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

TL;DR

This paper investigates how time-reversal symmetry acts in 2+1 dimensional quantum field theories, revealing modifications involving global symmetries and clarifying the dynamics of abelian and non-abelian gauge theories with fermions.

Contribution

It uncovers the modified action of time-reversal symmetry on complex operators and explores the implications for the dynamics of $U(1)$ and $SO(N)$ gauge theories with fermions.

Findings

Time-reversal acts as ${ m T}^2 = (-1)^F { m M}$ on certain operators.

QED with a single charge-2 fermion flows to a free theory plus a topological field theory.

New non-abelian symmetry involving time-reversal in $SO(N)$ gauge theories.

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.