Vertex Tensor Category Research Articles

Tensor Categories for Vertex Operator Superalgebra Extensions

Let V V be a vertex operator algebra with a category C \mathcal {C} of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A A be a vertex operator (super)algebra extension of V V . We employ tensor categories to study untwisted (also called local) A A -modules in C \mathcal {C} , using results of Huang-Kirillov-Lepowsky that show that A A is a (super)algebra object in C \mathcal {C} and that generalized A A -modules in C \mathcal {C} correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C \mathcal {C} -algebra and (under suitable conditions) of generalized A A -modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V V -modules to A A -modules is a vertex tensor functor. Two applications are given: First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z \frac {1}{2}\mathbb {Z} -graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z \mathbb {Z} -graded components, respectively. Second, we analyze parafermionic cosets C = C o m ( V L , V ) C=\mathrm {Com}(V_L,V) where L L is a positive definite even lattice and V V is regular. If the vertex tensor category of either V V -modules or C C -modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

Kazama–Suzuki coset construction and its inverse

We study the representation theory of the Kazama–Suzuki coset vertex operator superalgebra associated to the pair of a complex simple Lie algebra and its Cartan subalgebra. In the case of type A1, B.L. Feigin, A.M. Semikhatov, and I.Yu. Tipunin introduced another coset construction, which is “inverse” of the Kazama–Suzuki coset construction. In this paper we generalize the latter coset construction to arbitrary type and establish a categorical equivalence between the categories of certain modules over an affine vertex operator algebra and the corresponding Kazama–Suzuki coset vertex operator superalgebra. Moreover, when the affine vertex operator algebra is regular, we prove that the corresponding Kazama–Suzuki coset vertex operator superalgebra is also regular and the category of its ordinary modules carries a braided monoidal category structure by the theory of vertex tensor categories.

Direct limit completions of vertex tensor categories

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of [Formula: see text] and the [Formula: see text] super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.

The Vertex Algebras $$\mathcal {R}^{(p)}$$ and $$\mathcal {V}^{({p})}$$

The vertex algebras $$\mathcal {V}^{(p)}$$ and $$\mathcal R^{(p)}$$ introduced in Adamović (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra $$\mathcal {V}^{(p)}$$ (respectively, $$\mathcal {R}^{(p)}$$ ) is a large extension of the simple affine vertex algebra $$L_{k}(\mathfrak {sl}_{2})$$ (respectively, $$L_{k}(\mathfrak {sl}_{2})$$ times a Heisenberg algebra), at level $$k=-2+1/p$$ for positive integer p. Particularly, the algebra $$\mathcal {V}^{(2)}$$ is the simple small $$N=4$$ superconformal vertex algebra with $$c=-9$$ , and $$\mathcal {R}^{(2)}$$ is $$L_{-3/2}(\mathfrak {sl}_3)$$ . In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $$\mathcal {V}^{({p})}$$ and we decompose $$\mathcal {V}^{({p})}$$ as an $$L_{k}(\mathfrak {sl}_{2})$$ -module and $$\mathcal {R}^{({p})}$$ as an $$L_k(\mathfrak {gl}_2)$$ -module. The decomposition of $$\mathcal {V}^{({p})}$$ shows that $$\mathcal {V}^{({p})}$$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $$\mathcal {V}^{({p})}$$ is the logarithmic doublet algebra $$\mathcal {A}^{({p})}$$ introduced in Adamović and Milas (Contemp Math 602:23–38, 2013), while the reduction of $$\mathcal {R}^{({p})}$$ yields the $$\mathcal {B}^{({p})}$$ -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize $$\mathcal {V}^{({p})}$$ and $$\mathcal {R}^{({p})}$$ from $$\mathcal {A}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $$KL_{k}$$ of ordinary $$L_{k}(\mathfrak {sl}_{2})$$ -modules at level $$k=-2+1/p$$ is a rigid vertex tensor category equivalent to a twist of the category $$\text {Rep}(SU(2))$$ . This finally completes rigid braided tensor category structures for $$L_{k}(\mathfrak {sl}_{2})$$ at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are certain non-principal $$\mathcal {W}$$ -algebras of type A at boundary admissible levels. The same uniqueness result also shows that $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are the chiral algebras of Argyres-Douglas theories of type $$(A_1, D_{2p})$$ and $$(A_1, A_{2p-3})$$ .

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.

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.

Representation theory of $L_k(\mathfrak {osp}(1 \vert 2))$ from vertex tensor categories and Jacobi forms

The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let $L_k\left (\mathfrak {osp}(1 | 2)\right )$ be the simple affine vertex operator superalgebra of $\mathfrak {osp}(1|2)$ at an admissible level $k$. We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of $L_k(\mathfrak {sl}_2)\otimes \text {Vir}(p, (p+p’)/2)$ where $k+3/2=p/(2p’)$ and $\text {Vir}(u, v)$ denotes the regular Virasoro vertex operator algebra of central charge $c=1-6(u-v)^2/(uv)$. Especially, for a positive integer $k$, we get a regular vertex operator superalgebra, and this case is studied further. The interplay of the theory of vertex algebra extensions and modular data of the vertex operator subalgebra allows us to classify all simple local (untwisted) and Ramond twisted $L_k\left (\mathfrak {osp}(1 | 2)\right )$-modules and to obtain their super fusion rules. The latter are obtained in a second way from Verlinde’s formula for vertex operator superalgebras. Finally, using again the theory of vertex algebra extensions, we find all simple modules and their fusion rules of the parafermionic coset $C_k = \text {Com}\left (V_L, L_k\left (\mathfrak {osp}(1 | 2)\right )\right )$, where $V_L$ is the lattice vertex operator algebra of the lattice $L=\sqrt {2k}\mathbb {Z}$.

