SciPost Submission Page
Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients
by Shi Chen, Yuya Tanizaki
Submission summary
| Authors (as registered SciPost users): | Shi Chen · Yuya Tanizaki |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202503_00063v1 (pdf) |
| Date submitted: | March 31, 2025, 2:45 p.m. |
| Submitted by: | Chen, Shi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Originating from the topology of the path-integral target space Y, solitonic symmetry describes the conservation law of topological solitons and the selection rule of defect operators. As Ref.~\cite{Chen:2022cyw} exemplifies, the conventional treatment of solitonic symmetry as an invertible symmetry based on homotopy groups is inappropriate. In this paper, we develop a systematic framework to treat solitonic symmetries as non-invertible generalized symmetries. We propose that the non-invertible solitonic symmetries are generated by the partition functions of auxiliary topological quantum field theories (TQFTs) coupled with the target space Y. We then understand solitonic symmetries as non-invertible cohomology theories on Y with TQFT coefficients. This perspective enables us to identify the invertible solitonic subsymmetries and also clarifies the topological origin of the non-invertibility in solitonic symmetry. We finally discuss how solitonic symmetry relies on and goes beyond the conventional wisdom of homotopy groups. This paper is aimed at a tentative general framework for solitonic symmetry, serving as a starting point for future developments.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Best regards,
Shi Chen, Yuya Tanizaki
List of changes
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======================== Reply to Referee 1 ========================
Thanks very much for reviewing our paper with saying "this is a timely and important paper." We made corrections to clarify the point you raised.
(1) and (4) The typos were already corrected in the arXiv version (namely v2) in our first submission.
(2) Eq. (2.8) is correct in the current form. As an example, let us consider the U(1) gauge theory. The magnetic monopole chagre ∫M12πda is always defined on a 2-manifold M in any spacetime dimension. Because of this independence from the ambient spacetime, we can develop the theory of solitonic symmetries just focusing on the target space Y.
(3) We removed E-orientable following your suggestion.
======================== Reply to Referee 2 ========================
Thanks very much for reviewing our paper with saying "I recommend to publish it in SciPost." We made corrections to clarify the points you raised.
On Major points:
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Thanks for raising this point. We now think that the structure type on the operator manifold M may be chosen independently from the tangential structure of the spacetime manifolds. We thus modified a lot of contents in the paper to address the problems raised by this issue. Most of them are minor wording changes, and the major modifications are the following. i) We added one sentence to the last of the paragraph below Eq.(3.2). ii) We removed the leading paragraph of Sec. 3.3 and instead added a new paragraph.
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We agree that chiral Chern-Simons theories are not included in our general formal framework, and our discussions here were actually incomplete. Hence we are pleased to follow the referee's suggestion. i) We first added a new paragraph to the end of Sec. 4.2.2 to explain that our general framework precludes chiral TQFTs. ii) We then modified the construction of topological functional of our primary example of 4d CP1 sigma model in Sec. 5.2.2 using non-chiral Chern-Simons theories. Explicitly, we replaced the last two paragraphs of Sec. 5.2.2 with three new paragraphs.
3. - Issues about Sec. 4.2.2: Freed and Hopkins discussed about the topology on C× in their Sec. 5.3. In the paragraph above Remark 5.16, they said that ΣnIC× corresponds to discrete topology on C×. In the paragraph above Eq. 5.17, they said that Σn+1IZ(1) correpsonds to continuous topology on C×. Intuitively, continuous topology allows us to deform one theory to another, whereas discrete topology isolates each individual theory. Therefore, the use of continuous topology classifies the isomorphism classes, while the use of discrete topology classifies the deformation classes. We added Footnote 11 to clarify this point and cited Sec. 5.3 of Freed-Hopkins for readers who are interested in.
