The question about finiteness of the "landscape" is an important question because if someone claims that the probabitility distribution on the space of vacua may be approximated by the uniform distribution in the zeroth approximation (and the exact distribution is simply this number multiplied by another, less important function, or it is even exact?) - an assumption that I've always considered irrational - then an infinite number of vacua presents a big hurdle because there is no uniform distribution on infinite sets.
Why do I think that the assumption is irrational? The numbers like 10^{500} are effectively infinite and a rational argument simply should not break down if this huge number is replaced by the true infinity. If there exists any rule that decides about the probability of different vacua, it must undoubtedly be able to imply that a subclass of 10^{500} vacua - or an infinite subclass for that matter - has a vanishing probabilistic weight. Assuming that this is not possible is equivalent to repeating Zenon's error with Achilles and the turtle: Zenon also believed that Achilles could never catch up with the turtle essentially because the path separating Achilles and the turtle may be divided to infinitely many line intervals.
My personal prejudice is just the opposite one: I believe that the classes of similar vacua with too many elements - i.e. the less predictive ones - will be disfavored when we understand things better at the end. It is roughly because the vacua may be organized as a countable set and the populated classes will only appear at the end (close to the infinity) with a very high "ranking" or "meta-energy" and a small "meta-Maxwell-Boltzmann" statistical weight. This will reflect the desire of the theory to be as predictive as possible. But unlike the statistical counters, I admit that this is a pure prejudice, not a piece of science.
So is the number of vacua finite or infinite? Frankly speaking, the number of vacua has always been infinite. For example, the "AdS4 x S7" supersymmetric backgrounds come as an infinite class parameterized by the integer flux. The only way how we may eliminate some of them is to say that the seven-sphere is really far too large and we only want the backgrounds in which the size of the internal manifold is much smaller than the AdS curvature.
A similar situation occurs in the new class of DeWolfe, Giryavets, Kachru, and Taylor. They construct an infinite class of type IIA supersymmetric flux vacua in which the internal manifold may be arbitrarily large. Obviously, if we want to use the statistical reasoning that starts with the uniform probability measure, we must cut most of the vacua off - which means to ban all the vacua in which the compact manifold is "larger" than a certain bound, according to a certain criterion. However, on page 35 they admit the following:
- On the other hand, one can legitimately worry that the conclusions of any statistical argument will be dominated by the precise choice of the cut-off criterion since the regulated distribution is dominated by vacua with volumes close to the cut-off.
Do I think that this example should stop the growth of the anthropic and statistical treatment of the vacua? Yes, much like another blogger, I do think so. It is OK if a few people are thinking in this direction, but it is inappropriate if this approach became the most active subfield of string theory. Any conceivable prediction of the statistical framework depends on the exact statistical distribution - something we don't know yet (and we don't even know whether it is a good question to be asked). The uniform distribution is as far from the true one as any other random guess we may make. The uniform distribution is just a reflection of a prejudice which is as good as any other prejudice (for example, the assumption that the Hodge numbers of the internal manifolds must be as small as possible) - except that this particular prejudice (uniform distribution) cannot work at all because the number of vacua is infinite.
Once again, the goal of theoretical physics is to find the correct theories and correct models, not a randomly chosen probabilistic measure on the space of incorrect models. The latter is unphysical. The theories and models are physically attractive if they reproduce some features of the real world, and if they are mathematically elegant and robust. Concerning the flux vacua, I doubt both features. Nevertheless, it is clear that the "statistical" approach will continue. The authors summarized the main philosophical idea on page 32 of their paper - a refreshing and entertaining quote is included. The first sentence of section 6 also explains the relation of the anthropic principle to religion and it says:
- "To understand God's thoughts we must study statistics, for these are the measures of His purpose" - Florence Nightingale (a famous nurse, 1820-1910)
Stabilization
We at Harvard were looking forward to see this paper for another reason: it looked - and looks - like a truly concrete example of the families of completely stabilized vacua that we are supposed to think about. A calculable construction could allow one to identify the dual conformal field theory and sharpen many conceptual and technical questions. We thought that unfortunately, it was not the case. Their models only stabilize the geometric moduli and there remain other moduli that are not fixed. They are the periods of the three-form but we may call them axions.
But Wati Taylor has pointed out to me that my discussion was too negative: they show that one combination of the axions is stabilized, and they do present an example where this one axion (or combination) is the only axion. In this example, which is also an infinite family, all moduli are stabilized classically, as desired. Thanks, Wati, and also Shamit and Oliver, for your feedback!
Concerning the other classes of vacua with unstabilized extra axions: the separation of the moduli to geometric and non-geometric would be artificial. We have known for more than a decade that they can be related by various dualities and it is important for string theory that these two types are able to transmute into each other. Drawing thick lines between geometric and non-geometric moduli would mean to return physics 15 years to the past. But it is not just a philosophical question: the dual conformal field theory, whatever it is, simply cannot distinguish between these two types of moduli. If the theory is stabilized, it must be all of the moduli that are stabilized. Examples where some moduli are stabilized - and even examples where probably all of them are stabilized but some of them at uncalculable values - have been known for decades, too - and a heterotic example with the racetrack mechanism is mentioned as the first fast comment under this article.
When we now see this model to stabilize all the moduli, one may try to find the dual CFT. Sergei Gukov says that he has some ideas how to find it, for example.
