holomorphic net, and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] which is a categorification of [Formula: see text] and [Formula: see text] is braided equivalent to the Drinfel’d center [Formula: see text]. More generally, if [Formula: see text] is a completely rational conformal net and [Formula: see text] acts properly on [Formula: see text], then there is a unitary fusion category [Formula: see text] extending [Formula: see text], such that [Formula: see text] is given by the double cosets of the fusion ring of [Formula: see text] by the Verlinde fusion ring of [Formula: see text] and [Formula: see text] is braided equivalent to the Müger centralizer of [Formula: see text] in the Drinfel’d center [Formula: see text].