**The constructive part of the "landscape" is finite and under control**

James Gray, Alexander S. Haupt, and Andre Lukas posted a highly impressive yet amusing preprint on maths of string theory,

All Complete Intersection Calabi-Yau Four-Folds (arXiv)They looked for all possible eight-dimensional Calabi-Yau manifolds – we call them Calabi-Yau four-folds because it's sensible to count the "complex dimension" which is just four – of a certain constructive type, namely the complete intersections in products of projective spaces (CICYs).

Text with results, Mathematica results, C+Mathematica code (supplementary website)

At least in principle, this subclass is easily accessible because almost everyone knows what a projective space is, everyone can construct their Cartesian products, and everyone can write a set of complex polynomial equations that define submanifolds of these products of projective spaces. In total, 660 of these products ("ambient manifolds") harbor some CICYs.

Nevertheless, the work is formidable and it was done for the first time. The result is that there exist 921,497 topologically distinct complete intersection Calabi-Yaus. Each of them is given by certain matrices. You may download all these matrices: the compressed files have a few megabytes but they uncompress to hundreds of megabytes.

While Calabi-Yau three-folds may have both positive and negative values of the Euler characteristic \(\chi\) – in fact, mirror symmetry universally pairs manifolds with opposite values \(\pm \chi\) – the CICY Calabi-Yau four-folds have a non-negative Euler characteristic. In the complete list of CICYs, \(\chi\) is in between \(0\) and \(2610\).

Almost all of these Calabi-Yau four-folds are elliptic fibrations. It means that these 8-real-dimensional manifolds may be imagined as 6-dimensional real manifolds ("the base") with a 2-dimensional torus attached at each point. Physically, it means that they may be used as the hidden dimensions of generalized, nonperturbative type IIB string theory compactifications envisioned by Cumrun Vafa – i.e. as the F-theory compactifications. Four spacetime dimensions remain large; 6 dimensions of the base are compactified; and the 2 dimensions of the torus ("elliptic fiber", a fiber that is an "elliptic curve") are used in the F-theoretical way. Their complex structure (shape) \(\tau\) determines the type IIB dilaton-RR_scalar complex field which may depend on the location in the 6-dimensional based and which may undergo nontrivial \(SL(2,\ZZ)\) monodromies if you make a round trip around singular fibers.

Whether these CICY topologies form a substantial part of all Calabi-Yau four-folds is unknown but it's plausible that the answer is Yes. Note that F-theory on four-folds is the scenario in which the huge landscapes with \(10^{500}\) vacua is often being discussed. the number 921,497 is so much smaller – it could be analyzed by the world's grad students, one student for each topology, if you were able to teach some maths to grad students in the humanities as well – because it only counts different topologies. The vacua carry some extra decoration for each topology, namely the generalized electromagnetic fluxes, and the number of the integers that determined these fluxes is a power of a googol for a complicated enough topology.

Much of the irrational disgust by string theory is caused by people's widespread mathphobia. People think that if the number of possibilities or solutions to certain conditions is large, the topic ceases to be a science and it can't be analyzed. But as this paper and others show, it may often be analyzed and the possibilities may be, in fact, fully listed and classified. In principle, one may also find the right vacuum that describes the Universe around us if it exists in a given set.

By the way, there's another math-oriented stringy paper constructing a non-linear realization of \(E_8\) and connecting it with the bosonic fields in M-theory, including a dual gravitational potential of a sort.

## snail feedback (8) :

I just misused reading this exciting post to cool down and cheer me up, because I got very upset and annoyed about something at work and urgently needed to read something nice to not explode ... :-P

Cheers and thanks :-) !

Nice results.

Two questions by a string layman:

* Could the same techniques in principle be applied to classify the CY 6-manifolds ?

* What is the relevance of CY manifolds in string theory these days ? I thought that the focus has shifted to G2 manifolds.

Best.

Is it easy to limit four-folds to those giving three generations? As in, does the current list already have the needed information?

Do you mean 6-real-dimensional Calabi-Yau 3-folds or 12-real-dimensional manifolds?

I am sure that the research focusing on G2 compactifications remains negligible relatively to the Calabi-Yaus. In fact, I don't remember any G2 papers (in M-theory) from recent years - a pity, I would say. The complex numbers are powerful and people want to work with them because it's simpler.

Still, it's important to distinguish CY3 compactifications (of heterotic strings) and CY4 compactifications of F-theory. Both are Calabi-Yaus but 3-folds and 4-folds have almost as different properties as 3-folds and G2 holonomy manifolds.

It's doable, see e.g. this paper and its section 5

http://arxiv.org/pdf/0906.4672.pdf

and there are of course many papers like that. People doing F-theory know how to count generations - and many more things. There are many such models.

Nice. I hope that would bring the amount of interesting cases down to hundreds so you don't have to teach humanist grad students math ;)

Dear Lubos,

thanks for answering my questions.

I meant 6 real dimensions. What came to my mind was this (heroic) paper

http://arxiv.org/abs/hep-th/0002102

- but I haven't read it.

"In fact, I don't remember any G2 papers (in M-theory) from recent years - a pity, I would say." Interesting. A pity, indeed.

"The complex numbers are powerful and people want to work with them because it's simpler." - Complex numbers are powerful, yes, nevertheless, octonions are central stage in mathematics and very deep, e.g. closely related to E8, sporadic groups, etc.

Too me G2 appears much more natural and is more unique than is this Calabi-Yau "patchwork". But time will tell - hopefully - who is right ....

"Both are Calabi-Yaus but 3-folds and 4-folds have almost as different properties as 3-folds and G2 holonomy manifolds." Yep, that answers my question.

Best.

Just a minor comment:

"Whether these CICY topologies form a substantial part of all Calabi-Yau

four-folds is unknown but it's plausible that the answer is Yes."

It doesn't take away from this very ncie work, but I think the answer is almost certainly 'no'. In the threefold case, there are about 7000 CICY matrices, but about half a billion reflexive four-dimensional polytopes, which correspond to Calabi-Yau threefolds via Batyrev's construction (http://arxiv.org/abs/alg-geom/9310003). It's not known how many of these give topologically distinct manifolds, but there are well over 30,000 different pairs of Hodge numbers (compared to only 264 for the CICYs). If we generalise further to complete intersections in arbitrary higher-dimensional toric varieties, 'counting' becomes impossible. The situation will be even worse for fourfolds.

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