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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.
In 1806.01115 you show how K-theory emerges from double-dimensional reduction by taking $G=U(1)$. Is it obvious what happens when one instead takes $G=U(1)^7$, compactification to four dimensions?
Certainly not obvious to me. But in principle the same logic also applies in other cases, one would have to work it out…
Here’s something that confuses me about the result of that paper. In physics one sometimes sees that if something is classified in cohomology in $d$ dimensions, then there is usually some lift to $d+1$ where the classification now happens in K-theory (e.g. math/0611945, or even the literature of quantum cohomology lifting to quantum K-theory) (I also recall seeing some discussion of lifting to $d+2$ to some elliptic cohomology, but can’t find the reference rn). The obvious generalization of this (that should be proven) is that cocycles of an $S^k$ bundle $X$ with base space $M$ in the $n$th stage of the chromatic tower of $H\mathbb{C}$, then one obtains cocycles of $M$ in the $(n-k)$th stage. But here we are starting with a cocycle of $X$ in $MU$, so I would think that doing dimensional reduction to $M$ corresponds to a cocycle, let’s say, in the $\infty -1$ stage, again $MU$. So why does one obtain K-theory in this paper? Is it because we are taking the first-order approximation (in the sense of Goodwillie), so that a second-order approximation would give something in elliptic cohomology? Wouldn’t this imply that computing all approximations would again give a cocycle of $M$ in $MU$?
If this assumption that lifting in circle bundles allows to lift cocycles to stuff higher up in the chromatic tower is correct, then assuming Hypothesis K would imply 11d M-theory indeed corresponds to something elliptic (as Hisham suggested years ago), and this still leaves room for further lifting to higher dimensions, e.g. for F-theory and beyond. Now since there are some problems with Hypothesis K then would it be relatively easier to produce a hypothesis for the 4d theory and then use this idea to lift cocycles to 10d?
I know what you are thinking, but the usual argument with the chromatic tower is rather broad-brush, even as hand-waving goes. In that article we were trying a much more concrete analysis.
By the way for some reaon you write:
here we are starting with a cocycle of $X$ in $MU$
but this is not what “we” did:
We started in 11d with a cocycle in unstable 4-Cohomotopy, since this (but no abelian cohomology theory, be it elliptic or otherwise) can quantize the C-field flux in 11d.
Then we observe (in “Rational sphere valued supercocycles in M-theory”, arXiv:1606.03206) that double dimensional reduction turns such 4-cocycles into those of an unstable cohomology theory in 10d which, rationally, is a kind of truncation of twisted K-theory, in that it does not see the D8-brane.
This is actually quite satisfactory a result already, since also 11d sugra compactified to 10d does not actually give the D8. The D8 lives in massive type IIA string theory, instead, whose M-theory lift, if any, is rather obscure.
This major issue notwithstanding, the traditional “Hypothesis K”, without much ado, lumps the D8 into the mix in order to justify its conclusion.
What we tried in the arXiv:1806.01115 that you point to is to find a natural homotopy-theoretic “reason” for this step.
Independent of our particular answer, I want to highlight that the issue is crucially an “unstable” one which cannot possibly be properly resolved by looking at any tower of abelian (i.e.: Whitehead-generalized) cohomology theories.
Even though spectra, Whitehead-generalized cohomology and the chromatic tower are magnificently rich mathematical structures, they are nevertheless a very coarse approximation to the full non-abelian homotopy theory evidently governing M-theory.
Of course, one will still want to understand how and to which degree this approximation is useful. The general idea is that the approximation is given by unstable versions of the Boardman homomorphism, now going from an unstable sphere to a spectrum or to a bundle thereof, and thought of as approximating the unstable Cohomotopy by whatever it is that is still seen in that spectrum.
We comment on possible approximations of twisted 4-Cohomotopy by quasi-elliptic cohomology on p. 9 of Cyclification of Orbifolds. We recently had Zhen Huan visit us at NYUAD for a month, making heavy computations concerning this approach. We are planning to write this up, but it may take until after January that we get around to doing so.
