G2 manifold

"A few years ago, though, String Theorists declared that perhaps, after all, the dimension of the Universe is 11. To account for the difference between this and the four dimensions we see, the other 7 dimensions have to be 'rolled up' into a 7-dimensional manifold with a very small radius, of about 10-33cm. It turns out that the geometrical structure on this 7-manifold must be the exceptional holonomy group G2, one of the structures of which I had found the only known examples. And so, until it went out of fashion again six months later, a number of physicists were writing research papers about 'Joyce manifolds', which was nice while it lasted." - Dominic Joyce

M-theory, has 7 compact dimensions, not 6. Thus the compact geometry is not a 6-dimensional Calabi-yau, however, in "this case one considers 7-dimensional manifolds of G2 holonomy. (Six-dimensional Calabi-Yau's have SU(3) holonomy.) The math is very challenging. Also, there are lots of issues if one wants to get realistic physics this way." This quote is from one of my email conversations with John H. Schwarz.

Holonomy groups of a Riemannian manifold are a compact Lie group. They provide a local measure of the global curvature of the manifold.

A G2 manifold is:

7-dimensional
Ricci-flat
orientable
Spin manifold

These are 7-dimensional Riemannian manifolds. They are important in string theory. The original supersymmetry is reduced to 1/8 the original amount. For example, in M-theory, if we compactify with a G2 manifold, to yield a realistic model of the universe (11 = 7 + 4 dimensions), we are left with the realistic N=1 supersymmetry.

11-dimesional supergravity has 32 supercharges. These are independent generators of supersymmetry transformations. 11-dimesional supergravity also has 4 linear combinations that remain unbroken when spacetime is the product of a G2 manifold times R^{1,3}. This is 1/8. 4 supercharges in 4D is what people usually call 4D N=1 supersymmetry.

We of course need to consider what we mean by "realistic," when describing the N=1 supersymmetry. Besides gravity, you can get gauge theories, charged matter and Yukawa couplings. The details of how well this is understood and how much has been achieved is a rather long story.

Peter Horava

There are other drawbacks and difficulties to the G2 manifold approach. Peter Horava, a collaborator with Edward Witten and key contributor to M-theory, explains:

We cannot recover 4-dimensional physics by compactifying on “smooth” 7-dimensional manifolds.
Also, 7-dimensional manifolds, unlike the Calabi-yau are not complex (complex means even number of dimensions). Horava notes that complex manifolds are “much better behaved, much easier to understand, and much easier to work with.” Complex algebraic geometry is "easy" and allows you to write down lots of examples of Calabi-Yau manifolds explicitly. There is no such technique available for G2 that we know of.

Indeed, there is much to be learned about these 7-dimensional manifolds. However, it is far from clear that one can conclude that we unable to derive realistic 4-dimensional physics from models of compactification that involve g2 manifolds. The challenges and possibilities are more or less the same for all string constructions of (realistic) 4D N=1 vacua if you start asking detailed questions.

Edward Witten and Shing-Tung Yau have looked into developing something similar to the Calabi conjecture for G2 manifolds, however, have not gotten very far. This is one of the reasons M-theory is not as developed as superstring theory. The mathematics is not as well established.

A lot of things are still too difficult to describe and a lot of things one would like to compute we don't know how to tackle. The same can be said about other constructions of 4D N=1 vacua in string theory.

Andreas Braun

A very short wishlist for G2 (from email conversation with Andreas Braun) is:

- explicit construction of large classes/geometrical engineering of chiral gauge theories in compact models

- detailed description of their moduli spaces

- finding associative submanifolds/non-perturbative corrections

Braun also held that "a statement such as Calabi's conjecture does not hold in general for G2s, but there are certainly specific classes in which something similar holds. My best bet for progress is to continue using string dualities to better understand G2 manifolds. "

There is no real textbook, at least at the present, for G2 geometry. References could be made to the work of Simon Salamon and to Dominic Joyce. Salamon's work emphasizes the representation theoretic aspects of Riemannian holonomy. Joyce's work serves as both a text on Kahler and Calabi-Yau geometry as well as a monograph detailing Joyce’s original construction of compact manifolds with G2 Riemannian holonomy. Both the work of Salamon and of Joyce are excellent resources.

History

Marcel Berger

Jim Simons

Edmond Bonan

In 1955, Marcel Berger was the first to suggest that G2 might be the possible holonomy group for certain Riemannian 7-manifolds. In 1962, Jim Simons published a simplified proof of this conjecture. In fact, not a single example of such a manifold had yet to be discovered. However, Edmond Bonan, in 1966, makes a useful contribution. What Bonan proposed was that if such a manifold did exist, it would carry both a parallel 3-form and a parallel 4-form. It would also be Ricci -flat.

Robert Bryant

Simon Salamon

In 1984, Robert Bryant constructed the first local examples of manifolds with holonomy G2. His full proof appeared in the Annals of Mathematics in 1987.

In 1989, Bryant and Simon Salamon constructed complete (but still noncompact) 7-manifolds with holonomy G2.

