Friday, October 14, 2005

Geometric transitions

Anke Knauf from Hamburg and Maryland has made a nice lunch presentation of the geometric transitions, especially in the context of non-Kähler compactifications.


For the non-experts, let me mention that the six-dimensional Ricci-flat Calabi-Yau manifolds used in string theory to "hide" the six additional dimensions - such as the quintic (fifth order) hypersurface in the projective space CP^4 - may be deformed in such a way that a conical singularity develops within the manifold. When this singularity is present, it is not really a manifold anymore but rather a conifold - a generalization of the concept of a manifold where some patches may be diffeomorphic not to R^6 but rather to a cone. The relevant cone for the conifold is a cone over the product of two spheres, S^2 x S^3. Both of these spheres shrink to zero size at the tip of the cone. There are two ways to obtain a smooth Calabi-Yau by changing this conifold: either resolving it or deforming it which means that either S^2 or S^3 is blown up to a finite size.

The deformed conifold with N units of flux is equivalent to the resolved conifold with N D-branes on the dual cycle. Mirror symmetry morally exchanges the deformed conifold and the resolved conifold; it also interchanges D5-branes wrapped on S^2 and D6-branes wrapped on S^3 in the previous sentence because it exchanges type IIA and type IIB string theories. The Gopakumar-Vafa results about the topological partition sum for these backgrounds - and their equality - were described.

The resolved conifold is not literally a mirror of the deformed conifold, as a reader emphasizes. The simplest way to see it is that you won't find any candidate dual 3-cycles for your original 0-cycles.

Xi Yin initiated quite a debate whether this relation between two topologies and between the D-branes and fluxes - something that holds exactly for the BPS-protected quantities - is a full-fledged duality or just something that holds for the topological subsector. We mostly concluded it was exact. If you start with no branes or fluxes, the original two branches of the moduli space are connected in the singular point (conifold point in the moduli space) where the topology may change, as discussed in chapter 13 of The Elegant Universe. ;-) Nevertheless, the branches are sharply separated.

Let me emphasize that the full understanding of string theory near the conifold-like singularities is one of the huge technical victories of string theory. It proves, among other things, that topology of space can change according to string theory without introducing any disasters or inconsistencies. String theory is completely smooth around this point while general relativity itself would break down because the curvature invariants classically diverge. Moreover, shocking new dualities and identities between a priori different quantities follow from this relation as an extra bonus. Roger Penrose seems to miss this point completely. If you read this blog, Sir Penrose, your humble correspondent would like to ask you to look at the relevant papers that actually prove the power and amazing consistency of string theory; nowadays, they are definitely not examples of phenomena that can be used to question extra dimensions in general or string theory in particular!

When you include branes (or fluxes), the two branches actually get continuously connected - much like two intersecting lines that can be resolved into a hyperbola. Along this "hyperbola", you may identify regions where the D-brane or the flux description is better - these two descriptions have different spacetime topology. But the transition between them is continuous; you can't divide a hyperbola into a vertical and a horizontal part.

Anke studied a non-Kähler deformation of these conifold vacua with D-branes on it and roughly 6 dual descriptions of them. There has been a lot of debate going on whether her chain of dualities was the most efficient one, whether half-flatness was necessary for supersymmetry, whether the new backgrounds were actually supersymmetric, whether various forms had to be co-closed, whether there were any relations to generalized Kähler geometry and other modern concepts, and so forth. It would probably consume too much time to analyze all the points she raised and others discussed.


  1. Hello, thank you very much for your simple words to explain something complex, I have some questions:

    1) Ok, the base of the cone is S^2xS^3 but as we move away from the singularity it remains S^2xS^3 ?

    2) Change the smooth peak for something does not affect the reality of things, i.e the physics of what this happening?

    3) Why do you want to remove the singularity, apart from the fact that I want to be Calabi Yau Non compact, Ricci flat and Kahler, what is the benefit of removing the singularity? If the singularity is a coordinate and not only physical?

    thanks for replying, all the best :-)

  2. Dear Bárbara, thanks for your kind words.

    1) One may slice a finite neighborhood of the tip of the cone along the values of "distance from the tip", and the slices will be S2 x S3 up to some distance. Later, however, the "result of the manifold", whatever it is, will show up. It can have a complicated topology and not every singular manifold with a conifold singularity may be globally sliced to S2 x S3 slices, of course.

    2) Sorry, I don't understand this question at all. In general, you can't just replace pieces of geometry by something completely different. Geometry has to obey stringent conditions, essentially Einstein's equations (Ricci flatness) so while I am not sure what you exactly wanted to replace by what and why, I guess that it is not possible.

    3) One may easily prove that the conifold singularity isn't just a coordinate singularity. The Riemann tensor clearly goes to infinity. That's true for all cones that are not locally flat (the curvature radius is equal to the distance from the tip of the cone times a universal constant) and the S2 x S3 -based conifold is obviously not Riemann-flat.

    Otherwise I don't want to "remove it" at any cost. String theory really makes it possible for spacetime to be a conifold - it allows well-defined predictions even at this geometry that naively looks singular, from the viewpoint of "just the geometry", so classical GR would be ill-defined but string theory is OK on this space.

    All the best