A domain wall in twisted M-theory

1) The paper provides a conjecture for the twisted holography of a KK monopole in twisted M-theory, which is seemingly the last BPS defect that remained to be considered. Moreover, the conclusions pose further physics questions.
2) The paper uses techniques from recent developments in the formal study of operator algebras of QFT, including higher products.

1) Despite the presence of a clear introduction, containing a review of the status quaestionis, the paper refers with no explanation to a few technical mathematical results that are probably not familiar to most of the readers. It would be good to at least acknowledge this.
2) The presentation of the paper could be slightly improved.

The paper considers a well-defined question in the growing field of twisted holography, and solves it. The techniques used in solving the problem belong to parts of formal QFT that are deservedly receiving increasing attention by the community. There are a few typos and unclear parts, but subject to corrections to these minor points, I would recommend publication.

1) In Section 4.3, the authors find the quantum deformation of the algebra of operators on the domain wall by relying on a previous result by Costello. Whilst in the Introduction it is mentioned that Costello's result is mathematically rigorous, here the authors refer to it both as a proved and a conjectural isomorphism. Could they please clarify this point? More generally, since this is the concluding part of the paper, I would welcome a slower pace in the exposition, stating clearly what is mathematically proved and what is physically conjectured (perhaps in the same vein as the Introduction), especially since twisted holography is an area where results inspired by physics can be rigorously proved algebraically. Finally, it seems curious that the algebras for the M2 and for the KK monopoles are the same, and the authors could consider highlighting this more.

2) The protected subsector for the field theory associated to the KK monopole is described by a field theory in the same dimension as the original one. Is there a physical interpretation for this?

3) Some additional minor clarifications:
- p. 9 +9 "massless D6-D6' strings" I think that strings can be tensionless, and they may have massless spectrum. Do the authors mean the latter?
- p. 9 -11 as the authors point out, this secondary product is related to the descent procedure introduced by Witten. Here the product is not referred to as "secondary", whereas later on it is without referring to the introduction (p. 11 above (30)). I would suggest introducing the terminology "secondary" at p. 9 or drawing the relation at p. 11.
- p. 11 +5 I didn't find this description easy to understand. I would rephrase it as something like "We can construct it by first applying... in the Q-cohomology. The resulting $\mathcal{O}^{(1)}$ is not Q-closed... If we then wedge..."
- p. 12 caption to Fig 2. I think that $\mathcal{O}^{(1)}_2$ is a one-form, so it's $\Omega \mathcal{O}^{(1)}_2$ that is supported on $S^3$
- p. 13 below (35) I found the mention of the derived setting a bit off-hand: is that crucial or is that only a nod to the expert reader?
- p. 15 -3 Is it correct to say that the boundary conditions in (52) are equivalent? If so, could you mention it explicitly?

4) There are numerous typos and minor grammatical mistakes throughout the paper. Here is a list of some
- there should be a space between word and bracket, starting from page 2 line 10
- Kahler and hyperKahler should have a umlaut on the a
- p. 2 +14 "Twisted M-theory is an eleven-dimensional..."
- p. 2 -12 "M5 branes should wrap a four-cycle in the..."
- p. 3 +18 I think it would be clearer as "reducing along a circle in the $G_2$ manifold" (same at p. 5 +3, p. 5 + 15, p. 7 -7, p. 7 -6)
- p. 3 -12 something like "We will describe in more detail this set of operators..." (no about), same in p. 11 - 9, p. 23 -2
- p. 4 -14 "squares not to zero but to $\epsilon V$"
- p. 4 - 7 "The background induced by $\Psi_\epsilon$..." do the authors mean that it makes the dependence on $\mathcal{M}_7^T$ topological and $\mathcal{M}_4^H$ holomorphic?
- p. 4 footnote 2 "Appendix"
- p. 5 +6 "7d maximal SYM"
- p. 6 -10 "defects cannot engineer" (same elsewhere)
- p. 7 -12 "this new geometry $\widetilde{\mathcal{M}}_7^T$"
- p. 8 eqn (13) Usually "rotations" have determinant 1: is there a particular reason why the authors took det -1?
- p. 10 above (22) "one may act with $\mathbf{Q}$..."
- p. 10 -6 "complex one-dimensional holomorphic theory"
- p. 10 footnote 7 "distinguish it from the secondary product"
- p. 12 below (33) "The Bochner-Martinelli kernel..."
- p. 14 -6 "We call $dz \wedge dw \wedge J^{(1)}$ a higher current."
- p. 15 -8 " We need to make sure of the locality of the... let us vary the action obtaining... Using Stokes' theorem... boundary conditions which make the boundary variations (51) zero."
- p. 16 +2 "valued in the Lie algebra"
- p. 18 -6 "The reader who is interested in the detail of the calculation may..."
- p. 22 +8 "is reflected in the change of the original structure constants..."
- p. 23 +4 "isomorphism between the boundary algebra and"
- p. 23 +13 "The intrinsic degree..."
- p. 23 -2 "what we mean by..."