arXiv subjects

Robert McRae

Publications and source records attributed to Robert McRae.

At least 19 recordsLinked to original sources

arXiv2026

Cocompletions for non-abelian vertex tensor categories

It was recently shown by Huang that the category of $C_1$-cofinite modules for any vertex operator algebra $V$ admits a natural braided monoidal structure. Here, we show that this structure extends uniquely to a vertex algebraically natural braided monoidal structure on the completion of the category of $C_1$-cofinite $V$-modules under filtered colimits, within the ambient category of all generalized $V$-modules. A critical point here is that we do not assume the category of $C_1$-cofinite $V$-modules is abelian or that $C_1$-cofinite modules are compact in the cocompletion, since these properties are not known to hold in general. Our results have many applications in the representation theory of vertex operator algebra extensions, since many vertex operator algebras can be realized as objects in the filtered colimit completion of the category of $C_1$-cofinite modules for a vertex operator subalgebra. Generalizing from the specific vertex algebraic setting, we also establish existence and uniqueness for extensions of monoidal structures along a dense inclusion $\mathscr{C}_0 \to \mathscr{C}$ from an abstract, essentially small monoidal category into a well-structured cocomplete target.

math.QA↗

arXiv2025

A Tensor Category Construction of the $W_{p,q}$ Triplet Vertex Operator Algebra and Applications

For coprime $p,q\in\mathbb{Z}_{\geq 2}$, the triplet vertex operator algebra $W_{p,q}$ is a non-simple extension of the universal Virasoro vertex operator algebra of central charge $c_{p,q}=1-\frac{6(p-q)^2}{pq}$, and it is a basic example of a vertex operator algebra appearing in logarithmic conformal field theory. Here, we give a new construction of $W_{p,q}$ different from the original screening operator definition of Feigin-Gainutdinov-Semikhatov-Tipunin. Using our earlier work on the tensor category structure of modules for the Virasoro algebra at central charge $c_{p,q}$, we show that the simple modules appearing in the decomposition of $W_{p,q}$ as a module for the Virasoro algebra have $\mathrm{PSL}_2$-fusion rules and generate a symmetric tensor category equivalent to $\operatorname{Rep}\mathrm{PSL}_2$. Then we use the theory of commutative algebras in braided tensor categories to construct $W_{p,q}$ as an appropriate non-simple modification of the canonical algebra in the Deligne tensor product of $\operatorname{Rep}\mathrm{PSL}_2$ with this Virasoro subcategory. As a consequence, we show that the automorphism group of $W_{p,q}$ is $\mathrm{PSL}_2(\mathbb{C})$. We also define a braided tensor category $\mathcal{O}_{c_{p,q}}^0$ consisting of modules for the Virasoro algebra at central charge $c_{p,q}$ that induce to untwisted modules of $W_{p,q}$. We show that $\mathcal{O}_{c_{p,q}}^0$ tensor embeds into the $\mathrm{PSL}_2(\mathbb{C})$-equivariantization of the category of $W_{p,q}$-modules and is closed under contragredient modules. We conjecture that $\mathcal{O}_{c_{p,q}}^0$ has enough projective objects and is the correct category of Virasoro modules for constructing logarithmic minimal models in conformal field theory.

math.QA↗

arXiv2025

The non-semisimple Kazhdan-Lusztig category for affine $\mathfrak{sl}_2$ at admissible levels

