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Promising class of heterotic \(\ZZ_8\) orbifolds

Edward Witten is 62 today, congratulations!
After many twists and turns and the birth of roughly five competing classes of stringy compactifications, I still consider the heterotic strings to be the most realistic category of the stringy vacua we know. Heterosis is a powerful tool.

Most of the time we mention the heterotic strings, we usually think of a compactification on a smooth Calabi-Yau manifold. The world may be described by one of the 10,000 or so known topologies of six-real-dimensional Calabi-Yau three-folds (each of which has several continuous parameters, the moduli). Those require interacting (not free) world sheet theories and the experts who study them quantitatively usually have to be hardcore mathematicians – very good geometers whose home is a higher-dimensional space full of bundles and sheaves. Until recently, they would find it almost impossibly hard to find some allowed and realistic vector bundles but with the help of line bundles, they've made lots of progress in this technical hurdle during the recent years.

But the world may be simpler than that. The spacetime coordinates describing the Calabi-Yau directions may be fermionized which leads us to a large subclass of free fermionic heterotic models. And even if these degrees of freedom remain bosonized, we may obtain free world sheet theories because the Calabi-Yau geometry may be based on \(T^6\), the six-dimensional torus. We actually need to consider orbifolds to break the unrealistic gang of 16 supercharges to the phenomenologically viable 4 supercharges.

The realistic orbifolds are compactifications on \(T^6/G\) where the discrete group \(G\) may be \(\ZZ_2\times \ZZ_2\), \(\ZZ_2\times \ZZ_4\), \(\ZZ_{6-I}\), \(\ZZ_{6-II}\), \(\ZZ_{12-I}\), \(\ZZ_{12-II}\), and probably a few others. But a new German-Greek paper
MSSM-like models on \(\ZZ_8\) toroidal orbifolds by Stefan Groot Nibbelink, Orestis Loukas
focuses on a different group \(G=\ZZ_8\) that is, in some sense, more extreme than the previous groups. In their computer-aided search, they should have gone through something like 100 billion candidates but some of them form classes of equivalent copies and some symmetry considerations are used to reduce this number. This speedup is needed for practical reasons but it's probably not quite clean so they may be paying the price of omitting some genuinely different candidates. At the end, they are left with something like hundreds of models.

In all these models, closed strings – and heterotic strings must be closed because open strings' boundary conditions must relate the left-moving and right-moving degrees of freedom but they have qualitatively different types on a heterotic string so they can't be matched – may be twisted by an element of the orbifold group. The twist may include an \(SO(6)\) rotation of the six dimensions in \(T^6\). The allowed group of orbifold twists must actually be a subgroup of \(SU(3)\subset SO(6)\) because to preserve at least a glimpse of order and realism, we want to produce a supersymmetry-preserving Calabi-Yau manifold.

I would add that you may interpret the discrete group \(G\) as the holonomy group of the singular manifold – orbifold – which is actually a subgroup of the "generic" Calabi-Yau holonomy. In this case, we have the \(\ZZ_8\subset SU(3)\) holonomy group.

We mustn't forget that these orbifolds start from a torus \(T^6\) that is defined as the quotient \(\RR^6/\Gamma\) involving a lattice \(\Gamma\) and this lattice must have a \(\ZZ_8\) symmetry in the present case. A two-dimensional lattice couldn't have a \(\ZZ_8\) symmetry – recall that the allowed symmetries are \(\ZZ_2\), \(\ZZ_3\), \(\ZZ_4\), and \(\ZZ_6\) (although quasicrystalline symmetries such as \(\ZZ_5\) are interesting and allowed if we drop the requirement of the strict periodicity).

