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Recent papers: SFT, SUSY GUT, a new light gauge boson, \(p\)-adic string gravity, swampland

I plan to write a heuristic post about some exciting ideas of mine but let me dedicate a comment to recent papers on the arXiv. There were some rather interesting papers on that server yesterday and today.

Yesterday, Hiroyuki Hata (whom I know as a guy from the Kyoto group, a team behind the first string paper that I intensely studied 25 years ago) presented some analytic solutions for cubic string field theory that are supposed to describe an arbitrary integral number of spacetime-filling D25-branes. The work is meant to be a generalization of the \(KBc\) Prague work by Martin Schnabl (who is Czech) and Ted Erler (who is at most an honorary Czech).

The solution is of the form \(U Q_B U^{-1}\) where the product is a product of string fields and, believe it or not, one may write down string fields that behave as unitary matrices under the relevant product. So these folks conjugate the BRST operator by a unitary map and, depending on what the map exactly is, one gets unequivalent configurations that contain a different number of branes. I still haven't been given a good intuitive explanation why the mere unitary conjugation produces something else than an equivalent configuration. In non-commutative field theory, we were comparing kernels and images (the maps would be irreversible on one side) but I don't know what's the trick here.

Hata constructs the most general unitary string field \(U\) which depends on some function of two variables and perhaps some additional functions. It is not quite clear to me why this is the right amount of freedom to start with. He constructs the solution. Because one starts with heuristic solutions like that, one must pay some attention to the possibility to transform the formal solutions to something that makes sense, level-by-level, and that has a finite norm and energy density. He demands the correct tension and energy density and claims to have generalizations for any \(N\).

It's not quite clear whether he is sure that those are solutions and whether he has found all solutions of a certain class – his constraints may be too strong or too weak. Also, I find it surprising how irregularly the solutions behave as a function of \(N\). Something new happens for \(N=5\) and higher etc. (suddenly some \(\pi\) appears in the form of some required parameters) – it is just like explaining the grammar of Slavic numerals! ;-) You have "one/jedna minuta", "two-three-four/dvě-tři-čtyři minuty", and "five-or-more/pět-a-více minut". Only 2,3,4 are "truly plural" numerals and the numerals behave a bit like adjectives but 5,6,7... is too much, the individual minutes lose their separate existence, and all the numerals behave like nouns similar to "a dozen" in English! So it is a "dozen of something" – Slavic languages use an analogous construction for almost all numbers starting with five.



Yesterday, Depisch et al. also wrote a nice paper about SUSY GUT with the \(SO(10)\) group. That's arguably the nicest grand unified group. Quarks and leptons are unified in a nice spinorial 16-dimensional representation of \(SO(10)\). They perform some fits and take the neutrino mass matrices into account. The number of input parameters is about the same as the number of experimentally known parameters so there isn't a real prediction. But they may exclude the models where not enough freedom exists. In particular, models without threshold corrections are gone.

For some reasons, despite the sufficient number of parameters that can fit either anything or an elephant, they claim to pick preferred values of the supersymmetric parameter \(\tan\beta\), the ratio of two Higgs vevs in the supersymmetric Standard Model, and the two preferred values are around 38 or 50! That's pretty exciting. One doesn't have enough certainty which of these two magic values are correct if any.



I think that the number of such papers, like SUSY GUT, is low these days and that's one of the reasons why the hep-ph archive has become even more boring for me than it used to be. As you know, I think that some people are responsible for this deliberate suppression of exciting physics and they deserve at least the death penalty. Nevertheless, one more SUSY GUT paper made it to hep-ph today, in fact, a Korean paper on heterotic orbifolds. They obtain a realistic neutrino mass-and-mixing matrix from heterotic string theory on orbifolds which is cool! Quite a direct link between the heterotic string on orbifold, a stringy physics par excellence, and the PDG tables of measured particle properties.

Also today, Dan Hooper and three co-authors proposed an interesting solution to resolve a problem with a disagreement about the values of the Hubble constant from different measurements, a "global crisis" that you may have heard about in recent months. Their solution involves a very light new gauge boson, one whose mass is some \(20\MeV\) and that couples to the difference between the mu-type lepton number and the tau-type lepton number, \(L_\mu-L_\tau\). With such a boson, some early cosmology is modified, producing both Hubble observations, and for some value of the coupling constant of the new \(U(1)\) group, they also fix the anomaly in the muon magnetic moment.



