Given a commutative algebra $A$ in a braided monoidal category $C$, the category of local A-modules, $C_A^\mathrm{loc}$, is defined as a subcategory of the category $C_A$ of right $A$-modules in C. Pareigis showed that $C_A^\mathrm{loc}$, which is important for studying vertex operator algebra extensions, is a braided monoidal category under very general conditions. In this setting, I will present a criterion for $C_A^\mathrm{loc}$ to be a rigid monoidal category. When $C$ is pivotal/ribbon, I will also discuss when the category $C_A$ is pivotal and when $C_A^\mathrm{loc}$ is ribbon.

As an application, I will show that when $C$ is a modular tensor category and $A$ is a commutative simple symmetric Frobenius algebra in $C$, then $C_A^\mathrm{loc}$ is a modular tensor category. Furthermore, I will discuss methods to construct such commutative algebras using simple currents and the Witt group of non-degenerate braided finite tensor categories. This presentation is based on joint work with Kenichi Shimizu.