- Issues about Sec 4.3.2: Vect× and sVect× are the 0-truncation and the 1-truncation of IC×, respectively. The former fact is trivial, and we explained the latter fact around Eq. 4.28 in the present version. Meanwhile, Vect× and sVect× are also the 0-truncation and the 1-truncation of IC×MSpin, respectively. This is because the spin Thom spectrum MSpin is 1-connected. As for our SU, it is an alternative of sVect× when the target space is finite, and it is not the 1-truncation of IC×MSpin on its own. We rewrote many sentences in Sec. 4.3.2, to refine the descriptions of the relevant spectra; this passingly includes a reply to Minor Point 10.
On Minor points:
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We added some related references and changed the sentence accordingly.
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Here, we are not discussing the continuous deformation of the submanifolds. Rather, what we are discussing the continuous deformation of the field configurations on a fixed submanifold. We note that this is roughly equivalent to consider the continuous deformation of the base manifold while keeping its topology fixed. In this sense, we can answer your question that our continuous deformation excludes the case when the sphere is retracted to a point. As we are just mentioning the sufficient condition (not the necessary condition) for the values of topological functionals being nontrivial, we think "as long as" is the right word.
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We added references.
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We removed the leading paragraphs of Sec. 3.3 as we made a mistake due to our misunderstanding in the first version, as explained in the Major point 1.
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We added a footnote to make the remark on the difference from the conventional definition.
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Thanks for pointing it out. We added the following sentence: "For the claim of equivalence among several models, see Ref.~\cite{Barwick:2021}."
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We moved the part of previous Footnote 17 (present Footnote 20) to the place you suggested in Sec. 4.2.1 to explain the notion of universe. We, however, left previous Footnote 17 (present Footnote 20) partly, because we have not introduced Repn(Y) in Sec. 4.2.1 yet.
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We added a footnote for clarifications.
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We added comments with the reference to Lurie's paper for clarifications.
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We updated the discussion before Proposition 4.11 and clarified how we define SC (see also our answer for Major Point 3).
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We made the correction as suggested.
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As the referee pointed out, B3Z→X→B2Z→B4Z is a fragment of a homotopy fiber sequence, and each class ∈[B2Z,B4Z] prescribes a distinct space X (up to homotopy equivalence). Nevertheless, computing which class yields X≃π≤3CP1 requires additional work. We expanded the text and added a footnote to explain why the generator of [B2Z,B4Z]≃\Z yields X≃π≤3CP1.
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We added several references on the condensation operators.
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We agree with your point, but we also believe that the solitonic symmetry should be anomaly free even for continuous cases. We added a footnote there to justify our conjecture 6.1.
======================== Additional modification ========================
In addition to both referees' suggestions, we found other inappropreiate descriptions in the manuscript, and we also modfied them accordingly.
(1) We modified our speculation in present Footnote 17 (previous Footnote 14). The Brown-Comenetz dual of MSO or MSpin gives gravitational theta angles in terms of characteristic classes of tangent bundles (including Pontryagin, Stiefel-Whitney, and KO-theoretic classes). Our TQFTs are a priori defined on framed manifolds, and all the charactertic classes of a framed manifold have to be trivial. Hence our TQFTs do not cover those gravitational theta angles.
(2) In Sec. 3.1 "Example: spin structure", our previous decription of the f-action on ℓ was incorrect, and we now corrected it (see Eq. 3.9). We also added references about the mapping class group of S2×S1 there.
Current status:
Reports on this Submission
Report
While most issues have been satisfactorily resolved, I still have one remaining major concern that needs to be addressed before the paper can be recommended for publication.
Major concern:
Regarding Point 3 of my previous report, the authors' treatment of topology on morphism spaces in Vec (bottom of page 20, second bullet point) remains problematic. Specifically:
Path-connected space morphisms must be equivalent via higher morphisms in (∞,1) category. If Hom(-,-) has a path-connected topology as prescribed, all morphisms would need to be equivalent to the zero map. While Freed-Hopkins' discussion of topology makes sense for invertible phases (where invertible (higher) morphisms form a non-trivial homotopy), applying similar topology to the full space of morphisms including non-invertible ones would make it contractible. That is, the standard topology of matrix algebra would be that of Cm itself for some m.
