Local Modules in Braided Monoidal 2-Categories

TL;DR

This paper develops the theory of local modules in braided monoidal 2-categories, showing they form braided monoidal structures with duals, and explores their applications to topological phases of matter.

Contribution

It introduces the concept of local modules in braided monoidal 2-categories and proves their structural properties, including braided monoidal structure and duals, with applications to topological phases.

Findings

The 2-category of local modules admits a braided monoidal structure.

If the 2-category has duals, local modules also have duals.

Lagrangian algebras are characterized in the Drinfeld centers of fusion 2-categories.

Abstract

Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3+1)d topological phases of matter.