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There are many more flux vacua in string theory than you were told

From \(10^{500}\) to \(10^{272,000}\)

Around 2003, lots of string theorists were maximally excited about the multiverse and the anthropic principle. There were lots of vacua, a tiny portion of them has a small enough cosmological constant which is needed for life to exist, and we just happen to live in one of them.

The estimate \(10^{500}\) semirealistic vacua – googol to the fifth power – became a part of the popular culture. The number has turned into the most popular rallying cry for the anti-science crackpots who loved to repeat (and some of the most retarded ones still love to repeat) their absolutely fallacious critiques of string theory.

The exponent, five hundred, was approximately derived from the magnitude of the Betti or Hodge numbers in some manifolds that may be employed as the shapes of extra dimensions in string theory. Somewhat more precisely, the exponent is comparable to the maximum third Betti number \(B_3\) of some six-dimensional manifolds – roughly speaking, the number of topologically distinct independent 3-dimensional submanifolds or "3D holes" inside the manifold.

Note that \(500\) is greater than \(123\) so that assuming some quasi-random distribution, there are most likely many vacua whose cosmological constant is comparable to the observed \(10^{-123}\) Planck densities or even smaller.

On the other hand, \(500\) is not "qualitatively" larger than \(123\). If Nature has picked one of the \(10^{500}\) vacua, it is not yet a "miracle" comparable to a spontaneous emergence of Adam and Even out of a soup of amino acids (without evolution), if you allow me to pick a particular kind of a miracle that should be believed to be false by scientists. The miraculously low probability is just \(\exp(-10^{26})\) for \(10^{26}\) molecules to spontaneously combine into an Adam. And the landscape exponent \(500\) is much lower than the exponent needed for miracles such as \(10^{26}\).

However, a new beautiful paper today claims to change the numbers dramatically and moves the landscape exponent closer to the Adam territory.

At MIT, Washington Taylor and Yi-Nan Wang released their 20-page paper

The F-theory geometry with most flux vacua
which brutally increases the estimated number of the flux vacua in F-theory. On top of that, they claim that a super-overwhelming majority of these vacua arise on a particular Calabi-Yau four-fold with a particular gauge group and they claim to know everything about this manifold and this gauge group.

First, the number. The number of the flux vacua isn't \(10^{500}\) anymore. Note that this number was estimated using the methods of Douglas+Ashok+Denef. I actually don't see the number \(10^{500}\) in any of these three papers. It's surprising that such a popular number can't be easily traced to the scientific literature.

At any rate, this number \(10^{500}\) is said to be obsolete and Taylor and Yi-Nan claim that a more accurate estimate is\[

{\Large 10^{272,000}.}

\] It's ten to the power of two hundred and seventy-two thousand or so. Nice. Almost all of these F-theory flux vacua are derived from F-theory on a particular elliptic four-fold (eight-real-dimensional manifold) \({\mathcal M}_{\rm max}\) which is an elliptic fibration over a specific three-fold toric base \(B_{\rm max}\).

The manifold has been known since 1997 and 1998, from the papers by Candelas et et al. and Lynker et al.. I am absolutely stunned by this kind of mathematics. They have found this "maximal" manifold and determined that it is a generalization of a projective space\[

{\mathbb P}_5^{(1,1,84,516,1204,1806)}[3612]

\] or even\[

{\mathbb P}_{(1806,151662,931638,2173882,3260733)}[6521466]

\] and it has some impressive Hodge numbers, too:\[


\] Not bad. (Note that these are four-folds; for known or all Calabi-Yau three-folds, both \(h^{1,1}\) and \(h^{2,1}\) seem to be lower than \(500\) and even the sum \(h^{1,1}+h^{2,1}\) seems to be lower than \(350\) for most of the topologies.) The gauge group in four dimensions doesn't start as some modest \(SU(3)\times SU(2)\times U(1)\) that you know from low-brow physics. Instead, it is\[

G_{121328} = E_8^{2561} \times F^{7576}_4 \times G^{20168}_2 \times SU(2)^{30200}

\] which is broken, at generic places of the moduli space, to\[

E_8^9 \times F_4^8 \times (G_2\times SU(2))^{16}.

\] If you a surfer dude who loves the \(E_8\) group, there are nine big reasons to love this manifold (plus eight smaller reasons and sixteen bonus reasons). Lots of seemingly random integers appear all over the place but they're not quite random in algebraic geometry; recall the story about the 1729th yellow cab and the Picard number.

This technology is incredible and I have only mastered some "toy model" examples of that so I couldn't have verified any of the geometric claims above.

