The algebra of anomaly interplay

This paper is a valuable addition to the growing field of knowledge characterizing anomalies of quantum field theories in terms of mathematical concepts like bordism etc. I wholeheartedly recommend the publication of this paper in SciPost. I have a small complaint however:

In footnote 1, the authors say that the anomaly theory need not be invertible, citing the example of 6d (2,0) theory. Though this might be a sensible statement from the mathematical definition of "anomaly theory" used by the authors, this is not completely sensible from the standard definition used in physics literature. A 't Hooft anomaly is characterized by anomalous phase of the partition function in the presence of non-trivial symmetry background. However, the partition function is well-defined when there is no symmetry background turned on. On the other hand, for the (2,0) theory the partition function with no symmetry background (for the 2-form symmetry) turned on itself is ill-defined. Thus the (2,0) theory has a problem "without the introduction of symmetry into the system" and this is not traditionally referred to as an anomaly. The authors might want to consider clarifying this in the introduction and/or the footnote.

In this paper the authors give a modern reformulation (using bordism) of the old "Elitzur-Nair" method for computing global anomalies by embedding into a bigger groups with local anomalies. This was an important missing result in the recent developments in the subject, and the paper gives a very lucid and clear account of how the old developments should be rephrased in a modern (and better) way.

It is an interesting and well written paper, and I would be happy to recommend publication once the authors clarify some minor points (I am omitting the points that the other referee has already brought up in their review):

16. In pg. 10, in the last paragraph, it should be "... the vanishing of $Ext^1(\Omega_{d+1}^{H_2}, \mathbb{Z})$ implies $Ext^1(\Omega_{d+1}^{H_2}=0$..." (so $H_1\to H_2$).

17. At many points in the paper (starting in pg. 23, as far as I can tell) the authors write $H^6_{I\mathbb{Z}}(H)$ with $H$ the symmetry group, instead of $H^6_{I\mathbb{Z}}(MTH)$. Are these two the same object? If not, I would encourage the authors to switch to the second form everywhere (I think that the most natural interpretation of the first form would be the homology of the suspension spectrum of $H$, which is presumably not what the authors intend).

18. In the beginning of section 6 the authors write "To our knowledge, [ElitzurNair] was the first instance in which a local anomaly was essentially 'pulled back' to derive a global anomaly,...". In fact, Elitzur and Nair credit Witten with the initial idea, in his "Global Aspects of Current Algebra paper". It might be fair to cite this reference too.

In this paper the authors study how the anomaly of a fermion system with symmetry $G$ looks when only the subgroup $H\subset G$ is considered, when $G$ can have nonzero anomaly polynomials while $H$ can only have global anomalies. This question was first studied in the 80s by Elitzur and Nair using the homotopy group of $G$, but the authors revisit this question using the modern understanding of the anomalies based on bordism groups.

The manuscript is well presented and can be recommended for publication, once the following points are taken care of.

1. Referring to the (Anderson/Pontryagin) dual of the bordism group as the cobordism group is often done in recent physics literature but the referee considers it still as a non-standard usage which can lead to confusion. Historically, what are now called as bordism groups were called cobordism groups. (Two manifolds can be called cobordant or bordant, interchangeably.) Later, the math community shifted to call them bordism groups, most probably because cobordism groups are found to be generalized homology groups (rather than generalized cohomology groups) associated to Thom spectra. One could have then repurposed the name "cobordism groups" to the generalized cohomology theories associated to Thom spectra, but apparently it wasn't done very often.

2. The authors use the symbol $H_n$ for two closely related concepts concerning symmetry types, which the referee found too confusing. Namely, $H_n$ is a sequence of groups specifying the symmetry type, whose limit is $H$. Then, the authors use $H_1$ and $H_2$ to refer to limits of two such sequences $(H_1)_n$ and $(H_2)_n$ respectively. This makes it unclear what is meant when the author simply writes $H_i$. The referee would suggest the authors to use $H$ and $H'$ instead for the second purpose.

3. in p.5, the authors refer to "the Anderson dual of the sphere spectrum $I\mathbb{Z}$." Those who are unfamiliar with these matters won't be able to tell whether $I\mathbb{Z}$ is the sphere spectrum or the Anderson dual of that. It would be better to write the Anderson dual $I\mathbb{Z}$ of the sphere spectrum.

4. Later in p.5, the authors say gravitational Chern-Simons terms in even dimensions are obvious counterexamples, but why? In the 2d/3d case, a chiral 2d fermion can live on the boundary of the gravitational Chern-Simons term, and similarly in higher dimensions, at least for a particular choice of gravitational Chern-Simons terms corresponding to the $\hat A$ genus.

