String Condensations in 3+1D and Lagrangian Algebras

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.

7+17 pages, 16 figures. Comments are welcome

References in corpus (11)

• Condensate induced transitions between topologically ordered phases
• Genons, twist defects, and projective non-Abelian braiding statistics
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• A theory of topological edges and domain walls
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
• Layer construction of 3D topological states and string braiding statistics
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• Topological Quantum Computation with Gapped Boundaries
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point

Cited by in corpus (13)

• Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
• Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• SymTFT for (3+1)d Gapless SPTs and Obstructions to Confinement
• Fusion 3-Categories for Duality Defects
• Duality via Sequential Quantum Circuit in the Topological Holography Formalism
• Non-invertible symmetries in finite-group gauge theory
• Local Modules in Braided Monoidal 2-Categories
• Generalized Landau Paradigm for quantum phases and phase transitions
• Magic Boundaries of 3D Color Codes
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• Anomalies in mirror symmetry enriched topological orders
• Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries