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

by Clay Cordova, Po-Shen Hsin, Nathan Seiberg

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Submission summary

As Contributors: Clay Cordova · Po-Shen Hsin · Nathan Seiberg
Arxiv Link: (pdf)
Date accepted: 2018-06-08
Date submitted: 2018-04-05 02:00
Submitted by: Cordova, Clay
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Ontology / Topics

See full Ontology or Topics database.

Anomalies Dualities Gauge theory Global symmetries Monopoles Non-abelian symmetries Quantum electrodynamics (QED) Time-reversal symmetry Topological quantum field theories (TQFT)

Published as SciPost Phys. 5, 006 (2018)

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2018-6-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1712.08639v2, delivered 2018-06-01, doi: 10.21468/SciPost.Report.483


1. Addresses subtle details of time-reversal symmetry in three-dimensional gauge theories
2. Timely subject matter
3. Clarity of presentation


None, really.


The point of this paper is to work out various aspects of time-reversal symmetry and the exact global symmetry group for particular $T$-invariant Chern-Simons-matter theories in three dimensions. The basic question is exactly what bosonic symmetry does $T$ square to. In some examples (like 3d QED coupled to bosons) it squares to 1, and in others with charged fermions it squares to $(-1)^F$. Most interestingly, there are examples demonstrated by the authors where $T^2$ involves the monopole number symmetry.

More generally the authors work out in some detail the global symmetry group of these Chern-Simons-matter theories, see e.g. (1.15). The way they do it is straightforward enough to follow and reproduce. The non-trivial part is to deduce how the global symmetries act on monopole operators, and for this one simply looks at matter zero-mode creation/annihilation operators in the monopole background as around (2.3). The presentation is a little sparse on the details, but there is more than enough information given to work it out on one's own.

Beyond logical completeness, there is some payoff from these largely formal manipulations. The two I noticed were:

1. In examples where $T^2$ involves the monopole symmetry, one must now be careful about the computation of the time-reversal anomaly $\nu$. There may be mixed anomalies between $T$ and $(-1)^M$.

2. There has been a conjectured self-duality between 3d QED with 2 fermions, whereby the global symmetry is enhanced to $O(4)$ in the infrared. The authors find additional evidence for this conjecture by carefully tracking down how $T$ acts.

Requested changes

I have two extremely pedantic suggestions. The first is that the authors should explain in-text the notation for O(n) with a superscript and two subscripts in Table 1. The second is that in various places the authors refer to the "global symmetry algebra" when they mean the group.

  • validity: top
  • significance: good
  • originality: good
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2018-5-31 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1712.08639v2, delivered 2018-05-31, doi: 10.21468/SciPost.Report.480


-1- The questions investigated in the paper are subtle, and are treated with great care and detail.

-2- The paper is very well written, difficult issues are explained with great clarity.


-1- Just before eq. 2.13, the authors write "It follows that including charge conjugation extends U(1)M to the group Pin−(2)M". It would be nice to expand a bit the discussion, and explain in more detail the similarities/differences with previous analysis of the UV and IR global symmetry of QED with 2 flavors, of refs [30] and [21].


The paper considers various examples of Quantum Field Theories invariant under time-reversal, such that the time-reversal operator satisfies an interesting non-standard algebra.

Requested changes

-1- It looks like the variable M appearing in the r.h.s. of eq 1.10 is not defined in the introduction.

-2- I found a few typos:
- last line of sec 1.1: "their"
- 5th to last line of sec 1.2: "instance"
- eq 3.2: curly C vs standard C
- sec 3.1.1, 2nd line: "beginning"

  • validity: high
  • significance: high
  • originality: good
  • clarity: high
  • formatting: perfect
  • grammar: excellent

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