Today, he published

Supersymmetry, Ricci Flat Manifolds and the String Landscape.String theory and supersymmetry are "allies" most of the time. Supersymmetry is a symmetry that first emerged – at least in the Western world – when Pierre Ramond was incorporating fermions to the stringy world sheet. (In Russia, SUSY was discovered independently by purely mathematical efforts to classify Lie-algebra-like physical symmetries.) Also, most of the anti-string hecklers tend to be anti-supersymmetry hecklers as well, and vice versa.

On the other hand, string theory and supersymmetry are somewhat independent. Bosonic string theory in \(D=26\) has no SUSY – and SUSY is also broken in type 0 theories, some non-supersymmetric heterotic string theories, non-critical string theory, and more. Also, supersymmetry may be incorporated to non-gravitational field theories, starting with the Wess-Zumino model and the MSSM, which obviously aren't string vacua – because the string vacua make gravity unavoidable.

Some weeks ago, Alessandro Strumia was excited and told us that he wanted to become a non-supersymmetric stringy model builder because it was very important to satisfy one-half of the anti-string, anti-supersymmetric hecklers. It's a moral duty to abandon supersymmetry, he basically argued, so string theorists must do it as well and he wants to lead them. He didn't use these exact words but it was the spirit.

Well, string vacua with low-energy supersymmetry are rather well understood and many of them have matched the observed phenomena with an impressive (albeit not perfect, so far) precision – while those without supersymmetry seem badly understood and their agreement with the observations hasn't been proven too precisely. It's not surprising for many reasons. One of them is that supersymmetry makes physics both more stable, promising, and free of some hierarchy problems which is good phenomenologically; as well as full of cancellations and easier to calculate which is good from a mathematical viewpoint. Oh, SUSY, with a pictorial walking.

It is totally plausible that supersymmetry at low enough energies is an unavoidable consequence of string/M-theory – assuming some reasonably mild assumptions about the realism of the models. This belief was surely shared e.g. by my adviser Tom Banks – one of his prophesies used to be that this assertion (SUSY is unavoidable in string theory or quantum gravity) would eventually be proven. Acharya was looking into this question.

He focused on "geometric" vacua that may be described by 10D, 11D, or 12D (F-theory...) supergravity – which may

*then*be dimensionally reduced to a four-dimensional theory. Assuming that these high-dimensional supergravity theories are good approximations at some level, the statement that "supersymmetry is unavoidable in string theory" becomes basically equivalent to the statement that "manifolds used for stringy extra dimensions require covariantly constant spinors".

Calabi-Yau three-folds – which, when used in heterotic string theory, gave us the first (and still excellent) class of realistic string compactifications in 1985 – are manifolds of \(SU(3)\) holonomy. This holonomy guarantees the preservation of 1/4 of the supercharges that have existed in the higher-dimensional supergravity theory in the flat space because the generic holonomy \(SU(4)\sim SO(6)\) of the orientable six-dimensional manifolds is reduced to \(SU(3)\) where only 3 spinorial components out of 4 are randomly rotated into each other (after any closed parallel transport) while the fourth one remains fixed.

In table 1, Acharya lists all the relevant holonomy groups. If you forgot, the holonomy group is the group of all possible rotations of the tangent space that is induced by a parallel transport around any closed curve.

\(SO(N)\) is the generic holonomy of an \(N\)-dimensional real manifold. It would be \(O(N)\) if the manifold were unorientable. This transformation mixes the spinors in the most general way so there are no covariantly constant spinors. But there could nevertheless be Ricci-flat manifolds of this generic holonomy. The three question marks are written on that first line of his table because they exactly correspond to the big question he wants to probe in this paper.

Now, in real dimensions \(n=2k\), \(n=4k\), \(n=7\), and \(n=8\), one has the holonomies \(SU(k)\), \(USp(2k)\), \(G_2\), and \(Spin(7)\), respectively. All these special holonomies guarantee covariantly constant spinors i.e. some low-energy supersymmetry; and the Ricci-flatness of the metric, too. On the other hand, one may also "deform" the \(SU(k)\) and \(USp(2k)\) holonomies to \(U(k)\) and \(USp(2k)\times Sp(1)\), respectively, and this deformation kills both the covariantly constant spinors (i.e. SUSY) as well as the Ricci-flatness.

