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M-theory: \(G_2\) holonomy manifolds

This blog entry was written as a part of the math background for a new preprint by Kane, Kumar, Lu, Zheng that uses M-theory to predict the mass of the Higgs boson to be around 125 GeV. (We have previously seen that the prediction wasn't "made up" in recent weeks.)
I want to say some things about the group called \(G_2\), manifolds that have a "holonomy" equal to \(G_2\), and M-theory compactifications on these manifolds.

Symmetries and groups in mathematics: discrete and continuous ones

In mathematics, symmetries are codified by the concept of a "group". It's a set \(G\) of elements with an operation – called multiplication – that has to be associative, \((gh)j=g(hj)\), with an identity elements satisfying \(1g=g1=g\), and with the inverse element obeying \(gg^{-1}=g^{-1}g=1\) for each element \(g\) of the group \(G\).

One may view a group as a set of certain "reversible operations" with any "object". The left-right reflection, \(\ZZ_2\), is the simplest example of a group. The operations in a group may be mapped in a one-to-one way – so that
\[ \phi(gh) = \phi(g)\phi(h) \] – to operations acting on another group. The equation above is the key condition for \(\phi\) to be called an isomorphism (or "the-same-shapeness" in English): the property that the element \(gh\) is the product of \(g\) and \(h\) has to be preserved if you "translate" all these three elements \(g,h,gh\) by the dictionary.

Groups may be classified. For example, all "finite groups" – with a finite number of elements in them – have been classified in a collection of thousands of papers written by a whole generation of mathematicians. Such a classification knows everything about cyclic groups, symmetries of Platonic polyhedra, permutation groups, the group of operation you may do with a Rubik cube, groups of matrices whose matrix entries are taken to be from a discrete field, as well as 26 (or 27, if I count the Tits group) "sporadic groups" which defy all the simple rules how to classify them.

The monster group is the largest one among the sporadic group; it contains approximately \(8\times 10^{53}\) elements or "distinct operations" and some of its unbelievable numerical properties have been explained by maths behind string theory ("monstrous moonshine"). It's remarkable that maths gives this diverse structure of groups with all the sporadic exceptions but it's simply a fact of life, a property of the "universe of ideas" that has to be respected in our physical world as well as all others.

Lie groups

Aside from the finite groups (and other discrete groups), physicists especially need to discuss about continuous groups (or Lie groups for short). They include, among other things, the group of all rotations of a \(d\)-dimensional Euclidean space, \(SO(d)\). There also exists a generalization to the complex and quaternionic spaces, \(U(d)\) and \(USp(2n)\). These are the three key infinite sequences of groups in the classification of compact (=finite-volume) simple (=not reducible to simpler ones) Lie groups.

However, much like there exist 26 or 27 huge "sporadic groups" that defy any classification, there are 5 exceptional Lie groups,
\[ G_2, F_4, E_6, E_7, E_8. \] These are not some random labels that a human has made up. These are important universal facts about the world of mathematics. Each such group is a set of "selected transformations" acting on a real or complex Euclidean space of a sufficient dimension. The smallest dimensions – the dimensions of the "fundamental representation" – are the following numbers for the five groups:
\[ 7_\RR,\, 26_\RR,\, 27_{\CC},\, 56_\RR,\, 248_\RR. \] Note that \(E_6\) is the only one among the five groups that has a natural "complex fundamental representation": it is a subgroup of \(SU(27)\). All the other exceptional groups only possess "real representations". Each of the groups is composed of transformations that may be uniquely specified by \(d\) real numbers (analogous to e.g. 3 Euler angles for \(SO(3)\)). We call these numbers the "dimensions of the groups" and they coincide with the "dimension of the adjoint representation" (all these representations are real). For the exceptional groups, the values of \(d\) turn out to be
\[ 14, 52, 78, 133, 248.\] Note that for \(E_8\), the fundamental representation has the same dimension as the adjoint one: there's no simpler non-trivial representation of \(E_8\). Also, the group \(E_8\) contains all the previous exceptional groups – and even \(F_4\times G_2\) – as subgroups. I could call for hours but we wanted to choose a more specific topic: \(G_2\).

