String Condensations in 3+1D and Lagrangian Algebras

Understand

We present three Lagrangian algebras in the modular 2-category associated to the 3+1D $\mathbb{Z}_2$ 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.