String Condensations in 3+1D and Lagrangian Algebras

TL;DR

The paper develops a higher-categorical framework to describe gapped boundaries and domain walls in 3+1D topological order by constructing three Lagrangian algebras $A_e$, $A_1$, and $A_2$ inside the modular 2-category $\mathsf{TC}$ of the 3+1D $\mathbb{Z}_2$ toric code. It interprets these algebras through string condensation, module categories, and local modules, and provides explicit lattice realizations and a layered construction that connects to minimal modular extensions and anomaly-free 2d $\mathbb{Z}_2$ states. A braided autoequivalence exchanging $A_1$ and $A_2$ gives rise to an invertible 2d domain wall and links the smooth vs twisted-smooth boundaries to 2d $\mathbb{Z}_2$ SPT order. The work uncovers higher-dimensional algebraic structures (nontrivial 2-associators and 2-commutators) that enable end points of condensed strings to terminate at boundaries, and proposes that this framework extends to all 3d topological orders by the string-condensation viewpoint.

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.

Figures (16)

• Figure 1: Layered construction of the 3d $\undefined{Z}_2$ topological order: (a) Each blue line represents a layer of the $\undefined{Z}_2$ topological order and the numbers on the left label these layers. Each red circle represents a tensor product state between the electric charges of consecutive layers. The condensation of these tensor product states will introduce coupling between consecutive layers. (b) Besides the trivial string, the unconfined anyons are the $e$-particle and the $\mathsf{m}$-string.
• Figure 2: Boundary conditions from layered construction: The layer labeled by b is the boundary layer. (a) Besides the tensor product states of electric charges, $e_b$ itself is also condensed to make sure that the electric charges can be absorbed into the boundary. (b) Only the tensor product states of electric charges are condensed. Then one can readily see that the $\mathsf{m}$-string can end on the boundary. (c) Here the boundary layer is a double semion order instead of a $\undefined{Z}_2$ order. The related condensed tensor product state is $s\bar{s}\times e$. An $\mathsf{m}$-string can still end on this boundary as long as its end point on the boundary layer is either $s$ or $\bar{s}$.
• Figure 3: Trangulation of the cube
• Figure 4: Local operators, $A_{14}$ and $B'_{[2,10,14]}$, in the bulk of 3+1D toric code
• Figure 5: The $e$-particle (red dot) and $\mathsf{m}$-string (blue plaquettes) in 3+1D toric code

Theorems & Definitions (17)

• Remark 4.1
• Definition B.1
• Definition B.2
• Definition B.3
• Definition C.1
• Definition C.2
• Definition C.3
• Definition C.4
• Definition C.5
• Definition C.6