We show that Kazhdan and Lusztig's category $KL^k(\mathfrak{sl}_2)$ of modules for the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ at an admissible level $k$, equivalently the category of finite-length grading-restricted generalized modules for the universal affine vertex operator algebra $V^k(\mathfrak{sl}_2)$, is a braided tensor category. Although this tensor category is not rigid, we show that the subcategory of all rigid objects in $KL^k(\mathfrak{sl}_2)$ is equal to the subcategory of all projective objects, and that every simple module in $KL^k(\mathfrak{sl}_2)$ has a projective cover. Moreover, we show that the full subcategory of projective objects in $KL^k(\mathfrak{sl}_2)$ is monoidal equivalent to the category of tilting modules for quantum $\mathfrak{sl}_2$ at the root of unity $ζ=e^{πi/(k+2)}$. Using this, we establish a universal property of the tensor category $KL^k(\mathfrak{sl}_2)$, and as an application, we prove a weak Kazhdan-Lusztig correspondence, that is, we obtain an exact essentially surjective (but not full or faithful) tensor functor from $KL^k(\mathfrak{sl}_2)$ to the category of finite dimensional weight modules for the quantum group associated to $\mathfrak{sl}_2$ at the root of unity $ζ$. We also use the universal property to classify the categories $KL^k(\mathfrak{sl}_2)$ up to (braided) tensor equivalence and to obtain a tensor-categorical version of quantum Drinfeld-Sokolov reduction, that is, we construct a braided tensor functor from $KL^k(\mathfrak{sl}_2)$ to a category of modules for the Virasoro algebra at central charge $1-\frac{6(k+1)^2}{k+2}$.

math.QA↗

arXiv2024

$N=1$ super Virasoro tensor categories

We show that the category of $C_1$-cofinite modules for the universal $N=1$ super Virasoro vertex operator superalgebra $\mathcal{S}(c,0)$ at any central charge $c$ is locally finite and admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. For central charges $c^{\mathfrak{ns}}(t)=\frac{15}{2}-3(t+t^{-1})$ with $t\notin\mathbb{Q}$, we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge $c^{\mathfrak{ns}}(1)=\frac{3}{2}$, we show that this tensor category is rigid and that its simple modules have the same fusion rules as $\mathrm{Rep}\,\mathfrak{osp}(1\vert 2)$, in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges $c^{\mathfrak{ns}}(t)$ with $t\in \mathbb{Q}^\times$, we show that the simple $\mathcal{S}(c^{\mathfrak{ns}}(t),0)$-module $\mathcal{S}_{2,2}$ of lowest conformal weight $h^{\mathfrak{ns}}_{2,2}(t)=\frac{3(t-1)^2}{8t}$ is rigid and self-dual, except possibly when $t^{\pm 1}$ is a negative integer or when $c^{\mathfrak{ns}}(t)$ is the central charge of a rational $N=1$ superconformal minimal model. As $\mathcal{S}_{2,2}$ is expected to generate the category of $C_1$-cofinite $\mathcal{S}(c^{\mathfrak{ns}}(t),0)$-modules under fusion, rigidity of $\mathcal{S}_{2,2}$ is the first key step to proving rigidity of this category for general $t\in\mathbb{Q}^\times$.

math.QA↗

arXiv2024

Ribbon categories of weight modules for affine $\mathfrak{sl}_2$ at admissible levels

We show that the braided tensor category of finitely-generated weight modules for the simple affine vertex operator algebra $L_k(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ at any admissible level $k$ is rigid and hence a braided ribbon category. The proof uses a recent result of the first two authors with Shimizu and Yadav on embedding a braided Grothendieck-Verdier category $\mathcal{C}$ into the Drinfeld center of the category of modules for a suitable commutative algebra $A$ in $\mathcal{C}$, in situations where the braided tensor category of local $A$-modules is rigid. Here, the commutative algebra $A$ is Adamovi\'{c}'s inverse quantum Hamiltonian reduction of $L_k(\mathfrak{sl}_2)$, which is the simple rational Virasoro vertex operator algebra at central charge $1-\frac{6(k+1)^2}{k+2}$ tensored with a half-lattice conformal vertex algebra. As a corollary, we also show that the category of finitely-generated weight modules for the $N = 2$ super Virasoro vertex operator superalgebra at central charge $-6\ell-3$ is rigid for $\ell$ such that $(\ell+1)(k+2) = 1$.

