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    • CommentRowNumber1.
    • CommentAuthorEugenioLandi
    • CommentTimeMar 25th 2020
    • (edited Mar 25th 2020)

    Together with my colleague Ph.D. student Mattia Coloma and with Domenico Fiorenza we have written a short note on the graded Hori transform following the latest paper of Fei Han and Varghese Mathai. We show how, in the rational setting, the graded Hori map can be naturally seen as a “pull-iso-push” transform.

    You can find it here, any comment before we upload it on arxiv is welcome.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2020
    • (edited Mar 25th 2020)

    Interesting. Will have a look tomorrow. Just to point out that your reference FSS17 has been published, the full publication data is here: T-duality in rational homotopy theory via strong homomotopy Lie algebras

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2020
    • (edited Mar 26th 2020)

    Some random and trivial comments:

    \,

    p. 1: “We will construct this pull-iso-push transform using only…”

    Maybe there is room left to expand on explaining to the reader the relevance of the result presented. I gather the idea is that the proof in [HM20] of Theorem 2.2 there secretly repeats steps that are already known to be true, if written down more suggestively; and you are tidying up the scenario to bring this out.

    Since your article in total is, while short for an article, considerably longer than the proof of Theorem 2.2 in [HM20], it might help the reader to offer them some guidance as to what simplification you mean to point out.

    I think your point is as follows:

    Up to and including Section 3 you review topological T-duality in rational homotopy theory according to [FSS17, FSS18].

    Then you observe that in this form (as opposed to traditional form) a range of generalizations are fairly immediate:

    Section 4: Observation that, in this form, we may immediately generalize to Laurent series.

    Section 5: Observation that, in this form, we may easily tensor with further algebras, such as that of meromorphic functions.

    Section 6: Observation that, in this form, we may easily tensor with Jacobi forms.

    Right? If this is the logic of the article, it might be worthwhile to make this more explicit in the introduction.

    \,

    p.1 “we address the unfamiliar reader to [FSS17]”:

    probably better: “we point…” or similar

    \,

    p. 13 “a completed and detailed account”

    probably better: “a complete and detailed account”

    • CommentRowNumber4.
    • CommentAuthorEugenioLandi
    • CommentTimeMar 26th 2020
    • (edited Mar 26th 2020)

    Thank you, we have followed your suggestions. The latest version can be found at the same link.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeMar 26th 2020

    Some typos left:

    p. 1: “asscoaited”

    p. 1: “a manifestation of [a] canonical equivalence”