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16 out of half a billion: elite Calabi-Yau manifolds with a fundamental group

Heterotic phenomenology seems to converge to an excitingly sparse shortlist of candidates

Heterotic compactifications represent a convincing – if not the most convincing – class of superstring vacua that seem to pass the "first great exam" for producing a theory of everything. A week ago, I discussed \(\ZZ_8\) orbifolds but now we return to smooth Calabi-Yau manifolds, the nice creatures you know from the popular books and from the girl who has a 3D printer.

The reason is a new hep-th paper by He, Lee, Lukas, and Sun (of China, Korea, and Oxford):

Heterotic Model Building: 16 Special Manifolds

...Mathematica supplements... (will be posted later)
These 16 manifolds are really special; they seem to have something that the remaining more than half a billion of manifolds in a list don't possess.

What is it? I will answer but let me begin with a more general discussion.

Calabi-Yau three-folds are used in the heterotic string model building and they're 6-real-dimensional shapes whose holonomy group is \(SU(3)\), a nice midway subgroup of the generic potato manifolds' \(O(6)\) holonomy. (Holonomy is the group of all rotations of the tangent space at any point that you may induce by the parallel transport around any closed curve through the manifold.) They come in families – topological classes. The most important topological invariant of such a manifold is the Euler characteristic \(\chi\).

For Calabi-Yaus, the Euler character (let me shorten the term in this way) may be written in terms of more detailed quantities, the Hodge numbers, as\[

\chi = 2(h^{1,1}-h^{2,1})

\] where, roughly speaking, the two terms refer to the number of topologically distinct and independent, non-contractible 2- and 3-dimensional submanifolds, respectively. In two different ways, they generalize the "number of handles" on a 2-dimensional Riemann surface. Let me assume that you know some complex cohomology calculus or you're satisfied with the sloppy explanation of mine.

These Hodge numbers \(h^{1,1},h^{2,1}\) also dictate the number of moduli – continuous parameters that can be used to deform a given Calabi-Yau manifold without changing its topology. Each Calabi-Yau seems to have a "mirror partner" (the relation is "mirror symmetry") that acts as (among other changes) as\[

(h^{1,1},h^{2,1})\leftrightarrow (h^{2,1},h^{1,1})\quad \Rightarrow\quad \chi\leftrightarrow -\chi

\] on the Hodge numbers and (as a consequence) on the Euler character. In total, \(30,108\) distinct Hodge number pairs \((h^{1,1},h^{2,1})\) are known to be realized. The number of known Calabi-Yau topologies is therefore at least equal to this number of order "tens of thousands". The plot is nice and reproduced many times on this blog and on Figure 1 in the paper.

The largest Hodge numbers appear in the "extreme" Calabi-Yaus with \((h^{1,1},h^{2,1})=(491,11)\) and its mirror with \((11,491)\). Those produce \(\chi=\pm 960\) which is the current record holder and the probability is high (but not 100 percent) that no greater Euler character is mathematically possible for the Calabi-Yau threefolds (the basic "pattern" of the picture would seem to be violated if there were larger Euler characters). In this sense, heterotic string theory predicts that there are at most \(480\) generations of quarks and leptons. The prediction seems to be confirmed experimentally. ;-)

There are of course much more detailed predictions you can make if you consider specific models.

But let me first say that there are three main methods to construct Calabi-Yau manifolds:
  • CICYs, complete intersection Calabi-Yaus,
  • elliptically fibered Calabi-Yaus,
  • Calabi-Yau three-folds obtained from ambient four-folds coming with a reflexive polytope.
One needs to know some fancy higher-dimensional geometry and I am not a genuine expert myself but I could still give a short lecture that will have to be dramatically reduced here.

The first group, the complete intersections, are defined by sets of algebraic (polynomial) equations for homogeneous coordinates of products of projective spaces. Candelas, Dale, Lutken, and Schimmrigk began to probe this class in 1988.

The elliptically fibered ones were intensely studied e.g. by Friedman, Morgan, and Witten in 1997.

The paper we discuss now is all about the third group, the Calabi-Yau three-folds extracted from Calabi-Yau four-folds. In 2000, Kreuzer and Skarke listed \(473,800,776\) ambient toric four-folds with a reflective polytope. The list was "published" two years later. You know, it's not easy to print almost half a billion of entries in a journal. Just kidding, the full list was of course not printed. ;-)

This class obtained from four-folds seems to be most inclusive and it's this class which is enough to produce the examples with \(30,108\) pairs of the two Hodge numbers. While one can construct at least one Calabi-Yau three-fold from each four-fold, it's my understanding that the number of topologies of Calabi-Yau three-folds is vastly smaller than half a billion because there are very many repetitions once you "reduce" the four-folds to three-folds.

The new Asian/Oxford paper wants to focus on heterotic phenomenology. For such models to be viable, we need manifolds with a nontrivial (different than the one-element group \(\ZZ_1\): a clever notation, by the way, right?) fundamental group \(\Gamma\). This group counts non-contractible curves along the manifold that are needed for the symmetry breaking by Wilson lines, something that is apparently necessary to break the GUT group down to the Standard Model group in similar models (the GUT Higgs fields are totally circumvented, they probably have to be circumvented, and the Wilson-line-based stringy breaking seems more viable than GUT-scale Higgses, anyway).

In other words, we need four-folds that come in pairs. The pair includes an "upstair" manifold \(\tilde X\) that has an isometry \(\Gamma\) and the "downstair" manifold i.e. quotient \(X=\tilde X/\Gamma\).
To make the story short, they only found 16 four-folds whose order (number of elements) \(|\Gamma|\gt 1\) i.e. for which the fundamental group is non-trivial.
This is quite a reduction, from half a billion to sixteen. The basic topological data about these manifolds are listed in Table 1 on page 9 of the paper. The Euler characters of \(\tilde X\) and \(X\) belong to the intervals \(96-288\) and \(40-144\), respectively. The fundamental group \(\pi_1(X)=\Gamma\) is \(\ZZ_2\) in 13 cases, \(\ZZ_3\) in 2 cases, and \(\ZZ_5\) for 1 manifold.

There has to be quite a competition to get this small shortlist. One could argue that except for the \(16\) entries, the half a billion candidates don't allow life in the sense of the usual anthropic discussions. A non-trivial fundamental group seems to be more important for life than oxygen.

Now, for some esoteric Picard geometric reasons, they eliminate two candidates (with \(\Gamma=\ZZ_2\)) and they try to put line bundles on the remaining \(14\) manifolds and pick the manifold/bundle combinations that lead to three chiral families embedded in consistent supersymmetric models. This produces \(29,000\) models, still a pretty exclusive club. Most of them (\(28,870\)) have the \(SO(10)\) gauge group, my favorite one, when interpreted as grand unified theories broken by a Wilson line; there are \(122\) \(SU(5)\)-based models, too. If you believed that the right vacuum is a "generic" element of the list, it would be likely that the group is \(SO(10)\) which means that there exist right-handed neutrinos, among other things.

It's sort of impressive what progress has occurred in this heavily mathematical portion of string phenomenology in recent years. The advances were made possible by a combination of progress in algebraic geometry and in computer-aided algebra – and with the realization of the importance of holomorphic vector bundles that satisfy the Hermitian Yang-Mills equations as an efficient way to deal with the difficult problem of the gauge fields that have to have a profile on the compactification manifold.

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snail feedback (33) :

reader NumCracker said...

Dear Lubos. A layman's question : did they do their job by brute force computing? So, how to be sure they tested the physically most significant portion of the 10^500 vacua? Thanks

reader Andrei Patrascu said...

hopefully not ;)

reader Luboš Motl said...

Dear NumCracker,

the populist number 10^{500} has nothing whatsoever to do with these vacua. 10^{500} is some vague counting of F-theory flux vacua, a completely different set.

No one can be sure that they have tested a set of vacua that is guaranteed to contain the right one. But the argument that this is likely enough to be the case is based on the natural viability of the heterotic vacua.

The number of semirealistic 4D heterotic vacua is far lower than 10^{500}. try to read at least my blog entry to have some idea about the numbers.


reader Andrei Patrascu said...

It is an unfortunate habit that many experimentalists share, namely to believe that something can be proved right or wrong only after testing each and every possibility by brute force. This is obviously not the case. More often than not some aspects of a problem can be revealed just by looking at its global structure or making a far smaller number of tests than one might naively expect. In this sense there is very likely that there will never be necessary to test 10^500 possibilities to infer the "right" structure one needs! As this beautiful article shows (approximatively, of course)

reader lukelea said...

Heterotic phenomenology seems to converge to an excitingly sparse shortlist of candidates

Dear Lubos, Your headline reminds me of Einstein saying his quest for a unified field theory was motivated by a desire to know whether "the Old One" had any choice or not?

Would it be correct to say that, as a scientist, one is not supposed to have any preferences (prejudices?) concerning questions of this kind?

And yet we are all human beings too, not just scientists. Closure is a satisfying feeling, and closure there most certainly would be if it could be shown that the universe had to be the way it is. For those in search of closure any whittling down of the number of possibilities must seem like progress, a step in the right direction.

