Simple Vertex Operator Algebra Research Articles

Cohomological varieties associated to vertex operator algebras

Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras.

Classification of irreducible admissible modules for 𝒮-local vertex operator algebra V #H

For an [Formula: see text]-local vertex operator algebra [Formula: see text], where [Formula: see text] is a simple vertex operator algebra, and [Formula: see text] is the group algebra of a finite subgroup of [Formula: see text], Frobenius reciprocity is investigated. We give an explicit construction and classification of admissible [Formula: see text]-modules in terms of admissible [Formula: see text]-modules. We also give a complete set of irreducible inequivalent admissible [Formula: see text]-modules.

Energy bounds for vertex operator algebra extensions

Let V be a simple unitary vertex operator algebra and U be a (polynomially) energy-bounded unitary subalgebra containing the conformal vector of V. We give two sufficient conditions implying that V is energy-bounded. The first condition is that U is a compact orbifold V^G for some compact group G of unitary automorphisms of V. The second condition is that V is exponentially energy-bounded and it is a finite direct sum of simple U-modules. As a consequence of the second condition, we prove that if U is a regular energy-bounded unitary subalgebra of a simple unitary vertex operator V, then V is energy-bounded. In particular, every simple unitary extension (with the same conformal vector) of a simple unitary affine vertex operator algebra associated with a semisimple Lie algebra is energy-bounded.

Vertex operator algebras with positive central charges whose dimensions of weight one spaces are 8 and 16

We classify rational simple self-dual vertex operator algebras V of CFT and of finite type with positive central charges, where dV=dim⁡V1 is either 8 or 16, under the condition that the space spanned by characters of V-modules is equal to the solution space of some monic modular linear differential equation of the third order. It is shown that central charges are c=8 for dV=8, and c=4,16 for dim⁡dV=16, respectively. In fact, under the condition that V1 equipped with the 0th product of V are abelian Lie algebras for dV=8,c=8 and dV=16,c=16, the vertex operator algebras V are isomorphic to V2E8 and VΛ16, respectively. For dV=16 and c=4, V is isomorphic to the lattice vertex operator algebra VA2⊕A2.

Formula omitted] at level [formula omitted

Let Ll=L(sl2l+1,−l−12) be the simple vertex operator algebra based on the affine Lie algebra slˆ2l+1 at boundary admissible level −l−12.We consider a lift ν of the Dynkin diagram involution of A2l=sl2l+1 to an involution of Ll. The ν-twisted Ll-modules are A2l(2)-modules of level −l−12 with an anti-homogeneous realization. We classify simple ν-twisted highest-weight (weak) Ll-modules using twisted Zhu algebras and singular vectors for slˆ2l+1 at level −l−12 obtained by Perše.We find that there are finitely many such modules up to isomorphism, and the ν-twisted (weak) Ll-modules that are in category O for A2l(2) are semi-simple.

UNITARY REPRESENTATIONS OF THE mathcal{W} 3-ALGEBRA WITH c ≥ 2

We prove unitarity of the vacuum representation of the mathcal{W} 3-algebra for all values of the central charge c ≥ 2.We do it by modifying the free field realization of Fateev and Zamolodchikov resulting in a representation which, by a nontrivial argument, can be shown to be unitary on a certain invariant subspace, although it is not unitary on the full space of the two currents needed for the construction. These vacuum representations give rise to simple unitary vertex operator algebras. We also construct explicitly unitary representations for many positive lowest weight values. Taking into account the known form of the Kac determinants, we then completely clarify the question of unitarity of the irreducible lowest weight representations of the mathcal{W} 3-algebra in the 2 ≤ c ≤ 98 region.

Gluing vertex algebras

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed account of the canonical algebra construction in the Deligne product C⊠Crev. Especially, we show that if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C)X′⊠X is a commutative algebra, where X′ is a representing object for the functor HomC(•⊗CX,1C) (assuming X′ exists) and the sum runs over all inequivalent simple objects of U. Conversely, let A=⨁i∈IUi⊠Vi be a simple commutative algebra in a Deligne product U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. We show that if the unit objects 1U and 1V form a commuting pair in A in a suitable sense, then there is a braid-reversed equivalence between (sub)categories of U and V that sends Ui to Vi⁎.These results apply when U and V are braided (vertex) tensor categories of modules for simple vertex operator algebras U and V, respectively: Given τ:Irr(U)→Obj(V) such that τ(U)=V, we glue U and V along U⊠V via τ to create A=⨁X∈Irr(U)X′⊗τ(X). Then under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A has a simple conformal vertex algebra structure that (conformally) extends U⊗V. As examples, we glue suitable Kazhdan-Lusztig categories at generic levels to construct new vertex algebras extending the tensor product of two affine vertex subalgebras, and we prove braid-reversed equivalences between certain module subcategories for affine vertex algebras and W-algebras at admissible levels.

Tensor categories of affine Lie algebras beyond admissible levels

We show that if V is a vertex operator algebra such that all the irreducible ordinary V-modules are $$C_1$$ -cofinite and all the grading-restricted generalized Verma modules for V are of finite length, then the category of finite length generalized V-modules has a braided tensor category structure. By applying the general theorem to the simple affine vertex operator algebra (resp. superalgebra) associated to a finite simple Lie algebra (resp. Lie superalgebra) $$\mathfrak {g}$$ at level k and the category $$KL_k(\mathfrak {g})$$ of its finite length generalized modules, we discover several families of $$KL_k(\mathfrak {g})$$ at non-admissible levels k, having braided tensor category structures. In particular, $$KL_k(\mathfrak {g})$$ has a braided tensor category structure if the category of ordinary modules is semisimple or more generally if the category of ordinary modules is of finite length. We also prove the rigidity and determine the fusion rules of some categories $$KL_k(\mathfrak {g})$$ , including the category $$KL_{-1}(\mathfrak {sl}_n)$$ . Using these results, we construct a rigid tensor category structure on a full subcategory of $$KL_1(\mathfrak {sl}(n|m))$$ consisting of objects with semisimple Cartan subalgebra actions.

