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nLab > Latest Changes: group-completed configuration space of points

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2024

    some observations on Okuyama’s model for the group completion of configuration spaces of points in terms of configurations of open strings with charged endpoints.

    (previous discussion here)

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 24th 2024

    added (here) the previously missing argument that all framed links are cobordant to framed unknots.

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2024

    a pdf/LaTeX rendering of this discussion is now also in section 4.1 of the draft kept here

    diff, v11, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 31st 2024

    Behind the talk of miracles and the miraculous, can one conceive of an explanation?

    The humongous cancellations that happen to make this work seem no less short of a miracle, quite reinforcing the idea that 11d supergravity occupies a special point in the space of all field theories.

    (By the way, it sounds a little odd. Maybe, “…to make this work seem nothing short of a miracle…” or “…to make this work seem nothing less than a miracle…”)

    I remember André Joyal giving as an example of a miracle the fact that the algebraic closure of the reals just required a simple extension of degree 2.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2024

    Thanks for catching the broken wording, I have fixed it now.

    On the “miracles”: The point of using the word is to indicate that it is lacking an “explanation”!

    One expert told me that the reason the original authors didn’t check this was “probably because of an argument on counting of supersymmetries”. But I don’t see that this is a valid argument. So it remains a miracle that our mechanical computer algebra agrees after more than 10 minutes of crunching what must be millions of terms.

    We don’t mean to belabor the point in itself, but in this article we do need to explain why we check explicit super-embeddings where previous authors were content with looking at the usual equations of motion. It seems this way they missed the critical radius.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 31st 2024

    Ok, so I guess I’m wondering whether an explanation might be forthcoming. Presumably there are historical cases where what appeared to be meaningless calculations later can be seen to be explicable. But then how could one know in advance?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2024

    I don’t see that the miracle will be explained away. Doing these superspace computations is the closest I have come to feel interacting with an ulterior entity.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 16th 2024
    • (edited Aug 16th 2024)

    Thinking now about generalizing the result — which, under the equivalence of ΩΩ 2S 2\Omega \Omega^2 S^2 with the loop space of Okuyama’s configuration space Conf I(ℝ 2)Conf^I(\mathbb{R}^2), identifies the Hopf generator with the framed link that is the 1-framed unknot — to ΩΩ nS n\Omega \Omega^n S^n for n≥2n \geq 2.

    Since the Hopf generator becomes a 2-torsion element for n≥3n \geq 3, one simply needs to show in this case that the 2-framed unknot in ℝ n+1\mathbb{R}^{n+1} is framed cobordant to the 0-framed unknot.

    My guess is that this is probably (1.) a well-known statement which (2.) works essentially by the “belt trick” move. But I have not yet found it in the literature.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 28th 2024
    • (edited Aug 28th 2024)

    Today Shingo Okuyama kindly informed me by email about the existence of this set of slides, which I had missed before:

    • Shingo Okuyama: Configuration space of intervals with partially summable labels, talk at National Institute of Technology, Kagawa College (2018) [pdf]

    These slides show very similar pictures (p. 23-30 and pp. 52) and at the end claim an argument answering my question in #8 above.

    diff, v14, current

1 to 9 of 9

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