Ideally, I want the authors to clarify how their topology prescription can be made compatible with the categorical structure while properly distinguishing between invertible and non-invertible cases. Otherwise, since this is not the core part of the paper, making the sentence more vague remark would be acceptable.
Recommendation: I recommend acceptance after the authors address this remaining issue.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Strengths
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shows a kind of creativity with the subject
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shows an effort to connect to modern tools of homotopy theory and algebraic topology
Weaknesses
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a discernible new result is missing, while a high density of technical buzzwords may obscure this fact to the lay reader
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most of the article is but a review of standard material, especially so in the second half on extended TQFT, sprinkled with some ideosyncratic terminology.
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the presentation is most informal and handwavy. While superficially using notation common to mathematical physics (like numbered proposition environments) it is not remotely close to the standards of the field of mathematical physics.
Report
I find it is a mishmash of basic known facts, often not clearly highlighted as such, with a bit of ideosyncratic new terminology added, justified by handwaving and not leading to a discernible new insight.
At the same time, there are several paragraphs (see again the comments below) that superficially sound like making quite grandiose claims, while however consisting but of a soup of buzzwords that one wouldn't be surprised to see auto-generated from a mechanical language model. I am not suggesting that the authors used such text generation tools, but all the more would I urge them to work on their use of language: It is possible to speak about these matters in an intelligible way.
On the positive side I recognize that the authors looked up some modern concepts and tools of homotopy theory and algebraic topology, more so than what is often seen in the physics literature. This is commendable and would be promising if the authors can now adopt a more logical style of justifying claims and substantiating arguments.
At the same time, I worry that the reader not familiar with these matters may mistake several of the statements being recalled as being original claims by the authors, presented as they are as numbered "propositions" without citation of where they are taken from.
Of course, it is a long tradition in physics to be handwavy and vague about the subject, and the usual justification is that authors thus freed form the constraints of stringent logic can be more creative and thereby may happen upon valuable insights that would not otherwise have been found but can still be more rigorously justified after the fact. I sense that this is what the authors here are are after and I am willing to recognize and appreciate this when I see it. However, I do not see it happening with this article: I am not getting away from it with any non-tautological insight, and be it one of the "conjectures" offered.
What I seem to understand to be the main intended claim by the authors is to think of generalizing classifying spaces (hence infinity-groupoids) -- into which to map the fundamental higher groupoids of a domain manifold -- by monoidal (infinity,n)-categories, and the main justification for this seems to be the observation that the known classification of extended topological field theories says that this happens for TQFTs parameterized over a space.
But then, first I don't see how this is more than recalling this fact about extended TQFTs from the literature, and second it does not justify the claim advertized in abstract and introduction, that somehow homotopy groups get replaced by something "non-invertible":
No, the map from a space (hence: an infinity-group) into an (infinity,n)-category necessarily factors through the latter's groupoid core, which is again a space with usual homotopy groups.
In Lurie's arXiv:0905.0465 this is discussed starting on p. 43 (where that groupoid core is denoted by a tilde superscript, which reappears in the statement of Lurie's theorem 2.4.18 which the auhtors cite: That "C-tilde" there is a space with ordinatry homotopy groups).
In view of this fact I find all allusion to "non-invertible symmetry" in the text (incidentally an oxymoron, even if popular these days) either non-enlightning (in as far as it just refers to TQFTs which are not tensor invertible) or wrong (as concerns the claim that somehow homotopy groups are generalized to something "non-invertible").
And with this even a potential content of the article seems to collapse.
Requested changes
This is a list of comments/criticisms going linerly through the article.
p. 4: "the quantum field theory (QFT) is defined by the path integral"
Of course the "path integral" for QFT is famously ill-defined and provides not a definition but at best a heuristics!
p. 4: "We focus on the non-Grassmann sector of this path integral, and thus σ might be a scalar field, a gauge field, a higher-form gauge field"
But for (higher) gauge fields the Lagrangian famously needs a FP ghost sector and hence a "Grassmann sector", after all.Of course, such technical fine print doesn't matter here precisely because the "path integral" is not actually being used to define anything.
p. 4: "Conveniently, we can always find a topological space Y such that there exists a one-to-one correspondence,... .We shall refer to this topological space Y as the (homotopy) target space"
If this is meant as a general statement then it would need a citation (for instance section 3.2 of the Encyclopedia article "Flux Quantization" arxiv:2402.18473 where these spaces are called "classifying spaces", following an old tradition).
p. 4: "Proposition 2.1 In a d-dimensional QFT defined by a path integral,"
As stated, this is not a "proposition" in the sense of mathematics, since it refers to undefined objects. On the other hand, the reference to the "path integral" is also superfluous for what the authors actually want to express here, which is the same they already said in the sentence before. Therefore, just deleting this non-professional "Proposition 2.1"-label would take away nothing from the author's intent.
p. 5: Definition 2.2:
Since this is not the author's definition but taken from algebraic topology textbooks, it needs to come with a citation.
Just on the grammar: Proper English requires to say "*A* topolgical space..." and then better style may be to say "is *called* n-aspherical".
p. 5:
"we can always replace a general Y with its d-th Postnikov truncation."
Yes, but since this is not the author's insight but taken from the textbooks, it needs to be cited.
p. 5: "1. Y ≃ X" for a X-valued scalar field, where X is a topological space.
Better to call this a "non-linear sigma-model field", since a "scalar field" is really one for which Y is a vector space.
Even for a non-linear sigma model, the target space is typically not any topological space but a manifold.
Indeed, if the authors really allow any topological space already in this first example, then all the following examples would just be special cases!, which is hardly the intention.
p. 5 the further examples:
These examples are valid and good, but not new. The authors are clearly taking this from some literature and need to cite sources (such as the above encyclopedia article, or section 2 in https://doi.org/10.1142/13422 or some other source).
p. 5 section 2.2: "From a contemporary viewpoint of QFTs..."
Unfortunately, this "contemporary viewpoint" is but a folklore notorious for not providing actual definitions.
p. 6: "Proposition 2.3":
In serious mathematical physics texts one would not label such tautologies as numbered Propositions.
A numbered proposition is usually something that needs and is provided with a non-trivial and rigorous proof.Otherwise it is at best a Remark (which can also be numbered, if one does want to be able to refer to it.)
There is nothing close to mathematical proofs in this article, which makes its use of numbered propositions appear out of place.
p. 6, equation (2.7):
the symbol "sigma" is not declared (of course it is meant to be a map sigma : M -> Y, but this needs to be said!).
Also, the declaration of "U" on the left is lacking its dependence on sigma, which more important (and implies) the dependence on M, i nparticular if you want to claim to be interested in the "path integral".
p. 6 "operator (2.7) gives the universal construction of invertible solitonic symmetry"
That superficially sounds like a bold claim. If such a claim is really intended, an actual definition of "invertible solitonic symmetry" should be given, a provable characterization of what it means for it to be "universal", next a construction of a candidate and finally a proof that it satisfies this characterization. None of this seems to be happening in the article, even remotely.
p. 6: "In this paper, we shall discuss the most generalized connotation of topological functional and solitonic symmetry"
This has a similar ring to it as the previous sentence, and an analogous comment applies.
But also the semantics of this sentence seems broken: According to the English dictionary, a "connotation" is "an idea or feeling which a word invokes in addition to its literal meaning". I don't think that this is actually meant here and the sentence needs to be rewritten.
p. 6: Proposition 2.4
If the statement of the proposition is meant (as appears to be the case) to say just what the surrounding text already says then it is redundant and in any case a triviality.
p. 6, equation (2.8)
Again, if this says what it seems to say then it is a tautology.
p. 7, equation (2.9)
Same.
p. 7 "From this point of view, the algebraic structure of solitonic symmetry is supposed to be a sort of cohomology theory on the target space Y "
This statement is extremely vague and sounds almost like the result of machine auto-generated text.
p. 7 "a symmetry shows the presence of certain conserved charges."