Note that string/M-theory allows you to derive Einstein's equations of general relativity from a more fundamental starting point. In the absence of matter sources (i.e. in the vacuum), Einstein's equations reduce to Ricci-flatness i.e. \(R_{\mu\nu}=0\). This is relevant for the curved 4D spacetime that everyone knows. But it's also nice for the extra dimensions that produce the diversity of low-energy fields and particles.

So whether you find it beautiful or not, and all physicists with a good taste find it beautiful (and the beauty is very important, I must make you sure about this basic fact because you may have been misled by an ugly pundit), string/M-theory makes it important to study Ricci-flat manifolds – both manifolds including the 4 large dimensions that we know, as well the compactified extra dimensions. The former is relevant for 4D gravity we know; the latter is more relevant for the rest of physics.

Acharya divides the question "whether the Ricci-flat manifolds without covariantly constant spinors exist" into two groups:

* simply connected manifolds

* simply disconnected manifolds

In the first group, he doesn't quite find the proof but it seems that he believes that the conjecture that "no such compact, simply connected, Ricci flat manifolds without SUSY exist" seems promising.

In the second group, there exist counterexamples. After all, you may take quotients (orbifolds) of some supersymmetric manifolds – but the orbifolding maps the spinors to others in a generic enough way which breaks all of supersymmetry. So SUSY-breaking, Ricci-flat compactifications exist.

However, at the same moment, Acharya points out that all such simply disconnected Ricci-flat manifolds seem to suffer from an instability – a generalization of Witten's "bubble of nothing". It's given by a Coleman-style instanton that has a hole inside. The simplest vacuum with this Witten's instability is the Scherk-Schwarz compactification on a circle with antiperiodic boundary conditions for fermions (the easiest quotient-like way to break all of SUSY because when a constant is antiperiodic, it must be zero). The antiperiodic boundary conditions are perfect for closing a cylinder into a cigar (a good shape for Coleman-like instantons in the Euclideanized spacetime, especially because of Coleman's obsessive smoking) on which the spinors are well-behaved.

So the corresponding history in the Minkowski space looks like a vacuum decay – except that the new vacuum in the "ball inside" – which is growing almost by the speed of light – isn't really a vacuum at all. It's "emptiness" that doesn't even have a vacuum in it. The radius of the circular dimension – which is \(a\to a_0\) for \(r\to\infty\) – continuously approaches \(r=0\) on the boundary of Witten's bubble of nothing – basically on \(|\vec r|=ct\) where \(c\) is the speed of light – and it stays zero for \(|\vec r|\lt ct\) which means that there's no space for \(|\vec r| \lt ct\) at all.

Such instabilities are brutal and Acharya basically proves that these instabilities make all Ricci-flat, simply disconnected, non-supersymmetric stringy compactifications unstable. We see that our Universe doesn't decay instantly so we can't live in such a vacuum. Instead, the extra dimensions should either be supersymmetric and simply disconnected; or they should be simply connected. When they're simply connected, the conjecture – which has passed lots of tests and may be proven – says that these compactifications imply low-energy supersymmetry, anyway.

If this conjecture happened to be wrong, it would seem likely to Acharya – and me – that the number of non-supersymmetric, simply connected, Ricci-flat compact manifolds would probably be much higher than the number of the supersymmetric Ricci-flat solutions. If it were so, SUSY breaking could be "generic" in string/M-theory, and SUSY breaking could actually become a rather solid prediction of string/M-theory. (Well, the population advantage should also beat the factor of \(10^{34}\) to persuade us that we don't need to care about the non-supersymmetric vacua's hierarchy problem.) Note that with some intense enough mathematical work, it should be possible to settle which of these two predictions are actually being made by string theory.

Acharya has only considered "geometric/supergravity" vacua. It's possible that some non-geometric vacua not admitting a higher-dimensional supergravity description are important or numerous or prevailing – and if it is so, the answer about low-energy SUSY could be anything and Acharya's work could become useless for finding this answer.

But some geometric approximation may exist for almost all vacua - dualities indicate that there are often

*several*geometric starting points to understand a vacuum, so why the number should be zero too often? – and the incomplete evidence indicates that low-energy SUSY is mathematically needed in stable enough string vacua. When I say low-energy SUSY, it may be broken at \(100\TeV\) or anything. But it should be a scale lower than the Kaluza-Klein scale of the extra dimensions – and maybe than some other, even lower, scales.

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