Po-Shen Hsin, Ho Tat Lam, Nathan Seiberg
SciPost Phys. 6, 039 (2019) ·
published 29 March 2019
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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.
