Cancelling mod-2 anomalies by Green-Schwarz mechanism with $B_{μν}$

We are very grateful to the referees for constructive comments. We list our replies to the referees' comments and the improvements made to the draft. We hope that the article is now acceptable for publication.

Reply to Referee #1:

1. The typo was corrected by replacing "the relation" by "the relations".

2. We added a concluding section.

Reply to Referee #2:

The referee asked us a few questions within the "Report" section. Let us first respond to these.

• The first question was whether our method is applicable to $\mathbb{Z}_k$ anomaly for $k\neq 2$. Indeed, our method does not require that the anomaly is mod 2. The same analysis applies equally well to other mod k anomalies, and in fact to the perturbative anomalies as well, since the bordism formulation of anomalies is very general. We added a few comments on this issue in the new Sec. 6.

• The second question was whether we can treat the effects of discrete $p$-form gauge fields on the anomalies in the same manner. Indeed, our method is general in that we can analyze the effect of the introduction of discrete $p$-form fields to the anomaly by constructing a fibration $Q$ of $K(\mathbb{Z}_k,p)$ over $BG$ and studying the bordism group of $Q$. A few comments on this issue is now in Sec. 6.

• The referee then asked whether the introduction of discrete $p$-form gauge fields can cancel Witten's $SU(2)$ anomaly in 4d. This is an interesting question. There is a physics argument saying that Witten's SU(2) anomaly can never be canceled without additional massless fields, given in Sec. 5 of https://arxiv.org/abs/1710.04218 . This translates to mathematical predictions concerning bordism groups of fibrations $Q$ of $K(\mathbb{Z}_k,p)$ over $BSU(2)$ never vanishes. It is not hard to confirm this prediction. We added a new Appendix C to explain this point; we would like to thank the referee for asking this very interesting question.

• The referee's next question was about the relation to the shifted Wu structure and Wu-Chern-Simons theories used by Monnier, Moore and collaborators. We should mention that Monnier et al. used the shifted Wu structure and the Wu-Chern-Simons theories to formulate chiral $p$-form fields, whereas our intention in this paper is to study non-chiral $p$-form fields. As chiral $p$-form fields cannot be formulated by a path integral over a modified configuration space, the method we developed in the paper does not apply. Therefore, we say that there definitely is a relation, but the relation is distant. Again we made some comments in Sec. 6.

• The next comment was that our description of the generator of $\Omega^\text{spin}_5(BSU(2))=\mathbb{Z}_2$ at the beginning of Sec. 3 was too terse and also confused. We agree with the referee's assessment here. We added more discussions in Sec.2.1 when these bordism classes were first treated, and we referred to Sec.2.1 from Sec.3.

• The final question within the main "Report" section was the outlook for using the methods described in this paper to other situations within string theory or hep-th in general. This is addressed in the new concluding section, Sec. 6.

The referee also requested to change another confusing sentence at the end of Sec.2.3. This was corrected as suggested by the referee.