Tensor categories for vertex operator superalgebra extensions

Abstract

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.

133 pages, final version to appear in Mem. Amer. Math. Soc., incorporating referee suggestions; in this version, many lengthy tensor category arguments in Section 2 have been replaced with graphical calculus, and more details and motivation have been added in Section 4; references and discussion related to recent progress have also been added

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Cited by in corpus (51)

• W-algebras as coset vertex algebras
• Braided tensor categories of admissible modules for affine Lie algebras
• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• Trialities of $\mathcal{W}$-algebras
• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• Duality of subregular W-algebras and principal W-superalgebras
• On the tensor structure of modules for compact orbifold vertex operator algebras
• Gluing vertex algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
• Modularity of logarithmic parafermion vertex algebras
• A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation
• Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Cosets of the ${\mathcal{W}}^k({\mathfrak{sl}}_4,f_{\text{subreg}})$-algebra
• Cyclic orbifolds of lattice vertex operator algebras having group like fusions
• Logarithmic conformal field theories of type $B_n,\ell =4$ and symplectic fermions
• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• Modularity of Bershadsky-Polyakov minimal models
• $N=4$ superconformal algebras and diagonal cosets
• Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
• Fusion and positivity in chiral conformal field theory
• Unbounded field operators in categorical extensions of conformal nets
• Generalized parafermions of orthogonal type
• Quantum coordinate ring in WZW model and affine vertex algebra extensions
• Hopf algebra actions in tensor categories
• A general mirror equivalence theorem for coset vertex operator algebras
• Kazama-Suzuki coset construction and its inverse
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Representation theory of $L_k\left(\mathfrak{osp}(1|2)\right)$ from vertex tensor categories and Jacobi forms
• Symplectic level-rank duality via tensor categories
• Hecke Operators on Vector-Valued Modular Forms
• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• Tensor Categories arising from the Virasoro Algebra
• The Drinfeld-Kohno theorem for the superalgebra $gl(1|1)$
• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• Orbifolds and minimal modular extensions
• On rationality for $C_2$-cofinite vertex operator algebras
• Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson
• $N=1$ super Virasoro tensor categories
• Affine Lie algebras and tensor categories
• Orthosymplectic Feigin-Semikhatov duality
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Simple current extensions of tensor products of vertex operator algebras
• Vertex operator superalgebras and 16-fold way
• Vertex Algebras and Commutative Algebras
• Sigma involutions associated with parafermion vertex operator algebra $K({\mathfrak{sl}}_2,k)$
• Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras