Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients

We thank both referees for their valuable comments, questions, and suggestions. We made lots of modifications accordingly, which has led to a vast improvement of our manuscript. The following “List of changes” includes the changes we made along with our detailed replies to each point raised by the referees. We believe the present form of our manuscript is suitable for a publication on SciPost.

Best regards,
Shi Chen, Yuya Tanizaki

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Reply to Referee 1
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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 $\int_{M}\frac{1}{2\pi}d a$ 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.

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Reply to Referee 2
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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:

1. 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.

2. 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 $\mathbb{C}^{\times}$ in their Sec. 5.3. In the paragraph above Remark 5.16, they said that $\Sigma^{n}I\mathbb{C}^{\times}$ corresponds to discrete topology on $\mathbb{C}^{\times}$. In the paragraph above Eq. 5.17, they said that $\Sigma^{n+1}I\mathbb{Z}(1)$ correpsonds to continuous topology on $\mathbb{C}^{\times}$. 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:
$Vect^{\times}$ and $sVect^{\times}$ are the 0-truncation and the 1-truncation of $I_{\mathbb{C}^{\times}}$, respectively. The former fact is trivial, and we explained the latter fact around Eq. 4.28 in the present version. Meanwhile, $Vect^{\times}$ and $sVect^{\times}$ are also the 0-truncation and the 1-truncation of $I_{\mathbb{C}^{\times}}MSpin$, respectively. This is because the spin Thom spectrum $MSpin$ is 1-connected. As for our $\mathbb{SU}$, it is an alternative of $sVect^{\times}$ when the target space is finite, and it is not the 1-truncation of $I_{\mathbb{C}^{\times}}MSpin$ 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:

1. We added some related references and changed the sentence accordingly.

2. 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.

3. We added references.

4. 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.

5. We added a footnote to make the remark on the difference from the conventional definition.

6. Thanks for pointing it out. We added the following sentence: "For the claim of equivalence among several models, see Ref.~\cite{Barwick:2021}."

7. 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 $Rep^n(Y)$ in Sec. 4.2.1 yet.

8. We added a footnote for clarifications.

9. We added comments with the reference to Lurie's paper for clarifications.

10. We updated the discussion before Proposition 4.11 and clarified how we define $\mathbb{SC}$ (see also our answer for Major Point 3).

11. We made the correction as suggested.

12. As the referee pointed out, $B^3\mathbb{Z}\to X\to B^2\mathbb{Z}\to B^4\mathbb{Z}$ is a fragment of a homotopy fiber sequence, and each class $\in[B^2\mathbb{Z},B^4\mathbb{Z}]$ prescribes a distinct space $X$ (up to homotopy equivalence). Nevertheless, computing which class yields $X\simeq\pi_{\leq 3}\mathbb{C}P^1$ requires additional work. We expanded the text and added a footnote to explain why the generator of $[B^2\mathbb{Z},B^4\mathbb{Z}]\simeq\Z$ yields $X\simeq\pi_{\leq 3}\mathbb{C}P^1$.

13. We added several references on the condensation operators.

14. 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.

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Additional modification
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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 $\ell$ was incorrect, and we now corrected it (see Eq. 3.9). We also added references about the mapping class group of $S^2\times S^1$ there.