Commutative exact superalgebras and super-modular tensor categories

Abstract

We extend the theory of commutative exact algebras in braided finite tensor categories (Shimizu-Yadav) to superalgebras, motivated by extensions of vertex operator (super)algebras. For a braided category $\C$ let $\B=\C^{\gr}$ be the category of $\mathbb Z/2$-graded objects with the Koszul-signed braiding. It is the underlying category of the supercategory of Creutzig-Kanade-McRae. A supercommutative superalgebra $A=A_0\oplus A_1$ in $\C$ is the same as a commutative algebra $\bA$ in $\B$, and super-local modules are the local $\bA$-modules. Hence rigidity, pivotality and ribbon statements are instances of the ordinary theorems, with the ribbon condition $θ_{A_0}=\id$, $θ_{A_1}=\varepsilon\,\id$, where $\varepsilon=\pm1$ is fixed by the statistics of the odd part. For $\C$ non-degenerate and $\bA$ haploid and exact we prove that the category of super-local modules is a braided finite tensor category whose Müger center is $\mathsf{sVect}$. It is the centralizer of a fermion in a non-degenerate category obtained by condensing in $\C\boxtimes(\text{toric code})$. If $\C$ is ribbon and $\bA$ is symmetric Frobenius it is ribbon. We construct examples from graded simple currents, treat the free fermion and the symplectic fermion, and compare with Creutzig-Kanade-McRae and Creutzig-McRae-Shimizu-Yadav.