Most of the talks at Strings 2005 about the landscape have now taken place, although there’s at least one more this afternoon by Dine. Frederik Denef gave a survey talk entitled Constructions and distributions of string vacua. One amusing thing he does is note that even in toy models with these exponentially large numbers of states, counting the number of states with vacuum energy less than some bound is computationally an NP-hard problem. He describes a wide range of constructions that people have come up with to fix the moduli, concluding that you can “throw enough ingredients together to get sufficiently complicated potential, and this will fix moduli, at least at effective field theory level”, but that these constructions are “ugly”. He then goes on to survey various results about the statistical distributions of these states, and ends by announcing a workshop in Trieste next spring on “String Vacua and the Landscape.”
The talk on Is the number of string theory vacua finite? by Michael Douglas makes Denef’s survey of distributions of vacua kind of pointless. The number of such vacua is definitely infinite, which ruins ones ability to get a probability distribution by counting vacua. Douglas hopes that by putting in a cutoff on the diameter and volume of the compactification space, as well as the size of the vacuum energy, he can make the number of vacua finite. He explains this conjecture, for which the evidence is not very compelling.
Even if he gets the finiteness he hopes for after imposing these cutoffs, the problem then is that the distributions of vacua depend strongly on the cutoff and are peaked at the cutoff value. This is what happens in examples that Kachru talked about at the conference. Douglas is reduced to arguing that “it seems a priori plausible that cosmological selection could depend on the volume of the extra dimensions”, i.e., that somehow the Big Bang would get rid of the problem that his program is predicting large compactification spaces when he wants small ones. There seems to be no reason for this other than wishful thinking. One thing is clear though now: it makes no sense to spend time computing distributions of these vacua, since this gives a result you don’t want. In this game though, it’s not like you give up on your research program when it gives results that don’t look at all like the real world.