Non-invertible topological defects in 4-dimensional $\mathbb{Z}_2$ pure lattice gauge theory

TL;DR

This work identifies and constructs non-invertible topological defects in a 4D pure $Z_2$ lattice gauge theory with a 1-form center symmetry and a Kramers-Wannier-Wegner duality. Using the Aasen–Mong–Fendley framework, it explicitly builds a non-invertible KWW duality defect and accompanying $Z_2$ 1-form defects, together with defect junctions and crossing relations that close the defect algebra. The authors compute defect-configuration expectation values on manifolds such as $S^3$ and $S^2\times S^1$, revealing invariants under 4D embeddings and establishing a concrete set of crossing relations. The results illuminate a robust non-invertible symmetry structure in higher dimensions and suggest potential connections to continuum QFTs, including supersymmetric gauge theories like $N=4$ SU(2) SYM.

Abstract

We explore topological defects in the 4-dimensional pure $\mathbb{Z}_2$ lattice gauge theory. This theory has 1-form $\mathbb{Z}_{2}$ center symmetry as well as the Kramers-Wannier-Wegner (KWW) duality. We construct the KWW duality topological defects in the similar way to that constructed by Aasen, Mong, Fendley arXiv:1601.07185 for the 2-dimensional Ising model. These duality defects turn out to be non-invertible. We also construct the 1-form $\mathbb{Z}_{2}$ symmetry defects as well as the junctions among KWW duality defects and 1-form $\mathbb{Z}_{2}$ center symmetry defects. The crossing relations among these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.

Figures (26)

• Figure 1: A schematic illustration of lattices. Although it is depicted as 2-dimensional in this figure, the actual lattices treated in this paper are 4-dimensional ones. The black lattice represents the lattice $\Lambda$, and the blue lattice represents the lattice $\hat{\Lambda}$. They are dual to each other.
• Figure 2: Stereographic projection of a 16-cell into 3 dimensions. The black plaquette represents an active plaquette, and the blue plaquette represents an inactive plaquette. The 16-cell consists of 16 tetrahedrons, each of which contains one active link and one inactive link.
• Figure 3: A schematic illustration of the duality defect. The black dots represent the active lattice and the blue dots represent the inactive lattice. A duality defect is located at the boundary between the two regions. The active lattice (black dots) and the inactive lattice (blue dots) are swapped across the duality defect. A unit cell of the duality defect is a tetrahedral prism depicted as a green parallelogram in this figure.
• Figure 4: The building block of duality defects in the 4-dimensional $\mathbb{Z}_2$ lattice gauge theory. The 3-dimensional surface is composed of tetrahedrons each of which includes an active link and an inactive link. A tetrahedron on which a duality defect is supported is doubled and becomes a tetrahedral prism. An inactive link is put on the edge of a tetrahedron in this tetrahedral prism associated to the active link of the other tetrahedron and vice versa.
• Figure 5: A schematic illustration of a defect commutation relation. The circle represents a 16-cell and a green line represents a duality defect. The defect commutation relation implies that the value of the duality defect remains the same even if it is deformed without changing the topology.