Heterotic landscape: statistics

Much is known about the structure and statistics of the flux compactifications of M-theory and F-theory and even non-geometric flux vacua and non-Kähler compactifications. Acharya and others have thought about the landscape of G2 compactifications of M-theory, too. Some people have even considered the landscape of non-critical string theory with the dilaton stabilized by higher-order effects.

Still, many of the conservatives among us could think that the heterotic models based on the E8 x E8 gauge symmetry naturally predict many features of reality that are less natural in other frameworks, and this is why we should look more carefully at them, regardless of other, less phenomenological arguments. These models predict things like the right unified gauge groups, gauge coupling unification, realistic representations for fermions, see-saw neutrino masses, and an upper bound on the rank of the gauge group (the total rank equals 22 from now on).

Needless to say, the number of four-dimensional compactifications of the heterotic string has been known to be rather high for a long time, too. But do we have an idea about the demography of these vacua?

A talk by Keith Dienes - based not only on this preprint - offers you a positive answer; you can start with the first page and click at "Next page" about 80 times. It's a good time investment!

He and his collaborators have looked at the models that I had loved so much as an undergrad and I still like them: the four-dimensional free fermionic heterotic string models. Take the usual (1,0) heterotic string and keep the four spacetime bosons to describe our superficially four-dimensional spacetime both on the left-moving side as well as the right-moving side. Fermionize all remaining degrees of freedom and imagine that all fermions are free.

On the bosonic side, you will get fermions from 26-4 = 22 bosons, i.e. 44 fermions, while on the supersymmetric side, you get the 8 usual light cone fermions plus the fermionization of the 6 compact coordinates i.e. 12 extra fermions. Recall that one real boson is equivalent to two real fermions. The total is 20 fermions on the supersymmetric side.

Together, you deal with 64 physical fermions in the light cone gauge. Find all possible models where you impose various GSO-like projections in a modular invariant way. Include those without spacetime supersymmetry. The resulting models inherit many great realistic properties from the E8 x E8 ten-dimensional superstring but they have some additional virtues, too.

For example, there is a natural GSO-like orbifold group of projections that smoothly leads to three generations of quarks and leptons. This set of allowed boundary conditions and corresponding GSO projections is called the NAHE set. NAHE is an acronym of the discoverers and it means "beautiful" in Hebrew and "naked" in Czech. ;-)

What I liked about the free fermionic models is that they are, in principle, incredibly simple to deal with in perturbation theory. They describe the same kind of four-dimensional physics as generic Calabi-Yau compactifications. Nevertheless, they are still captured by free field theories on the worldsheet and all the additional structure enters through the allowed boundary conditions which is a discrete, combinatorial addition to your task.

Now, I would be the first one to say that Nature doesn't care whether our calculations are going to be easy. Human laziness is something different than mathematical beauty or rigidity. ;-) But still, it is conceivable that the network of transitions and dualities is so dense that the free models could always be used as a starting point to describe a realistic model. Alternatively, you might conjecture that a deeper scientific principle wants to keep the Kähler moduli of the hidden dimensions near the self-dual values where the free fermionic description becomes very natural.

Why did these models become less attractive in the middle 1990s? The reason was nothing else than the duality revolution: their strong coupling behavior is largely obscure and they don't fit into the simplest "geometric" duality networks of string/M-theory.

But imagine that you look not only at the special choices of the boundary conditions that include the NAHE set but all possible modular invariant choices for the boundary conditions of those 64 fermions. Some of these choices may be special points in the moduli spaces of heterotic strings on the Calabi-Yau manifold. Most of them will probably correspond to some non-geometric asymmetric orbifolds.

The basic computer codes to generate the models and test their modular invariance have been available since the early 1990s. I remember trying to play with these scripts using the oldest versions of Netscape.

Today, they take the machinery to an entirely new level. They divide the different vacua according to their "shatter" - something like the number of factors in the gauge group. In a certain set, there are 123,573 models but if you categorize them according to their gauge groups, you only find 1301 choices for the full gauge group. It actually becomes very hard to increase the number of different full gauge groups well above 1301 - you would have to include a huge number of new models to "randomly" get new gauge groups.

