Local Modules in Braided Fusion 2-categories | BIMSA-Tsinghua Quantum Symmetry Seminar

Abstract

Given a braided algebra in a braided fusion 2-category, under certain rigidity condition, itsmodules form a monoidal 2-category. Refining the notion of module to local module, we prove thatlocal modules over a separable braided algebra form a braided multifusion 2-category. Meanwhilelocal modules and free modules centralise each other (generalising the 1-categorical setting), andsatisfy a type of reciprocity (which is a new phenomenon emerging in the 2-categorical setting).

By analogy with Lagrangian algebras in braided 1-categories, we define a Lagrangian algebra in abraided fusion 2-category as a connected separable braided algebra whose local modules form atrivial 2-category 2Vect. Lagrangian algebras play an important role in classifying topologicalboundaries of (3+1)D topological orders. l will comment on how to address the parallel question inmathematics, that is the classification of (bosonic) fusion 2-categories, via Lagrangian algebras inthe Drinfeld center of a strongly fusion 2-category 2Vect%

This talk is based on arXiv:2307.02843 (joint with Thibault Dcoppet), arXiv:2403.07768 and an on-going project.