However, if the lattice is six-real-dimensional, \(\ZZ_8\) is possible and just fine. The generator of \(\ZZ_8\) acts on the three complexified directions of \(T^6\) by adding three phases,\[

(z_1,z_2,z_3) \mapsto (e^{2\pi i v^1} z_1, e^{2\pi i v^2} z_2, e^{2\pi i v^3} z_3)

\] and the coefficients \((v_1,v_2,v_3)\) encode some basic data of the orbifold twist. It turns out that there are two inequivalent choices for this vector \(v\), namely\[

v=\frac 18 (1,-3,2), \qquad v=\frac 18 (1,3,-4),

\] which define the classes of orbifolds called \(\ZZ_{8-I}\) and \(\ZZ_{8-II}\), respectively. The number \(8\) in the denominator means that each of the three complex planes are rotated by a multiple of 45 degrees by any element of the orbifold group. For the two choices of \(v\) above, the first two complex coordinates are rotated by any multiple of 45 degrees in one of the sectors. The third complex coordinate is only rotated by multiples of 90 degrees or multiples of 180 degrees in the two inequivalent classes of \(\ZZ_8\) models.

Note that the sum of the three coordinates of \(v\) is equal to zero, modulo one, which is needed to preserve the supersymmetry – it's needed for the \(S\) in the \(SU(3)\) holonomy group, if you will.

The rotation of the six compactified bosonic coordinates is just the "most nontrivial part" of the orbifold action. There can also be an action that shifts these coordinates by a translation and that shifts or rotates the 32 fermions responsible for the \(E_8\times E_8\) symmetry we start with. Those are the "gauge shifts" and "Wilson lines" that are highly non-unique and have to be classified with a help of a computer program.

It turns out that there are three 6-dimensional lattices defining a six-torus that allow the \(\ZZ_{8-I}\) orbifold, namely\[

SO(9)\times SO(5), \quad SU(4)\times SU(4), \quad \text{non-Lie lattice}.

\] The first two lattices are root lattices of the eponymous groups while the third choice is a lattice not associated with any Lie algebra. Similarly, the \(\ZZ_{8-II}\) orbifolds may be built from root lattices of\[

SO(9)\times SU(2)^2,\quad SO(10)\times SU(2)

\] and there's no other (e.g. non-Lie-algebra-based) lattice in this case. So the authors go through the whole conceivable set of choices for the gauge shifts and discrete Wilson lines (a finite number) in each of these \(3+2=5\) subgroups of the orbifolds and look for realistic models.

At the end of their selection, they require the Standard Model group as a part of the unbroken gauge group, the correct GUT normalization of the hypercharge, the presence of at least three generations of quarks and leptons, at least one pair of Higgs doublets (a vector-like Higgs doublet), and the absence of exotics – more precisely, all Standard-Model-group-charged particles that differ from the usual content of the Standard Model must come in vector-like representations (full, uniformly charged Dirac spinor for fermions, for example) which allows the pairs to combine and get very massive (and therefore invisible to us).

In some counting, before they impose all the "realistic" filters, they see that 87% of their models contain exactly 3 generations of quarks and leptons! You could say that with the IPCC methodology and confidence level, the \(\ZZ_8\) orbifolds predict exactly three generations of quarks and leptons! This is far from a rock-solid, indisputable prediction of a theory but it is an incredibly intriguing feature of this class of string vacua.

In the two classes out of five I have mentioned (\(3+2=5\)), namely in the \(SU(4)\times SU(4)\) and non-Lie lattices, the percentage of three-generation models is really overwhelming (49 of 49; and 78 of 81 models, respectively). They also find exactly two models in the whole search that produce the exact MSSM spectrum and nothing else.

For the models that contain additional states beyond the MSSM, they statistically evaluate how many states of various kinds tend to be present. Bifundamental particles are rare (they would probably be omnipresent in braneworlds) while triplets are almost omnipresent. Charged and neutral singlets appear in various reasonable percentages of the models that are captured by numerous histograms in the paper.

It's also interesting to look at the hidden sector gauge groups in these models. These groups may be \(SU(8)\), \(SU(7)\), \(SU(6)\times SU(2)\), and other 24 possibilities (Table 6).

We're clearly far from "unique solutions" but I can't get rid of the feeling that the heterotic vacua (and perhaps some other subclasses of vacua), especially this class, tend to naturally predict certain patterns seen in the Standard Model spectrum but unexplained by the Standard Model or at least correlations between the patterns, at least in an unexpectedly large percentage of the vacua. In my opinion, this has always been a legitimate reason to think that such string vacua are more promising than other string vacua for which the realistic features are much less likely (the powers-of-a-googol of the F-theory flux vacua, for example).

