Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d (1812.04716v1)

Abstract: 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.

• The paper examines one-form global symmetries, particularly The paper explores The paper analyzes one-form global symmetries, focusing on The paper presents a framework for understanding The paper analyzes one-form global symmetries, particularly \( \mathbb{Z}_N \) symmetries, revealing how 3d TQFTs with such symmetries can factorize based on symmetry lines and their properties.
• It details how gauging these symmetries is possible when they are anomaly-free and extends the theory to 4d SU(N) and PSU(N) gauge theories, linking symmetries to confinement and oblique confinement.
• The work predicts non-trivial phenomena at interfaces in 4d gauge theories with varying parameters, demonstrating how liberated quarks on these interfaces can exhibit anyonic behavior due to the interplay of topology and anomalies.

Analysis of One-Form Global Symmetries and Their Gauging in 3D and 4D

The paper explores the concepts of one-form global symmetries in three-dimensional (3d) and four-dimensional (4d) systems. Within this work, the authors examine the implications of these symmetries and elucidate the complexities involved in their gauging. The focus is primarily on $\mathbb{Z}_N$ one-form symmetries, which are a special class of symmetries that act on line operators.

Key Theoretical Contributions

1. Factorization and ’t Hooft Anomalies in 3D TQFTs: The authors highlight that a 3d topological quantum field theory (TQFT) $T$ with a $\mathbb{Z}_N$ one-form symmetry can be structured through the presence of $N$ characteristic lines generating these symmetries. The braiding and spin characteristics of these lines are defined by an integer $p$ modulo $2N$, and if the greatest common divisor (gcd) $(N, p) = 1$, the TQFT can be expressed as $T = T' \otimes A_{N,p}$. Here, $T'$ is a decoupled TQFT neutral under global symmetries, and $A_{N,p}$ represents a minimal TQFT that carries the $\mathbb{Z}_N$ one-form symmetry with a specified label $p$.
2. Gauging and Anomalies: When $p = 0 \mod 2N$, the one-form symmetry is anomaly-free, allowing for gauging. Otherwise, gauging is infeasible unless coupled to a 4d bulk where gauge fields permeate the bulk. This theory naturally extends to considerations of SU(N) and PSU(N) 4d gauge theories, where the dynamics pertain to confinement and oblique confinement.
3. Interfaces and Anomalies in 4D Gauge Theories: The theory predicts non-trivial scenarios, particularly when a 4d gauge theory has spatially varying parameters leading to interfaces. On these interfaces, the liberated quarks exhibit anyonic behavior—a significant result that emerges from the interplay of the topology across the interfaces and the anomaly structure of the theory.

Notable Results and Claims

• The examination of the symmetry lines in a general discrete one-form symmetry illuminates their dependence on a symmetric integral matrix, capturing both diagonal and off-diagonal components in the case of Abelian groups.
• Through canonical duality and applying a three-step gauging process, the authors present nuanced ways of analyzing and transforming TQFTs under the operation of gauging these symmetries.
• The paper elucidates how phenomena like confinement in the SU(N) gauge theory are linked to such one-form symmetries, with robust predictions about domain walls and dual theories.

Theoretical and Practical Implications

The implications of this research extend to both the theoretical understanding of TQFTs and practical computational approaches in gauge theories. The results provide a powerful framework to harness one-form symmetries in understanding confinement phenomena in non-Abelian gauge theories, potentially informing the development of computational models that bridge quantum field theories and condensed matter systems.

Speculations on Future Directions

The intricate work of analyzing symmetry lines, anomaly matching conditions and the factorization of TQFTs suggest future investigations may explore non-Abelian cases or applications in quantum computing and information science. Further studies could explore the relevance of these concepts in the evolving landscape of quantum technologies where topology and symmetry play critical roles. Moreover, the application of these theories in enhancing our understanding of the phases of matter and changes under symmetry is a promising direction, especially in high-energy physics contexts where duality and anomaly considerations become paramount.

Given the complexity and depth of this exploration in higher-form symmetries, the paper lays a robust groundwork for advancing both theoretical frameworks and application-driven developments in physics and computation.