The Landscape of M-theory Compactifications on Seven-Manifolds with \(G_2\) Holonomyby David Morrison and James Halverson. The most important classes or descriptions of superstring/M-theory compactifications (or solutions) that produce realistic physics are

- heterotic \(E_8\times E_8\) strings on Calabi-Yau three-folds, string theorists' oldest promising horse
- heterotic Hořava-Witten M-theory limit with the same gauge group, on Calabi-Yaus times a line interval
- type IIB flux vacua – almost equivalently, F-theory on Calabi-Yau four-folds – with the notorious \(10^{500}\) landscape
- type IIA string theory with D6-branes plus orientifolds or similar braneworlds
- M-theory on \(G_2\) holonomy manifolds

Halverson and Morrison focus on the last group, the \(G_2\) compactifications, although they don't consider "quite" realistic compactifications. To have non-Abelian gauge groups like the Standard Model's \(SU(3)\times SU(2)\times U(1)\), one needs singular seven-dimensional \(G_2\) holonomy manifolds: the singularities are needed for the non-Abelian enhanced group.

They are satisfied with smooth manifolds whose gauge group in \(d=4\) is non-Abelian, namely \(U(1)^3\).

Recall that \(G_2\) is the "smallest" among five simple exceptional Lie groups – the others are \(F_4,E_6,E_7,E_8\). \(G_2\) is a subgroup of \(SO(7)\), the group rotating a 7-dimensional Euclidean space, but instead of allowing all 21 \(SO(7)\) generators, \(G_2\) only allows 2/3 of them, namely 14, those that preserve the multiplication table between the 7 imaginary units in the Cayley algebra (also known as octonions).

It's a beautiful structure. The preservation of the multiplication table, the antisymmetric tensor \(m_{ijk}\) where \(i,j,k\in \{1,2,\dots , 7\}\), is equivalent to the preservation of a spinor \(s\) in the 8-dimensional real spinor representation of \(Spin(7)\). After all,\[

m_{ijk} = s^T \gamma_{[i} \gamma_j \gamma_{k]} s.

\] And it's this conservation of "one spinor among eight" that is responsible for preserving one eighth of the original 32 real supercharges in M-theory. We are left with 4 unbroken supercharges or \(\NNN=1\) in \(d=4\).

Pretty much all the other groups deal with six-dimensional compact manifolds of the "hidden dimensions". In the M-theory case, we have eleven dimensions in total which is why the \(G_2\) holonomy manifolds are seven-dimensional. So the dimensionality is higher than for the 6-dimensional manifolds in string theory.

You may say that having a "higher number of dimensions", like in M-theory, means to "do a better job in translating the physics to geometry". We are geometrizing a higher percentage of the physical properties of the geometrization – which some people could consider to be a "clear aesthetic advantage". And the \(G_2\) compactifications treat this maximum number of (seven) compactified dimensions on equal footing which may be said to be "nice", too. More physical properties deciding about the particle spectrum are encoded in the geometric shapes of the compatified 7 dimensions; fewer of them are carried by "matter fields" or "branes" living on top of the compactified dimensions. All these comments are mine but I guess that string theorists including the authors of this paper would generally endorse my observations.

(The type IIB vacua may also be viewed as "12-dimensional" F-theory on 8-dimensional manifolds, Calabi-Yau four-folds, and in some sense, because of these 8 extra dimensions, F-theory geometrizes even a "higher fraction of physics" than M-theory. It may translate some fluxes to a topology change of the 8-dimensional manifold. But unlike M-theory's 7 dimensions, the 8 dimensions in F-theory are not treated on completely equal footing – two of them have to be an infinitesimal toroidal fiber.)

These differences have an impact on the counting of the number of vacua. You have heard that the type IIB flux vacua lead to \(10^{500}\) different solutions of string theory. They are built by adding fluxes and branes to the compactified dimensions. The fluxes and branes are "decorations" of a geometry that is given to start with. But the number of topologically distinct 6-dimensional manifolds used in these games is of order 30,000 (at least if we assume that each choice of the Hodge numbers \(h^{1,1}\) and \(h^{1,2}\) produces a unique topology which I believe is close to the truth because if there were a huge excess, almost all arrangements of these small enough Hodge numbers would be realized by a known topology which is known not to be the case), even though this upper class may be built in many ways, sometimes from millions of building blocks. On the other hand, the decoration may be added on top of the geometry in "googol to the fifth or so" different ways.

As I said, M-theory has "more dimensions of the underlying geometry" and "fewer decorations". Instead of 30,000 different topologies, they show that some recent construction produces something like 500 million different topologies, i.e. half a billion of allowed seven-dimensional manifolds that are so qualitatively different that they can't be connected with each other continuously, through non-singular intermediate manifolds. But there's nothing much to add (matter fields' backgrounds, fluxes, branes) here, so this is pretty much the final number of the vacua. (The four-form fluxes \(G_4\) over 4-cycles of the manifold may be nonzero but for any allowed compactification, its cousin with \(G_4\) equal to zero is allowed, too. And nonzero values of \(G_4\) qualitatively change the story on moduli stabilization – by adding a superpotential term that most researchers seem to find unattractive, at least now.)

