Comments on one-form global symmetries and their gauging in 3d and 4d
Po-Shen Hsin, Ho Tat Lam, Nathan Seiberg
SciPost Phys. 6, 039 (2019) · published 29 March 2019
- doi: 10.21468/SciPostPhys.6.3.039
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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.
TY - JOUR
PB - SciPost Foundation
DO - 10.21468/SciPostPhys.6.3.039
TI - Comments on one-form global symmetries and their gauging in 3d and 4d
PY - 2019/03/29
UR - https://scipost.org/SciPostPhys.6.3.039
JF - SciPost Physics
JA - SciPost Phys.
VL - 6
IS - 3
SP - 039
A1 - Hsin, Po-Shen
AU - Lam, Ho Tat
AU - Seiberg, Nathan
AB - 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.
ER -
@Article{10.21468/SciPostPhys.6.3.039,
title={{Comments on one-form global symmetries and their gauging in 3d and 4d}},
author={Po-Shen Hsin and Ho Tat Lam and Nathan Seiberg},
journal={SciPost Phys.},
volume={6},
pages={039},
year={2019},
publisher={SciPost},
doi={10.21468/SciPostPhys.6.3.039},
url={https://scipost.org/10.21468/SciPostPhys.6.3.039},
}
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Authors / Affiliations: mappings to Contributors and Organizations
See all Organizations.- 1 2 Po-Shen Hsin,
- 2 Ho Tat Lam,
- 3 Nathan Seiberg
- 1 California Institute of Technology [CalTech]
- 2 Princeton University
- 3 Institute for Advanced Study [IAS]