Monday, April 25, 2005

The volume of the haystack

We've been asking various questions to Frederik Denef and Michael Douglas who are visiting us.

One of the things one typically imagines is that the volume of the haystack (formerly known as the landscape) is very large. How large is large?

Take the quintic hypersurfaces in CP^4. They have 101 complex structure moduli. Construct the 101-dimensional moduli space, determine its Kahler metric from the kinetic terms in type II string theory, and measure its volume. What will you get? Something like
• 1 / 5^24 times...
well, that's a pretty small number, but it's not the worst factor, so let me continue:
• 1 / (5^24 times 120!)
Yes, it's the factorial of 120 in the denominator. That's a wicked small number, something like 10^{-250}. A typical example of my thesis that the "very interior" of the haystack (or the "configuration space") has a small volume. Nevertheless, in this small volume, one is supposed to find googols of vacua. That's because the estimated density of the vacua
• det (R - omega)
where "R" is the curvature and "omega" is the Kahler form (in dimensionless units) really does not contain the factor (1/120!). But still, don't you find it a bit strange that there is a density of 10^{350} metastable vacua per unit volume? We don't have a real emotional intuition how "density" in a very-high-dimensional space should behave, but we should probably try to learn it. I feel that these (especially the de Sitter) vacua cannot be quite isolated. There are just many other vacua nearby (virtually all of them) into which one should be able to decay. KKLT only consider one Coleman-DeLuccia instanton, without an enhancement, and I feel it can't be the whole story.