At any rate, reliable enough people claim that they have found a very complicated particular eight-real-dimensional topology that may be used as the compactified dimensions for F-theory and it is maximal in a very particular sense. If you count the number of flux vacua that may be built by "decorating" this background geometry with fluxes allowed by string theory, you get about \(10^{272,000}\) string vacua and this group becomes an overwhelming majority of the flux vacua.

If you pick the second most populous manifold, the third one, and all the remaining ones, and you sum up the number of vacua from them, the first one will still have an overwhelming majority. The number of elements in this group will be at least \(10^{3,000}\) times larger than the number of elements in all the minorities combined!

This is possible because the numbers of vacua in this business are exponentially large and the exponent from the "winning" manifold is simply large and substantially larger – by several thousand – than the exponents of all the competitors. So the competitors just don't matter.

Let us call the manifolds with the "winning" formula the "populist Calabi-Yau four-fold", the "populist base" etc.

You might think that the enhancement of the exponent from \(500\) to \(272,000\) doesn't change the story qualitatively because the number was already large, anyway. But I think that the story does change. With the higher new number, a point that I've been making for more than a decade becomes spectacularly obvious: there exist majorities so overwhelming that if you believe in the "typicality" – the assumption that the prior probabilities of different vacua are the same or at least comparable – you will unavoidably deduce that we must live in such a majority.

Concerning the populist manifold, there are three possible options:
  1. We are living in one of the flux vacua upon the populist manifold.
  2. We are living in one of the other vacua.
  3. String theory doesn't describe the Universe around us.
The third option is extremely far-fetched but I've included it for the sake of completeness.

You know, I would like to claim that the stringy anthropic principle predicts the option (1). It seems plausible that we may actually falsify this whole option because all these vacua may have some phenomenologically unacceptable properties.

If that's so, and if we can derive that the option (2) is right, then I would claim that the "anthropic principle" – or any notion of "typicality" – has been falsified.

If their claims about the "super-overwhelming majority of vacua" – and even if they managed to misidentify the most populous topology – and if this most populous group may be shown not to describe Nature (but some of those must still admit some sort of "different life"), we may see far-reaching consequences for any attitude to the vacuum selection problem.

In particular, some people – including pretty much your humble correspondent – believe that some processes in the very early, Planckian cosmology basically imply some probability distributions or probabilities \(p_i\) for all the individual vacua. The "typicality" people would love to choose\[

p_i \approx \frac{1}{N_{\rm vacua}}

\] which amounts to the absence of any mechanism to study. I've never believed that. Instead, I have weakly believed just the opposite assumption, a "misanthropic" principle of a sort, namely that the elements of the most populous groups \(G\) are punished and they are less likely even when you combine the probabilities of all the elements. So\[

p_i\ll \frac{1}{N_{\rm group}}

\] for a large group of vacua. In other words, the laws of physics heavily favor some special vacua. This assumption may be said to be analogous to the observation that highly excited states of a physical system are much less (exponentially) likely at low temperatures than the unique or special ground state.

You may see that the work by Taylor and his collaborator points in this direction. We could actually gain robust evidence in favor of the misanthropic principle. If there's some Hartle-Hawking-like or similar mechanism that assigns the probabilities to the compactification manifold, it may be absolutely necessary for this mechanism to heavily punish compactifications on the populist manifold. There may exist a nearly rock-solid argument that the "uniform measure" has to be wrong at least by 3,000 orders of magnitude. ;-)

The vacuum selection mechanism may of course have some features that set the probability of the populist compactifications strictly to zero. But why? It's a bit hard to imagine that those manifolds are strictly forbidden while the "qualitatively analogous" compactification we inhabit remains allowed. It's more likely that (almost) all the probabilities are nonzero numbers but many of them have to be extremely tiny.

Other people could try to think within alternative working hypotheses.

But I think that this semi-qualitative increase of the "number of the flux vacua" changes the picture and forces you to think differently and abandon certain assumptions that you should have arguably abandoned a decade ago. And again, yes, I believe that either
  1. most of your string phenomenology research should focus on the populist manifold (and even in the huge class of the populist manifold vacua, those with some "typical" distributions and properties are by far dominating)
  2. or you should abandon the "typicality" assumptions altogether; or you should prove that the populist manifold is incompatible with the intelligent life
  3. or conclude that string theory has been falsified.
I obviously choose to "abandon the typicality altogether" – and, in fact, one should begin to favor the "anti-typicality" assumption, I think, and look for the most special, most exceptional, most isolated (and perhaps "simplest" or "lowest Hodge numbers") vacua. But I don't really discard the other options (except for the last one), either. It's possible that we live on the populist manifold and string theory correctly predicts it. And some of the large numbers that happen to describe the topology of the populist manifold may be helpful to solve some hierarchy problems etc.

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