Tuesday, February 20, 2007

Generalised Geometries and Flux Compactifications

For two days, I have attended a workshop at Hamburg on Generalised Geometries and Flux Compactifications. Even though the talks have been generally of amazingly high quality and with only a few exceptions have been very interesting I refrain from giving you summaries of the individual contributions. If you are interested, have a look at the conference website and check out the speakers most recent (or upcoming) papers.

I still like to mention two things though: Firstly there are the consequences of having wireless network links available in lecture halls: By now, we are all used to people doing their email during talks if one is less interested in what is currently going on on stage. Or alternatively, you try not to be so nosy as to try to read the emails on the laptop of the person in the row in front of you. But what I have encountered for the first time is that the speaker attributes a certain construction to some reference and then somebody from the audience challenging that reference to be the original source of that construction and than backing up that claim with a quick Spires HEP search.

In this case, it was Maxim Zabzine talking and Martin Roceck claiming to know an earlier reference to which Maxim replied "No, Martin, don't look it up on your laptop!". But it was already too late....

The other things is more technical and if you are not interested in the details of flux compactifications you can stop reading at this point. I am sure, most people are aware of this fact and also I had read about it but in the past it had never stroke me as so important: In traditional compactifications without fluxes on Calabi-Yaus say, the geometry is expressed in terms of J and Omega which are both covariantly constant and which fulfill J^3=Omega Omega-bar = vol(CY). Both arise from fierzing the two covanriantly constant spinors on the CY. Now, it's a well defined procedure to study deformations of this geometry: For a compact CY and as the Laplacian (or the other relevant operator to establish a form is harmonic) is elliptic, the deformations can be thought of to come from some cohomology class which is finite dimensional. So, effectively one has reduced the infinite dimensional spaces of forms on the CY to a finite dimensional subspace one in the end arrives at a finite number of light (actually massless) fields in the 4d low energy theory.

Or even more technical: What you do is to rewrite the 10d kinetic operator (some sort of d'Alambertian) as the sum of 4d d'Alambertian and a 6d Laplacian. The latter one is the elliptic operator and one can decompose all functions on the CY in terms of eigenfunctions of this Laplacian. As a result, the eigenvalues become the mass^2 of the 4d field and since the operator is elliptic, the spectrum is discrete. Any function which is no harmonic has a KK-mass which is parametrically the inverse linear dimension of the CY.

If you now turn on fluxes, the susy conditions on the spinors is no longer that they are covariantly constant (with respect to the Levi-Civita connection) but that they are constant relative to a new connection where the flux appears as torsion, formally Nabla' = Nabla + H. As a consequence one only has SU(3) structure: One can still fierz the spinors but now the resulting forms J and Omega are no longer harmonic. Thus it no longer makes sense to expand them (and their perturbations) in terms of cohomology classes. Thus the above trick to reduce the deformation problem to a finite dimensional one now fails and one does no longer have the separation into massless moduli and massive KK-states. In principle, one immediately ends up with infinitely many fields of all kinds of uncontrollable masses (unless one does not assume some smallness due to the smallness of the fluxes one has introduced). This is just because there is no longer a natural set of forms to expand things in.

However, today, Paul Koerber reported on some progress in that direction for the case of generalised Kähler manifolds: He demonstrated that one can get in that direction by the consideration of the cohomology with respect to d+H, the twisted differential. But still, this is work in progress and one does not have these cohomologies under good control. And even more, quasi by definition these do no longer contain the 'used to be' moduli which now obtained masses due to flux induced superpotential. Those are obviously not in the cohomology and thus still have the same status as the massive KK-modes which one would like to be parametrically heavier to all this really make sense.

There were many other interesting talks, some of them on non-geometries, spaces pioneered by Hull and friends where one has to use stringy transformations like T-dualities when going from one coordinate patch to another. Thus at least they are not usual geometries but maybe as T-duality has a quasi-field theoretical description there, might be amenable to non-commutative geometry.