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I am giving fiber integration in K-theory a dedicated entry.
One section In operator K-theory used to be a subsection of fiber integration in generalized cohomology, and I copied it over.
Another section In terms of bundles of Fredholm operators I have now started to write.
The map $i_{!}$ does not generally satisfy properties such as being injective, right? I’m confused about why the classification of D brane charges in terms of K theory of X is meaningful, what does one make of the nontrivial classes in the kernel of $i_{!}$? I imagine some of this is refined in the differential picture but even there I suppose there one has a nontrivial kernel?
To my mind — by the general logic of flux quantization — the fundamental meaningful quantity (in the case of RR-flux quantization in topological K-theory) is the K-theory of the spacetime domain.
In this picture there is no a priori notion of well-identified individual D-brane: All D-branes present contribute charge to a single K-theory class on the spacetime domain, in which their sourced RR-flux contributions are smeared and superposed and intermingle.
Of course, in simple situations it is clear what the individual D-brane charge contributions should be (taken to be): For instance, in the case of spacetime domains which are just flat Euclidean space times a sphere, one may reasonably attribute their K-theory (hence that of the sphere factor, hence the K-theory of the point in the corresponding degree) with the charge of a single “stack” of D-branes at the “center” of the sphere (either singular or solitonic branes, as discussed behind the above link).
Given this, one may feel emboldened to identify further, “less flat” configurations of individual branes imprinted in the K-theory of their ambient spacetime.
The push-forward of a class over a brane’s supposed worldvolume submanifold into the ambient spacetime is a plausible such notion. But by what I just said, it should be regarded from the end result of this process: Given any K-class on a spacetime domain, we are entitled to associate a “sourcing D-brane worldvolume” with it if the class happens to equal such a push-forward.
However, there is no intrinsic meaning to the class on the worldvolume itself being pushed-forward (despite the intuitive urge one may have to assign one) since – by the general logic of flux quantization – it is ultimately the classes on the spacetime domain which reflect all brane charges.
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