Also, I feel that these constructions look like simple enough theories and it might be interesting to order some grad students (and others) to study some of these models, e.g. the two pure MSSM models in this \(\ZZ_8\) search, really carefully because they could have certain properties that actually automatically solve some problems that many of us count as nearly unsolvable, e.g. the cosmological constant problem. These heterotic vacua are incredibly rigid; nothing can be adjusted about them. So if they're right, they must simply solve the C. C. problem and other problems without any additional fudges or assumptions. I actually find it plausible that some of them do (or does) despite the fact that we're obviously ignorant about the mathematical reason that does the trick.

BTW Willie Soon sent me a transcript of this very long, 5-part 1966 interview with Richard Feynman. Some readers may like it.

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snail feedback (8) :

reader Dimension10 (Abhimanyu PS) said...

That sounds sad, actually : (

All the work that phenomenolgists have done are going to go to waste : (

How sure is it that we'll get the; right 125 GeVw higgs in this compactificaiton as well?

reader Luboš Motl said...

Huh, what exactly sounds sad here?

reader Dimension10 (Abhimanyu PS) said...

The sad thing is precisely that the Higgs mass hasn't been calculated yet, and as you said, it's a hard thing to do. Of course, everyone would hope that it would be calculated finally.

Wouldn't all the work on about M-theory on G2 manifolds, Heterotic E8 X E8 on CY manifolds be wasted if this is true? Or am I just confused?

I of course *do* feel "exhilerated" about this but it'd be scary if the Higgss mass wouldn't turn out right. And I certainly didn't say this should be discarded! Please don't ban me!

And, by the way, I was not making "terrorist" attacks against Physics.SE. I was only joking when I told Dilaton to make a sockpuppet : ) I obviously wouldn't actually encourage someone to do so.

reader Luboš Motl said...

Dear Dimension10, obviously at most one string vacuum - and I think that exactly one - is directly relevant for the visible part of the Universe.

This single string vacuum may have several mutually dual descriptions, however, so it's not necessarily true that the correct vacuum only occurs in one class of the descriptions (such as heterotic E8 x E8 description). There may be many models that provide us with expansions around different places that can "reach" to the correct point in the configuration space.

All the other solutions to string theory's equations are "allowed states of matter of being" and they exist "somewhere out there" so I surely don't consider them "trash" just like I don't use the word "trash" for some isotopes of nuclei just because they're absent on Earth.

reader Dimension10 (Abhimanyu PS) said...

Phew, my iq is at least 50 : ), thankfully. Or not.

Thanks, I see that the answer to my quwestion was that I am just confused : )

So, you mean to say that a vacua of M-theory compactified on a G2 manifold or Type HE on a CY can be dual or equivalent to a vacua of M=theory, -compactified this way? On a simple toroid?

reader Luboš Motl said...

Dear Dimension, even 9-10-dimensional vacua are enough to connect all of superstring/M vacua by dualities.

This translates to dualities for 4D realistic vacua, too. Heterotic E8 x E8 strings become heterotic M-theory at strong coupling. Also, the compactification of heterotic strings on T2 fibers is equivalent to type II string theory on K3 fibers so many realistic heterotic vacua have type II duals. You can view these type II vacua as type IIB flux vacua or type IIA braneworlds, sometimes related by T-duality, while the latter - type IIA braneworlds - may be sent to strong coupling and given an alternative description using G2 holonomy manifolds of M-theory.

All of those classes are connected which doesn't mean that every vacuum in each class has a (known or any) dual in all the other major classes.

reader Dilaton said...

Hi Dimension10

I think these models here are very nice and a good progress towards putting more and more SM features on a nice string theoretical foundation.

It does not make me sad, but will make me sleep with a happy smile on my face to further recover from my surgery :-). It went well so far and I'll probablybe back home tomorrow.


reader Luboš Motl said...

Dear Dilaton, thanks for telling us and me about the good news! Keep on recovering. Yours LM