SciPost Submission Page
Straightening Out the Frobenius-Schur Indicator
by Steven H. Simon and Joost K. Slingerland
Submission summary
| Authors (as registered SciPost users): | Steven Simon |
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| Preprint Link: | scipost_202209_00037v1 (pdf) |
| Date submitted: | 2022-09-19 12:07 |
| Submitted by: | Simon, Steven |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
Amplitudes for processes in topological quantum field theory (TQFT) are calculated directly from spacetime diagrams depicting the motion, creation, annihilation, fusion and splitting of any particles involved. One might imagine these amplitudes to be invariant under any deformation of the spacetime diagram, this being almost the meaning of the "topological" in TQFT. However, this is not always the case and we explore this here, paying particular attention to the Frobenius-Schur indicators of particles and vertices. The Frobenius-Schur indicator is a parameter $\kappa_a=\pm 1$ assigned to each self-dual particle $a$ in a TQFT, or more generally in any tensor category. If $\kappa_a$ is negative then straightening out a timelike zig-zag in the worldline of a particle of type $a$ can incur a minus sign and in this case the amplitude associated with the diagram is not invariant under deformation. Negative Frobenius-Schur indicators occur even in some of the simplest TQFTs such as the $SU(2)_1$ Chern-Simons theory, which describes semions. This has caused some confusion about the topological invariance of even such a simple theory to space-time deformations. In this paper, we clarify that, given a TQFT with negative Frobenius-Schur indicators, there are two distinct conventions commonly used to interpret a spacetime diagram as a physical amplitude, only one of which is isotopy invariant --- the non-isotopy invariant interpretation is used more often in the physics literature. We clarify in what sense TQFTs based on Chern-Simons theory with negative Frobenius-Schur indicators are isotopy invariant, and we explain how the Frobenius-Schur indicator is intimately linked with the need to frame world-lines in Chern-Simons theory. Further, in the non-isotopy-invariant interpretation of the diagram algebra we show how a trick of bookkeeping can usually be invoked to push minus signs onto the diagrammatic value of a loop (the ``loop weight''), such that most of the evaluation of a diagram does not incur minus signs from straightening zig-zags, and only at the last step minus signs are added. We explain the conditions required for this to be possible. This bookkeeping trick is particularly useful in the construction of string-net wavefunctions, where it can be interpreted as simply a well-chosen gauge transformation. We then further examine what is required in order for a theory to have full isotopy invariance of planar spacetime diagrams, and discover that, if we have successfully pushed the signs from zig-zags onto the loop weight, the only possible obstruction to this is given by an object related to vertices, known as the ``third Frobenius-Schur indicator''. We finally discuss the extent to which this gives us full isotopy invariance for braided theories.
Current status:
Reports on this Submission
Report
There may be some nice observations specific to Chern-Simons theory in Section V, semion theory in Section VI, and string-net models in Section VII, but I was not able to evaluate them due to the serious theoretical errors in Section I-IV, IX, and Appendix B.
A minor gripe is that some remarks made in this paper perpetuate an unhelpful narrative that mathematicians working in topologically ordered systems don’t understand the physical interpretation and that physicists don’t understand the mathematics. For the core audience of this article neither stereotype will be true. One really needs to take seriously the mathematics to get the physics correct.
Strengths
Making effort to interpret the notion of Frobenius-Schur indicators in the authors' way
Weaknesses
Lacking mathematical understanding of basic notions of tensor categories such as spherical structure, dualities and ribbon structures. The opinion on Frobenius-Schur indicators presented in the paper is inconsistent with the applications of modular tensor categories in TQFT. In essence, misunderstanding of Frobenius-Schur indicators is elaborated throughout this paper.
Report
file attached.
Strengths
1. This paper presents a clear and thorough discussion of the Frobenius-Schur indicator, which is accessible to a relatively wide audience of physicists (and mathematicians who may be interested in understanding its physical implications).
2. It fills a gap in the existing physics literature, which to the best of my knowledge does not contain a simple, intuitive explanation of what the FS indicator is, or the physical meaning of different conventions for keeping track of it.
Weaknesses
1. The paper is slightly lacking in its references to the existing literature, particularly of string net models. This can be corrected in a future version.
2. While this topic is one that I personally have wondered about on multiple occasions, it remains quite specialized, and will be of interest to a relatively small subset of the community.
Report
I enjoyed this article, which discusses the physical interpretation of the Frobenius-Schur indicator, and clarifies the relationship between various conventions, such as flags, cups and caps, that describe it. The article presents both a straightforward relationship between these conventions and choices of framing for particle world-lines, which is generally more familiar to physicists. This nice picture resolved, for me, a nagging question that I have long had about just how one should think about the Frobenius-Schur indicator.
Overall, I find the work clearly presented and accessible, which is especially commendable given the highly technical nature of the topic, and I recommend it for publication with the changes suggested below. Some of these are cosmetic; however, I feel that the distinction between FS indicators in fusion categories (the input for a string net model) and in unitary modular tensor categories (i.e. anyon models) is important and should be clarified. In particular, framing is not something that I have seen discussed in the context of fusion categories, which are 2-dimensional (or 0+1 dimensional) theories that do not admit a braiding structure.
Requested changes
1. P. 3, left column: “mutate a vertex" -> "rotate a vertex” ?
2. Beginning of 1a: I suggest adding a reference to e.g. a textbook or other seminal reference that uses the same conventions, allowing readers familiar with the material to skip this section and know which notation is being used.
3. String nets: the original construction by Levin and Wen assumed certain symmetries of the F symbols (specifically, tetrahedral invariance). There are subsequent papers (Lan and Wen, Kitaev and Kong, Lin and Levin, Lin Levin and Burnell) that treat the general situation. In these constructions, you can get the correct quantum double with what would amount to a different flag convention (but the string net wave function is not isotopy invariant).
4. "Since switching between the two conventions is a non-local procedure, the resulting topological orders will generally be different.” I don’t understand this, and it doesn’t seem consistent with the subsequent discussion of the two ways to view the doubled semion wave function, and the fact that they are related by a local unitary transformation on the lattice. The authors should clarify this point.
5. Unitary vs non-unitary: I think this is not so mysterious. On the lattice the loop is not an inner product, so it can take on any value. Indeed, I suggest clarifying the following: the signs in the loops are associated with non-unitarity of the UMTC, because these closed loops are associated with inner products in the Hilbert space. In the string net construction, however, the diagrams should really be interpreted as diagrams in a fusion category, in which case a negative sign in the loop values is not related to unitarity. In the string nets, it is unitarity of the F symbols that is important, as it is required for hermiticity of the string net Hamiltonian. (This is discussed, for example, in Ref. 18).
6. In a similar vein, one might consider the implications of FS indicators in 3D string-net type (or Walker-Wang) models. Here I do not know of a treatment in the literature that details whether these can be removed by a local unitary transformation, for example. If the authors have any insight on this it would be a nice addition to the text, though I don't think it's necessary for publication.
7. The fact that in string nets, a gauge transformation can move the Z_2 FS indicator between zig-zags and loops was discussed in Ref. 18.
8. I enjoyed the discussion of Z_2 grading, which I have not seen elsewhere in the literature (though I am not very familiar with the mathematics literature on this subject). Personally, I would mention explicitly in the main text one of the examples from Appendix C1 (and certainly a reference to this appendix is appropriate on p. 15). I would also suggest that the authors list some of the insight gained in Appendix A, in terms of specifically which classes of theories admit a FS grading, in the main text. (Or do the authors feel that there is no structure to comment on in this list?)