Twisted K-theory

9 Responses to Twisted K-theory

1. Urs Schreiber says:

   July 15, 2004 at 1:05 pm

   Hi again –

   many thanks for your reply!

   I should have made myself clearer. I think my question concerned a slightly different issue. I’d be grateful if you had a look at this refined question and drop me a note on what you think about it.
2. Urs Schreiber says:

   July 15, 2004 at 11:59 am

   Hi Peter –

   since you are an expert on loop space: Could you help me with the question that I ask here?
3. Urs Schreiber says:

   July 15, 2004 at 5:26 am

   I was referring to my old problem of putting fermions on the lattice.There Kahler-Dirac fermions are basically Kogut-Susskind fermions, and disentangling a single spinor (an ideal in the Clifford algebra) is tricky.

   Ah, you are thinking of Dirac-Kähler on the lattice. This has problems even for a flat ‘metric’. I have once thought about this a little together with Eric Forgy, when writing section 5.2 of our notes math-ph/0407005 on discrete differential geometry (where we used the loop space deformation techniques that describe superstring backgrounds and applied them to the point particle limit). Maybe Dirac-Hestenes spinors are the best choice for the lattice.

   What happens when lattice QFT is done with something that is not exactly a spinor in a single ideal? For small lattice spacing the undesireable mixing should disappear. Wouldn’t that be sufficient? Or does a finite mixing survive the continuum limit?

   Another way of getting spinors from forms is by picking a complex structure. Then spinors are (up to a twist by a square root of the top dim form) just the holomorphic exterior algebra, and this is preserved by the Levi-Civita connection when your metric is Kahler.

   Yes, when the metric is Kähler. Alternatively, if spinors are already modeled in left ideals on the exterior bundle, then a Kähler background allows one further split of degrees of freedom and admits to increase N=1 susy to N=2. When twisting one of these two susys one obtains the topological string and all kinds of major wonders begin to happen.
4. Peter says:

   July 14, 2004 at 1:26 pm

   When I was quoting Witten I was referring to my old problem of putting fermions on the lattice.There Kahler-Dirac fermions are basically Kogut-Susskind fermions, and disentangling a single spinor (an ideal in the Clifford algebra) is tricky.

   In your case presumably what you say works. Another way of getting spinors from forms is by picking a complex structure. Then spinors are (up to a twist by a square root of the top dim form) just the holomorphic exterior algebra, and this is preserved by the Levi-Civita connection when your metric is Kahler.
5. Urs Schreiber says:

   July 14, 2004 at 1:04 pm

   Hi Peter –

   you wrote:

   As you might guess, I’m interested in this model as a 1+1 d QFT, not a string theory.

   That’s fine with me. If one wants, for instance, to think of the WZW model as not describing a string in a group manifold, but as describing a group valued field on some low-dimensional spacetime, that doesn’t change any of the results, of course.

   I remember Witten pointing out to me “Yeah, but what you want are spinors, not the exterior algebra”, which is the heart of the problem.

   Hm, now I am surprised. Of course it is true that we are interested in spinors and not in differential forms. But…

   The point is that the supercharges on the worldsheet simply are Dirac-Kähler operators. We are not choosing them to be such.

   So there is a puzzle, and it does have a resolution:

   The puzzle is: How can it be that the string has spinors in its spectrum, when the supercharge is really the Dirac-Kähler operator?

   As was noted already by Kähler himself, his equation models spinors on flat spacetime, but no longer on a curved spacetime. That’s because the connection on the exterior bundle does not respect the spinor ideals, but mixes them. (I have once tried to give a hands-on physicist-like way to see how this works here.)

   So how can it be that superstrings have spinors in their spectrum even on curved spacetime, if, as I claim, their supercharges are Dirac-Kähler operators?

   Well, the reason is, for superstrings there are conditioons on the background fields. You cannot just arbitrarily curve the background. Something has to provide the energy-momentum density for this curvature and the background equations of motion have to be satisfied. These extra fields will enter into the Dirac-Kähler operator as additional connection-like terms, and these terms will cancel the problkematic mixing effect.