Shing-Tung Yau and two co-authors excitingly study \(p\)-adic strings producing perturbative stringy quantum gravity. These are mathematicians and they arguably have a much better imagination when it comes to the exotic, seemingly unphysical, structures such as \(p\)-adic numbers. So everything works roughly like for regular bosonic strings. There is a world sheet, it has some scaling symmetry, you may write expressions for plane waves of gravitons etc.

However, their world sheet isn't \(\RR^2\) but rather \(p\)-adic. If you don't have a clue what \(p\)-adic numbers are, they are roughly the generalization of rational numbers in the opposite, infinite direction than the real numbers. ;-) Normally, you generalize rational numbers such as 3.14 to 3.14159... – they may continue indefinitely to the side of millionths and vigintillionths etc. – those where the digits matter less and less. On the other hand, \(p\)-adic numbers are like ...55553.14, they are terminating on the right but continuing indefinitely to the left. You can calculate with these left-wing monsters. They induce a new topology that is intuitively less comprehensible to us, the right-wingers, but still demonstrably consistent at some level.

Fine. So it's almost the same gravity except that some real world sheet coordinates become this \(p\)-adic crazy. Some extra consequences change as well. The adelic (some holistic property linking \(p\)-adic objects for all prime values of \(p\)) spectrum of the bosonic string corresponds to the nontrivial zeroes of the Riemann zeta function. I've known about the relationship for a long time (in the \(p\)-adic amplitude business, the omnipresent \(\Gamma(s)\) function from Veneziano-Euler-Virasoro-Shapiro-like string perturbative amplitudes emerges as a ratio of two zeta functions, a ratio you know from the "symmetry" of the zeta function) and I still believe that a theorem about \(p\)-adic string theory (and its spectrum) may provide us with a proof of the Riemann Hypothesis. It's one of the 10 basic approaches to RH I have tried.

I don't know which of the authors of that paper were the most important ones but with a neutral assumption, I am amazed how versatile Shing-Tung Yau (and even the other two) must be because \(p\)-adic string theory surely looks like a different part of mathematical physics than "ordinary" Calabi-Yau manifolds. Look how different their pictures of the \(p\)-adic world sheet (all the pictures look like trees branching at many levels!) are relatively to the smooth Calabi-Yau manifolds of assorted topology.

Also on hep-th today, Renata Kallosh released two papers. One of them is about the perfect square. The other one, co-written with Linde, an Irishman, and an Italian mafioso is a new criticism of the swampland. You can see an explicit difference in the tools that the two sides of the swampland conflict trust. Kallosh and friends from Team Stanford trust 4D effective theories that are being constructed by expanding over the extra (six or so) spacetime dimensions and by calculating the effect of wrapped branes etc. on the 4D physics.

On the other hand, in papers by Team Vafa, a bigger role is played by 10-dimensional field theory as an intermediate approximate description. It's unsurprising because Vafa, the father of F-theory (Father Theory, so it makes Vafa a grandfather of a sort), is good at analysing local physics in higher-dimensional geometries. Kallosh et al. think that the lessons from that 10D description are bad, inconclusive, politically incorrect, misleading etc.

Needless to say, I agree that 10D field theory can't be mindlessly believed – but 4D field theory can't be, either. String/M-theory is more accurate and special than both, you know. So at the technical level, the swampland wars may be seen as a game in which two demonstrably inaccurate approximate schemes – with a different number of spacetime dimensions – fight for the trustworthiness. I am really open-minded here – my odds are comparable to 50-50. At the end, when done correctly, the full physics may be described as a clarification of 4D field theory as well as 10D field theory and I don't know which of the exact steps in the increasingly accurate understanding of physics of some vacua or non-vacua is more helpful.

This uncertainty is a relatively big question – if Cumrun is right, literally thousands of Stanford-connected papers are plagued by a lethal disease. On the other hand, I think that both sides already recognize that the technical arguments are subtle and some chance exists for both sides to be right. This isn't really a deep war where the sides differ about "everything" when it comes to string theory. For example, you may see that Kallosh et al. use the agreement of their reasoning with the weak gravity conjecture (of Nima, Cumrun, me, and Alberto) as one of their arguments! It could have happened that they also determine that the weak gravity conjecture must also be rubbish because it's "a conjecture from the swampland corners" that they generally dislike. But they apparently do believe that the weak gravity conjecture isn't nonsense, to say the least.

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