Non-Negative Integral Level Affine Lie Algebra Tensor Categories and Their Associativity Isomorphisms

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\mathfrak{g},\ell)$, the category of finite-dimensional modules for the Zhu's algebra $A(L_{\hat{\mathfrak{g}}}(\ell,0))$, which is a quotient of $U(\mathfrak{g})$. Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in $\mathbf{D}(\mathfrak{g},\ell)$ induced from the associativity isomorphisms constructed by Huang and Lepowsky in $L_{hat{\mathfrak{g}}}(\ell,0)-\mathbf{mod}$. This construction shows that $\mathbf{D}(\mathfrak{g},\ell)$ is closely related to the Drinfeld category of $U(\mathfrak{g})[[\hbar]]$-modules used by Kazhdan and Lusztig to identify categories of $\hat{\mathfrak{g}}$-modules at irrational and most negative rational levels with categories of quantum group modules.

Braided Tensor Categories and Extensions of Vertex Operator Algebras

Let $V$ be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of $V$ and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of $V$-modules are equivalent.

Tree algebras: An algebraic axiomatization of intertwining vertex operators

We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a tree algebra. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\mathbb{C}$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\mathbb{Q}$.

DIFFERENTIAL EQUATIONS AND INTERTWINING OPERATORS

We show that if every module W for a vertex operator algebra V = ∐n∈ℤV(n) satisfies the condition dim W/C1(W)<∞, where C1(W) is the subspace of W spanned by elements of the form u-1w for u ∈ V+ = ∐n>0 V(n) and w ∈ W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a result used to establish the existence of such systems. Using these systems of differential equations and some additional reductivity conditions, we prove that products of intertwining operators for V satisfy the convergence and extension property needed in the tensor product theory for V-modules. Consequently, when a vertex operator algebra V satisfies all the conditions mentioned above, we obtain a natural structure of vertex tensor category (consequently braided tensor category) on the category of V-modules and a natural structure of intertwining operator algebra on the direct sum of all (inequivalent) irreducible V-modules.

INTERTWINING OPERATOR SUPERALGEBRAS AND VERTEX TENSOR CATEGORIES FOR SUPERCONFORMAL ALGEBRAS, I

We apply the general theory of tensor products of modules for a vertex operator algebra (developed by Lepowsky and the first author) and the general theory of intertwining operator algebras (developed by the first author) to the case of the N=1 superconformal minimal models and related models in superconformal field theory. We show that for the category of modules for a vertex operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex operator superalgebras associated to the N =1 Neveu–Schwarz Lie superalgebra, the intertwining operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of a braided tensor category structure on the category of finite direct sums of minimal modules of central charge [Formula: see text] for the N = 1 Neveu–Schwarz Lie superalgebra for any fixed integers p, q larger than 1 such that p - q ∈ 2ℤ and (p - q)/2 and q relatively prime to each other.

Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II

We construct the intertwining operator superalgebras and vertex tensor categories for the N = 2 N=2 superconformal unitary minimal models and other related models.

Intertwining operator algebras and vertex tensor categories for affine Lie algebras

We apply the general theory of tensor products of modules for a vertex operator algebra developed in our papers hep-th/9309076, hep-th/9309159, hep-th/9401119, q-alg/9505018, q-alg/9505019 and q-alg/9505020 to the case of the Wess-Zumino-Novikov-Witten models and related models in conformal field theory. We show that for the category of modules for a vertex operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex operator algebras associated to affine Lie algebras, the intertwining operators among the modules have the associativity property, the category has a natural structure of vertex tensor category, and a number of related results hold. We obtain, as a corollary and special case, a construction of the previously-studied braided tensor category structure on the category of finite direct sums of standard (integrable highest weight) modules of a fixed positive integral level for an affine Lie algebra.

A theory of tensor products for module categories for a vertex operator algebra, III

This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a “vertex tensor category” structure on the category of modules for a suitable vertex operator algebra. The notion of vertex tensor category is essentially a “complex analogue” of the notion of symmetric tensor category, and in fact a vertex tensor category produces a braided tensor category in a natural way. In this paper, we focus on a particular element P( z) of a certain moduli space of three-punctured Riemann spheres; in general, every element of this moduli space will give rise to a notion of tensor product, and one must consider all these notions in order to construct a vertex tensor category. Here we present the fundamental properties of the P( z)-tensor product of two modules for a vertex operator algebra. We give two constructions of a P( z)-tensor product, using the results, established in Parts I and II of this series, for a certain other element of the moduli space. The definitions and results in Parts I and II are recalled.