At the other extreme, at least by my reading, are those in the anthropic camp, for whom the more possibilities the merrier. Blind chance over "intelligent design" at all costs, they seem to be thinking, not realizing that for others the greater the number of possibilities the bigger the "miracle" of the universe we actually live in (a miracle being defined according to Webster as an extremely unlikely event of extraordinary human significance.

As one born with a religious temperament (it's in my genes!) let me say I have no preference in the matter. All I care about is whether there is a kind of symmetry between pleasure and pain such that in the end every sentient being gets what he/she/it deserves (sum total of pleasure = sum total of pain) This is strictly a biological/neurological issue as far as I can tell, which may or may not be accessible to empirical investigation. See here for example.

Thus I remain as always a tortured agnostic, a state of being over which I have no choice.

reader Luboš Motl said...

I completely agree!

Too bad that not only experimenters but even some very fancy string theorists have promoted the wrong viewpoint you discussed, e.g.


It's clearly not the case that one has to test things one by one. The set of vacua may be fragmented into tons of subsets by various features, we may live in some vacuum that has lots of highly selective features, and of course that one can't prove that we won't be able to find the right one by clever tricks. The only way to "prove" that the brute force is the only possible path is to assume what we want to prove - equivalently speaking, that all the manually untested vacua will always look equally likely. But this isn't the case.

reader Luboš Motl said...

Dear Luke, good questions. As an honest scientist, you can't say that some claims about the Old One - or anyone or anything else - have been established if they haven't (except for one's prejudices).

On the other hand, one is allowed to make bets when he decides what to think about. In fact, it's usually needed to make such bets. The only problem is that Nature doesn't guarantee that your bet is the right one. In many cases, other scientists may already know that and why it's a wrong one. ;-)

I would surely prefer to see a final theory explaining all observed details and parameters of the Universe around us but I have no guarantee it will ever happen.

reader Andrei Patrascu said...

which makes me think ( for some personal reasons I don't wish to further underline here ;) ) at the P vs. NP problem ...

reader Luboš Motl said...

LOL, right.

If they could really rigorously prove - circumventing all the clever structure of string theory vacua - to prove that the search for the vacuum is an NP problem with a large N, and I don't think that they really proved it or could prove it at the mathematical level, then I would view their proof as evidence supporting P = NP because the problem has no reason to be impossible to solve.

reader Dimension10 (Abhimanyu PS) said...

Exciting indeed!

Just 16 manifolds, wow.

reader kashyap vasavada said...

Some naive questions. Are all of these 16 compatible with SUSY? Do they differ very much in particle spectrum. so can you rule out some based on observed spectrum?

reader Luboš Motl said...

Dear Kashyap, these are 16 Calabi-Yau three-folds, i.e. real 6-dimensional shapes. A defining property of a Calabi-Yau - or a property equivalent to other defining conditions - is that it preserves 1/4 of SUSY whenever a SUSY theory is compactified on it.

So of course that all these models preserve (more or less) low-energy N=1 SUSY in 4D. SUSY is an inseparable part of the heterotic phenomenology.

Now, what they have isn't 16 models. They have about 30,000 models - the 16 shapes (well, 14 viable out of them) have to be decorated with gauge bundles. This set of 30,000 was already selected with the requirement that there are three fermion generations and other things; otherwise the number of models would be much higher. If something is wrong about their particle spectrum, it's something subtler that will be studied in future papers by this group.

reader Peter Golian said...

Dear Lubos,

I am engineer developing special effects for
movies. By one experiment I develope 3d fractal RethreeDiG. At below video second part of it is something similar like your above video and I
am working on Calabi Yau surface from it. Sory this video is not best presentation for this page because I am using it for graphical purposes and preparing better one. I am not expert and I would like to know your opinion?

Thank you, Peter

reader Andrei Patrascu said...

I should not lose hope ;)

reader Andrei Patrascu said...

Honestly, I would like it far more to have some all encompassing principle that tells us what we should chose and how in order to get the physical world that we see and not to have to "fit to observations"... at least this is what I think a "beautiful" theory should do. Now, of course, nature may have its own options and maybe there were some "choices" for the old one... :) I tend to become again philosophical but then who knows? The fact that they made so much progress may bring us to some ideas about a "global principle"... P.S. I don't really like anthropic arguments... I always have the impression one can stuff everything in that kind of arguments: why is the cosmological constant so small? It's "anthropic selection"! Why is the Higgs mass as it is? "anthropic selection", etc. etc. :(

reader Dilaton said...

