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I-folds, anomalies, and how string theory agrees with "pure thought" when it comes to the allowed ranks

At Harvard, Miguel Montero and Cumrun Vafa were fighting for their new and exciting paper

Cobordism Conjecture, Anomalies, and the String Lamppost Principle
to be the first one in the hep-th listings today. To win, your paper has to be the earliest one posted after 18:00:00 UTC. And they got 18:00:01 UTC. Too bad, it was only enough for the silver medal today. Try harder next time, Gentlemen. ;-)

It's a paper within the Swampland program that is refining the evidence in favor of the String Lamppost Principle. In plain English (i.e. when the technical language of cobordisms and anomalies is removed), the principle says the following:
If you don't like that string theory is absolutely needed to define a consistent theory of quantum gravity, then you are a dick or an aßhole that should go fudge himself or herself.
You may protest that there is a complementarity problem here: if you are a dick, you may apply the verb on others, but you can't be the passive element at the same moment. And on the contrary, if you are an aßhole, you can be the passive one but not the active one. Indeed, complementarity poses a problem for these individuals but it's their problem, not a problem for those who sort of know that the principle is right.

OK, the funny thing is that the set of string vacua is constrained but it is still extremely rich. String/M-theory produces lots of patterns saying which combinations of the spectrum, ranks, symmetries, and spacetime dimensions are possible and which are not. You may learn most of these patterns and correlations by "constructively producing" some string compactifications – and by passively studying the patterns in these vacua as if you were a practical man.

If you assumed that string theory may be ignored, these "recommended patterns" would be pretty much uncorrelated to the "general truth". However, in reality, an increasing percentage of these patterns and correlations may be derived from more general principles than "constructive string theory" – and almost from the pure and minimal requirement of consistency.

One of the broader Swampland principles, or perhaps a requirement of consistency, is the 2019 McNamara-Vafa Cobordism Conjecture (McNamara on TRF). It is a generalization of the absence of global symmetries in quantum gravity, something that has been a part of the lore for a few decades and that has become much more quantitative after 2000 or so. The principle demands all cobordism classes (in the space of configurations) to be trivial. Quantum gravity really unifies all configurations so there shouldn't be any "decoupled components" of the configuration space. The superselection sectors from different values of global symmetries' charges or the cobordism classes would be the "separated components" and this lack of unity just doesn't rhyme with a theory of quantum gravity (or a unifying theory that really wants to unify all objects and configurations).

As they repeat (and exploit) here, it means that if there is any apparently nontrivial cobordism class in a vacuum of quantum gravity, it must be actually trivial because a new object that trivializes it exists. The objects that do this job end up being generalizations of the orientifolds. An orientifold is the orbifold-like fixed (hyper)plane of a mirroring procedure that tells the strings that they and/or "they with their images" must exist symmetrically on both sides of the (hyper)plane; in the case of the orientifold, the orientation of the string must be the opposite on the opposite side.

Montero and Vafa call these generalized orientifold planes "inversion-folds" or I-folds. Note that the orientifold planes are defined according to their effect on the fundamental strings so the orientifolds need a weakly coupled, perturbative string theory for them to be well-defined. Away from the weak string coupling, you may have similar objects but they're at most generalizations of the orientifolds.

So the reasoning based on the consistency (well, the Cobordism Conjecture) leads you to postulate the existence of a new, higher-dimensional object. However, what most laymen don't understand is that this isn't some random addition of stuff that just makes a theory more messy and you can't always do it. When an I-fold is obliged to exist in a vacuum, it doesn't mean that the I-fold exists. Instead, in many cases, other consistency criteria imply that this I-fold cannot exist and therefore the required vacuum is impossible as a whole! String theory is extremely constraining – it often requires objects to exist while candidate theories (would-be string vacua) just can't fulfill these conditions and their hopes are killed.

In the case of these I-folds, it is the old-fashioned "anomaly cancellation on them" which prevents a large fraction of them from existing. They look at the spectra on these objects and their contributions to the global gravitational and global gauge anomalies. And the anomaly cancellation may be translated to constraints on the rank of the gauge group. In \(d=9\) large dimensions, even ranks 0-16 are prohibited (red), odd ranks 1,9,17 are demonstrably possible (green), and the rest is uncertain so far (yellow).

In \(d=8\) large dimensions, the rules follow from the \(d=9\) rules. Odd ranks 1-17 are prohibited, ranks 2,10,18 are demonstrably possible, and everything else is uncertain so far. However, in \(d=7\), where new compactifications (distinct from higher-dimensional theories on a circle) are possible, no no-go theorems are known so far while the ranks 3,5,7,11,19 are known to exist. These red-yellow-green traffic lights (Table 1 in the paper) may be obtained both from the explicit stringy constructions (including compactifications of the heterotic string, CHL string [M-theory on the Möbius strip is an equivalent definition], and the Dabholkar-Park background are used) and from the broader consistency considerations (including the anomaly cancellation plus the Cobordism Conjecture).

There is a very powerful new way of thinking presented here. The objects in string theory – fundamental strings, D-branes, other branes but also orbifold planes, orientifolds, and their generalizations – must be allowed because of various old consistency conditions (in the spacetime or the world sheet, often both are enough to derive the conclusion) and/or somewhat newer Swampland-style principles. None of them is truly "optional". They really must be allowed in the physical world, otherwise the theory suffers from some inconsistencies. The classification of the possible vacua (or the possible ranks, gauge groups, tables describing the field content of a vacuum) is then analogous to the classification of simple compact Lie groups or anything of this kind (in pure mathematics). The number of options is relatively high but it is still heavily constrained, especially in higher spacetime dimensions.

As you go from \(d=12,11,10\) to lower dimensions, the simplest constraints (on the rank etc.) tend to disappear. Correspondingly, you may produce many string compactifications that cover most of the a priori possible qualitative classes of effective field theories. But there still exist some subtle correlations in the spectrum and the list of interactions that persist even in the string vacua such as those in \(d=4\). The understanding gets more complex as we climb down from the simplest and most unique spacetime dimensions, \(d=12,11,10\), but \(d=4\) is really just a finite number of dimensions away from these master higher-dimensional spacetimes.

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