String condensations in $3+1D$ and Lagrangian algebras
Jiaheng Zhao, Jia-Qi Lou, Zhi-Hao Zhang, Ling-Yan Hung, Liang Kong, Yin Tian
Abstract
We present three Lagrangian algebras in the modular 2-category associated to the 3+1D Z 2 topological order and discuss their physical interpretations, connecting algebras with gapped boundary conditions, and interestingly, maps (braided autoequivalences) exchanging algebras with bulk domain walls.A Lagrangian algebra, together with its modules and local modules, encapsulates detailed physical data of strings condensing at a gapped boundary.In particular, the condensed strings can terminate at boundaries in non-trivial ways.This phenomenon has no lower dimensional analogue and corresponds to novel mathematical structures associated to higher algebras.We provide a layered construction and also explicit lattice realizations of these boundaries and illustrate the correspondence between physics and mathematics of these boundary conditions.This is a first detailed study of the mathematics of Lagrangian algebras in modular 2-categories and their corresponding physics, that brings together rich phenomena of string condensations, gapped boundaries and domain walls in 3+1D topological orders.Contents 1 Introduction 2 A physicist's sketch of the modular 2-category associated to the 3d toric code model 3 Lagrangian algebras in TC and their lattice realizations 3.1 The algebra A e and the rough boundary . . . . . . .