The number of gauge group factors is most likely to be around 13 if it is odd, and it is most likely to be around 20 if it is even. This different behavior according to the number mod 2 reminds me of some results that Gordon Ritter showed me yesterday. He drew the fraction of the number of reducible d-dimensional representations of "SU(n)" groups as a function of "d", and because this function is quasiperiodic with period 16 (not sure whether for any "n" or for "n=16"), the resulting graph looks like 8 DNA double-helix strands. ;-)

If the number of factors is around 12, you get the highest possible diversity for the gauge groups you can obtain. That's natural that the result 12 is in the middle: if there is only one factor, there are not too many choices and you obtain something like SO(44). If there are too many factors, they must be small groups and you don't get too much freedom either. From this perspective, you maximize the possibilities in the middle of the range of possibilities which means roughly 12 factors. Note that this is not necessarily inconsistent with reality because the Standard Model is not necessarily everything there is.

They have made a lot of statistics of this kind. The probability that a model contains the required SU(2) and/or SU(3) factors in the gauge group is of order 10% in various cases. In 99.81% of the models, the number of U(1) factors matches or exceeds the number of SU(N) factors. If you care how "generic" things are and you use the naive democracy as your measure, additional U(1) factors of the gauge group simply are generic predictions in this class much like the additional scalars and moduli are generic in geometric compactifications.

U(1), SU(2), and SU(3) contribute to the total rank significantly only if you assume that there are many factors. Otherwise, most of the rank is obtained from larger groups.

Among the models that have U(1), SU(2), and SU(3) simultaneously, most of them have about 18 factors in the gauge group. There are many other fascinating probabilistic statements, for example about the correlation between the appearance of different factors in the gauge group.

Dienes et al. show that in their non-supersymmetric heterotic models (SUSY breaking at the string scale), only several partition sums can occur at the one-loop level; this contribution is only nonzero because you break spacetime supersymmetry at the string scale. So only a few values of the cosmological constant can appear. They also conjecture that this result holds beyond one loop and maybe exactly - that there are "flat directions in the non-supersymmetric landscape". That sounds like a very strong and hard-to-believe hypothesis. I don't understand why they believe that the property of these simple one-loop results can be extended so much further.

In their ensemble, 73% of the models have a positive cosmological constant. The distribution of the cosmological constant kind of has a peak at small positive values, near zero. ;-) This graph on the page 62.html is kind of fun, too. You could hypothesize that the probabilistic distribution of different values of the cosmological constants is not really uniform in between -M_{Planck}^4 and +M_{Planck}^4 or so. Instead, it could be naturally peaked around Lambda=0, resembling a normal distribution obtained via the central limit theorem. If someone showed that the models tend to add up the cosmological constant to something near zero and the distribution is a Gaussian whose width is close to the observed cosmological constant, your humble correspondent could start to believe that our vacuum might be a random one (with a quasi-democratic measure), after all. ;-)

A problem with the hope in the previous paragraph is that the width of the distribution is expected to be of order the square root of the number of light particle species that contribute, not 10^{-120}. ;-)

The cosmological constant is statistically shown to be in linear relationship with the rank: small ranks typically mean a small cosmological constant, suggesting that the vector contributions dominate the vacuum energy and the scalars' vacuum energy tends to cancel against the spin 1/2 fermions. The cosmological constant also statistically seems to be inversely proportional to the number of factors of the gauge group, suggesting that a small value of the C.C. means an expected large number of factors.

Whether or not someone finds a non-anthropic selection mechanism in this class of vacua or others or, on the contrary, whether or not the generic and seemingly Godless vacua will be increasingly believed to be relevant for reality, I think that it doesn't hurt to know something about the demography of the possible solutions that are consistent in perturbation theory: it gives us a better idea about the measure on the low-energy parameter space and a new refinement of the notion of naturalness. This is why their work is kind of fascinating.