5. In p.7 footnote 7, the authors say that the appearance of line bundles is what relates the fermion anomaly to K theory, but this is doubtful. Many of the discussions in [24] or [3] applies to any combination of the boundary theory and the anomaly theory, resulting in the fact that the partition function of the boundary theory is a section of a line bundle which is specified by the anomaly theory. This is independent of whether the theory in question is a free fermion theory or not. Yes the K-theory is defined in terms of vector bundles but it requires not just line bundles but vector bundles with general dimensions.

6. This is simply a comment. In p.12, the authors say that Elitzur and Nair preceded the modern understanding of global anomalies via bordisms. But Elitzur-Nair is from 1984 while Witten's "Global Gravitational Anomalies" is from 1985, and the latter essentially contains the statement that the global anomalies are captured by bordism invariants whose prime examples are eta. So it is not perfectly correct to say that the understanding of (global) anomalies via bordisms is modern. It is just that its popularization is modern.

7. In Sec 4 and later, the authors provide many explicit examples. There is one general problem in the presentation, when the authors describe the generators of the free part of the extension sequence, which is of the form $\mathbb{Z}^n$. This is done by the authors by finding $n$ anomaly polynomials which correctly integrate to integers. But this argument only guarantees that the authors found a subgroup $A\subset \mathbb{Z}^n = \mathrm{Hom}(\pi_{d+2}(MTH),\mathbb{Z})$ such that $A\simeq \mathbb{Z}^n$, but $A$ can be a proper subgroup.

The referee believes this did not actually caused the problem, because of two reasons.
First, the pull-back of this (possibly proper) subgroup $A$ was enough to reproduce the global anomaly in most of the cases (except in the 2d case where the $\mathbb{Z}_2$ anomaly of Majorana-Weyl fermion was not covered).

Second, the group $A$ found by the authors was actually equal to the full $\mathrm{Hom}(\pi_{d+2}(MTH),\mathbb{Z})$ in all the cases in this paper, but this is something one needs to be checked.

8. In p.14, the authors might want to say that the splitting $\Omega(X)=\Omega(pt)+\tilde\Omega(X)$ is canonical.

9. In p.16, the authors describe a generator of $\mathbb{Z}_8$ as a mapping torus and then find its null bordism. The referee wants to mention that the null bordism of $\mathbb{RP}^3$ where a $\mathbb{Z}_2$ connection is embedded in $U(1)$ is also standard. Namely, $\mathbb{RP}^3$ is the asymptotic infinity of the simplest type of the ALE space (originally found by Eguchi and Hanson), which is topologically a disk bundle over $S^2$. See e.g. Sec.4.1 of http://arxiv.org/abs/hep-th/9605184 . There, the integral $\int F\wedge F/(2\pi)^2=-1/4$ is also given in (4.6); note that the normalization of $F$ is different by a factor of two there.

10. The referee did not understand the footnote 19 in p.18. Isn't it that the author considered a U(1)-charged Weyl fermion? Then there are only charged right-movers. If so, when $U(1)$ is restricted to $\mathbb{Z}_2$, it seems to simply give two copies of $\mathbb{Z}_2$-odd Majorana-Weyl fermions.

11. In p.20, could you clarify how $\mathcal{F}$, $f$ and $F$ are related? There can be multiple conventions imaginable. Also, it was not clear why the authors use $(1/2)(f/2\pi)$ instead of $f/(2\pi)$ for the first Chern class.

12. In p.22, a related issue is that the authors' human language explanation of the $U(1)$ flux on the hemisphere was too ambiguous.
When the authors say a monopole flux on $S^2$, do they consider a spherically symmetric configuration, or a configuration concentrated on a single hemisphere? Namely, the referee could not figure out whether the authors intended to have $\int_{S^2} f=\int_{D^2} f$ or $\int_{S^2}=2\int_{D^2} f$. Also, the referee could not figure out whether the minimal Dirac flux corresponds to \[
\int_{S^2?\ \text{or}\ D^2} (\frac{f}{2\pi}\ \text{or}\ \frac12\frac{f}{2\pi}) = 1.
\]
Please clarify.

13. In p.37, footnote 37, the authors say that $\sigma$ doesn't correspond to a chiral fermion. But $\sigma-21\Phi_8$ is the anomaly of a gravitino, which is a chiral fermion of spin $3/2$.

14. In p.45, (A.13), please say that $B(\mathbb{Z}/2^m))^{2\xi}$ is the Thom space of the bundle $2\xi$.

15. In p.51, the URL for [48] was listed twice.