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Relative Anomalies in (2+1)D Symmetry Enriched Topological States
by Maissam Barkeshli, Meng Cheng
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Submission summary
| Authors (as registered SciPost users): | Maissam Barkeshli · Meng Cheng |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1906.10691v1 (pdf) |
| Date submitted: | 2019-07-19 02:00 |
| Submitted by: | Barkeshli, Maissam |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Certain patterns of symmetry fractionalization in topologically ordered phases of matter are anomalous, in the sense that they can only occur at the surface of a higher dimensional symmetry-protected topological (SPT) state. An important question is to determine how to compute this anomaly, which means determining which SPT hosts a given symmetry-enriched topological order at its surface. While special cases are known, a general method to compute the anomaly has so far been lacking. In this paper we propose a general method to compute relative anomalies between different symmetry fractionalization classes of a given (2+1)D topological order. This method applies to all types of symmetry actions, including anyon-permuting symmetries and general space-time reflection symmetries. We demonstrate compatibility of the relative anomaly formula with previous results for diagnosing anomalies for $\mathbb{Z}_2^{\bf T}$ space-time reflection symmetry (e.g. where time-reversal squares to the identity) and mixed anomalies for $U(1) \times \mathbb{Z}_2^{\bf T}$ and $U(1) \rtimes \mathbb{Z}_2^{\bf T}$ symmetries. We also study a number of additional examples, including cases where space-time reflection symmetries are intertwined in non-trivial ways with unitary symmetries, such as $\mathbb{Z}_4^{\bf T}$ and mixed anomalies for $\mathbb{Z}_2 \times \mathbb{Z}_2^{\bf T}$ symmetry, and unitary $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry with non-trivial anyon permutations.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2019-9-9 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1906.10691v1, delivered 2019-09-09, doi: 10.21468/SciPost.Report.1158
Strengths
1- Not only their new results, it also contains a concise and readable summary of the framework, including why the symmetry fractionalization classes are a torsor over $H^2_{\rho}(G,\mathcal{A})$.
Weaknesses
1- The authors gave the formula for the relative anomaly (43) but did not show that it is actually a cocycle. (The referee understands that the authors mentioned that "however we do not pursue this further here". But it is a weakness.)
Report
In this paper, the authors obtained a universal formula for the change of the anomaly valued in $H^4(G,U(1))$ of a 2+1d TQFT under the shift of the symmetry fractionalization class by an element of $H^2_{\rho}(G,\mathcal{A})$. This universal formula was then applied to many concrete cases, reproducing many results previously obtained in other papers by a case-by-case analysis.
The referee found the paper clearly written and containing interesting results, thus worth publishing on SciPost.
The referee has one question and one suggestion:
1- The relative anomaly is (more or less) a map from $ H^2_{\rho}(G,\mathcal{A})$ to $H^4(G,U(1))$. What is it? $M_{ab}$ provides a bilinear map $M:\mathcal{A}\times\mathcal{A}\to U(1)$, so there is a natural quadratic pairing $M(t,s)\in H^4(G,U(1))$, given $t,s\in H^2_{\rho}(G,\mathcal{A})$. Maybe $I(t)$ is a quadratic refinement of $M(t,s)$, in the sense $I(t+s)=I(t)I(s)M(t,s)$?
2- The authors might want to comment on https://arxiv.org/abs/1805.02738 in your section VI. There, the $\mathbb{Z}_2^T$ anomalies of abelian anyon systems were extensively studied. Your $H^2_{\rho}(\mathbb{Z}_2^T,\mathcal{A})$ was denoted by $C=\mathrm{Ker}(1-T)/\mathrm{Im}(1+T)$, and the total anomaly was identified as $\mathrm{Arf}(q)$ where $q(a)=\theta(a)\eta(a)$ is considered as a function on $C$. Since $\mathrm{Arf}(q)$ has the well-known property $\mathrm{Arf}(tq)=q(t) \mathrm{Arf}(q)$, this implies your relative anomaly formula. Also, in this case at least, $q(t+s)=q(t)q(s) M(t,s)$.
Requested changes
All the requested changes are extremely minor typos:
The fist line of I. Introduction:
The last few years ... has seen major progress $\to$ The last few years ... have seen major progress
some lines below it:
an important class of invertible states are $\to$ an important class of invertible states is
(22):
$|a,b;c,\nu\rangle$ on the RHS should probably be $|^ga,{}^gb;{}^gc,\nu\rangle$
some lines above (29):
Eq. (28) should be Eq.~(28) in the LaTeX source file; a period following a small-case letter is known to automatically produce a wider space, and this feature needs to be suppressed here.
two lines above (65):
a (absolute) vison $\to$ an (absolute) vison
The first line of VIII.B:
Sec. II, III should be Sec.~II, III
three lines above (110):
We now given an ... $\to$ We now give an ...
one line above (119):
The sentence "The $\mathbb{Z}_2$ charge conjugation symmetry $C:a\to N-a$." lacks a verb.
In (125):
The first line there are $R^{ta} R^{at}$ which are converted to $M_{ta}$ in the second line. The second line also contains $M_{tb}$ but there is no corresponding $R^{tb}R^{bt}$ in the first line.