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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
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)
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?
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.
But so the connection to the discussion in say 0410293 hasn’t been expanded on yet?
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 $\mathbb{C}P^1 \times \mathbb{C}P^1$. 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.
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?
The Bianchi identity for the NS5-brane flux ($d H_7 = - \tfrac{1}{2} R_4 \wedge R_4 + R_6 \wedge R_2$) is the genuinely non-linear component in (the Whitehead $L_\infty$-algebra of) $\mathcal{L} S^4$, whose non-linearity cannot be absorbed into the twisting/slicing by $H_3$. 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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