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Domenico Fiorenza started a page for the thesis of his student, Alessandra Capotosti: From String structures to Spin structures on loop spaces.
Am I right in thinking the main innovation is the passage from the map
$\mathbf{B}Spin_{conn} \rightarrow {\mathbf{B}}^2({\mathbf{B}}U(1)_{conn})$to
$\mathbf{B}Spin \rightarrow {\mathbf{B}}^2({\mathbf{B}}U(1)_{conn})?$Is that likely to work more generally, e.g., can one do something similar with
$\mathbf{B}String_{conn} \rightarrow {\mathbf{B}}^7 U(1)_{conn}?$Am I right in thinking the main innovation is the passage from
The roots of this reasoning go back quite some time. That particular statement (passage) appeared before in Higher geometric prequantum theory.
I would say the main innovation here is to efficiently use all the tools we had developed to get an elegant quick derivation of this result. The main technical point is to see that instead of just homming $[S^1,-]$ into the differentially unrefined $\mathbf{B} Spin \to \mathbf{B}^3 U(1)$, which yields an extension that is bigger than one is looking for, one is to build in differential connection data in just the right way to make the result come out accurately.
Is that likely to work more generally,
Yes, absolutely. One could have stated a corollary to this entent. The problem is maybe only sociological: since nobody had previously defined spin structures on mapping spaces $[\Sigma_5,X]$ before, there is now no statement that something is being reproduced. Instead the two things to be compared are both generated by the theory now.
I have
added a few more hyperlinks to keywords in the entry;
cross-linked it from string structure and differential string structure;
gave it the marker “category:reference” .
Re #2,
nobody had previously defined spin structures on mapping spaces…
I think I was wondering about fivebrane structures on some manifold, $X$, being compared to string structures on something related to $X$, but I probably haven’t understood the general mechanism.
@David C
A “spin structure” on a loop space is rather passing to the next stage of the Whitehead tower of the frame bundle of the loop space–that is, the (level-1) central extension of the loop group, which is the 2-connected cover. The correct analogue would then be to consider the next stage of the Whitehead tower of the structure group of the frame bundle of the mapping space $[\Sigma_5,X]$. Since we are free now to consider all sorts of 5-manifolds $\Sigma_5$, this is not so simple as the loop space case, and as Urs pointed out, people haven’t really considered this yet. A lift to this next stage of the Whitehead tower would then be the analogue of the “spin structure”, and this is what one should compare to the Fivebrane structure on $X$.
In the abstract general, one can just make the comparison, but we don’t know concretely what is going on. One could consider for instance $\Sigma_5 = S^5$ as a fairly canonical choice, or perhaps some other general family of simply-connected 5-manifolds. According to the manifold atlas project, closed, oriented, smooth, simply-connected 5-manifolds are completely and explicitly classified.
Indeed, in dimension 5 smooth classification is governed by classical algebraic topology, namely, two simply connected 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving linking form and the second Stiefel–Whitney class. Moreover any such isomorphism is induced by some diffeomorphism. -MAP, 5-manifold
The case of spin 5-manifolds is even easier, apparently. This would be a good place to start, methinks. (The non-simply-connected case is outright impossible…)
Notice that it is still essentially open which use there is of spin structures on mapping spaces over higher structures down on base spaces.
It is still an open conjecture that the Witten genus is rigorously the index of a suitably defined Dirac operator on a loop space equipped with spin structure. What has been proven is instead that the Witten genus is a certain index of a something defined on the base space equipped with string structure (namely the “string orientation of tmf”).
Hence, while it is useful for computation to know how spin structures on loop space are related to string structures on base space (which is analogous to how a bundle gerbe expresses a 2-bundle on a base space in terms of a 1-bundle on some richer spaces) it is at the moment open whether spin structures on loop spaces are relevant beyond this relation.
For that reason it seems premature to worry too much about spin structures on $[\Sigma_5, X]$. Because, judging from what is known, it seems unlikely that there is a construction of an index in Calabi-Yau cohomology as an index of a Dirac operator on $[\Sigma_5,X]$. It seems much more likely that instead such an index comes from a “fivebrane orientation” on a universal CY-cohomology theory which is built from fivebrane structure down on base space $X$.
And that, of course, is a key motivation for studying higher structures in the first place. Lower structures don’t cut it.
@Urs,
I agree. The trouble one (or rather Konrad) has to go to in order to get something in 1-bundle land actually equivalent to bundle gerbe is considerable. The case of anything higher is daunting. The only reason all the fusion stuff might be useful, at least to my mind, is the similarity to conformal nets, and perhaps being able to get one from the fusion bundle data.
That’s my impression, too. Konrad’s fusion structure seems to go in the direction of making the Dirac operator on loop space be defined in terms of 2d SCFT. Which is of course what Witten started with in the first place! After all, he suggested that it is the worldsheet 0-mode of the supercharge of a 2d SCFT that behaves as if it were a Dirac operator on loop space. But locality (and that’s what all the fusion is about) makes it unlikely that just this zero mode without its higher modes is sufficient data. So Witten’s index could be read more as a motivation for rigorously defining 2d SCFT sigma-models on curved backgrounds than for defining Dirac operator on loop spaces.
Anyway, whatever the answer will be, it is good that people are exploring all possibilities. I just wanted to remark that unless something unexpected happens, then there is not too much motivation behind considering spin structures on $[\Sigma_5,X]$, instead of fivebrane structures on $X$.
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