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Heckman, Vafa: QG bounds the number of hierarchy-like problems

Every competent physicist knows that fine-tuning is a kind of a problem for a theory claimed to be a sufficiently fundamental description of Nature.

Fundamental physicists have wrestled with the cosmological constant problem, the Higgs hierarchy problem,... and perhaps other problems of this kind. Fine-tuning is a problem because assuming that the fundamental "theory of everything" works like a quantum field theory and produces the couplings of the low-energy effective field theories via renormalization group flows, the observed hierarchies between the scales etc. seem extremely unlikely to emerge.

In principle, there could be arbitrarily many couplings and even fine-tuned couplings which could cause an infinite headache to every theorist. In a new paper, Cumrun Vafa – the Father of F-theory and the Swampland Program (where this paper belongs) – and Jonathan Heckman, a top young research on both topics, present the optimistic evidence that in string/M-theory and/or quantum gravity, the infinite fine-tuning worries are probably unjustified:

Fine Tuning, Sequestering, and the Swampland (just 7 pages, try to read all)
What's going on? Effective field theories outside quantum gravity may be built by "engineers". You may apparently always add new fields, new sectors, and they allow you to tune or fine-tune many new couplings. There doesn't seem to be a limit.



String/M-theory is more predictive and chances are that even if there were another consistent theory of quantum gravity, it would be more predictive, too. In particular, as they say, the number of couplings that can be independently fine-tuned to unnatural values is finite.

I have a feeling that they count the moduli among the couplings that can be "fine-tuned", even if they correspond to physical fields. But that doesn't invalidate their statement because they say that the number of moduli is bounded, too.



Moreover, the bound is a fixed finite number for every choice of the number of large dimensions and the number of supercharges. Fine, what's the evidence?

First, the number of the Minkowski, flat spacetime solutions in string/M-theory seems to be finite. Also, the number of Calabi-Yau topologies seems to be finite. The latter statement hasn't been quite proven but propositions that are very close have been proven. For example, if you restrict the manifolds to be elliptically fibered and the base to be a toric geometry, it's been proven that Calabi-Yau three-fold topologies form a finite set. It seems very likely that the manifolds that cannot be represented like that are a "minority", so even the number of all Calabi-Yau topologies should be finite.

Their first full-blown discussion is in 6D field theories. Conformal field theories have either \((1,0)\) or \((2,0)\) supersymmetry; \((1,1)\) cannot be conformal. Infinitely many classes of such theories with lots of deformations exist as CFTs. But if you want to couple them to gravity, you see restrictions. The cancellation of anomalies requires the total number of tensor multiplets to be 21 which is a particular finite number. In fact, all stringy 6D CFTs only allow deformations that result from operators that exist in the theory. In this 6D case, their new principle largely reduces to the anomaly cancellation.

In another related example, the total rank of some gauge group is 22. Perturbative string theory obviously restricts these ranks by the central charge – the rank cannot be too high for the same reason why the spacetime dimension cannot be arbitrarily high. Well, the central charge is also a gravitational anomaly – on the world sheet.

They discuss a few more rather specific examples – so their paper has many more equations and inequalities than what is actually needed for their main claims. But the overall new swampland principle has ramifications. In particular, if you imagine many sequestered or hidden sectors in artificially engineered apartheid-style models of particle physics, all their couplings seem to be independent, and could therefore admit independent fine-tuning.

According to Heckman and Vafa, if the number of such sectors is too high, quantum gravity actually implies some correlation between the fine-tunings. At the level of effective field theory without gravity, many parameters \(g_i\) could be independently adjusted and very small. But if you require that the theory may be coupled to quantum gravity, it already follows that there are equations that correlate almost all these constants \(g_i\), up to a finite (pre-determined) number of exceptions.

Sometimes people express their doubts about the reasoning involving naturalness and the disfavoring of fine-tuned theories. Indeed, the thinking based on quantum field theories is ultimately imprecise and incomplete and has to be adjusted. But "just ignore all the fine-tuning problems" isn't a scientifically valid adjustment to the problem. The problems cannot be completely ignored because they're implied to be problems by a rather specific, successful framework of physics that we use all the time – quantum field theory – combined with the probability calculus. To ignore the problem would mean to cherry-pick what we like about the framework – quantum field theory – and what we don't.

Instead, the adjustment to the fine-tuning rules must have the form of "quantum field theory isn't an exact description of Nature and the correct framework differs in respects A,B,C, and these differences also imply different predictions concerning the fine-tuning". This new Heckman-Vafa swampland may be counted as an actual scientific way to go beyond the existing rules about the naturalness and fine-tuning in effective field theories. The paper tells us how string/M-theory actually modifies the semi-rigorously proven intuition or lore about the fine-tuning in our effective field theories.

The modification primarily says that the couplings are automatically more constrained than naively indicated by the low-energy effective field theory analysis. In other words, string/M-theory is – in a new specific sense – more predictive than quantum field theory. It shouldn't be surprising because quantum gravity needs to reconcile the low-energy behavior with the high-energy behavior – where the particle spectrum must gradually merge with the black hole microstates whose entropy is again dictated by a low-energy effective field theory (including Einstein's gravity). When you're playing with the low-energy couplings, quantum gravity actually tells you that you have to aim at and hit several targets for the trans-Planckian behavior of the theory to remain consistent (with gravity).

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