math.QA↗

arXiv2024

Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

Let $A$ be a commutative algebra in a braided monoidal category $\mathcal{C}$; e.g., $A$ could be an extension of a vertex operator algebra (VOA) $V$ in a category $\mathcal{C}$ of $V$-modules. We study when the category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ and its subcategory $\mathcal{C}_A^{\text{loc}}$ of local modules inherit rigidity from $\mathcal{C}$, and then we find conditions for $\mathcal{C}$ and $\mathcal{C}_A$ to inherit rigidity from $\mathcal{C}_A^{\text{loc}}$. First, we assume $\mathcal{C}$ is a braided finite tensor category and prove rigidity of $\mathcal{C}_A$ and $\mathcal{C}_A^{\text{loc}}$ under conditions based on criteria of Etingof-Ostrik for $A$ to be an exact algebra in $\mathcal{C}$. As a corollary, we show that if $A$ is a simple $\mathbb{Z}_{\geq 0}$-graded VOA with a strongly rational vertex operator subalgebra $V$, then $A$ is strongly rational, without requiring the categorical dimension of $A$ as a $V$-module to be non-zero. Next, we assume $\mathcal{C}$ is a Grothendieck-Verdier category, i.e., $\mathcal{C}$ admits a weaker duality structure than rigidity. We first prove $\mathcal{C}_A$ is also a Grothendieck-Verdier category. Using this, we prove that if $\mathcal{C}_A^{\text{loc}}$ is rigid, then so is $\mathcal{C}$ under conditions such as a mild non-degeneracy assumption on $\mathcal{C}$, an assumption that every simple object of $\mathcal{C}_A$ is local, and that induction from $\mathcal{C}$ to $\mathcal{C}_A$ commutes with duality. These conditions are motivated by free field-like VOA extensions $V\subseteq A$ where $A$ is often an indecomposable $V$-module, so our result will make it more feasible to prove rigidity for many vertex algebraic monoidal categories. In a follow-up work, our result will be used to prove rigidity of the category of weight modules for the simple affine VOA of $\mathfrak{sl}_2$ at any admissible level.

math.QA↗

arXiv2024

Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge

Let $\mathcal{O}_c$ be the category of finite-length modules for the Virasoro Lie algebra at central charge $c$ whose composition factors are irreducible quotients of reducible Verma modules. For any $c\in\mathbb{C}$, this category admits the vertex algebraic braided tensor category structure of Huang, Lepowsky, and Zhang. Here, we begin the detailed study of $\mathcal{O}_{c_{p,q}}$ where $c_{p,q} = 1-\frac{6(p-q)^2}{pq}$ for relatively prime integers $p, q \geq 2$; in conformal field theory, $\mathcal{O}_{c_{p,q}}$ corresponds to a logarithmic extension of the central charge $c_{p,q}$ Virasoro minimal model. We particularly focus on the Virasoro Kac modules $\mathcal{K}_{r,s}$, $r,s\in\mathbb{Z}_{\geq 1}$, in $\mathcal{O}_{c_{p,q}}$ defined by Morin-Duchesne, Rasmussen, and Ridout, which are finitely-generated submodules of Feigin-Fuchs modules for the Virasoro algebra. We prove that $\mathcal{K}_{r,s}$ is rigid and self-dual when $1\leq r\leq p$ and $1\leq s\leq q$, but that not all $\mathcal{K}_{r,s}$ are rigid when $r>p$ or $s>q$. That is, $\mathcal{O}_{c_{p,q}}$ is not a rigid tensor category. We also show that all Kac modules and all simple modules in $\mathcal{O}_{c_{p,q}}$ are homomorphic images of repeated tensor products of $\mathcal{K}_{1,2}$ and $\mathcal{K}_{2,1}$, and we determine completely how $\mathcal{K}_{1,2}$ and $\mathcal{K}_{2,1}$ tensor with Kac modules and simple modules in $\mathcal{O}_{c_{p,q}}$. In the process, we prove some fusion rule conjectures of Morin-Duchesne, Rasmussen, and Ridout.

