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TimeReversal Symmetry, Anomalies, and Dualities in (2+1)$d$
by Clay Cordova, PoShen Hsin, Nathan Seiberg
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Submission summary
Authors (as registered SciPost users):  Clay Córdova · PoShen Hsin · Nathan Seiberg 
Submission information  

Preprint Link:  http://arxiv.org/abs/1712.08639v2 (pdf) 
Date accepted:  20180608 
Date submitted:  20180405 02:00 
Submitted by:  Córdova, Clay 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We study continuum quantum field theories in 2+1 dimensions with timereversal 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 nontrivial 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 timereversal 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 longdistance 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 twoindex 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 nonabelian symmetry involving timereversal.
Published as SciPost Phys. 5, 006 (2018)
Reports on this Submission
Anonymous Report 2 on 201862 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1712.08639v2, delivered 20180601, doi: 10.21468/SciPost.Report.483
Strengths
1. Addresses subtle details of timereversal symmetry in threedimensional gauge theories
2. Timely subject matter
3. Clarity of presentation
Weaknesses
None, really.
Report
The point of this paper is to work out various aspects of timereversal symmetry and the exact global symmetry group for particular $T$invariant ChernSimonsmatter 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 ChernSimonsmatter theories, see e.g. (1.15). The way they do it is straightforward enough to follow and reproduce. The nontrivial part is to deduce how the global symmetries act on monopole operators, and for this one simply looks at matter zeromode 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 timereversal anomaly $\nu$. There may be mixed anomalies between $T$ and $(1)^M$.
2. There has been a conjectured selfduality 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 intext 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.
Anonymous Report 1 on 2018531 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1712.08639v2, delivered 20180531, doi: 10.21468/SciPost.Report.480
Strengths
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.
Weaknesses
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].
Report
The paper considers various examples of Quantum Field Theories invariant under timereversal, such that the timereversal operator satisfies an interesting nonstandard 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"