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A huge landscape with 16 supercharges

Jester at Resonaances writes about a project of Edward Witten who is spending some time at CERN, apparently bringing shock and awe to some slightly less famous string theorists over there. ;-)

I understand where the fear comes from but I think it is counterproductive and irrational, after all. And as a citizen of EU, I have to urge the CERN people to defend the good name of the European hospitality rather than the image of CERN as a very cold place (1.9 Kelvin).

So please, participate at some lunches with Edward Witten and listen not only to his cool ideas but also to some precious observations why some of your work or approaches suck, please: he is a very pleasant person, after all. Thanks! :-) But that's not what we want to talk about here.

How large is the landscape?

Recall that a substantial body of evidence has accumulated since 2003 that implies that there exists a countable but huge number of semirealistic stringy vacua in four dimensions. They're parameterized by the topology of the internal dimensions, the information about the branes, fluxes, Wilson lines, and many other things.

The existence of the extensive landscape is not really new: people have known that string theory admits a large number of four-dimensional backgrounds for more than two decades. However, during the last 5 years, the number of these vacua has been estimated as 10^{500} or more. (Some people, e.g. Wolfgang Lerche, have been mentioning even higher number as early as in the 1980s.)

Many people dislike the very existence of the landscape but their emotional gestures are probably the only thing they can do against this mathematical fact.

An entirely different question is concerned with the rules that choose the right compactification. The anthropic people want to believe that Nature has chosen a random vacuum from the landscape, in order to make the cosmological constant small and to make a few dozens of other low-energy parameters "life-ready".

For this interpretation, they need the number of a priori conceivable vacua to be qualitatively similar to one googol, something like 10 to the power of a few hundreds. There's no real proof that the total number of vacua to start with is of this form: no upper bound is known. In the anthropic ideology, the existence of our world required a random event in which our vacuum was chosen from 10^{500} candidates or so. That reduces our ability to predict the world by "pure thought" but the information you need to extract from the real world to identify the right vacuum and to restore predictivity is only of the order of "500 x log(10)" which is considered acceptable by the anthropic people.

An infinite landscape?

Of course, if the number of a priori conceivable vacua were infinite, it would be impossible to assume that all vacua are equally likely: there is no uniform yet normalizable probability distribution at infinite sets. So all the anthropic arguments that our world is generic need to assume that the landscape is finite. And in fact, it shouldn't really be much greater than a small power of one googol, otherwise the "generic" vacua could have low-energy parameters that are much less random than what they seem to expect.

For example, among 10^{10^{120}} vacua, virtually all of them could have a cosmological constant almost equal to the constant we observe.

In 2006, Michael Douglas and Bobby Acharya articulated the "anthropic" assumptions of the landscape. The landscape of the four-dimensional vacua is finite as long as you require that the cosmological constant is lower than a certain bound, the lightest Kaluza-Klein tower is heavier than a certain bound, and the compactification volume is similarly smaller than a certain bound.

This conjecture has been pretty much proven. As far as I understand, Witten's construction below doesn't give us a counterexample to the Douglas-Acharya conjecture; their having a universal bound on the compactification volume is a very strong assumption, after all. But it changes the numbers so substantially that it makes the Douglas-Acharya theorem "morally wrong".

Witten's AdS orbifold vacua

We are going to follow Witten and Jester and construct a huge class of four-dimensional AdS vacua with sixteen supercharges (and a negative cosmological constant). These vacua allow you to count instantons by various quiver theories and enjoy the new Gaiotto-Witten mirror symmetry relating the Higgs and Coulomb branches. But we will focus on their existence and implications for the interpretation of the landscape.

Start with the maximally supersymmetric AdS4 x S7 background, the near-horizon geometry of M2-branes in M-theory. And orbifold it. What you want to study are orbifolds of the type

AdS4 x S7 / Zm x Zn.
Fine. So I must explain how the Z_m, Z_n groups act. They rotate the 8 transverse dimensions of spacetime by elements in SO(8). The generators of both Z_m and Z_n are chosen to preserve the same one half of supersymmetry; at least I hope it is possible to embed them in SO(8) in this way: some other choices of Z_m and Z_n would probably break the SUSY down to one quarter. But when an SO(4) x SO(4) subgroup of the transverse SO(8) is considered (because of triality, note that there are 1 vectorial plus 2 spinorial qualitatively different ways to embed it), the Z_m group belongs to one SO(4) factor while the Z_n group is a subgroup of the other SO(4).