This of course holds famously for infinitesimal symmetries to which Noether's theorem applies. That or whether it holds for what the authors have in mind would need an argument or reference.
p. 7: "excise its infinitesimal neighborhood"
Just to note that "infinitesimal neighbourhood" is not the correct technical term. The authors mean tubular neighbourhood and to allude to the fact that its "size" is irrelevant as long as it is positive while small enough.
p. 7 "Locally, SN ≃ N × S^{d−p−1}). "
Indeed, since this in general holds only locally, the equivalence is wrong as stated! A correct form to say what the authors mean to say here is: "Locally on a patch U we have SN|U ≃ U × S^{d−p−1}".
p. 8: Top paragraph.
Again, this text is so vague that it reads as if auto-generated.
p. 8, between (2.12) and (2.13)
This sounds like the authors believe that a soliton must be point-like, which is not the case.
p. 10 "From the contemporary perspective..."
Such a qualifier seems suitable for a text in the social sciences. In natural sciences and especially in physics it seems out of place. But here it probably does indicate a reliance on fashionable folklore. I know that this has become common in parts of the community, but it is not scientific.
p. 10: "Unfortunately, in most interesting cases, the π0Diff(M )-action on [M, Y ] results in vast degeneracies":
Indeed, the problem here arises due to insistence of passing to gauge equivalence classes, which is not the right thing to do in (higher) gauge theory: Instead of just the connected components [M,Y] = pi0 Map(M,Y) of the mapping space, the full homotopy type Map(M,Y) captures the topological field configurations including their (higher) gauge transformations. Next, implementing diffeomorphism covariance on such as space then corresponds to forming not the plain quotient but the homotopy quotient by the diffeomorphism group Map(M,Y) // Diif(Y) (aka "Borel construction" or "stacky quotient"), cf. for instance
Dul (2023) [doi:10.1007/s11005-023-01653-3]
p. 14 "We regard this ansatz as a complete characterization of topological functionals."
This seems odd: Either it's just an Ansatz or it's a "complete characterization". And the claim of a "complete characterization" appears most loose, beginning with the definition of the terms, not to speak of the justification of the claim (which appears to be missing).
p. 15: "We are going to reveal the universal algebraic structure of solitonic symmetry"
This is a grandiose-sounding announcement for which I see little to no substantiation in the article. One way to prove my impression wrong would be to follow the claim by pointer to precise statement and proof.
pp. 16, section 4:
This section appears to be a kind of review of standard material from the literature on extended TQFTs. It remains unclear what contribution the authors mean to be advertising here.
p. 22, Definition 4.3
The term "Y -enriched" is not defined. In the context of (higher) category theory that the authors are speaking in here, "enrichment" is an established technical term, but it cannot be what the authors mean here.
The authors seem to mean what traditionally is called "Y-parameterized TQFT" or "TQFT over Y".
Granting this, then the content of Def. 4.3 still seems both empty and ill-stated. Empty, because granting the choice of codomain, this is just the standard definition well established now since almost 2 decades. And ill-stated because if the authors mean to say they want to restrict attention to the stated codomains, then they must say so intelligibly.
p. 23 Definition 4.5:
First, the authors are far from the first to consider definitions of higher representations, contrary to the impression that the reader will get here. (Some general pointers to the literature may be found at ncatlab.org/nlab/show/infinity-representation).
Second, since Y here is a space, hence an (infiinity,0)-category, the proposed functors in (4.15) factor through the underlying space of the codomain (infinity,n)-category (their groupoid "core", indicated by a tilde superscript in Lurie's text which the authors cite). This means that the structure defined by Def. 4.5 ends up being equivalent to just maps between spaces, and hence does not provide a "non-invertible" generalization of these.
p. 26, Prop. 4.9:
The "proposition" starts out saying "Consider a d-dimensional theory defined by a path integral..." which however is famously not a definition. It only gets less precise from here on.
It seems what the authors mean here is not a "proposition" as commonly understood, but something more like a "guiding principle" which may be added to the folklore (and this comment applies to the text at large).
Recommendation
Reject