Dominic Joyce

The first to propose a compact 7-manifold with G2 holonomy was Dominic Joyce in 1994. This is why compact G2 manifolds are sometimes known as "Joyce manifolds." This is especially in the physics literature.

This is a snapshot from the table of contents of Joyce's Compact Manifolds with Special Holonomy. (2000)

Sema Salur

In 2013, M. Firat Arikan, Hyunjoo Cho and Sema Salur publish an article with the name "Existence of Compatible Contact Structures on G2-manifolds."

The paper showed the existence of (co-oriented) contact structures on certain classes of G2 manifolds. These two structures are compatible in different ways.

What is a contact structure? I asked Sema Salur this very question in an email, she told me "A contact structure on a 7 manifold is a 6-plane distribution in the tangent bundle of the G2 manifold. This distribution is given as

the kernel of a differential 1-form alpha such that alpha \wedge (d \alpha)^3 is always nonzero. Almost contact structure is also a 6-dimensional distribution in the tangent bundle of G2 as the kernel of alpha but in this case alpha \wedge (d\alpha)^3 can be zero. Here wedge means exterior product and d \alpha means the exterior derivative."

Any 7-manifold with a spin structure, therefore any manifold with a G2 structure, admits an almost contact structure.

They also construct "explicit almost contact metric structures on manifolds with G2 structures."

She has also told me herself in email conversations that g2 manifolds are "7 dimensional spaces which satisfy some important geometric properties." She also said "We call them manifolds with special holonomy. In my paper (joint work with F. Arikan and H. Cho) "Existence of Compatible Contact Structures on G2-manifolds", we showed that G2 manifolds carry (almost) contact structures. This is interesting because I have this program where I use the (almost) contact structure on 7-manifolds to hunt for the topological description of G2 holonomy manifolds."

Johannes Nordstrom

In 2015, Alessio Corti, Mark Haskins, Johannes Nordström and Tommaso Pacini wrote an article titled "G2-Manifolds and associative submanifolds via semi-fano 3-folds."

Nordstrom has said in an email conversation "that "Semi-Fano" is not very standard terminology. It's a slightly stronger condition than "weak Fano", which usually means a projective variety whose anticanonical bundle is nef and big (although some authors use it in slightly different ways). In some papers with Alessio Corti, Mark Haskins and Tommaso Pacini, we found it useful to focus on the special case of weak Fanos where the morphism defined by the anticanonical bundle is in addition "semi-small". In the case of 3-folds, that means it can contract curves to points and divisors to curves, but must not contract divisors to points (in other words, the fibres have dimension at most."

In this article was constructed "new topological types of compact G2-manifolds, that is, Riemannian 7-manifolds with holonomy group G2." The twisted connected sum construction first developed by Kovalev (2003) extended. It is applied to "to the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds constructed previously by the authors. "

Nordstrom has also said in an email conversation that "Kovalev's construction provided the second class of examples of closed G2-manifolds, after Joyce's flat orbifold resolutions. Like Joyce's examples they come in families that exhibit a kind of degeneration, but of a different kind. Joyce's degenerate back to the original orbifold, while Kovalev's develop a long neck (so if you rescale so that the diameter is constant, then the volume goes to zero). The twisted connected sums also turn out to have more computable topology than examples from the orbifold construction. "

From their abstract of the article itself:

"any of the G2-manifolds we construct contain compact rigid associative 3-folds, which play an important role in the higher-dimensional enumerative geometry (gauge theory/ calibrated submanifolds) approach to defining deformation invariants of G2-metrics. By varying the semi-Fanos used to build different G2-metrics on the same 7-manifold we can change the number of rigid associative 3-folds we produce."

Spiro Karigiannis

Dominic Joyce, along with Spiro Karigiannis, propose a paper titled A new construction of compact G2-manifolds by gluing families of Eguchi-Hanson spaces. This paper was submitted in 2017 and is in it's introduction, admits that there is much in common between g2 manifolds and the better understood Calabi-yau manifolds. They say "In particular, they are both Ricci-flat compact Riemannian manifolds, and they both admit parallel spinors, making them candidates for supersymmetric compactification spaces in physics." They go on, "Both Calabi–Yau metrics, which are of holonomy SU(n), and holonomy G2 metrics on compact manifolds, are in some sense transcendental objects, in the sense that explicit formulas for such metrics are not expected to exist. "

The work also notes that Calabi-yau manifolds can be fruitfully studied using algebraic geometry. Such tools are not available for the 7-dimensional g2 manifolds. Also, there is no analogue for the Calabi-yau theorem in g2 geometry. They say "Because of this, there are far fewer known examples of compact G2 holonomy manifolds."

Xenia de la Ossa

Xenia de la Ossa, Magdalena Larfors and Matthew Magill, in 2021, proposed a paper titled Almost contact structures on manifolds with a G2 structure.

They admit "In M-theory they [g2 manifolds] provide four-dimensional N = 1 Minkowski vacua, and in type II or heterotic string theory they give rise to three-dimensional vacua with non-positive cosmological constant. These topics have been studied for a number of years, and important results have been established. "

They also say "A G2 structure exists whenever the seven manifold is orientable and spin. "

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