Orbifold Vertex Operator Research Articles

Fusion rules for the orbifold vertex operator algebras L𝔰𝔩2 ̂(k,0)K

For the Klein group [Formula: see text] and a positive integer [Formula: see text], irreducible modules of the orbifold vertex operator algebra [Formula: see text] have been classified and constructed in [C. Jiang and B. Wang, Representations of the orbifold VOAS [Formula: see text] and the commutant VOAS [Formula: see text], Bull. Malays. Math. Sci. Soc. 44(1) (2021) 353–373]. In this paper, we determine completely the fusion rules of [Formula: see text].

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

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

Representations and Fusion Rules for the Orbifold Vertex Operator Algebras Lsl2^(k,0)ℤ3

For the cyclic group [Formula: see text] and a positive integer [Formula: see text], we study the representations of the orbifold vertex operator algebra [Formula: see text]. All the irreducible modules for [Formula: see text] are classified and constructed explicitly. Quantum dimensions and fusion rules for [Formula: see text] are completely determined.

The large N limit of orbifold vertex operator algebras

We investigate the large N limit of permutation orbifolds of vertex operator algebras. To this end, we introduce the notion of nested oligomorphic permutation orbifolds and discuss under which conditions their fixed point VOAs converge. We show that if this limit exists, then it has the structure of a vertex algebra. Finally, we give an example based on mathrm {GL}(N,q) for which the fixed point VOA limit is also the limit of the full permutation orbifold VOA.

Representations and fusion rules for the orbifold vertex operator algebras

For the cyclic group (p is a prime) and a positive integer k, we study the representations of the orbifold vertex operator algebra . All the irreducible modules for are classified and constructed explicitly. Quantum dimensions and fusion rules for are completely determined.

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

Let $L$ be an even (positive definite) lattice and $g\in O(L)$. In this article, we prove that the orbifold vertex operator algebra $V_{L}^{\hat{g}}$ has group-like fusion if and only if $g$ acts trivially on the discriminant group $\mathcal{D}(L)=L^*/L$ (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 $\Lambda$, we prove a conjecture proposed by G. H\"ohn. In addition, we also discuss a construction of certain holomorphic vertex operator algebras of central charge $24$ using the the orbifold vertex operator algebra $V_{\Lambda_g}^{\hat{g}}$.

On the tensor structure of modules for compact orbifold vertex operator algebras

Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact group $G$ acting on a simple abelian intertwining algebra $V$. We show that if all irreducible $V^G$-modules contained in $V$ live in some braided tensor category of $V^G$-modules, then they generate a tensor subcategory equivalent to the category $\mathrm{Rep}\,G$ of finite-dimensional representations of $G$, with associativity and braiding isomorphisms modified by the abelian $3$-cocycle defining the abelian intertwining algebra structure on $V$. Additionally, we show that if the fusion rules for the irreducible $V^G$-modules contained in $V$ agree with the dimensions of spaces of intertwiners among $G$-modules, then the irreducibles contained in $V$ already generate a braided tensor category of $V^G$-modules. These results do not require rigidity on any tensor category of $V^G$-modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when $V^G$ is $C_2$-cofinite but not necessarily rational. When $V^G$ is both $C_2$-cofinite and rational and $V$ is a vertex operator algebra, we use the equivalence between $\mathrm{Rep}\,G$ and the corresponding subcategory of $V^G$-modules to show that $V$ is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge $1$ admits a braided tensor category structure equivalent to $\mathrm{Rep}\,SU(2)$, up to modification by an abelian $3$-cocycle.

Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

Abstract 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 generalized Kac–Moody algebra of rank 14

We construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12. The BRST-cohomology group of this vertex algebra is a new generalized Kac–Moody algebra of rank 14. We determine its root space multiplicities and a set of simple roots.

Representations of the vertex operator algebra [formula omitted

The rationality and C2-cofiniteness of the orbifold vertex operator algebra VL2A4 are established and all the irreducible modules are constructed and classified. This is part of classification of rational vertex operator algebras with c=1.

On Generalized Modular forms and their Applications

AbstractWe study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

Induced modules for orbifold vertex operator algebras

Let V be a simple vertex operator algebra and G< Aut V a finite abelian subgroup such that VG is rational. We study the representations of V based on certain assumptions on VG-modules. We prove a decomposition theorem for irreducible V-modules. We also define an induced module from VG to V and show that every irreducible V-module is a quotient module of some induced module. In addition, we prove that V is rational in this case.

Induced modules for orbifold vertex operator algebras

Let V be a simple vertex operator algebra and G< Aut V a finite abelian subgroup such that VG is rational. We study the representations of V based on certain assumptions on VG-modules. We prove a decomposition theorem for irreducible V-modules. We also define an induced module from VG to V and show that every irreducible V-module is a quotient module of some induced module. In addition, we prove that V is rational in this case.

Fusion Rules for the Free Bosonic Orbifold Vertex Operator Algebra

Fusion rules among irreducible modules for the free bosonic orbifold vertex operator algebra are completely determined.