Symmetry-Enriched Quantum Spin Liquids in (3+1)d(3+1)d

We use the intrinsic one-form and two-form global symmetries of (3+1)$d$ bosonic field theories to classify quantum phases enriched by ordinary ($0$-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the $0$-form symmetry on the lines and surfaces of the theory. From these data we classify the couplings to the background gauge field by the 0-form symmetry defects constructed from the higher-form symmetry defects. For trivial two-form symmetry the classification coincides with the classification for symmetry fractionalizations in $(2+1)d$. We also provide a systematic method to obtain the symmetry protected topological phases that can be absorbed by the coupling, and we give the relative 't Hooft anomaly for different couplings. We discuss several examples including the gapless pure $U(1)$ gauge theory and the gapped Abelian finite group gauge theory. As an application, we discover a tension with a conjectured duality in $(3+1)d$ for $SU(2)$ gauge theory with two adjoint Weyl fermions

Non-Invertible Defects in Nonlinear Sigma Models and Coupling to Topological Orders

Nonlinear sigma models appear in a wide variety of physics contexts, such as the long-range order with spontaneously broken continuous global symmetries. There are also large classes of quantum criticality admit sigma model descriptions in their phase diagrams without known ultraviolet complete quantum field theory descriptions. We investigate defects in general nonlinear sigma models in any spacetime dimensions, which include the "electric" defects that are characterized by topological interactions on the defects, and the "magnetic" defects that are characterized by the isometries and homotopy groups. We use an analogue of the charge-flux attachment to show that the magnetic defects are in general non-invertible, and the electric and magnetic defects form junctions that combine defects of different dimensions into analogues of higher-group symmetry. We explore generalizations that couple nonlinear sigma models to topological quantum field theories by defect attachment, which modifies the non-invertible fusion and braiding of the defects. We discuss several applications, including constraints on energy scales and scenarios of low energy dynamics with spontaneous symmetry breaking in gauge theories, and axion gauge theories.Comment: 39 pages, 3 figures; v2: updated references and corrected typo

Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d

We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on $\mathbb{Z}_N$ one-form symmetries. A 3d topological quantum field theory (TQFT) $\mathcal{T}$ with such a symmetry has $N$ special lines that generate it. The braiding of these lines and their spins are characterized by a single integer $p$ modulo $2N$. Surprisingly, if $\gcd(N,p)=1$ the TQFT factorizes $\mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}$. Here $\mathcal{T}'$ is a decoupled TQFT, whose lines are neutral under the global symmetry and $\mathcal{A}^{N,p}$ is a minimal TQFT with the $\mathbb{Z}_N$ one-form symmetry of label $p$. The parameter $p$ labels the obstruction to gauging the $\mathbb{Z}_N$ one-form symmetry; i.e.\ it characterizes the 't Hooft anomaly of the global symmetry. When $p=0$ mod $2N$, the symmetry can be gauged. Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with gauge fields extended to the bulk. This understanding allows us to consider $SU(N)$ and $PSU(N)$ 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement -- probe quarks are confined. In the $PSU(N)$ theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent $\theta$-parameter, which leads to interfaces. Typically, the theory on the interface is not confining. Furthermore, the liberated probe quarks are anyons on the interface. The $PSU(N)$ theory is obtained by gauging the $\mathbb{Z}_N$ one-form symmetry of the $SU(N)$ theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the $PSU(N)$ theory.Comment: 56 pages, 3 figures, 5 table

On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases

The moduli space of gapped Hamiltonians that are in the same topological phase is an intrinsic object that is associated to the topological order. The topology of these moduli spaces is used recently in the construction of Floquet codes. We propose a systematical program to study the topology of these moduli spaces. In particular, we use effective field theory to study the cohomology classes of these spaces, which includes and generalizes the Berry phase. We discuss several applications to studying phase transitions. We show that nontrivial family of gapped systems with the same topological order can protect isolated phase transitions in the phase diagram, and we argue that the phase transitions are characterized by screening of topological defects. We argue that family of gapped systems obey a version of bulk-boundary correspondence. We show that family of gapped systems in the bulk with the same topological order can rule out family of gapped systems on the boundary with the same topological boundary condition, constraining phase transitions on the boundary.Comment: 37 pages, 2 figure