Universal wavefunction

The topological conference was partly dedicated to Raoul Bott who passed away last month. Hirosi Ooguri spoke about chiral baby Universes, Greg Moore about torsion, and so forth.

Robbert Dijkgraaf was the last speaker - and he had the opportunity to discuss topics that are closer to the dreams of physics. First, he helped the mathematical portion of the audience and defined a physicist as a supersymmetric hero (also known as a superhero) who can save the Universe. ;-)

After the beginning, he was explaining things that I find completely uncontroversial. If you fully compactify all spatial dimensions of string theory (compactification on "M9 x R" where "R" is time), you obtain cosmology where everything is connected and all transitions may occur as long as you have a finite energy.

Robbert believes that the right Hilbert space for this completely compactified string theory is a symmetric product of another space "H" - which corresponds to the fact that you may have disconnected Universes. This symmetric product may also be rewritten as a sum over superselection sectors labeled by different fluxes and charges.

He continued with explanations of OSV and OVV - the Hartle-Hawking stuff. OVV is based on an analogy of the cylinder diagrams in string theory that can be interpreted in the open string channel or the closed string channel. The path integrals compute a sort of index that only depends on a few parameters. A conclusion that Dijkgraaf offered is that the index counts various things like the ground states of some quantum mechanical models whose presence prefers a certain kind of Calabi-Yau manifolds with symmetries.

Robbert's ultimate goal was nothing smaller than to reveal the grand structure that underlies all of string theory - i.e. also all of mathematics and physics. A part of this task is to understand the overall space of all Calabi-Yau manifolds. Robbert promoted the program of Miles Reid: every Calabi-Yau's can be obtained by gluing "J" copies of an "S3 x S3" via conifold-like transitions. This procedure should be analogous to the way how you obtain genus "g" manifolds by gluing tori. The simpler manifolds appear as degenerations at the boundary of the moduli space of the really complicated manifolds. Dijkgraaf discussed how this can be reconciled with the picture of a Calabi-Yau manifold as a T3-fiber over a three-sphere. The fiber is degenerate on special points of the base that look like a graph (imagine a skeleton of a tetrahedron).

As soon as the discussion started, Michael Atiyah proposed that Reid's program can be combined with the observations of Atiyah-Witten that the cone over "S3 x S3" is a manifold of G2-holonomy which would allow one to acquire a more unified seven-dimensional perspective on this whole geometric problem.

At any rate, the point of Robbert's talk was that many people should try to study the properties of the Hartle-Hawking wavefunction in string theory.

Concerning this general problem, I started to believe that the small volumes of the moduli spaces of complicated Calabi-Yau manifolds mean something. For example, consider the quintic. The volume of its complex structure moduli space is a number that has some large positive factors in the numerator - but they are beaten by a gigantic number of order "120 factorial" in the denominator.

The volume is tiny. You should not be too impressed by this tiny number because all moduli spaces with dimensions of order 100 tend to have such tiny volumes. However, I feel that the natural Hartle-Hawking measure on such moduli spaces will be more or less uniform, and therefore the probability will be roughly proportional to the volume. If this is true, it means that the backgrounds with a large number of massless (or light) fields will be heavily suppressed and the Hartle-Hawking picture will prefer configurations with a minimal number of light fields (I am thinking about the scalar fields right now).

The relative weights of Calabi-Yau manifolds with different dimensions of their moduli space should be determinable by a proper analysis of the region near the critical transition that connects them, I think.

No one yet knows how to do these calculations right and quantitatively, but I definitely share the belief that people should try to look at these questions and understand the behavior of all relevant factors while avoiding prejudices about the philosophical conclusions.