Non-invertible symmetries and boundaries in four dimensions

TL;DR

The paper addresses boundary physics in four dimensional theories with non invertible symmetries by analyzing a Z_2 lattice gauge theory at criticality and implementing KWW duality defects. It develops a lattice construction with three boundary types N, D and ~D, enforces boundary defect commutation relations to fix boundary weights, and computes D^3 expectation values that connect boundary data to g functions. Using these D^3 results, it derives the key g function relation \frac{1}{2}g_D = \frac{1}{\sqrt{2}}g_N = g_{~D} and shows that certain boundary RG flows are forbidden in four dimensions. The findings provide a nonperturbative, symmetry based framework to constrain boundary RG behavior in 4D QFTs with non invertible symmetries and hint at generalizations to other theories with similar dualities.

Abstract

We study quantum field theories with boundary by utilizing non-invertible symmetries. We consider three kinds of boundary conditions of the four dimensional $\mathbb{Z}_2$ lattice gauge theory at the critical point as examples. The weights of the elements on the boundary is determined so that these boundary conditions are related by the Kramers-Wannier-Wegner (KWW) duality. In other words, it is required that the KWW duality defects ending on the boundary is topological. Moreover, we obtain the ratios of the hemisphere partition functions with these boundary conditions; this result constrains the boundary renormalization group flows under the assumption of the conjectured g-theorem in four dimensions.

Figures (11)

• Figure 1: Derivation of the relation between the g-functions. The gray disks are hemispheres on which the Ising CFT lives. The black boundary represents the fixed boundary condition $+$ and the blue boundary represents the free boundary condition $0$. The green circle is the KW duality defect. The KW duality defect can act on the boundary with the $+$ boundary condition, and change the boundary condition to the $0$ boundary condition (the left-hand side). On the other hand, a circular KW duality defect that does not contain operators inside can be replaced by its quantum dimension $\sqrt{2}$ (the right-hand side).
• Figure 2: The configuration that we consider in this paper to derive the relations between the g-functions. The gray disk is a four-dimensional hemisphere on which the $\mathbb{Z}_2$ gauge theory lives. The black boundary represents D or $\widetilde{\mathrm{D}}$, and the blue boundary represents N. The green line represents a KWW duality defect on $D^3$ that has $S^2$ edges on $S^3$ at the boundary of the hemisphere.
• Figure 3: A schematic picture of a quarter 16-cell. The black plaquette represents a plaquette on the boundary and the blue square dot represents a link in the bulk.
• Figure 4: A schematic picture of a cubic cone. The black dots and lines represent sites and links on the boundary, respectively. The blue dot represents a site in the bulk.
• Figure 5: A schematic picture of a building block of KWW duality defects on a boundary. The unit is defined on a doubled square pyramid. Each square pyramid includes a boundary plaquette and a bulk site closest to the plaquette.