How many string theories with more than one supersymmetry?

[MTd2]

In fact the role of K-theory in string physics is somewhat overrated and doesn't play an important role beyond cohomology, so I would advise anybody not to waste time by studying papers on K-theory, unless she really wants to understand subtleties in defining D-brane charges and similar.

It was later argued that K-Theory is not enough, you have to consider elliptic-cohomology, which is one step further in complexity, since it deals with curves on a torus, or lattice. According to what I sent you, K-theory cannot classify 3-form fields with torsion.

And if I don't classify what the fields are, how can I find what is m-theory?

 

[suprised]

It was later argued that K-Theory is not enough, you have to consider elliptic-cohomology, which is one step further in complexity, since it deals with curves on a torus, or lattice. According to what I sent you, K-theory cannot classify 3-form fields with torsion.

And if I don't classify what the fields are, how can I find what is m-theory?

Well I have seen claims but right now I don't know of any concrete use of elliptic cohomology in M-theory. It sounds more like a wish list for math concepts to be applied somewhere in physics (and yes, there are plenty). I studied a few papers of Sati et al but really couldn't get anywhere.

I know somewhat better the K-theory story in relation to D-branes, and you could say the same thing: how could one possibly understand D-branes without K-theory? However indeed K-theory doesn't play an important role beyond cohomology. In other words, cohomology is about RR-charges of branes and open strings (which are certainly important) but essentially what K-theroy adds for physicists is torsion pieces to charges, for example Z_2 factors (concretely, this boils down for example that certain orientifold planes are labelled by some extra signs, +/-). This by itself isn't just particularly important.

My suspicion is that many people believe that K-theory must be important because some renowned people worked on it and it sounds cool. But in all the many years since it has been introduced in physics, I wouldn't know about any single truly important application. In fact K-theory is just the poor man's version of derived categories, and this is a much better way to understand brane-antibrane annihilation, tachyon condensation and so on. For example, all D0 branes are equal in K-theory, which just knows about their total charge; but in the category also the location of the D0 branes play a role, and so can distinguish if two D0 branes sit on different places or are on top of each other (and can bind or whatever).

So while all of this seems interesting in one way or other, and certainly is of some conceptional relevance, I would think that there are some more down-to-earth things to study which would be more worthwhile to study!

 

[MTd2]

Well I have seen claims but right now I don't know of any concrete use of elliptic cohomology in M-theory. It sounds more like a wish list for math concepts to be applied somewhere in physics (and yes, there are plenty). I studied a few papers of Sati *et al* but really couldn't get anywhere.

But, have you been following "et al"? One of those is Urs Schreiber! That's why a large number of n-category cafe posts is about this. I guess that's why I came across this kind of thing.

But what kind of down to Earth thing do you mean?