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 $\tau:\text{Irr}(\mathcal{U})\rightarrow\text{Obj}(\mathcal{V})$ such that $\tau(U)=V$ to create $A=\bigoplus_{X\in\text{Irr}(\mathcal{U})}X'\otimes\tau(X)$. 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.