Construction and classification of holomorphic vertex operator algebras

Abstract

We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of V 1 -structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds. arXiv:1507.08142v3 [math.RT]

Orbifolds of lattice vertex operator algebras at d = 48 and d = 72

“…What we observe in this article is that in conjunction with [EG18], the cyclic methods of [vMS17] can be applied not just to cyclic orbifold groups, but to a much larger class of non-Abelian groups: We define G to be an effectively cyclic orbifold group if for every pair (a ∈ G, b ∈ C a ), we can find a cyclic subgroup C < G such that a, b ∈ C. For such a G we can compute all twisted twining characters by considering orbifolds by various cyclic groups C. A large class of effectively cyclic groups that we orbifold with are of the form…”

“…For the cases we consider however, it turns out that ω is determined by its restriction to various cyclic subgroups. To test for anomalies of these cyclic orbifolds, we instead use the approach of [vMS17], which we describe now.…”

“…Cyclic Orbifolds. For cyclic groups Z N this was worked out explicitly in [vMS17]. Let a be a generator of Z N .…”

“…For many cases, it is not necessary to go through this whole procedure. If G is cyclic, then one can of course simply use the general theory of cyclic orbifolds worked out in [vMS17], which simplifies things considerably: First of all, it is then unnecessary to construct the twisted modules and their twining characters, as it is enough to know the twining characters of the untwisted module, from which all other twining characters can be obtained by modular transformations [vMS17,GK19]. Second, the condition…”

On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras

“…For V with central charge divisible by 24, the characters of the irreducible V Gmodules ch W (i,j) (τ ) = tr W (i,j) q L(0)−c/24 are holomorphic on the upper half-plane and modular of weight 0 for Γ 0 (N) (Theorem 5.1 [29]).…”