- The probability that the cosmological evolution will end up as a Universe with a particular shape of the hidden dimensions (and particular values of the fluxes) is determined by the (exponentiated) entropy of a corresponding black hole whose geometry flows via the attractor mechanism to the given shape of the Universe near the horizon. Note that this contrasts sharply with the "anthropic principle" - which itself is not a principle, rather a lack of principles. In the anthropic principle, the corresponding probabilistic weight is determined by the ability of the Universe to support intelligent life.
However, an infinitesimal change of the reference point in the moduli space induces the so-called holomorphic anomaly which a slight, exactly understood dependence on Xbar that can be locally visualized as an infinitesimal Fourier transform. By an infinitesimal Fourier transform, I mean the conversion of a wave function "psi(x)" to "psi'(x')" where "x' = x+epsilon.p" - you see that we are mixing "x,p," the coordinates on the phase space. Therefore it is more appropriate to talk about the partition sum as a vector in a Hilbert space rather than a well-defined function.
Also, the partition sum is a complex number whose squared absolute value has been shown to have many physical interpretations in various recent papers that had Cumrun Vafa among its authors; it has been related to black hole entropy as well as the partition sum of two-dimensional gauge theory. That's another reason to imagine that the partition sum is a "wave function".
In the new paper, this concept is taken very seriously. "The wave function" is interpreted as nothing else than the Hawking-Hartle wave function of the Universe. You know, the Hawking-Hartle wave function is something like a wave functional of quantum gravity that solves the Wheeler-deWitt equation (a sophisticated definition of the quantum equation
- H.psi = 0
In the case of string theory, we may be afraid that the Hawking-Hartle state is a functional of a much broader collection of fields, including excited strings. Ooguri, Vafa, and Verlinde only consider a restricted set of degrees of freedom - a minisuperspace. They only consider the Hartle-Hawking wavefunction to be the function of a few parameters that describe the shape of the Universe - namely the complex structure moduli of the Calabi-Yau manifold. This truncation should be enough to calculate all holomorphic, SUSY protected quantities. The Universes they study have the form of AdS2 x S2 x CY3 and the corresponding Hartle-Hawking state is defined on S1 x S2. This may look like an oversimplification, but the advantage is that the simplified Hartle-Hawking state may be calculated including all the higher-order corrections - it's calculated as the partition sum of the topological string theory on the same Calabi-Yau three-fold.
The results may be viewed as something that gives you a "natural" probabilistic measure for the Universe to have different particular values of the complex structure moduli. Of course, their context only deals with AdS2 - a space that is geometrically equivalent to dS2 - and they're using both directions of this space in the role of time in various viewpoints that they relate to each other. In other words, they apply a double-Wick-rotation on their AdS2, and they're slicing it in two different ways, in analogy with the open-closed duality seen for the cylindrical worldsheets in string theory.
You may argue that the supersymmetric AdS2 backgrounds are not a good description of our Universe, and you may be right. But there could exist some generalization that will turn out to be relevant for the vacuum selection problem sometime in the future.
At any rate, I like their brave attempt to replace the naive "vacuum counting" strategy - a prejudice that each single "vacuum" of string theory is supposed to be equally likely - by something more realistic and potentially justifiable by quantum cosmology. This attempt is one of the first ones that uses the Hartle-Hawking wave function; I was informed by a reader-insider about an earlier paper by Firouzjahi, Sarangi, and Tye from the last summer that tries to phenomenologically apply the Hartle-Hawking state on the KKLMMT compactifications.
The Hartle-Hawking states are very natural and important tools that may eventually shed light on the vacuum selection problem and the evolution of the Universe a Planck time after the beginning ;-), and therefore I think it is a good idea to try to give the Hartle-Hawking wave function a well-defined meaning in string theory. Ooguri, Vafa, and Verlinde have certainly made many new steps towards this kind of goal.
The measure as they formulate it today kind of gives the different Universes a weight proportional to the entropy of the corresponding black hole. Although one should not be comparing different topologies of Calabi-Yau manifolds, you can nevertheless do it, and your conclusion will be that the Calabi-Yau manifolds with large fluxes - corresponding to large black holes - will be dominating the ensemble even more dramatically than in the "democratic counting".
I don't like this result too much and I hope that it will eventually go away, but this inconvenient preliminary result can't diminish my sympathy for the thesis that the attempts to define and calculate the Hartle-Hawking state are more scientific than the ad hoc assumptions that "every vacuum has the same weight".