\(G_2\) group and octonions

When you become a schoolkid, they tell you what the real numbers \(\RR\) are. As repeatedly emphasized on this blog, the complex numbers \(\CC\) of the form \(x+iy\) where \(x,y\in\RR\) are, in some sense, even more natural and more fundamental than the real numbers. They're an essential part of quantum mechanics. Mathematically, you can do "pretty much anything" with complex numbers. Every natural equation may be solved by complex numbers. You may even compute the square root of any complex number. The "business-as-usual" works as a truly flawless industry. Relatively to the complex numbers, their larger extensions have much more limited importance and applications.

They include quaternions and octonions.

All these "number systems" including complex numbers \(\CC\), quaternions \({\mathbb H}\) called after Hamilton who discovered them, and octonions \({\mathbb O}\) may be viewed as linear combinations of some "imaginary units" with real coefficients. They respect the usual vector-like rules of addition; and the distributive law for addition and multiplication. However, they differ by the number of "imaginary units" as well as – the much more detailed - multiplication tables for these units. The condition is that each number except for \(0\) has to have an inverse with respect to multiplication. With these conditions, \(\RR, \CC, {\mathbb H}, {\mathbb O}\) are the only four possibilities (up to isomorphisms, i.e. renaming of the elements).

Aside from isomorphisms, one may also talk to "isomorphisms to themselves", nontrivial permutations of the elements of these number systems that preserve the multiplication rules.

Complex numbers \(\CC\) are based on the imaginary unit \(i\) obeying \(i^2=-1\). In fact, \((-i)^2\) is equal to \(-1\) as well so there is some ambiguity about the question which of the units is \(+i\) and which of them is \(-i\). This ambiguity is the reason behind the \(\ZZ_2\) automorphism group of the complex numbers \(\CC\). This automorphism group contains two transformations: the identity which does "nothing" and has to be an element of any group; and a non-trivial element that, in this case, permutes \(+i\leftrightarrow -i\).

Every nonzero complex number \(a+bi\) has an inverse,
\[ \frac{1}{a+bi} = \frac{a-bi}{a^2+b^2}. \] You may easily check this identity because
\[(a+bi)(a-bi) = a^2 + b^2 \] because the mixed terms cancel. Note that \(a^2+b^2\) is positive for any nonzero complex number.

Quaternions use 3 imaginary units \(i,j,k\) with the usual,
\[i^2=j^2=k^2=-1 \]
and the other multiplication rules
\[ \qquad ij=-ji=k, \quad jk=i=-kj, \quad ki=-ik = j. \] Note that two different imaginary units anticommute with each other and the three parts of the equation above are cyclic permutations of each other. With
\[ i_a=i,j,k, \qquad a=1,2,3,\] we may unify the multiplication rules to
\[ i_a i_b = -\delta_{ab}\cdot 1 + \sum_{c=1}^3 \epsilon_{abc} i_c. \] The tensors \(\delta,\epsilon\) on the right hand side are preserved by rotations of the three basis vectors \(i,j,k\) that belong to an \(SO(3)\) group. That's why \(SO(3)\) is the automorphism group of the quaternions. Every quaternion has an inverse:
\[ \frac{1}{a+bi+cj+dk} = \frac{a-bi-cj-dk}{a^2+b^2+c^2+d^2}. \] Now, the new \(ij,jk,ki\) mixed terms canceled because the imaginary units anticommute.

The last division algebra is composed of generalized "numbers" which have 8 components, therefore called the octonions (also Cayley numbers). Aside from the real unity \(1\), it has 7 imaginary units
\[i,j,k,A,B,C,D.\] I introduced some ad hoc notation for the new four. As the notation \(i,j,k\) suggests, quaternions are a subset of the octonions much like complex numbers are a subset of quaternions. However, there are many ways to embed quaternions to octonions. The "simplest to write" examples involve the following triplets of the octonionic imaginary units. Each of them behaves as the triplet \(ijk\) of the quaternions:
\[ ijk, iAB, iDC, jAC, jBD, kCB, kAD. \] Similarly to the quaternions, we have
\[ i_a i_b = -\delta_{ab}\cdot 1 + \sum_{c=1}^7 f_{abc} i_c. \] Each of the units squares to \(-1\) and the product of two units is always well-defined. Note that in the previous displayed equation (list), I enumerated 7 triplets. Each of them tells you 3 rules for the product of a pair of imaginary units (plus the rule for the same pair in opposite order, with the opposite sign). In total, we have \(7\times 3=21\) rules for multiplication which happens to agree with \(7\) choose \(2\), so the product of any pair of imaginary unit is well-defined. You may see that there is a sufficient number of rules because of a simple reason: the new imaginary units \(A,B,C,D\) may be divided to pairs in 3 different ways and each of them is linked to one of the old, quaternionic imaginary units \(i,j,k\).

