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# Advances in Theoretical and Mathematical Physics

## Volume 27 (2023)

### Number 2

### String condensations in $3+1D$ and Lagrangian algebras

Pages: 583 – 622

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n2.a5

#### Authors

#### Abstract

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. 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.

Published 12 October 2023