- What is K-theory bad for and what it does not classify?

- Embeddings
- Homotopy
- Homology
- K-homology
- S-covariant K-homology

Jarah has explained many examples in which the neighboring structures differ. For example, one-cycles on a genus g surface may be understood as embeddings. In that case, a D1-brane and a different D1-brane nearby cannot annihilate - this pair is distinguished in the structure called "embeddings". Then you may reduce the set of possible embeddings to homotopy; the first homotopy on the genus g surface is a non-Abelian group. Its abelianization gives you the homology; homology is a smaller group. I won't write anything else about the difference between embeddings and homotopy.

Instead, let's continue with the difference between homotopy and homology. Jarah has discussed various special cases - such as the branes on RP^7 x S^3. Homology gives you Z_2 for H_1, H_3, as well as H_5, and you might think that the natural group is Z_2 cubed - three independent numbers are added modulo 2. However, K-homology gives you a different answer. It shifts the Z_2's relatively to each other, so that Z_2 cubed is actually replaced by Z_8. This means that if you annihilate two 3-branes, instead of nothing you obtain a 1-brane, and so forth. He also defined K-theory, twisted K-theory, and gave several other examples.

We have had many discussions led by Jarah about the ability to refine K-theory in such a way that it is invariant under all dualities or satisfies similarly big constraints. Jarah's S-covariant K-theory is not invariant under T-duality, for example, simply because he classifies possible fluxes of the 3-form H, but he does not classify the possible topologies (geometry) even though these topologies may be T-dual to the H-fluxes. My feeling is that something that would classify all these conserved objects and respected all dualities would have to know - more or less - about the whole landscape, and therefore knowing this ultimately refined K-theory is almost equivalent to knowing the whole "theory of everything".

There are issues about these refined K-homologies being groups or just semigroups or nothing like that. It's intuitively clear that one expects a group structure for objects that can be thought of as small perturbations of a background. For example, K-theory should count all generalized D-branes - everything that contributes to the total energy/action by an amount proportional to 1/g_{string}. At weak coupling, that's much smaller than the tension/action of a nontrivial closed string background (e.g. NS5-branes) that goes like 1/g_{string}^2, and therefore it is natural that we get a structure of an Abelian group as K-theory. Once we add both NS-NS H-fluxes as well as R-R fluxes, it's not shocking to learn that the classifying set won't behave as a group. The condition of the objects being small perturbations of the same background does not hold anymore: similarly, we also don't expect the set of Calabi-Yau topologies to be a naturally definable group because we have no canonical way how to "add" topologies! ;-)

At any rate, Jarah's talk was very interesting and I am now very tired.

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