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nLab > Latest Changes: D-brane charge quantization in K-theory

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2020
    • (edited Feb 12th 2020)

    am finally giving this its own entry, to be split off (not done yet) from D-brane charge and to be in parallel with K-theory classification of topological phases of matter

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2021

    finally expanded out the Idea-section a little more.

    Changed page name from “K-theory classification of D-branes” (which sounds like there is a classification theorem, which there is not) to “D-brane charge quantization in K-theory” (which hopefully sounds more like a prescription than a theorem)

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 31st 2021

    Now some more material in place. I could go on, but I leave it at that for the moment.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorperezl.alonso
    • CommentTimeSep 10th 2023

    If one tries to construct the equivalent to Remark 4.9 in 1606.03206 for the IIB string, where does the problem of S-duality show up?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2023
    • (edited Sep 10th 2023)

    By the way, S-duality in terms of these super-cocycles but without the twisting is paragraph 4.3 in arXiv:1308.5264.

    This mixes the cocycles for the F-string and the D1-brane. The problem with S-duality and twisted K-theory is that the latter assign distict roles to these two cocyles:

    The F1 cocycle serves as the twist while the D1-cocycle is among those being twisted. Therefore S-duality is not a symmetry on twisted K-theory as usually understood.

    And it’s not just that it were a transformation which fails to be invertible, but it just doesn’t act on the structure: To twisted K-theory the NS B-field and the RR-fields are apples and oranges, but S-duality would have to relate them to each other.

    • CommentRowNumber6.
    • CommentAuthorperezl.alonso
    • CommentTimeSep 10th 2023
    • (edited Sep 10th 2023)

    But so the connection to the discussion in say 0410293 hasn’t been expanded on yet?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2023
    • (edited Sep 10th 2023)

    Not that I am aware of. But, I think, another attack on this question was the “mysterious triality” papers: In the mysterious duality by Iqbal et al., S-duality is witnessed as the exchange of the two factors in ℂP 1×ℂP 1\mathbb{C}P^1 \times \mathbb{C}P^1. So if one knew how these (blow-ups of) del Pezzo surfaces correspond to the classifying spaces appearing in Hypothesis H, then one could maybe see how S-duality wants to act on some form of cohomotopy.

    • CommentRowNumber8.
    • CommentAuthorperezl.alonso
    • CommentTimeSep 11th 2023

    Noted.

    On a similar subject, in 1806.01115, what does it really mean that the NS5 brane cocycle drops out when the D6 and D8 cocycles appear?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2023

    The Bianchi identity for the NS5-brane flux (dH 7=−12R 4∧R 4+R 6∧R 2d H_7 = - \tfrac{1}{2} R_4 \wedge R_4 + R_6 \wedge R_2) is the genuinely non-linear component in (the Whitehead L ∞L_\infty-algebra of) ℒS 4\mathcal{L} S^4, whose non-linearity cannot be absorbed into the twisting/slicing by H 3H_3. Since no non-linear terms appear in the Whitehead bracket of a spectrum, this term has to disappear under fiberwise stabilization, and it does. Physically this is the common (though rarely or never explicitly stated) fact that this NS5-brane charge is indeed ignored in the K-theoretic quantization of D-brane charge.

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