## 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 \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 $\pm 1$ depending on the magnetic charge. We study in detail U(1) $U\left(1\right)$ 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 $2$ 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) $SO\left(N\right)$ 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.

- Authors:

- Institute for Advanced Study, Princeton University
- Princeton University

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1461012

- Grant/Contract Number:
- SC0009988

- Resource Type:
- Published Article

- Journal Name:
- SciPost Physics Proceedings

- Additional Journal Information:
- Journal Name: SciPost Physics Proceedings Journal Volume: 5 Journal Issue: 1; Journal ID: ISSN 2542-4653

- Publisher:
- Stichting SciPost

- Country of Publication:
- Netherlands

- Language:
- English

### Citation Formats

```
Cordova, Clay, Hsin, Po-Shen, and Seiberg, Nathan. Time-reversal symmetry, anomalies, and dualities in (2+1)$d$. Netherlands: N. p., 2018.
Web. doi:10.21468/SciPostPhys.5.1.006.
```

```
Cordova, Clay, Hsin, Po-Shen, & Seiberg, Nathan. Time-reversal symmetry, anomalies, and dualities in (2+1)$d$. Netherlands. doi:10.21468/SciPostPhys.5.1.006.
```

```
Cordova, Clay, Hsin, Po-Shen, and Seiberg, Nathan. Fri .
"Time-reversal symmetry, anomalies, and dualities in (2+1)$d$". Netherlands. doi:10.21468/SciPostPhys.5.1.006.
```

```
@article{osti_1461012,
```

title = {Time-reversal symmetry, anomalies, and dualities in (2+1)$d$},

author = {Cordova, Clay and Hsin, Po-Shen and Seiberg, Nathan},

abstractNote = {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 ± 1 depending on the magnetic charge. We study in detail U(1) 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 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) S O ( 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.},

doi = {10.21468/SciPostPhys.5.1.006},

journal = {SciPost Physics Proceedings},

number = 1,

volume = 5,

place = {Netherlands},

year = {2018},

month = {7}

}

DOI: 10.21468/SciPostPhys.5.1.006