   We know this has to be true in principle (it is essentially nothing but the chiral splitting on the worldsheet) but it is probably instructive to see how it works in specific examples. The best example I know is just the super WZW model. Here we have target space curvature (target space is a Lie group manifold with the standard Killing metric). But there is also a 2-form field on target space. This 2-form field induces a torsion term in the Dirac-Kähler operator. This is precisely the parallelizing torsion on the group manifold. But this means that it exactly cancels the Levi-Civita connection in the inavriant orthonormal basis. This way the mixing effect is removed and we have true spinors described by the WZW Dirac-Kähler operator! 🙂

   I discuss this and all the details in section 2.1.4 of hep-th/0311064. (Unfortunately this paper is not very readably written, I have to admit.)

   So the Dirac-Kähler operator is indeed at the heart of it, but there is no problem! 🙂
6. Peter says:

   July 14, 2004 at 11:33 am

   Yes, this is the worldsheet Hamiltonian. As you might guess, I’m interested in this model as a 1+1 d QFT, not a string theory.

   The Dirac-Kahler stuff is very interesting, especially in its relation to TQFT, as in Witten’s original Morse theory paper. I actually first got interested in this way back in grad school when I was thinking about fermions on lattices and Dirac-Kahler. I remember Witten pointing out to me “Yeah, but what you want are spinors, not the exterior algebra”, which is the heart of the problem.
7. Urs Schreiber says:

   July 14, 2004 at 11:06 am

   Hi Peter –

   I think that unless something weird happens looking at one chirality is fully sufficient, since both chirality sectors are decoupled and identical.

   I would like to know what precisely you mean by the Hamiltonian picture. Probably and hopefully this refers to the worldsheet Hamiltonian, so that you are perhaps thinking of looking at the WZW QFT in the Schrödinger picture (operators do not depend on worldsheet time) instead of the usual Heisenberg picture of CFT (where they do)? That would be nice, because this is exactly the point of view which leads from superstrings to differential geometry on loop space.

   Namely loop space is of course the configuration space of the string, which only makes an autonomous appearance in the Schrödinger picture. In other words, loop space is the midi-superspace (super in the sense of Wheeler, not in the sense of supersymmetry) of the worldsheet gravity theory.

   My original motivation to look into strings at all was that I had found that all kinds of supersymmetric field theories, and in particular N=1, D=3+1 SUGRA, look like Dirac-Kähler systems (i.e. systems governed by (d + del)) when formulated in Schrödinger picture on their configuration space.

   Since it is hard to make progress with this insight in 3+1 dimensional gravity, I applied it to 1+1 dimensional gravity and arrived at the string. Turns out that the Dirac-Kähler point of view here works wonders, in my opinion.

   But it is also an unfamiliar point of view for physicists. It had been hinted at by Witten in his ‘Morse theory’ paper, but that’s about it. I am hoping to convice some people that it is a fruitful point of view.

   The comments you made suggest that this might in fact be easier in more mathematically inclined circles, where the notion of differential geometry on loop space is something that people can associate interesting results with.

   So if you find the time to sketch some ideas by Mickelsson and others, you are guaranteed to have at least one interested reader.
8. Peter says:

   July 14, 2004 at 10:42 am

   I’ve always been somewhat confused by the relation between what Mickelsson does and supersymmetric WZW. For one thing, he’s in a Hamiltonian picture, and only appears to be looking at one chirality.

   At some point in the next few weeks I plan to get back to working on this and maybe the references you give will help me sort it out.
9. Urs Schreiber says:

   July 14, 2004 at 6:20 am

   So what is it Mickelsson is doing with the supersymmetric WZW model?

   In the intro it says (and more I haven’t read so far)

   We also discuss the construction of twisted K-theory classes by families of supercharges for the supersymmetric Wess-Zumino-Witten model.

   The generic supercharges of the WZW model are just the left- and rightmoving worldsheet supercharges, which are (Kähler-)Dirac(-Ramond) operators on the loop group with respect to a connection which is Levi-Civita plus/minus the parallelizing torsion (as discussed for instance in hep-th/9310187 and hep-th/0311064). For certain groups there might be more supercharges (like when the group is complex Kähler – is that possible?).

   So are these twisted K-theory classes constructed from Dirac operators, in general? Or how else are they related to supercharges?

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