Just like in the case of quaternions, you may define the multiplicative inverse for any nonzero octonion. You just have 8 terms in the numerators and denominators instead of 4 terms. Quaternions were not commutative and octonions are not commutative, either. However, octonions also fail to be associative (with respect to multiplication). This prevents you from defining general "groups of matrices" with octonionic entries (because we usually want matrix multiplication to be associative) and that's one of the reasons why the importance of octonions is even lower than that of the quaternions; they're much more special in their impact.

What's interesting is the automorphism group of the octonions. It must be a group of transformations that preserve the tensors \(\delta, f\) in the multiplication table above. Because \(f_{abc}\) is no longer just \(\epsilon_{abc}\) if there are 7 distinct imaginary units, the rotations preserving this nontrivial tensors no longer include all rotations in \(SO(7)\) – which are those that preserve the metric given by \(\delta_{ab}\). Instead, the group preserving this antisymmetric cubic invariant \(f_{abc}\) is a subgroup of \(SO(7)\) known as \(G_2\).

As I have mentioned, its dimension is 14. A general rotation of the 7 octonionic imaginary units that preserves the multiplication table is specified by 14 real parameters. That's fewer than the dimension of \(SO(7)\) which is 21: the number of pairs inside a set of 7 elements. Why is the dimension of \(G_2\) equal to two thirds of the dimension of \(SO(7)\)? Well, think about all rotations that preserve any chosen imaginary unit, e.g. \(i\). This choice of \(i\) divides the remaining 6 imaginary units into pairs, in this case \(jk, AB, DC\). In \(SO(7)\), all the rotations in the \(jk,AB,DC\) planes would be allowed and independent; in \(G_2\), one only allows the combined rotations in these three planes whose total angle vanishes. That's why you have only 2-out-of-3 parameters labeling the rotations, and this reduction applies to all possible generators of \(SO(7\) and \(G_2\).

You see that the multiplication table of the octonions is kind of clever and sensitively depends on things you may do with 7 imaginary units. You won't be able to find any "bigger" multiplication table with similar cute properties.

That was enough to introduce the \(G_2\) symmetry. Let me mention that whoever talks about octonions but isn't really led to a \(G_2\) group is cheating. Whenever you really employ octonions as octonions, you are forced to consider their multiplication table – which is what really makes them octonionic – and the group \(G_2\) is the symmetry group of this table. No \(G_2\) means no genuine octonionic maths.

Manifolds and holonomy

That was enough maths about \(G_2\). Now we want to get to \(G_2\)-holonomy manifolds. Our ultimate goal is a possible theory of the physical world based on M-theory. M-theory has an 11-dimensional spacetime. Because we only observe a 4-dimensional spacetime, 11-4=7 dimensions have to be hidden. That's why seven-dimensional curved shapes are an important topic to be studied whenever you want to use M-theory as a theory of the physical world around us.

As the "\(G_2\) holonomy" terminology suggests, the group \(G_2\) we have presented will have a special role: it will be the "holonomy group" of the manifold (of the 7-dimensional shapes). Now, we must understand what the term "holonomy group" actually means.
It's simple: it's the group of all transformations (rotations) of the tangent space that you may induce by a parallel transport around a closed curve in the manifold.
What is the parallel transport? You just transport the arrows encoding the axes in the "straightest possible way" along the curve, making sure that you don't rotate the axes in any way. Think about gyroscopes that remember the axes and that are just being transported.

A few examples make the concept clear. Take a flat space. Pick these axes or gyroscopes and send them to a round trip. They always point in the "same" direction, so when you return back, they haven't changed at all! That's why the holonomy group of a flat space is always trivial, \(\{1\}\): it only contains the identity element (which "does nothing").

The opposite extreme situation involves totally randomly, chaotically curved manifold. When you send the gyroscopes on a round trip along a complicated enough curve, they rotate by an arbitrary rotation. The holonomy group will be all of \(SO(d)\) where \(d\) is the dimension of the generic curved manifold. And if the manifold is unorientable, you will really get \(O(d)\) and not just \(SO(d)\).