This description of the orbifold makes it clear that there will be two, A_{m-1} and A_{n-1}, singularities in the compactification. They will intersect one another. SU(m) and SU(n) gauge theories live on the singularities. Note that both of them are real codimension four loci inside the seven-sphere: they are three-dimensional (times the AdS4). The geometries supporting the gauge theories are AdS4 x S3 / Z_m or Z_n, respectively.

In seven dimensions, two three-manifolds behave similarly as a pair of strings in three dimensions: you may entangle them in "knots" and these "knots" can carry additional information analogous to Wilson lines. More explicitly, the Z_m and Z_n groups above may be chosen to act not only on the 8 transverse coordinates but they can also act on the "other" SU(n) or SU(m) gauge group. The combined action of the orbifold generators requires you to define elements of SU(n) and SU(m) whose m-th and n-th power is the identity, respectively.

Seeing why the sets are gigantic

If you write the eigenvalues of such SU(m) and SU(n) matrices, it's easy to see that the number of choices is very large even for moderate m,n. The number actually is
(n+m-1 choose n) x (n+m-1 choose m)
Wow. Taking m=n for simplicity, it is
(2m-1 choose m)2
These combinatorial numbers grow rapidly with "m". Qualitatively they go like "m!", the factorial of m.

A small googolplex of vacua

Think about the bounds on the compactification volumes. If you allow the compactification volume to be comparable to 10^{10} in the eleven-dimensional Planck units - an estimated gap between the fundamental Planck scales and the scales accessible to the accelerator - the construction above admits something like
(1010)! = 10^{10^{10}} or so
vacua. In the counting of the landscape, we have essentially replaced the popular exponent 500 by several billions. These vacua with 16 supercharges are unrealistic - too many supercharges and too high cosmological constants - but it is conceivable that similar constructions may be developed for vacua with 4 or 0 supercharges, too. The realistic Standard Model-like spectrum could perhaps be restored by adding stuff to Witten's construction and the excessive cosmological constant could perhaps be canceled.

(I don't know how to extend the construction above to smaller supersymmetries because in this form, it is linked to S^7 and its isometries too tightly.)

If you deal with 10 to the billion of vacua and treat them "democratically", as the anthropic people like to do, I think that the expected statistical distributions for low-energy parameters will no longer be uniform. On the contrary, the "density of vacua" in the low-energy parameter space is almost guaranteed to resemble "exp(L)" where "L" is a large, non-constant function of the parameters, taking values comparable to a billion. With such an Ansatz, almost all the vacua sit at the maximum (or a few maxima) of "L", by a "saddle point approximation".

Note that if the number of vacua were even more gigantic than considered previously, the predictivity would actually return to physics if you assumed that we live in a generic vacuum within a pre-defined set.

On the other hand, I think that such a substantial increase in the estimated number of the vacua also makes it more manifest that the "uniform" probability distribution on the vacua can't be the correct one. If you believed in genericity, the large sets of vacua in Witten's case would be dominated by the vacua with the maximum allowed values of m,n, i.e. the cutoff. While all values of the fluxes in KKLT vacua could have looked like "equally fundamental", Witten's construction makes the vacua with small m,n feel "more fundamental" and the "first among peers". It seems pretty manifest that in all sensible statistical distributions, the vacua with small values of m,n would be "more likely".

At any rate, Witten's enhanced landscape would falsify the comments that "physics is over" and "physicists should give up any research of the vacuum selection". Even the simplest assumptions about the probability distributions on the landscape would be highly predictive. Moreover, everyone should see that it is not the most naive, but the most physically natural and viable distributions that should be considered. The early cosmology and the vacuum selection problem provide us with a lot of legitimate scientific challenges that haven't yet been solved but that can be solved in the future.

And that's the memo.

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