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Classifying viable supergravity theories in 8D, those with \({\mathfrak g}_2\) didn't make it

The Swampland program is a clever filter that eliminates a big portion of the effective field theories with gravity which are not good enough because they violate certain conditions that are necessarily to find a consistent realization of these theories as theories of quantum gravity, i.e. a realization within string/M-theory. In a new paper

8d Supergravity, Reconstruction of Internal Geometry and the Swampland,
Cumrun Vafa (a renowned professor) and Yuta Hamada (a Harvard postdoc) use the existing tools to say "you're fired" to many supergravity theories in 8 (large) spacetime dimensions. Note that 11 (spacetime) dimensions is the maximum dimension of a flat spacetime admitting a supersymmetric quantum theory of gravity, M-theory. 12 dimensions is still possible to some extent, like in F-theory, but two dimensions must be compactified on a torus (or perhaps something else?). Supersymmetric theories above 12 dimensions must be "even more subtle or less physical" if they exist and I want you to assume that they can't exist. In 10 large (spacetime) dimensions, there are 5 different consistent supersymmetric Minkowski-based theories of gravity, all of them are string theories (type I, IIA, IIB, heterotic \(E_8\times E_8\), heterotic \(SO(32)\)).

As you reduce the number of large spacetime dimensions, the diversity of the possible vacua increases (because so does the complexity of the possible geometries of the compactified dimensions, and decorations added inside these manifolds) but the laws of string/M-theory seem to guarantee that the number of stabilized vacua remains finite (or at least countable). What about the theories with 8 large (spacetime) dimensions?

You may obtain them by compactifying 3 (spatial) dimensions in M-theory or 2 (spatial) dimensions in 10-dimensional string theories. There are various shapes of the extra dimensions (aside from the torus, you already have a cylinder, Klein bottle, Möbius strip... to use, and K3 if you start from F-theory) and some extra choices may be made. However, the construction of the approximate effective field theory description proceeds very differently and seems to produce a greater number of choices.

They take various previously justified swampland criteria into account and remind you that the rank of the gauge group – the maximum number of \(A_\mu\) \(U(1)\) gauge potentials that you can embed into the field spectrum – must be 2 or 10 or 18 for the UV completion to exist. Those can be geometrically realized by type II on a torus or Klein bottle, Möbius strip, and a cylinder (with 0,1,2 boundaries), respectively; or as K3 with 2, 1, 0 frozen singularities, respectively. The rank changes by 8 (eight is the rank of \(E_8\) which lives on an M-theory boundary and it is no coincidence, either) for each boundary or extra singularity. As you may expect, all the consistent vacua have the rank of 2, 10, or 18.

The rank's being even may be justified by the global gravitational anomaly. Similar considerations about the anomalies in some spacetime or world volume directions produce additional constraints. The 1-brane unitarity bounds the rank from above. They review similar facts.

Their new constraints come from 3-branes. You know, a supergravity theory in 8 (spacetime) dimensions must have 3-branes. Why? Because the supersymmetry unavoidably adds a superpartner field to the graviton which may be written down using a 2-form gauge potential \(B_{\mu\nu}\). String theory (except for type I) is an example of that; this \(B\)-field is coupled to the fundamental strings (type IIB has D-strings as well, and a corresponding second \(B\)-field in the Ramond-Ramond sector). Well, this gauge potential is a 2-form and its field strength is a 3-form (higher by one). Because we are in 8 large (spacetime) dimensions, the (electro-magnetic) Hodge dual is a 5-form field strength (eight minus three) which is the "d" of a 4-form potential (five minus one). And 4-form potentials couple to 4-dimensional world-volumes, so the number of spatial dimension of the magnetic objects has to be 3. They are called (magnetic) 3-branes.

Now, such 3-branes must really be possible in a consistent theory approximated by an 8-dimensional supergravity theory. Why? Because this supergravity is just an extension of Einstein's general relativity, you may find classical black \(p\)-brane solutions for \(p=3\) i.e. black holes boringly extended in 3 extra spatial dimensions which are sourcing the magnetic field (five-form field strength). There must exist some effective world volume theory on these magnetic 3-branes and these two Harvard Gentlemen look at it carefully. Well, in particular, they look at the instanton 3-branes and via the swampland "cobordism" conjecture, they have to be connected (via the one-dimensional Coulomb branch on the brane, if you want to know).

These are highly constraining conditions and they end up with "dozens or hundreds" of possible constructions or vacua. Their gauge groups are diverse but some are visibly missing. In particular, one may construct 8-dimensional supergravity theories with the \({\mathfrak g}_2\) (gee-two) gauge algebra but it seems to be forbidden by the swampland argument: it belongs to the swampland.

This \({\mathfrak g}_2\) is a wonderful group, the smallest one among the five basic simple exceptional Lie groups. It is the automorphism group of the octonions. You may extend "2D" complex numbers to "4D" quaternions which are non-commutative and then "8D" octonions which are also non-associative. The octonions have 7 imaginary units and just like there is a \(\ZZ_2\) symmetry between \(+i\) and \(-i\) for complex numbers (and the \(SO(3)\) symmetry rotating the \(i,j,k\) quarternionic imaginary units), there is a symmetry, a subgroup of \(SO(7)\), between the 7 octonionic imaginary units that preserves (the addition and more nontrivially) the multiplicative table.

In grand unification (phenomenology in 4 dimensions), \({\mathfrak g}_2\) is no good as a gauge group because it has real representations only. In fact, the 7-dimensional representation encoding the "7 imaginary units of the octonions" (which is clearly a real rep) is a fundamental representation and all others may be built from its tensor powers. Well, in fact, the algebra \({\mathfrak e}_6\) yields the only exceptional group that has complex reps which is what turns it into an important grand unified group.

Here, the \({\mathfrak g}_2\) is forbidden because of the cobordism constraints – which are really still another generalization of the lore that "there are no global symmetries in a theory of quantum gravity". Many options remain viable, others are eliminated. Vafa and Yamada are really climbing down to lower number of large (spacetime) dimensions. If they climbed down from 11 or 12 to 4 which is the apparent number of spacetime dimensions around us, they would deal with a lot of options, lots of topologies, singularities in K3-like manifolds and their intersections and combinations, and at the end, they also want to break SUSY which makes things much less straightforward because one can no longer study "just the well-behaved vacua" where many basic things are determined by the BPS bounds – in the spacetime and the world volumes. But quite generally, one may reasonably believe that "do properly analyze, classify, and understand possible choices in D spacetime dimensions" is a viable way to find the right theory of everything (in phenomenology), crackpot movement's shouting to the contrary notwithstanding.

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