The most interesting cases will be those in between.

But one more example. Take the 2-dimensional surface of the Earth. What is the holonomy group? Imagine that you start on the equator, longitude zero (South of Greenwich). The arrows point to the North (1) and to the West (2). Go to the North pole. The parallel transport clearly means that (1) was pointing to the North, to the front of your journey, and (2) was pointing to the West, as your left hand, all the time.

But now continue along the path that goes along the 90° longitude meridian, somewhere in Asia. Because we're talking about the parallel transport, we're not allowed to rotate the arrows just because the path changed the direction. So the "formerly North-bounded" arrow (1) is now pointing to the left and the "formerly West-bound" arrow (2) is pointing to the back. We get to the equator and this claim still holds. When we return to the original point along the equator, it's still true that (1) points to the East and (2) points to the North: both axes got rotated by 90°. More generally, if we encircle the area \(S\cdot R^2\) where \(S\) is the solid angle and \(R\) is the Earth's radius, the arrows get rotated by the angle \(S\) – which was \(\pi/2\) in our example of a journey encircling 1/8 of the Earth's surface.

What are the other possible holonomies? Are there interesting examples in between \(\{1\}\) and \(SO(d)\)? Yes, you bet.

It's useful to think about the "intermediate holonomies" that preserve an object. If some \(N\) vectors don't get rotated at all, the holonomy group decreases from \(SO(d)\) to \(SO(d-N)\). Not a big shock: only the remaining \(d-N\) axes may get rotated to each other.

However, we may also study how the parallel transport acts on spinors, not just vectors. And the monodromies that preserve some spinors are very important because the corresponding spinor may be "covariantly constant" and it may define an unbroken supersymmetry generator, even if the manifold is curved and not globally parallelizable.

Four-dimensional real manifolds may have the holonomy group equal to \(SU(2)\) which is smaller than \(SO(4)\sim SU(2)\times SU(2)\) holonomy of a generic curved four-dimensional manifold. The corresponding manifolds are said to be hyper-Kähler manifolds and the K3 manifolds are the only compact smooth (but curved) representatives of this class. They only induce rotations in one of the \(SU(2)\) factors which is really why these manifolds preserve one-half of the original supersymmetry.

Generic six-dimensional manifolds have the \(SO(6)\) monodromy group. However, \(SO(6)\) is (locally) isomorphic to \(SU(4)\) which has an obvious subgroup \(SU(3)\): only parallel-transport rotations of the 6 real coordinates of the tangent space that preserve their pairing to 3 complex numbers are allowed on manifolds of the \(SU(3)\) monodromy. You may see that 3/4 of the SUSY is broken: 1 component of the 4-dimensional complex spinor is preserved. In other words, these manifolds preserve 1/4 of the original supersymmetry. When you study whether you have heard about such manifolds, the answer is Yes: they're the well-known Calabi-Yau manifolds. In the case of 6 real dimensions, they're also called Calabi-Yau three-folds.

Calabi-Yau four-folds have the \(SU(4)\subset SO(8)\) holonomy group, they may be analogously used as compactification manifolds for a (formally) 12-dimensional F-theory, and they preserve 1/8 of the original supersymmetry. There also exist eight-dimensional manifolds with a \(Spin(7)\) holonomy.

Spinors and when the holonomy is \(G_2\)

However, we want to focus on 7-dimensional manifolds. Because 7 is an odd number, their coordinates won't be naturally complex. In fact, complex calculus isn't immediately useful for the \(G_2\)-holonomy manifolds. Still, much of the mathematics (based on real representations) is analogous to the Calabi-Yau case.

The \(G_2\)-holonomy manifolds preserve 1/8 of the original supersymmetry. Why? Well, the minimal spinor for a 7-dimensional Euclidean space has 8 real components. Imagine that you demand that one spinor \(s\) in this 8-dimensional space is demanded to be preserved under the parallel transport. How does it restrict the possible holonomy?

