Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)$d$

TL;DR

This work analyzes time-reversal symmetry in 2+1d gauge theories, revealing that on monopole sectors the would-be $T^2=(-1)^F$ can be modified to include a magnetic symmetry $M$, yielding $T^2=(-1)^F M$ in certain cases. Focusing on $U(1)$ and $SO(N)$ gauge theories with fermions of various charges and tensor representations, the authors derive precise monopole quantum numbers, anomaly structures (notably $ u\

Abstract

We study continuum quantum field theories in 2+1 dimensions with time-reversal symmetry $\cal T$. The standard relation ${\cal T}^2=(-1)^F$ is satisfied on all the "perturbative operators" i.e. polynomials in the fundamental fields and their derivatives. However, we find that it is often the case that acting on more complicated operators ${\cal T}^2=(-1)^F {\cal M}$ with $\cal M$ a non-trivial global symmetry. For example, acting on monopole operators, $\cal M$ could be $\pm1$ depending on the magnetic charge. We study in detail $U(1)$ gauge theories with fermions of various charges. Such a modification of the time-reversal algebra happens when the number of odd charge fermions is $2 ~{\rm mod}~4$, e.g. in QED with two fermions. Our work also clarifies the dynamics of QED with fermions of higher charges. In particular, we argue that the long-distance behavior of QED with a single fermion of charge $2$ is a free theory consisting of a Dirac fermion and a decoupled topological quantum field theory. The extension to an arbitrary even charge is straightforward. The generalization of these abelian theories to $SO(N)$ gauge theories with fermions in the vector or in two-index tensor representations leads to new results and new consistency conditions on previously suggested scenarios for the dynamics of these theories. Among these new results is a surprising non-abelian symmetry involving time-reversal.

Figures (5)

• Figure 1: The phase diagram of $U(1)$ gauge theory coupled to a Dirac fermion with charge two, together with a double monopole perturbation. The monopole perturbation splits the free Dirac point into two Majorana points. These transitions separate TQFTs.
• Figure 2: The phase diagram of $U(1)$ gauge theory coupled to a Dirac fermion of charge four, with a minimally charged magnetic monopole perturbation. In the two dual descriptions the Majorana fermion $\chi$ couples to the $\mathbb{Z}_2$ gauge field of $O(2)$ by the transformation $\chi\rightarrow -\chi$. The low energy TQFT is the T-Pfaffian theory.
• Figure 3: The phase diagram of $SO(N)$ gauge theory coupled a symmetric tensor fermion $S$. The infrared TQFTs, together with relevant level-rank duals are shown along the bottom. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. Each of these transitions can be described by a dual theory with adjoint fermions, in which the transition can be seen at weak coupling. These dual theories cover part of the phase diagram. These figures are identical to those in Gomis:2017ixy, with the map of the $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$ unitary global symmetry determined from Cordova:2017vab.
• Figure 4: The phase diagram of a $Spin(N)$ gauge theory coupled to a fermion $S$ in the two-index symmetric tensor representation. It can be obtained from $SO(N)$ gauge theory by gauging the magnetic symmetry in the UV, which corresponds to gauging the diagonal ${\cal CM}$ in the long-distance TQFT. The blue dots indicate the transitions from the semiclassical phase to the quantum phase. Each of these transitions can be described by a dual theory with adjoint fermions, in which the transition can be seen at weak coupling. These dual theories cover part of the phase diagram. These figures are identical to those in Cordova:2017vab.
• Figure 5: The phase diagram of a $Spin(N)$ gauge theory coupled to a fermion $\lambda$ in the adjoint representation. The phase transitions are visible at weak coupling in a dual theory with symmetric tensor fermions. This can be derived from the $SO(N)$ gauge theory by gauging the UV magnetic symmetry $\mathcal{M}^{\text{UV}}$, which corresponds to gauging the symmetry ${\cal C^{\text{IR}}M^{\text{IR}}}$ in the long-distance TQFT Cordova:2017vab.