Jarah's K-theory refinements

Although Jarah Evslin is an anarchist, his talk at the postdoc journal club was extremely organized and useful. His topic was

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

We immediately figured out a couple of possible answers - such as the hamburgers, and so forth. However, Jarah started and everything suddenly made much more sense. Jarah is a very mathematically skillful guy, but nevertheless, category theory is too hardcore even for him. Nevertheless, he wanted to solve similar tasks as category theory and classify various conserved brane charges and fluxes, and outlined the following sequence of mathematical objects:

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

These sets are increasingly refined. Everytime you fall one step lower, your space only classifies a subspace of the upper one, and moreover makes some identifications (quotienting). Therefore, every new entry in the list above is a set which is smaller "by two contributions" than the previous one, given a natural definition of "smaller". Actually, every new entry is not quite obtained as a quotient of the subset of the previous one because it can "twist" the previous entry so that Z_2 cubed is replaced by Z_8, as an example below will indicate. Also, every new entry is "more conserved" - its charges (elements) are invariant under a broader class of physical processes than the elements of the previous classifying set.



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.