math.QA↗

arXiv2023

A general mirror equivalence theorem for coset vertex operator algebras

We prove a general mirror duality theorem for a subalgebra $U$ of a simple conformal vertex algebra $A$ and its commutant $V=\mathrm{Com}_A(U)$. Specifically, we assume that $A\cong\bigoplus_{i\in I} U_i\otimes V_i$ as a $U\otimes V$-module, where the $U$-modules $U_i$ are simple and distinct and are objects of a semisimple braided ribbon category of $U$-modules, and the $V$-modules $V_i$ are semisimple and contained in a (not necessarily rigid) braided tensor category of $V$-modules. We also assume $U=\mathrm{Com}_A(V)$. Under these conditions, we construct a braid-reversed tensor equivalence $τ: \mathcal{U}_A\rightarrow\mathcal{V}_A$, where $\mathcal{U}_A$ is the semisimple category of $U$-modules with simple objects $U_i$, $i\in I$, and $\mathcal{V}_A$ is the category of $V$-modules whose objects are finite direct sums of the $V_i$. In particular, the $V$-modules $V_i$ are simple and distinct, and $\mathcal{V}_A$ is a rigid tensor category. As an application, we find a rigid semisimple tensor subcategory of modules for the Virasoro algebra at central charge $13+6p+6p^{-1}$, $p\in\mathbb{Z}_{\geq 2}$, which is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. Consequently, the Virasoro vertex operator algebra at central charge $13+6p+6p^{-1}$ is the $PSL_2(\mathbb{C})$-fixed-point subalgebra of a simple conformal vertex algebra $\mathcal{W}(-p)$, analogous to the realization of the Virasoro vertex operator algebra at central charge $13-6p-6p^{-1}$ as the $PSL_2(\mathbb{C})$-fixed-point subalgebra of the triplet algebra $\mathcal{W}(p)$.

math.QA↗

arXiv2023

Deligne tensor products of categories of modules for vertex operator algebras

We show that if $\mathcal{U}$ and $\mathcal{V}$ are locally finite abelian categories of modules for vertex operator algebras $U$ and $V$, respectively, then the Deligne tensor product of $\mathcal{U}$ and $\mathcal{V}$ can be realized as a certain category $\mathcal{D}(\mathcal{U},\mathcal{V})$ of modules for the tensor product vertex operator algebra $U\otimes V$. We also show that if $\mathcal{U}$ and $\mathcal{V}$ admit the braided tensor category structure of Huang-Lepowsky-Zhang, then $\mathcal{D}(\mathcal{U},\mathcal{V})$ does as well under mild additional conditions, and that this braided tensor structure is equivalent to the natural braided tensor structure on a Deligne tensor product category. These results hold in particular when $\mathcal{U}$ and $\mathcal{V}$ are the categories of $C_1$-cofinite $U$- and $V$-modules, if these categories are closed under contragredients, in which case we show that $\mathcal{D}(\mathcal{U},\mathcal{V})$ is the category of $C_1$-cofinite $U\otimes V$-modules. If $U$ and $V$ are $\mathbb{N}$-graded and $C_2$-cofinite, then we may take $\mathcal{U}$ and $\mathcal{V}$ to be the categories of all grading-restricted generalized $U$- and $V$-modules, respectively. Thus as an application, if the tensor categories of all modules for two $C_2$-cofinite vertex operator algebras are rigid, then so is the tensor category of all modules for the tensor product vertex operator algebra. We use this to prove that the representation categories of the even subalgebras of the symplectic fermion vertex operator superalgebras are non-semisimple modular tensor categories.

