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.