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}_NℤN one-form symmetries. A 3d topological quantum field theory (TQFT) \mathcal{T}𝒯 with such a symmetry has NN special lines that generate it. The braiding of these lines and their spins are characterized by a single integer pp modulo 2N2N. Surprisingly, if \gcd(N,p)=1gcd(N,p)=1 the TQFT factorizes \mathcal{T}=\mathcal{T}'\otimes \mathcal{A}^{N,p}𝒯=𝒯′⊗𝒜N,p. Here \mathcal{T}'𝒯′ is a decoupled TQFT, whose lines are neutral under the global symmetry and \mathcal{A}^{N,p}𝒜N,p is a minimal TQFT with the \mathbb{Z}_NℤN one-form symmetry of label pp. The parameter pp labels the obstruction to gauging the \mathbb{Z}_NℤN one-form symmetry; i.e. it characterizes the ’t Hooft anomaly of the global symmetry. When p=0p=0 mod 2N2N, 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(N) and PSU(N)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)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 \thetaθ-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(N) theory is obtained by gauging the \mathbb{Z}_NℤN one-form symmetry of the SU(N)SU(N) theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the PSU(N)PSU(N) theory.
Highlights
In Appendix C, we demonstrate that every Abelian topological quantum field theory (TQFT) corresponds to a unitary chiral Rational Conformal Field Theory (RCFT)
For Abelian TQFTs, we provide a construction of a corresponding unitary chiral RCFT in Appendix C
One of the main points in our discussion is that since the bulk lines are trivial in any 3d correlation functions, we find it natural to identify them with the trivial line and Table 3: Gauging an Symmetry Protected Topological (SPT) phase of N one-form symmetry with a boundary supporting a 3d TQFT T leads to a 4d-3d system
Summary
Point operators can be charged under an ordinary internal global symmetry. Extended operators can be charged under a higher-form global symmetry [1]. When a one-form global symmetry is unbroken the spectrum includes charged strings. If they are broken, the low-energy dynamics reflects the broken symmetry. These anomalies can be used, just like ’t Hooft anomaly matching of ordinary global symmetries, to constrain the IR behavior of a theory and to check duality between distinct theories. Such an anomaly in a higher-form symmetry can flow from a bulk to a defect in the bulk. Are these quarks liberated, they have nontrivial braiding, i.e. they are anyons! Below we will give an intuitive physical argument explaining why they are anyons
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