To me it seems that applying the brute force method would not lead to more insight about why a particular (or small number of) vacuum describes our universe than giving up on a deeper understanding of the structure of the landscape and throwing in the towel as the antrhropic people do ...

reader NumCracker said...

Andrei, on the other hand, without testing every single possible solution one needs a really sound mathematical proof of unicity of the physical ground-state. Still, it will be required to deliver SM with its already well-determined parameters. Without that, it is no better (physical) result than phenomenologists can provide us by traditional bottom-up model-building ;-)

reader Andrei Patrascu said...

It is of course not the end of the story... still, I do prefer as said, an all encompassing principle... despite of that I could live with a multitude of possible vacua to chose from. My opinion is that these two options are a result of our somehow classical or semi-classical way of looking at the problem. I believe there are several aspects missing. From the "political" point of view, I must admit that I am afraid that many people will apply the principle "if there is something to be published then it will be published, no matter how little sense it makes"... and this option of "anthropic" argumentation opens a sort of "Pandora's box" for this kind of people.

reader lukelea said...

At the 0.47 mark the soundtrack starts to sound like the dance of the seven veils.

reader Luboš Motl said...

Dear Peter, almost nothing is happening in the video, I am used to action! ;-)

Are you Czech or just a descendant of Czechs? Petr Veselý would be a highly frequent combination of first and last names here.

reader Peter Golian said...

Dear Lukelea,

I don´t understand what do you intend to.
First scene is setup for 3d fractal RethreeDiG (REflection Dimensional Graphical) .It is new kind of 3d fractal. 3d scena only with one object with reflections only one same pattern or part of it.
Sorry about scene but I used it for my painting. I am preparing
presentation with other object as sphere and other reflections as
“cardioid“. My idea was from this video, but make it with physics
of light.
And if I saw reflection “cardioid“ I saw relation with Mobius Strip and this video below
So I made experiment, setup test scene and render it. And result is Mobius Strip in Calabi Yau.
I know it is strange heart, Mobius and so on, even I do not have good feeling from it use as science, but it is too interesting and there are interesting results.

Kind regards, Golian

reader Peter Golian said...

Dear Lubos,

I am Slovak nationality. Vesely is my artistic pseudonym. My original name is Golian.

I have also another presentation, but unfortunately I uploaded incorrect file with grammar mistakes and it had fifty views so I didn´t uploaded correct one. Have a look
By the time I optimized render and decrease render time to 1/10.

Thank you, Peter

reader Dimension10 (Abhimanyu PS) said...

I think this "Peter Golian" here maybe a crack... On one of his video, he says he is going to "prove string theory" by simulating "strings in atom" . . . : )

reader Dimension10 (Abhimanyu PS) said...

Do you know of any such extension to M-theoretical vacua? Can this be extended to G(2) manifolds ? Better still, has this been done? .

reader Peter Golian said...

Dear Dimension10
(Abhimanyu PS),

I know that it sounds too crazy, but I would like to prove this experiment, I think that rotating object in animation is simulated atom with strings. I am working on better quality animation. After that I would identify, quarks, electron, forces … After that I know how to prove thats correct.

regards, Golian

reader Peter Golian said...

Dear Lubos,

I am Slovak nationality. Vesely is my artistic pseudonym. My original name is Golian.

I have also another presentation, but unfortunately I uploaded incorrect file with grammar mistakes and it had fifty views so I didn´t uploaded correct one. Have a look on this please.
By the time I optimized render and decrease render time to 1/10.

Thank you, Peter

reader Peter Golian said...

Lubos this please delete this comment, Thank you Golian

reader Alejandro Rivero said...

Is this guy, Yang-Hui, related to the early mathematician Liu-Hui? http://donwagner.dk/Pyramid/Pyramid.html

reader alejandro rivero said...

Hmm, first I wondered if Mr. He surname could be related to the chinese mathematician Liu Hui,


but of course, it is in honour of the later mathematician Yang Hui.


It is amazing. Are chinese scholars rebaptized after they get his PhD, or were Mr. He's parents already drawing the career path of their son?

reader Luboš Motl said...

Very interesting, Alejandro, but since I am not familiar with the Chinese culture to this extent, I can't really say whether it's too interesting. ;-) Maybe we should ask the author - or someone who knows everything about the Chinese culture, like Tony Zee.

reader Justin Glick said...

Lubos, slightly off topic. Someone uploaded to youtube what must be Ed Witten's famous lecture at Strings 95.


reader Luboš Motl said...

Wow, going to watch the whole thing. ;-)

It was incidentally a happy love-based month for me in Prague.

reader loco said...