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My question is motivated by a statement in this chapter (emphasis added, off-topic statements about supersymmetry elided):

Compactifying on tori ... is very interesting for its simplicity ... but not so much for phenomenology, as it is unable to mimic most of the physics observed in nature.

What are these failure modes for compactifying on tori?

Background: To explain the complex 4D behavior we observe, it is common to posit some simpler behavior in a number of as-yet-unobservable small (compact) extra dimensions. For example, Kaluza-Klein theory involves a compact fifth dimension often represented as associating a Planck-scale circle with each spacetime point; string theory extends this to many more compact dimensions. Compactification describes this strategy and the extraction of 4D observations from these higher-dimensional spaces.

I am aware of the classic Kaluza-Klein result in which compactification appears to unite gravity and electromagnetism, yet predicts a mass of the electron on the order of the Planck mass ($1.2\times 10^{19}$ GeV) versus the observed 0.51 MeV. This seems a nice solid failure.

So: what are the main (most easily grasped, please) observed physics that compactification on tori fails to reproduce? And what do we learn from this failure in terms of intuition about our physics?

The quote above is from a treatise on string theory, but to the extent that compactification and string theory are separable, I am asking about the compactification part. For a related question on compactification on tori try here, and for those who wish to pull on the supersymmetry thread I've suppressed, try here.

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  • $\begingroup$ Actually, toroidal compactifications with "magnetic flux" seem capable of getting many desired qualitative features, e.g. groundai.com/project/… $\endgroup$ Mar 4, 2021 at 4:07
  • $\begingroup$ That's good to know. I'm interested in how they fail. And without at all denigrating these results you've linked, which I'm not familiar with, one could say that Kaluza-Klein theory got the qualitative fact (electrons have mass) right, while absolutely blowing the quantitative fact (the actual electron mass). I'm asking about the quantitative results. $\endgroup$
    – biochemist
    Mar 4, 2021 at 4:22
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    $\begingroup$ I will remark that current models with extra dimensions do not give mass to the electron in that way, they use a Higgs field. Also, none of the forces are obtained in the classic KK way. Witten's paper "Search for a realistic Kaluza–Klein theory" is often regarded as the endpoint of the classic KK approach. $\endgroup$ Mar 4, 2021 at 6:00
  • $\begingroup$ Also helpful. I see a through line from KK to at least some versions of toroidal compactification, but I'd like to understand this better. $\endgroup$
    – biochemist
    Mar 4, 2021 at 6:43

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I'll start with a summary:

  • Toroidal Kaluza-Klein compactifications will only give you abelian forces like electromagnetism, but not nonabelian forces like the nuclear forces
  • More general Kaluza-Klein compactifications can give you nonabelian forces, but can't make the interactions chiral (dependent on the handedness of the matter particle), as is required for the weak force
  • String theory has extra dimensions, and even has realistic models based on toroidal compactification. However, string theory does not obtain the observed forces in the old Kaluza-Klein way, as resulting from symmetries of the extra dimensions

Now, the details:

I think it important to first emphasize that this is just one part of a broader topic, compactification of extra dimensions. A torus is just one option, for the shape of the extra dimensions. There is also the simpler option of a (hyper)sphere, as well as endless more complex options (e.g. the famous Calabi-Yau spaces).

Another important fact is that a central part of the old Kaluza-Klein vision is not pursued any more. In contemporary language, we would say that this is about getting gauge fields (i.e. strong and electroweak forces) from the symmetries (isometries) of the compact space. This is how people thought, up until string theory. In string phenomenology, the observed forces are not obtained that way. The Kaluza-Klein excitations do exist in string theory, but they are just one kind of excitation among many, and are not expected to explain the observed forces. (I do know one person who has proposed pursuing classic Kaluza-Klein phenomenology within string theory; but they are not a string theorist.)

So when you ask about the limitations of toroidal compactifications, it makes some difference as to whether you're talking about string theory, or about classic Kaluza-Klein. The book you link to is a string theory text, and so when they talk about the limits of toroidal compactifications, it is from a string theory perspective. Their stated concern is that the toroidal compactifications don't break supersymmetry enough. Their main concern would be, that to match reality, one needs the theory to be "chiral" (more on this below), and that requires supersymmetry with "N" less than or equal to 1.

Now, it is in fact possible to get realistic models from toroidal compactifications of string theory. But you need something extra. Quoting a 2015 paper:

The standard model is a chiral theory. Thus, the key point to realize the standard model is how to realize a chiral theory. Toroidal compactification is simple, but it can not realize a chiral theory unless introducing additional backgrounds. Orbifold and Calabi-Yau compactifications can lead to a chiral theory. Toroidal compactification with magnetic fluxes can also lead to a chiral theory. Here, we study such a background. That is, our key ingredients are the multiple U(1) magnetic fluxes inserted into SO(32) gauge group.

I mentioned that string phenomenology does not obtain the observed nongravitational forces via the classic Kaluza-Klein mechanism. This paper is an example. Their string theory already starts out with an SO(32) gauge field, and then the magnetic fluxes in the extra dimensions break the symmetry group to something smaller. That's how the strong and electroweak forces arise in their model - not as a Kaluza-Klein echo of gravity in the compact dimensions.

They call the resulting model realistic, because it has all the necessary kinds of particles and forces. We could attempt to discuss what problems remain to be solved, but I think your interests lie more in the direction of classic Kaluza-Klein, so let's turn to that topic now.

I would say that the zenith of classic Kaluza-Klein theory is found in Witten's 1981 paper "Search for a realistic Kaluza-Klein theory". Perhaps one should call this modern Kaluza-Klein theory (dating from the 1970s revival of higher dimensions in theoretical physics, owing to supergravity), reserving classic Kaluza-Klein theory for the 1920s work of Kaluza and Klein. But either way, we're talking about the pre-superstring stratagem of obtaining gauge forces from the symmetries of the extra dimensions.

If we do restrict ourselves to purely toroidal compactifications in this context, one problem is that we won't be able to get the strong and weak forces, because their symmetry groups are "nonabelian", but a torus can only give us an "abelian" symmetry of the form "U(1)n". U(1) refers to the group of 1x1 complex unitary matrices, i.e. complex numbers of modulus 1, i.e. the group of rotations of a circle. It's the symmetry possessed by Klein's periodic fifth dimension, and it is the symmetry group of electromagnetism. Topologically, an n-dimensional torus is a product of n circles, and its symmetry group consists of n independent rotations, thus, U(1)n. But for the weak force you need SU(2) (2x2 matrices), and for the strong force you need SU(3), so you would need something other than a torus to get those groups.

Witten doesn't restrict himself to tori, and so he can identify an entire family of seven-dimensional manifolds with SU(3) x SU(2) x U(1) symmetry. Great, you can get the forces of the standard model that way. However, the real problem arises when considering the matter of the standard model.

"The standard model is a chiral theory", said the Japanese string theorists in the paper quoted above. This means that particles come in left-handed and right-handed forms, and sometimes the response to a force differs according to handedness. And it's that different response, that classic Kaluza-Klein is unable to produce. There's no apparent way to make the interaction in the extra dimensions sensitive to the handedness of the matter field in the macroscopic dimensions. "It treats four-dimensional left- and right-handed fermions in the same way", writes Witten. He considers various ways out, and I'm sure other people have too, but overall, this is the single big issue which halted the classic Kaluza-Klein program in its modern form.

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