math.QA↗

arXiv2023

An $\mathfrak{sl}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

math.QA↗

arXiv2022

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We show that the category of finite-length generalized modules for the singlet vertex algebra $\mathcal{M}(p)$, $p\in\mathbb{Z}_{>1}$, is equal to the category $\mathcal{O}_{\mathcal{M}(p)}$ of $C_1$-cofinite $\mathcal{M}(p)$-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since $\mathcal{O}_{\mathcal{M}(p)}$ includes the uncountably many typical $\mathcal{M}(p)$-modules, which are simple $\mathcal{M}(p)$-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical $\mathcal{M}(p)$-modules. We also introduce a tensor subcategory $\mathcal{O}_{\mathcal{M}(p)}^T$, graded by an algebraic torus $T$, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of $\mathfrak{sl}_2$ at a $2p$th root of unity. We compute all tensor products involving simple and projective $\mathcal{M}(p)$-modules, and we prove that both tensor categories $\mathcal{O}_{\mathcal{M}(p)}$ and $\mathcal{O}_{\mathcal{M}(p)}^T$ are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ are non-semisimple modular tensor categories, and we confirm a conjecture of Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

math.QA↗

arXiv2022

Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$

We establish rigid tensor category structure on finitely-generated weight modules for the subregular $W$-algebras of $\mathfrak{sl}_n$ at levels $ - n + \frac{n}{n+1}$ (the $\mathcal{B}_{n+1}$-algebras of Creutzig-Ridout-Wood) and at levels $- n + \frac{n+1}{n}$ (the finite cyclic orbifolds of the $βγ$-vertex algebra), as well as for their Feigin-Semikhatov dual principal $W$-superalgebras of $\mathfrak{sl}_{n|1}$. These categories are neither finite nor semisimple, and in the $W$-algebra case they contain modules with infinite-dimensional conformal weight spaces and no lower bound on conformal weights. We give complete lists of indecomposable projective modules in these tensor categories and fusion rules for simple modules. All these vertex operator (super)algebras are simple current extensions of singlet algebras tensored with a rank-one Heisenberg algebra, so we more generally study simple current extensions in direct limit completions of vertex algebraic tensor categories. Then our results for $W$-(super)algebras follow from the known ribbon category structure on modules for the singlet algebras. Our results include and generalize those of Allen-Wood on the $βγ$-vertex algebra, as well as our own on the affine vertex superalgebra of $\mathfrak{gl}_{1|1}$. Our results also include the first examples of ribbon category structure on all finitely-generated weight modules for an affine vertex algebra at a non-integral admissible level, namely for affine $\mathfrak{sl}_2$ at levels $-\frac{4}{3}$ and $-\frac{1}{2}$.

math.QA↗

arXiv2022

On semisimplicity of module categories for finite non-zero index vertex operator subalgebras

Let $V\subseteq A$ be a conformal inclusion of vertex operator algebras and let $\mathcal{C}$ be a category of grading-restricted generalized $V$-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which $\mathcal{C}$ inherits semisimplicity from the category of grading-restricted generalized $A$-modules in $\mathcal{C}$, and vice versa. The most important condition is that $A$ be a rigid $V$-module in $\mathcal{C}$ with non-zero categorical dimension, that is, we assume the index of $V$ as a subalgebra of $A$ is finite and non-zero. As a consequence, we show that if $A$ is strongly rational, then $V$ is also strongly rational under the following conditions: $A$ contains $V$ as a $V$-module direct summand, $V$ is $C_2$-cofinite with a rigid tensor category of modules, and $A$ has non-zero categorical dimension as a $V$-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that $A$ is a vertex operator superalgebra.