Formula omitted]-code vertex operator algebras

We study a simple, self-dual, rational, and C2-cofinite vertex operator algebra of CFT-type whose simple current modules are graded by Z2k. Based on those simple current modules, a vertex operator algebra associated with a Z2k-code is constructed. The classification of irreducible modules for such a vertex operator algebra is established. Furthermore, all the irreducible modules are realized in a module for a certain lattice vertex operator algebra.

Permutation orbifolds of 𝔰𝔩2 vertex operator algebras

We analyze two types of permutation orbifolds: (i) S2-orbifolds of the universal level k vertex operator algebra Vk(𝔰𝔩2) and of its simple quotient Lk(𝔰𝔩2), and (ii) the S3-orbifold of the level one simple vertex operator algebra L1(𝔰𝔩2). We determine their structures and discuss related W-algebras.

Fusion rules for ℤ2-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(sl2,k) associated to the integrable highest weight modules for the affine Kac—Moody algebra $$A_1^{(1)}$$ is the building block of the general parafermion vertex operator K( $$K(\mathfrak{g},k)$$ ,k) for any finite-dimensional simple Lie algebra $$\mathfrak{g}$$ and any positive integer k. We first classify the irreducible modules of ℤ2-orbifold of the simple affine vertex operator algebra of type $$A_1^{(1)}$$ and determine their fusion rules. Then we study the representations of the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k). The quantum dimensions, and more technically, fusion rules for the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k) are completely determined.

A duality theorem for vertex operator algebras

In this paper, we prove that (V#H)#H⁎≅M(n,V) as S-local vertex operator algebras, where V is a simple vertex operator algebra, and H is the group algebra of a finite subgroup of Aut(V), and n=dim⁡H.

Vertex Algebras Associated with Hypertoric Varieties

Abstract We construct a family of vertex algebras associated with a family of symplectic singularity/resolution, called hypertoric varieties. While the hypertoric varieties are constructed by a certain Hamiltonian reduction associated with a torus action, our vertex algebras are constructed by (semi-infinite) BRST reduction. The construction works algebro-geometrically, and we construct sheaves of $\hbar $-adic vertex algebras over hypertoric varieties, which localize the vertex algebras. We determine when it is a vertex operator algebra by giving an explicit conformal vector. We also discuss the Zhu algebra of the vertex algebra and its relation with a quantization of the hypertoric variety. In certain cases, we obtain the affine ${\mathcal{W}}$-algebra associated with the subregular nilpotent orbit in $\mathfrak{s}\mathfrak{l}_N$ at level $N-1$ and simple affine vertex operator algebra for $\mathfrak{s}\mathfrak{l}_N$ at level $-1$.

On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automorphism group. Furthermore, we prove that the full vertex algebra automorphism group of the moonshine module over the field of real numbers is the Monster.

Classification of irreducible weak modules over Virasoro vertex operator algebras

The classification of irreducible weak modules over the Virasoro vertex operator algebra [Formula: see text] is obtained in this paper. As one of the main results, we also classify all irreducible weak modules over the simple Virasoro vertex operator algebras [Formula: see text] for [Formula: see text] [Formula: see text], where [Formula: see text] are coprime positive integers.

Simple current extensions beyond semi-simplicity

Let [Formula: see text] be a simple vertex operator algebra (VOA) and consider a representation category of [Formula: see text] that is a vertex tensor category in the sense of Huang–Lepowsky. In particular, this category is a braided tensor category. Let [Formula: see text] be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that [Formula: see text] is either a VOA or a super VOA. If the representation category of [Formula: see text] is in addition ribbon, then the categorical dimension of [Formula: see text] decides this parity question. Combining with Carnahan’s work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are [Formula: see text]-cofinite and non-rational are then given and induced modules listed.

Vertex operator algebras generated by Ising vectors of $$\sigma $$ σ -type

We prove the uniqueness of the simple vertex operator algebra of OZ-type generated by Ising vectors of $$\sigma $$ -type. We also prove that the simplicity can be omitted if the Griess algebra is isomorphic to the Matsuo algebra associated with the root system of type $$A_n$$ .

SCHUR–WEYL DUALITY FOR HEISENBERG COSETS

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}\left( {H}, {V}\right)$ be the coset of ${H}$ in ${V}$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of ${C}$ are found and every simple ${C}$-module is shown to be contained in at least one ${V}$-module. A corollary of this is that if ${V}$ is rational and $C_2$-cofinite and CFT-type, and $\text{Com}\left( {C}, {V}\right)$ is a rational lattice vertex operator algebra, then so is ${C}$. These results are illustrated with many examples and the $C_1$-cofiniteness of certain interesting classes of modules is established.

From Vertex Operator Algebras to Conformal Nets and Back

We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A_V acting on the Hilbert space completion of V and prove that the isomorphism class of A_V does not depend on the choice of the scalar product on V. We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W\mapsto A_W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A_V. Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c=1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and J\"{o}rss gives back the strongly local vertex operator algebra V from the conformal net A_V and give conditions on a conformal net A implying that A= A_V for some strongly local vertex operator algebra V.

On orbifold theory

Let V be a simple vertex operator algebra and G a finite automorphism group of V such that VG is regular. It is proved that every irreducible VG-module occurs in an irreducible g-twisted V-module for some g∈G. Moreover, the quantum dimensions of irreducible VG-modules are determined and a global dimension formula for V in terms of twisted modules is obtained. In particular, the orbifold theory conjecture is completely solved if G is solvable.