Unfolding $E_{11}$

1. The paper contains new results
2. The unfloded formulation is clearly and nicely explained
3. The paper is well-written and the results are clearly explained

1. A few statements are conjectural
2. In some cases the conjectural nature of the claims in not clearly stated

The paper deals with the unfolded formulation of the equations for the fields that occur in the E11 non-linear realisation. This gives rise to an infinite number of fields that do not belong to E11, and the authors conjecture a possible origin of such fields as representations of E11. Using the unfolded formulation, one constructs gauge invariant quantities which the authors use to propose at linearised level an infinite set of duality relations between the fields in E11. I am concerned about how the duality relations arise in E11 already at the level of the 3-form and 6-form. These relations are consistent with the E11 symmetry, but as far as I understand are not really derived from the dynamics, but rather imposed. I would like the authors to make more clear comments on this issue, which I think is very important and would be beneficial for the paper.

1. On the third paragraph of the introduction, I would like the authors to explain in more detail how the duality relations can be read or derived in the non-linear realisation.
2. I would like the authors to motivate more the statement below eqs. (4.45), (5.10) and (5.29) that the proportionality coefficients in the duality relations are fixed by E11 symmetry.

Ask for minor revision

1. fascinating subject matter generally
2. interesting open question addressed

1. introduction too roundabout, not to the point
2. presentation of results tends to be somewhat imprecise

The aim of the article, first alluded to on its p 6, is to clarify aspects of the (Hodge-)duality structure of fields in and beyond the E11-formulation of 11D supergravity by "proposing" suitable duality relations in an "unfolded" formulation of the (expected) linearized equations of  motions for these fields.

The bulk of the article goes iteratively through low-level examples of the field components, each time (1.) "introducing a set of variables", then (2.) stating "unfolded" equations of motion for them  and (3.) "proposing" duality relations. My understanding is that thisn each case the implied second-order EoMs are found to match expectations obtained from elsewhere, which thus justifies, a posteriori, the Ansatz of "variables" and the proposed duality relations. on them.

But the precise rules of the game which the authors play and the conclusions which they draw could be stated more explicitly and more up-front: Sections 1 & 2 are mainly a broad review of the E11 program in general, of the kind one might write for a grant proposal, rather than a concrete motivation of and introduction to the bulk of the article.  Much work is done in the text by pointing the reader instead to reference [17].

While therefore I cannot quite commit to judging the success that the authors have with their program (though I have no reason to doubt it), I can attest that the broader program of "unfolded EoMs subject to duality relations" is indeed most central for 11d SuGra due to the following result (not mentioned in the manuscript):

With the full field content of 11D sugra considered (hence crucially including also the gravitino, which the present authors seem to disregard) and thus working on super-spacetime, it turns out that the full (not just linearized) unfolded EoMs of the superized 3-form field (in other parts of the literature  known as the "duality-symmetric" formulation of the C-field Bianchi identity) already implies *all* of the SuGra equations of motion, to all order and including the duality between the 3-form and the 6-form. [Thm . 3.1 of doi:10.1007/JHEP07(2024)082]

For this reason I am sure that the manuscript under consideration is onto something important and eventually worth publishing. I would ask the authors to add a clearer description of their accomplishment: The text should make it clearer: What exactly is assumed, what exactly does it mean to "propose" in this context and how exactly is these proposals are being vindicated.

Besides the points raised in the report, here is a list of minor comments, going linearly through the text:

p. 5:  grammar: "on empty column"

p. 6: "In this paper..." This might better be said much earlier

p. 6: "we apply the unfolded formalism [55,56]": 
not sure that the terminology "unfolded" already appears in [55,56].  Better to also point to [63,64] or similar, for clarity.

p. 7: grammar: "the E11"

p. 7:  "we write \Psi_{4|3,2,2,1}" the last ":
"1" seems to be a typo and should be omitted

pp. 8, 9: the statements about the fields (such "is a higher dual", or "plays a role in gauge supergravity") should be referenced

p. 9: Sentences like 
  "The full non-linear equations for the fields follow uniquely from the non-linear realisation."
do not quite parse. An equation can follow from another equation or generally from another logical statement, but not from a mere thing (like a coset space aka "non-linear realization").

p. 13 "although it is very well known":
still, give a reference

p. 14: "The first two unfolded equations are...":
It may be worth saying *why* it is the case that this "are the first two equations". I gather it is an Ansatz that is justified by its implications. Generally, the rules of the game of "unfolding", as used here, would be worth stating more explicitly.


p. 18 equation (3.40):
here it may be worth re-amplifying that on the right the equation for F7 is only shown to linear order: To higher order it is not F7 itself but  only the non-linear combination F7 - 1/2 A3 F4  that has a 6-form potential.

p. 18 equation (3.24):
here and from looking at reference [17], it seems that the objects F^(n) must be understood as Taylor coefficients around a chosen spacetime point.(?)
Which would mean that the claims about reproducing the (linearized) equations of motion of SuGra pertain only to a formal neighbourhood of any one point.
This needs clarification.

p. 20 "its descendants do not need to be completely traceless on-shel" 
Best to add the argument or reference for how this claim comes about.

Ask for major revision