Comments on one-form global symmetries and their gauging in 3d and 4d

Journal Article · Fri Mar 29 00:00:00 EDT 2019 · SciPost Physics Proceedings

DOI:https://doi.org/10.21468/SciPostPhys.6.3.039· OSTI ID:1504279

Hsin, Po-Shen ^[1]; Lam, Ho Tat ^[2]; Seiberg, Nathan ^[3]

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1. California Institute of Technology, Princeton University
2. Princeton University
3. Institute for Advanced Study, Princeton University

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 ${\mathbb{Z}}_N$ one-form symmetries. A 3d topological quantum field theory (TQFT) \mathcal{T} $𝒯$ with such a symmetry has N $N$ special lines that generate it. The braiding of these lines and their spins are characterized by a single integer p $p$ modulo 2N $2N$ . Surprisingly, if \gcd(N,p)=1 $gcd(N,p)=1$ the TQFT factorizes \mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p} $𝒯=𝒯\prime \otimes {𝒜}^{N,p}$ . Here \mathcal{T}' $𝒯\prime$ is a decoupled TQFT, whose lines are neutral under the global symmetry and \mathcal{A}^{N,p} ${𝒜}^{N,p}$ is a minimal TQFT with the \mathbb{Z}_N ${\mathbb{Z}}_N$ one-form symmetry of label p $p$ . The parameter p $p$ labels the obstruction to gauging the \mathbb{Z}_N ${\mathbb{Z}}_N$ one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When p=0 $p=0$ mod 2N $2N$ , 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) $SU\left(N\right)$ and PSU(N) $PSU\left(N\right)$ 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement – probe quarks are confined. In the PSU(N) $PSU\left(N\right)$ theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a space-dependent \theta $\theta$ -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) $PSU\left(N\right)$ theory is obtained by gauging the \mathbb{Z}_N ${\mathbb{Z}}_N$ one-form symmetry of the SU(N) $SU\left(N\right)$ theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the PSU(N) $PSU\left(N\right)$ theory.

• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)
• Grant/Contract Number: SC0009988; SC0011632
• OSTI ID: 1504279
• Alternate ID(s): OSTI ID: 1596477
• Journal Information: SciPost Physics Proceedings, Journal Name: SciPost Physics Proceedings Journal Issue: 3 Vol. 6; ISSN 2542-4653
• Publisher: SciPost Foundation
• Country of Publication: Netherlands
• Language: English