The 500-million class of seven-dimensional \(G_2\) compactifications was constructed by Kovalev (and those who fixed some of his errors and extended the method). The method is known as TCS, the "twisted connected sum". One starts with two Calabi-Yau three-folds times a circle, twists them, and glues them in such a way that the final result is guaranteed to have \(G_2\) holonomy. It's probably no coincidence that 500 million is very close to the number of "subsets with two elements" of a set with 30,000 elements. The information carried by the topology of a \(G_2\) holonomy manifold

*could be*very close (the same? Probably not) to the information carried by two Calabi-Yau three-folds.

It seems to me that this method betrays some dependence on the complex geometry and Calabi-Yaus. This is sort of an undemocratic situation. The laymen often dislike complex numbers and fail to realize that complex numbers are more fundamental in natural mathematics (e.g. calculus and higher-dimensional geometry) than e.g. real numbers. However, professional mathematicians do not suffer from this problem, of course. They do realize that complex numbers are more natural. And I would argue that the complex geometries and other things may even be "overstudied" relatively to other things.

My feeling is that the pairing of the dimensions into "complex dimensions", something that is a part of the Calabi-Yau manifolds, is intrinsically linked to the supersymmetry as realized in perturbative string theory because the field \(B_{\mu\nu}\) coupled to the fundamental strings' world sheets has two indices, just like the metric tensor. That's why they're naturally combined into some complex tensors with two indices and why the spacetime dimensions end up paired. The "complex-like" character of D-brane gauge groups, like \(U(N)\), is probably a sign of the same character of perturbative string theory that loves bilinear invariants and therefore complex numbers. Lots of things are known and solvable.

On the other hand, M-theory likes compactifications with holonomies like \(G_2\) that also has a cubic (or quartic, it's equivalent) invariant, a higher-than-bilinear one. Enhanced gauge groups from singularities are not just \(U(N)\)-like, membranes are coupled to a 3-form gauge potential \(C_{\lambda\mu\nu}\), not a 2-form gauge potential \(B_{\mu\nu}\), and this is one of the sources of the cubic and higher-order structures. That's a heuristic argument why exceptional Lie groups and other things that go "beyond complex numbers" are more frequently encountered in M-theory but not in perturbative string theory.

The exceptional Lie groups and cubic invariants are perhaps "more exceptional" and "harder to solve" which is why our knowledge of the M-theory compactifications and \(G_2\) holonomy manifolds is probably less complete than in the case of the Calabi-Yau manifolds. Perturbative methods are usually inapplicable because there's no solvable "zeroth order approximation": M-theory wants couplings of order one and the dilaton or the string coupling is no longer adjustable (because the extra dimension whose size dictates the dilaton has been used as one dimension, some material to construct the 7-dimensional geometry from which it cannot be separated).

And we usually reduce the \(G_2\) manifolds – which are odd-dimensional and therefore obviously not complex – to some complex manifolds. Couldn't we discuss these manifolds without any reference to the complex ones? One may have a feeling that it should be possible, for the sake of democracy – the classes of the vacua have the same \(\NNN=1\) supersymmetry and may be expected to be treated more democratically – but this feeling may be wrong, too. Maybe the exceptional groups and M-theory should be viewed as some "strange derivatives" of the classical groups and string theory based on bilinear things, after all, and the democracy between the descriptions is an illusion.

Back to the paper.

They discuss these 500 million \(G_2\) manifolds and various membrane instantons and topological transitions in between them, along with the spectra of the models and the Higgs vs Coulomb branches. Most of this work deals with non-singular \(G_2\) manifolds that produce Abelian gauge groups in \(d=4\) only but in this context, it's natural to expect that insights about the compactifications that allow the Standard Model or GUT gauge groups, for example, are "special cases" of the constructions above – or "special cases of a generalization" of the TCS construction that also allows singularities.

Concerning the right compactification, I think it is "very likely" that our Universe allows a description in terms of one of the five compactifications mentioned at the top. The anthropic people think that the class with the "overwhelming majority of the solutions", the type IIB flux vacua, is almost inevitably the most relevant one. But I completely disagree. There's no rational argument linking "the number of solutions in a class" with the "probability that the class contains the right compactification".

I think it is much more rational to say that each of the five classes above has about 20% probability of being relevant. That is my idea about the fair Bayesian inference: each comparably simple, qualitatively different hypothesis (in this case, each of the 5 classes of stringy compactifications) should be assigned the same or comparable prior probability, otherwise we are biased. (The correct vacuum may also allow two or many different dual descriptions.) If the "last 20%" are relevant and our world is a \(G_2\) compactification of M-theory, it may ultimately be sufficient to eliminate about 500 million compactifications in total and pick the right one that passes some tests. It's plausible that the right compactification could be found if that's true. It may end up looking very special for some reason, too.

We don't know yet but I think it's obviously right to try to get as far as we can because the 5 descriptions of superstring/M-theory mentioned at the top seem to be the contemporary physicists' only strong candidates for a theory of Nature that goes beyond an effective quantum field theory and this situation hasn't changed for a decade or two.

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