Prelude: low energy description of Symmetry Protected Topological (SPT) phases

It is known [1] that the low energy effective description of SPT phases, protected by a group $G$ is an invertible field theory (iQFT). Namely, if $A$ is a $G$-background gauge field, the iQFT partition function $Z[A]$ has an inverse, $\overline{Z[A]}$, such that $$ Z[A]\; \overline{Z[A]} = 1.$$

For the case of bosonic SPT phases the iQFT is ungauged Dijkgraaf-Witten theories, and this is essentially what allowed Wen et al. to classify bosonic SPT phases through group cohomology [2]. The more general version initiated the cobordism classification (proved finally in [3]).

Motivation for low energy description of Symmetry Enriched Topological (SET) phases

In a recent paper [4] the classification of both SPT phases and SET phases was given, using a categorical approach. From this viewpoint, restricting an SET phase to an SPT is very natural and the new classification of SPTs coincides with the group cohomology for 1 and 2 dimensions and goes beyond group cohomology for higher dimensions. It might be possible to arrive to a classification of SET phases by their low energy QFT-like description, mimicking the older SPT classifications and see whether it coincides with the categorical description of [4], or whether it is compatible with restricting to SPTs afterwards etc. Or at least if such a classification is out of reach, it should be interesting to see where it fails and thus why is the categorical approach necessary.

A different motivation is the study of anomalies in QFT. Since iQFTs have a one-dimensional Hilbert space, an iQFT on an open manifold will support an anomalous theory on its boundary, and thus conversely anomalous theories are captured by invertible theories — and hence by SPT phases — in one-higher dimension. Is there a similar argument for SET phases?

My questions

  1. What is the low energy QFT-like effective description of SET phases?
  2. Is there an anomaly-on-the-boundary (or generalised anomaly, or anything that vaguely resembles anomalies) point of view of this description?


[1] D. S. Freed, Short-range entanglement and invertible field theories, arXiv:1406.7278

[2] X.Chen, Z.C.Gu, Z.X.Liu and X.G.Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, Phys. Rev. B 87 15, 155114 (2013), doi:10.1103/PhysRevB.87.155114, [arXiv:1106.4772]

[3] K. Yonekura, On the Cobordism Classification of Symmetry Protected Topological Phases. Communications in Mathematical Physics 368, 1121 (2019), doi: 10.1007/s00220-019-03439-y, [arXiv:1803.10796]

[4] L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, H. Zheng, Classification of topological phases with finite internal symmetries in all dimensions, arXiv:2003.08898


1 Answer 1


I'm not sure exactly what you're looking for in terms of a "low energy QFT-like description", but the generalization of an iQFT is a topological quantum field theory (TQFT) coupled to a background gauge field; these are also a generalization of TQFTs without background gauge field, which describe non-invertible topological phases without any symmetry. If your goal is to escape from category theory, though, I have bad news: the only (as far as I know) general and rigorous way to characterize a TQFT, let alone a TQFT coupled to background gauge field, is expressed in terms of category theory. A good reference on that is the first paper you cited, by Dan Freed, where he talks about the general categorical definition of TQFT before specializing to invertible TQFTs.

You can't expect to have describe a non-invertible phase with symmetry just in terms of a partition function $Z[A]$. The physical interpretation of the result for invertible phases is that you couple to a background gauge field and then integrate out the dynamical degrees of freedom (which you are allowed to do since they are gapped and trivial), leaving only an action for the background gauge field. Such an operation would be illegitimate for non-invertible phases because there are non-trivial topological degrees of freedom which can't be integrated out. In some cases, you might be able to write down an action based on coupling the background gauge fields to internal dynamical fields. Even for topological phases without symmetry though (i.e. no background gauge fields), it's not clear whether all topological phases can be described this way, and it's not really a good starting point for a classification because there is no good way to get a handle on what kind of dynamical fields you should be considering.

On the question of anomalies on the boundaries of non-invertible TQFTs: I don't think the general story here is that well understood, but you might be interested in the following recent paper:


  • $\begingroup$ Thanks for the answer! For question 1 I guess the main question lies in which TQFTs can be realised as low energy descriptions of SETs? Said differently I would expect a non-invertible generalisation of ungauged DW models. As for the partition function, surely I don't expect a partition function in a closed form, but you can certainly define a formal path integral (as is also reviewed on the paper you linked), and turn on background fields by considering symmetry defects. About anomalies I was hoping for a more field-theoretic approach but this paper does indeed look very interesting! $\endgroup$ Commented May 19, 2020 at 12:33
  • 2
    $\begingroup$ @idiot I guess the TQFTs that can be realized in lattice models are (conjectured to be) precisely those that can be constructed by a Turaev-Viro or Reshetikhin-Turaev construction (and indeed, these constructions give you lattice models via Levin-Wen/Walker-Wang models respectively), or generalizations in higher dimensions. The G-equivariant versions of these will describe G SETs. I think this is classification is probably equivalent to the one proposed by Kong et al, though it’s not obvious. $\endgroup$ Commented May 19, 2020 at 13:21
  • $\begingroup$ Thank you! Do you have a reference [or multiple if you're in the mood :)] for this guess/conjecture? $\endgroup$ Commented May 19, 2020 at 13:49
  • 1
    $\begingroup$ @idiotstroszek Since the input to the Reshetikhin-Turaev construction is a modular tensor category (MTC), this conjecture basically amounts to saying that topological phases are classified by MTC. The physical picture for that would be that the MTC precisely captures the braiding and fusion of anyons. See for example the legendary Appendix E of arxiv.org/pdf/cond-mat/0506438 This is all in (2+1)-D, similar statements presumably hold in higher dimensions but precise statements are much more open. $\endgroup$ Commented May 20, 2020 at 11:26
  • 1
    $\begingroup$ @idiotstroszek To get the G-equivariant version of these statements (corresponding to SETs) you replace MTC with a G-crossed MTC, the physical interpretation of which is discussed in: arxiv.org/abs/1410.4540 $\endgroup$ Commented May 20, 2020 at 11:28

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