Time-reversal symmetry, anomalies, and dualities in (2+1)d

Abstract

We study continuum quantum field theories in 2+1 dimensions with time-reversal symmetry T . The standard relation 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 T 2 = (-1)FM with M a non-trivial global symmetry. For example, acting on monopole operators, M could be ±1 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 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.