Noninvertible duality defects in $3+1$ dimensions

For any quantum system invariant under gauging a higher-form global symmetry, we construct a noninvertible topological defect by gauging in only half of the spacetime. This generalizes the Kramers-Wannier duality line in $1+1$ dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in $3+1$ dimensions, and determine the fusion rule. From a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. We give an explicit realization of this duality defect in the free Maxwell theory, both in the continuum and in a modified Villain lattice model. The duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides. We further construct the duality defect in non-Abelian gauge theories and the ${\mathbb{Z}}_N$ lattice gauge theory.

Figure 1

The spacetime manifold is divided into two regions along the interface which is depicted as a red line. The arrow on the interface indicates that the interface is generally oriented.

Figure 2

In the shaded region the system is coupled to a ${\mathbb{Z}}_N^{(q)}$ gauge theory, with Dirichlet boundary conditions imposed on the boundaries.

Figure 3

In the $1+1\mathrm{D}$ critical Ising CFT, when the duality line crosses the order operator $\sigma$, it becomes a disorder operator $\mu$ attached to the ${\mathbb{Z}}_2^{(0)}$ line. Here the red line denotes the duality line $\mathcal{D}$ and the dotted line denotes the ${\mathbb{Z}}_2^{(0)}$ line.

Figure 4

As the three-dimensional duality defect $\mathcal{D}$ (shown in red) sweeps past a ${\mathbb{Z}}_N^{(1)}$-charged line operator (shown in black), the latter is attached to the ${\mathbb{Z}}_N^{(1)}$ topological two-surface $\eta$ (shown in blue) on the other side.

Figure 5

By moving the duality defect $\mathcal{D}$ around the charged line operator, the ${\mathbb{Z}}_N^{(1)}$ topological surfaces $\eta$ can be freely generated and be absorbed by the duality defect, i.e., $\eta \times \mathcal{D}=\mathcal{D}\times \eta =\mathcal{D}$.

Figure 6

Renormalization group flow commutes with the discrete gauging. Suppose the low-energy phase is trivially gapped. It becomes a SPT phase protected by the one-form global symmetry ${\mathbb{Z}}_N^{(1)}$ when we activate the background gauge fields for ${\mathbb{Z}}_N^{(1)}$. This ${\mathbb{Z}}_N^{(1)}$-SPT phase is compatible with the duality defect if it is invariant under gauging ${\mathbb{Z}}_N^{(1)}$.

Figure 7

The red and blue interfaces are defined by the Dirichlet boundary conditions of the ${\mathbb{Z}}_N^{(0)}$ gauge fields. We can deform the red interface to the blue interface by integrating out ${\widehat{m}}_0$ and gauge fixing ${\widehat{n}}_1={\widehat{n}}_2={\widehat{n}}_3=0$. Keeping track of the appropriate normalization factors, we find that this deformation is topological. Here we omit the superscripts that indicate the form degrees.

Figure 8

Duality defect is inserted along a cut between a lattice and its dual lattice. Fields along the cut are coupled by the blue dots as in (5.14).