A six-member team from Columbia Universe, Cambridge U.K., and Haiti that includes Brian Greene has submitted a very interesting preprint:
In the "moduli space" of the Calabi-Yaus - a space of possible shapes of Calabi-Yau manifolds of the same topology - you usually find some "conifold points". At those points in the space of shapes, the Calabi-Yau manifold becomes singular by itself. It develops a "conifold singularity" that is locally diffeomorphic to a cone with an S^2 x S^3 base instead of a patch of the smooth space.
Such points in the moduli space can typically be regularized in several ways. If you move away from such points in the moduli space, you may obtain several possible topologies as the smooth versions of the conifold. These topologies are related by "conifold transitions." In fact, one of the topologies may typically be obtained in several reshuffled ways that are related by a simpler "flop transition".
Now, the conifold transition is very interesting mathematically and what's happening around this point has been studied for mathematical reasons. Physics of string theory is totally smooth, regular, well-defined, and predictive even in the presence of these singularities. The D3-branes that may wrap the shrinking three-sphere give us new massless/light degrees of freedom in four dimensions. This advanced stuff in algebraic geometry has actually been explained by The Elegant Universe to millions of readers.
The topology-changing phase transitions near the conifold points are arguably able to connect "almost all Calabi-Yau topologies" into one network. And all these things are very interesting.
However, there has been one "subdiscipline" in which the conifold points haven't played almost any role in the papers so far, namely dynamics (including time dependence) - especially cosmology, landscape, and tunneling. One could always invent hand-waving, sloppy arguments why the conifold points wouldn't be relevant. They're very special points and the landscape - and the tunneling in landscape - only cares about the generic points, doesn't it?
Well, using the formal algebraic as well as numerical methods, the new paper shows that it doesn't.
The tunneling wants to go right through - or near - the conifold points. So if your vacuum is changing from one Calabi-Yau to another, or if it is rearranging the values of the fluxes and the numbers of wrapped branes, you should imagine that a singular Calabi-Yau conifold appears somewhere in the middle. The authors actually don't make a sharp decision whether the tunneling goes "exactly" through the conifold. But it "grazes it". They try to coin another word, "conifunneling", and I predict that it won't be used too much by others, despite its similarities to "Brangelina" which became popular.
Also, the authors argue that the existence of the tunneling process would look impossible to you if you neglected the warping of the geometry. The Randall-Sundrum-like warping of the geometry, especially in the very vicinity of the conifold singularity of the Calabi-Yau shape, is essential for a correct calculation whose conclusion is that the tunneling can work.
Does it mean that it becomes more likely that the internal manifolds boast some (approximate) conifold singularities and warped geometry? In particular, does it increase the chances that the Randall-Sundrum models are correct at an accessibly low energy scale? I don't know. But it's likely that one has to go through calculations that are at least as comprehensive as those in this paper to answer such questions. However, there's still no well-defined principle that would relate the interesting questions about the vacuum selection to the settled or calculable features of the Calabi-Yau geometry.
And that's the memo.