Thanks for the comment, Urs. Regarding this remark about instability of yours, can this not be applied to Hypothesis K already? That is, can one not consider “unstable K-theory”, which I’m guessing would be about maps to some $U(n)$, e.g. $U(5)$? It’s funny because rationally one can take $U(5)$ as a product of odd-dimensional spheres $S^1\times S^3\times S^5\times S^7\times S^9$, which sounds somewhat reminiscent to D-brane content.
Yes, one absolutely can consider forms of unstable K-theory. The cyclic loop space of the 4-sphere classifies such an unstable twisted K-theory which, rationally, knows everything about the NS/D-brane content obtained by dimensional reduction from 11d SuGra. This is what Hypothesis H gives verbatim upon dimensional reduction to type IIA. I guess you are asking whether this answer of Hypothesis H is all there is, and whether the traditional “Hypothesis K” that this answer should be stabilized is ultimately just wrong. Could be! :-)
Regarding the observation about $U(5)$: Interesting that you mention this, Hisham was urging to push in this direction. But I wasn’t and am not sure what to do about it. Maybe I am being dense.
It would be fun to find physical interpretations to the image of the cocycles under the various quotients involving $U(5)$, e.g. the SES $1\to SU(5)\to U(5)\to U(1)\to 1$ and $U(6)/U(5)\cong S^{11}$, and of course ultimately $U/U(5)$. But I’m sure there are more interesting things to do there. I wonder what particular ideas Hisham had in mind in this topic.
Since rationally not much is going on with $U(5)$, what one would probably need to check is whether there are nontrivial maps $X \to U(5)$ which become trivial after composition to $X \to U(5) \to \underset{\longrightarrow_{\mathrlap{n}}}{\lim} U(n)$ and yet are discernible as reflecting some expected torsion charge in IIB.
added brief comment on unstable k-theory around Sen’s conjecture based on
Thanks, that’s good.
I have moved the paragraph out of “Checking Sen’s conjecture” into a new subsection “Unstable K-theory?” (here)
and prefaced it by a couple of remarks, highlighting that, to start with, the assumption of a would-be stable Chern-character of RR-flux forms is at best subtle (as it does not follow from 11d-sugra, instead needs massive IIA-theory with all its subtleties).
Thanks, a couple of questions.
Here why do twisted K-theory cocycles as maps to $BU\times \mathbb{Z} / BU(1)$ produce the RR-fields for IIA and not IIB?
And more importantly, what is the expectation of double-dimensional reduction of the super-cocycles presented in 2403.16456? among other things the $\phi$ and $P$ fields in Section 3 of Cremmer et al 98?
On the first question: I may be missing what exactly you are after. So let me say some things that you probably know already:
The classifying space for $K(-) = KU^0(-)$ is $\mathbb{Z} \times B \mathrm{U}$ or just $B \mathrm{U}$ for the reduced case (e.g. here) – with universal Chern classes in every (positive) even degree, as befits the type IIA RR-flux densities.
Accordingly the classifying space for $KU^1(-)$ is $U$, with universal Chern classes in every odd degree, as befits the RR-flux densities in type IIB.
In either case, the homotopy quotient by $B\mathrm{U}(1)$ classifies the twist by the degree-3 ordinary cohomology of a background B-field classified by $B^2 \mathrm{U}(1)$
On the second question: The double dimensional reduction of the C-field flux density and its dual is reviewed in Ex. 2.13 of here. An early clear account is section 4.2 in arXiv:hep-th/0312033.
The dilaton $\phi$ does not appear this way, it appears from the gravitational sector.
Re#21 so why say in Witten’s paper and even in the table here is it stated that it is $KU^0$ for IIB and $KU^1$ for IIA?
Oh, now I see what you have in mind:
That switch of degrees happens when people talk about “D-brane charge” as opposed to their “RR-flux density”
and then tacitly focus on “solitonic branes” instead of “singular branes”.
This is sorted out in Remark 4.5 on p. 24 here.
[edit: and I have fixed it in that table]
Got it, thanks!
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