Gluing vertex algebras

Abstract

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 $\mathcal{C}⊠{\mathcal{C}}^{\text{rev}}$: if $\mathcal{C}$ is semisimple but not necessarily finite or rigid, then ${⨁}_{X\in \text{Irr}(\mathcal{C})}{X}^{\prime }⊠X$ is a commutative algebra, with $X^{\prime }$ a representing object for ${\text{Hom}}_{\mathcal{C}}(\bullet {\otimes }_{\mathcal{C}}X,1_{\mathcal{C}})$. Conversely, let $A={⨁}_{i\in I}{U}_i⊠V_i$ be a simple commutative algebra in $\mathcal{U}⊠\mathcal{V}$ 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 $V_i^{\ast }$. When $\mathcal{U}$ and $\mathcal{V}$ are module categories for simple vertex operator algebras $U$ and $V$, we glue $U$ and $V$ along $\mathcal{U}⊠\mathcal{V}$ via a map $\tau :\text{Irr}(\mathcal{U})\to \text{Obj}(\mathcal{V})$ such that $\tau (U)=V$ to create $A={⨁}_{X\in \text{Irr}(\mathcal{U})}{X}^{\prime }\otimes \tau (X)$. Thus under certain conditions, $\tau$ 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.

arXiv comment58 pages, final version incorporating referee comments, abstract has been expanded

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• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• On semisimplicity of module categories for finite non-zero index vertex operator subalgebras
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• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
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• Ordinary modules for vertex algebras of ${\mathfrak{osp}}_{1|2n}$
• On the structure of W-algebras in type A
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• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Cosets from equivariant W-algebras
• Duality via convolution of W-algebras