This afternoon, I tried to read through this paper claiming to "present $\mathcal{O}(10^{15})$ string compactifications with the exact chiral spectrum of the Standard Model of particle physics".

As someone still learning the very basics of string theory, it was a rather dense read, but I think I gathered that they had found a class of compactifications that yielded a low-energy effective field theory with the gauge group of the minimal supersymmetric standard model (MSSM), as well as matter representations corresponding to the matter content of the MSSM.

If it's possible to summarise how this is done (i.e. how to determine the gauge groups and matter content of a compactification), I would love to know, though I gather it's not an easy thing to explain, so my main question is this:

Does the fact that they've found the correct gauge groups and matter representations in this context immediately imply that they've actually found the standard model, or even the MSSM? In particular, as far as I could tell, there wasn't much of a discussion about the breaking of supersymmetry or the masses of the fermions or weak bosons.


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If it's possible to summarise how this is done (i.e. how to determine the gauge groups and matter content of a compactification), I would love to know

It's done differently in different branches of string theory. I will give a description of what I remember about how F-theory works, but it may be inaccurate in places, hopefully someone will correct me if I make a mistake.

You may have read about strings in 10 dimensions, with 4 macroscopic dimensions and 6 microscopic dimensions in the shape of a Calabi-Yau space, and then perhaps some "D-branes" that occupy all the macroscopic dimensions and some subset of the 6 microscopic dimensions.

In the case of F-theory, one actually considers 8 microscopic dimensions, and then different ways that that the metric of this 8-dimensional space can become degenerate or singular. The possible 4-dimensional defects in the metric correspond to 7-branes, the two extra dimensions correspond to the strength of an "axio-dilaton" field which becomes singular at a 7-brane, and the gauge group for strings attached to that brane corresponds to the "Kodaira classification" of the defect.

Furthermore, the intersections of the defects (geometrized branes) in the 8-dimensional compact space, phenomenologically manifest as interactions of the gauge superfields inhabiting the defects. Just to repeat, each 7-brane has some gauge superfield in its interior; because it's a superfield, you have bosons and fermions. Then, when two 7-branes intersect, that means that in the effective theory, their gauge superfields interact; when supersymmetry is broken, that means you have the matter fields from one 7-brane interacting with the gauge field from the other 7-brane. And finally, when three 7-branes intersect at a point, that gives the yukawa couplings like those in the standard model, in which two chiral matter fields are coupled via a Higgs field.

It's quite elegant and remarkable that branes, intersections of two branes, and intersections of three branes, naturally give you the basic fields, the gauge couplings of the matter fields, and the yukawa couplings of the matter fields. This is how "intersecting braneworld" models work in general, but the geometric F-theory approach allows for branes with exceptional gauge groups like E6, E7, E8, something which isn't possible in simpler D-brane models.

Again, I want to emphasize that I am not an "F-theorist"; there may be inaccuracies or outright errors in the description I've given.

In particular, as far as I could tell, there wasn't much of a discussion about the breaking of supersymmetry or the masses of the fermions or weak bosons.

The masses weren't mentioned because no one is presently capable of calculating them. The paper you read mentions that the yukawa couplings (which determine the masses) are "moduli-dependent". The moduli are the parameters that characterize the size of the 8 compact dimensions (basically the size of each of the topologically distinct compact submanifolds). The 8-dimensional Calabi-Yau space is a dynamic geometry - the moduli can vary; to calculate the yukawa couplings associated with a particular F-theory Calabi-Yau, one would need to know the moduli values of the lowest-energy state of the manifold, the state in which its shape and size are stable. And that's something which is presently beyond calculation.

As you notice, they also say nothing about how supersymmetry is broken. But most likely the idea is, that it's broken by condensation of gauginos in one of the 7-branes, and then this is transmitted to the other 7-branes via gravitational interactions. (This would be a form of "gravitationally mediated dynamical supersymmetry breaking".) That's how it works in the "G2-MSSM", a kind of braneworld model from M-theory.

Just to be complete, I should also note that gauge-field flux along the submanifolds of the 8d Calabi-Yau also play a role - see the references in the paper to "G4-flux".

So to sum up, what they did was to identify a quadrillion Calabi-Yau spaces with flux, each corresponding to a different ground state of F-theory, in which the fields and interactions of the supersymmetric standard model should exist as a subset of the overall effective theory. That much could be inferred from the geometry of their constructions. But the parameters of all these "standard models" should vary widely, depending on how the moduli of their parent Calabi-Yaus settle down (if they settle down); there's no guarantee at all that any of them will match our observed reality. Nor did they seriously address whether and how supersymmetry gets broken, and again, it would need to be fairly badly broken to have a chance of matching observed reality.

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    $\begingroup$ In other words... it's not even wrong? $\endgroup$ Oct 24, 2022 at 7:01
  • $\begingroup$ Do you know if they addressed the cosmological constant? Often getting a positive cosmological constant is a barrier to "simple" string compactifications producing something phenomenologically viable. $\endgroup$
    – Andrew
    Oct 29, 2022 at 14:59
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    $\begingroup$ @Andrew They don't mention it... You know, Vafa, the founder of F-theory, seems to prefer quintessence over vacuum energy, as an explanation of dark energy. Meanwhile his student Heckman arxiv.org/abs/1811.01959 wants to get expansion from a runaway modulus... It's easier in M-theory since you can use the Bousso-Polchinski method of tuning the vacuum energy... I have no particular opinions about the right way to proceed. $\endgroup$ Nov 1, 2022 at 6:22
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    $\begingroup$ @FlatterMann It's as if the ancient Greeks had a quadrillion different ideas for what everything is made of. Hopefully it won't take 2000 years to reach testability this time. $\endgroup$ Nov 1, 2022 at 6:28
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    $\begingroup$ @MitchellPorter I'll drink to that but I am not having my hopes up that we will get an answer in my lifetime. Also no offense to the theorists, but I think we are simply still missing a few important observations. They key to the universe were indeed as you say not 2000 years of thought but two coarsely ground glass disks in Galileo's telescope. $\endgroup$ Nov 1, 2022 at 6:37

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