math.QA↗

arXiv2022

Gluing vertex algebras

We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for $\mathcal{C}$ a braided tensor category, we give a detailed construction of the canonical algebra in $\mathcal{C}\boxtimes\mathcal{C}^\text{rev}$: if $\mathcal{C}$ is semisimple but not necessarily finite or rigid, then $\bigoplus_{X\in\text{Irr}(\mathcal{C})}X'\boxtimes X$ is a commutative algebra, with $X'$ a representing object for $\text{Hom}_\mathcal{C}(\bullet\otimes_\mathcal{C}X,\mathbf{1}_{\mathcal{C}})$. Conversely, let $A=\bigoplus_{i\in I}U_i\boxtimes V_i$ be a simple commutative algebra in $\mathcal{U}\boxtimes\mathcal{V}$ with $\mathcal{U}$ semisimple and rigid but not necessarily finite, and $\mathcal{V}$ rigid but not necessarily semisimple. If the unit objects of $\mathcal{U}$ and $\mathcal{V}$ form a commuting pair in $A$, we show there is a braid-reversed equivalence between subcategories of $\mathcal{U}$ and $\mathcal{V}$ sending $U_i$ to $V_i^*$. When $\mathcal{U}$ and $\mathcal{V}$ are module categories for simple vertex operator algebras $U$ and $V$, we glue $U$ and $V$ along $\mathcal{U}\boxtimes\mathcal{V}$ via a map $τ:\text{Irr}(\mathcal{U})\rightarrow\text{Obj}(\mathcal{V})$ such that $τ(U)=V$ to create $A=\bigoplus_{X\in\text{Irr}(\mathcal{U})}X'\otimesτ(X)$. Thus under certain conditions, $τ$ extends to a braid-reversed equivalence between $\mathcal{U}$ and $\mathcal{V}$ if and only if $A$ is a simple conformal vertex algebra extending $U\otimes V$. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and $W$-algebras at admissible levels.

math.QA↗

arXiv2021

On rationality for $C_2$-cofinite vertex operator algebras

Let $V$ be an $\mathbb{N}$-graded, simple, self-contragredient, $C_2$-cofinite vertex operator algebra. We show that if the $S$-transformation of the character of $V$ is a linear combination of characters of $V$-modules, then the category $\mathcal{C}$ of grading-restricted generalized $V$-modules is a rigid tensor category. We further show, without any assumption on the character of $V$ but assuming that $\mathcal{C}$ is rigid, that $\mathcal{C}$ is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of $V$ is semisimple, then $\mathcal{C}$ is semisimple and thus $V$ is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to $V$. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that $C_2$-cofinite affine $W$-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such $W$-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of $C_2$-cofiniteness for the coset. That is, given a vertex operator algebra inclusion $U\otimes V\hookrightarrow A$ with $A$, $U$ strongly rational and $U$, $V$ a pair of mutual commutant subalgebras in $A$, we show that $V$ is also strongly rational provided it is $C_2$-cofinite.

math.QA↗

arXiv2021

On ribbon categories for singlet vertex algebras

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $\mathcal{M}(p)$, $p\geq 2$. The first category consists of all finite-length $\mathcal{M}(p)$-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra $\mathcal{W}(p)$. We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.

math.QA↗

arXiv2021

Tensor categories for vertex operator superalgebra extensions

Let $V$ be a vertex operator algebra with a category $\mathcal{C}$ of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let $A$ be a vertex operator (super)algebra extension of $V$. We employ tensor categories to study untwisted (also called local) $A$-modules in $\mathcal{C}$, using results of Huang-Kirillov-Lepowsky showing that $A$ is a (super)algebra object in $\mathcal{C}$ and that generalized $A$-modules in $\mathcal{C}$ correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a $\mathcal{C}$-algebra and (under suitable conditions) of generalized $A$-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 $V$-modules to $A$-modules is a vertex tensor functor. We give two applications. First, we derive Verlinde formulae for regular vertex operator superalgebras and regular $(1/2)\mathbb{Z}$-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and $\mathbb{Z}$-graded components, respectively. Second, we analyze parafermionic cosets $C=\mathrm{Com}(V_L,V)$ where $L$ is a positive definite even lattice and $V$ is regular. If the category of either $V$-modules or $C$-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.

math.QA↗

arXiv2021

Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$

We show that the Kazhdan-Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure, and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in $\mathcal{O}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in $\mathcal{O}_k^{fin}$. Then using Knizhnik-Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

math.QA↗