Looking for structure in the cobordism conjecture

Dear Editor,

We would like to thank both referees for the time they invested in carefully reading the manuscript, and for their valuable feedback.

Below we discuss our main modifications to the draft in order to address each of the points raised by the referees. We also tried to make every change to the original version easily identifiable by marking it with "%arXiv_v2" in the new tex file (available at https://arxiv.org/abs/2204.00021).

*** Referee report 1

1) Our definition of the bordism group \Omega_k^{j_m} indeed only captures the charges of the currents j_m. To clarify this, we added two sentences starting "Hence by definition..." at the end of the paragraph which introduces \Omega_k^{j_m}.

2) The referee is correct in their interpretation, and we agree that our explanation should be improved. As pointed out by the referee, the key point is indeed the possible multiplication by integers. To clarify the reasoning, we then added on page 26 the words "first consider only F with int_W \d F = n_j, and we", as well as "(with \int_W \d F = n_j)", and then rewrote most of the text between the Figure (4.14) and the result (4.16).

3) It was not our intention to imply that there is only one structure QG satisfying the cobordism conjecture. For example, on page 3 we called it a "necessary condition on the unknown structure QG". To make it explicit that QG may not be unique, we changed "structure" to "structure(s)", and we added the following sentence at the end of that paragraph: "Note that the cobordism conjecture does not make a statement on whether or not \Omega_k^{QG} = 0 has more than one solution."

Finally, to the last suggestion of the referee: we thank the referee for this remark. We agree that the presentation of Section 2.1 contains some fairly technical elements. In fact we worry that this review section is already quite long, and we would prefer not to add further explanations. To motivate the discussion of tangential structures, we had included the paragraph about fermions and spin on page 7; and we are confident that readers interested in the cobordism conjecture are familiar with G-bundles, e.g. from their role in the standard model of particle physics.

*** Referee Report 2

1) We agree with the referee that the explanation about gauging in Section 2.3.2 is not clear. We rephrased it along the lines suggested by the referee. In particular, we did not introduce any additional current j_{k+1}.

2) Indeed, there is a typo in the draft at the top of page 18. The correct expression should be 1/6 p_2(TM). (1/2 p_1(TM) is the obstruction for lifting to a string structure, while 1/6 p_2(TM) is the obstruction for lifting to a fivebrane structure.)

3) We thank the referee for the valuable remark. A first comment is that the Kaluza-Klein monopole is viewed here as a stringy object, as D-branes are, and as such, it equally needs to be predicted by the cobordism conjecture. The question is then: what bordism group should it kill? We agree with the referee that at first sight, since it is a pure gravitational object, one may think of a bordism group with a structure that appeared already for pure geometries, like spin structure, etc. Whether this is true or not, it remains a fair question to ask what is precisely this group, or in other words, how this object gets predicted, and this is the meaning of the question asked in (4.17). We call it a "problem", because as emphasized in the beginning of the section, the fact there is no obvious U(1)-gauge flux, contrary to other branes, makes it less clear what bordism group to consider. This is the motivation.

It turns out that viewing it as a 6-dimensional stringy object, in a 10d spacetime, makes (more) manifest the presence of a U(1)-bundle (partially) in its transverse dimensions. As argued at length in the section, this is what drives us to consider a bordism group with a structure including a U(1)-bundle, and not just a spin structure. This is in addition supported by T-duality. This remains of course a proposal that could be challenged in future works.

To clarify the situation, we added the following two sentences after (4.18):

"As will be explained, viewing the KKm as a 6-dimensional stringy object in a 10-dimensional spacetime makes manifest the presence of a U(1)-bundle. It will lead us to consider the above bordism group, making it eventually closer to other branes than initially thought."

- at page 3, we changed "structure" to "structure(s)", and we added the following sentence at the end of that paragraph: "Note that the cobordism conjecture does not make a statement on whether or not \Omega_k^{QG} = 0 has more than one solution."

- at page 3, we added reference [20];

-at page 13-14, we modified the second part of section 2.3.2, from "We now turn to gauging" until the end of the section;

-at page 18, we fixed a typo: 1/2 p_2 --> 1/6 p_2;

-at page 26, we added two sentences starting "Hence by definition..." at the end of the paragraph which introduces \Omega_k^{j_m};

-at page 26-27, we rewrote parts of the text from "We first discuss a property of the bordism group..." until to the end of the section;

-at page 28, we added two sentences after (4.18) ;