Well, you may "translate" the spinor to tensors. If \(s\) is preserved, the following things have to be preserved as well:
\[ s^T s, \quad s^T \gamma_\mu s, \quad s^T \gamma_{\mu\nu} s, \dots \] We just insert gamma matrices in between two copies of the real spinor. The first bilinear expression is just some "squared length" of the spinor and of course it is conserved by any orthogonal rotations. It doesn't restrict the spinor at all. The "expectation value of \(\gamma_\mu\) vanishes because the gamma matrices in the 7-dimensional Euclidean space are antisymmetric. And so are their products \(\gamma_{\mu\nu}\). So we should go on:
\[ s^T \gamma_{\lambda\mu\nu} s, \qquad s^T\gamma_{\kappa\lambda\mu\nu} s \] The products of 3 or 4 different gamma matrices are symmetric matrices but these two options are related to each other by "Hodge duality", i.e. by the translation using the fully antisymmetric tensor with 7 indices. So only one of them is a new possibility. What's my point? My point is that there exists a spinor \(s\) for which you get
\[ s^T \gamma_{abc} s = f_{abc} \] So the expectation value of the three-index gamma matrix in the spinor is nothing else than the tensor \(f\) encoding the multiplication table for those 7 imaginary units. So the preservation of the octonionic structure \(f_{abc}\) is the same thing as the preservation of a particular spinor – one-dimensional axis in an 8-dimensional spinor space. Seven-dimensional spaces that preserve 1/8 of the supersymmetry naturally know about the octonions and their \(G_2\) symmetry fingerprint.

That's great because we want to preserve 1/8 of the supersymmetry. In 11 dimensions, M-theory has 32 real supercharges which is the same amount as \({\mathcal N}=8\) supersymmetry in four spacetime dimensions. Killing 7/8 of it gives you the good old \({\mathcal N}=1\) supersymmetry in \(d=4\), the phenomenology's paradise.

Why the viable manifolds have to be singular

There exist smooth manifolds of \(G_2\) holonomy: Oxford mathematician Dominic Joyce constructed some of them as the "beautified" smooth resolutions of some orbifolds of the 7-torus which is why the term "Joyce manifolds" has sometimes been used even for all \(G_2\) holonomy manifolds.

However, smooth manifolds are no good for phenomenology. Why? It boils down to the discrete symmetries such as parity.

The Standard Model, especially its weak nuclear force, violates these discrete symmetries. In particular, the left-handed spinor components of various particles behave differently than their right-handed comrades. The (light) right-handed neutrinos don't exist at all. And the left-handed and right-handed spinors behind the electron's Dirac field carry different hypercharges, to mention an example of such differences.

This left-right-asymmetric structure of the Standard Model codifies a correlation between the handedness and the sign of the hypercharge (and other things). The handedness is given by the question whether we talk about \({\bf 2}\) or \(\overline{\bf 2}\) representation of \(SL(2,\CC)\sim SO(3,1)\). These two complex representations are complex conjugate to each other but they're not equivalent to each other (by ordinary matrix conjugation).

The electroweak theory, a key part of the Standard Model, may say that the lepton fields in the \({\bf 2}\) representation must always have one sign of the hypercharge while their complex conjugates must have the opposite sign of the hypercharge.

However, this correlation wouldn't be possible with smooth \(G_2\)-holonomy manifolds. Why? Because the spinors in the 10+1-dimensional as well as 7-dimensional (Euclidean) space are real. That means that even if you study fields of a fixed value of the hypercharge (including the sign), there will be a 2-dimensional space of possibilities (the real and imaginary part of the representation which is, for a nonzero hypercharge, inevitably complex). That's why you always end up with the "mirror fields" as well.

That doesn't mean that \(G_2\)-holonomy manifolds and M-theory isn't a viable scenario for the world. The required correction is that the 7-dimensional manifolds have to be singular. As explained by Acharya, Witten, and perhaps others, the chiral fermions don't live in the whole 7-dimensional manifold but only at some even-dimensional (usually point-like) singular place of the manifold. When the locus where the spinorial fields may live becomes even-dimensional, you may get chiral fermions as easily as you get them in the bulk of the 10-dimensional heterotic string theory's spacetime.

These were some basics as they entered physics a decade ago. During the last decade, realistic M-theory compactifications became an impressive industry led by masterminds such as Gordon Kane. If the papers written about these compactifications are right, M-theory phenomenology predicts a pretty specific pattern for new physics and supersymmetry breaking – which is much more specific than if you talk about a "generic garden-variety supersymmetry model" – and that seems pretty compatible with the existing observations from the LHC and other experiments.

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