Cyclic orbifolds of lattice vertex operator algebras having group-like fusions

Abstract

Let L be an even (positive definite) lattice and g ∈ O(L). In this article, we prove that the orbifold vertex operator algebra Vĝ L has group-like fusion if and only if g acts trivially on the discriminant group D(L) = L * /L (or equivalently (1 − g)L * < L).We also determine their fusion rings and the corresponding quadratic space structures when g is fixed point free on L. By applying our method to some coinvariant sublattices of the Leech lattice Λ, we prove a conjecture proposed by G. Höhn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge 24 using the the orbifold vertex operator algebra Vĝ Λg .where wt(·) denotes the conformal weight of the module. Moreover, R(V Lg ) is isomorphic to D(L g ) = L * g /L g as a quadratic space. By some recent results on coset constructions [CKM1, Main Theorem 2] (see also[KMi15] and [Lin17]), it is known that the VOA W also has group-like fusion and R(W ) forms a quadratic space isomorphic to (R(V Lg ), −q), where the quadratic form is defined by conformal weights modulo Z. Since V Lg ⊗ W is a full subVOA of V and V is holomorphic, the VOA V defines a maximal totally singular subspace of R(V Lg ) × R(W ); hence it induces an anti-isomorphism of quadratic spaces ϕ :2 Conversely, let ϕ : (R(V Lg ), q) → (R(W ), −q ′ ) be an isomorphism of quadratic spaces. Then the set {(M, ϕ(M)) | M ∈ R(V Lg )} is a maximal totally singular subspace of R(V Lg ) × R(W ) and hence U = M ∈R(V Lg ) M ⊗ ϕ(M) has a structure of a holomorphic VOA.

Automorphism groups of cyclic orbifold vertex operator algebras associated with the Leech lattice and some non-prime isometries

“…We assume that the conjugacy class of g is 4C, 6E, 6G, 8E or 10F . Then g = id on Λ * g /Λ g and by [La20a], all irreducible V ĝ Λgmodules are simple current modules. It implies that the set Irr(V ĝ L ) of all (isomorphism classes of) irreducible V ĝ L -modules forms a (finite) abelian group under the fusion product.…”

“…Later, we will specify the labeling by using their conformal weights and the fusion products as in [La20a]. By Remark 3.5, we obtain the following lemma.…”

“…In this subsection, we deal with the irreducible V ĝ L -modules which appear as V ĝ L -submodules of irreducible ĝi -twisted V L -modules for some 1 ≤ i ≤ n − 1. We adopt the notation in [La20a] and use the descriptions of the irreducible ĝi -twisted V L -modules in [ALY18, Sections 3.2 and 3.3] (see also [Le85,DL96,BK04]). Let 1 ≤ i ≤ n − 1 and set f = g i .…”

“…Transitivity for the cases 6G and 10F . In this subsection, we assume that g ∈ We now recall the quadratic space structure of Irr(V ĝ Λg ) from [La20a]. In order to do it, we have to describe certain vectors u and h in Λ * g ∩ (1/2)Λ g as in [La20a, Section 5.2.2]; in the following, we recall how to choose u and h. By [ATLAS] Let u be a minimal norm vector in (1/2)Λ g p \ E. Then (u|u) = 3/2.…”

Fourvolutions and automorphism groups of orbifold lattice vertex operator algebras

“…For L = BW 16 , we have (1 − g)L * = L for any fourvolution g. By [La20], all irreducible modules of V ĝ L are simple current modules. Moreover, the set of inequivalent irreducible modules Irr(V ĝ BW 16 ) of V ĝ BW 16 forms an abelian group under the fusion rules and it has a quadratic form q defined by conformal weights modulo Z.…”