# Masses from modular symmetry

One of the interesting new ideas in BSM model building began with this paper:

https://arxiv.org/abs/1706.08749
Are neutrino masses modular forms?
Ferruccio Feruglio

A modular form is a kind of two-dimensional analogue of the periodic trigonometric functions. Just as trigonometric functions can describe standing waves on a circle, modular forms can describe harmonics of two-dimensional surfaces. They come from the theory of the modular group in complex analysis, a group of transformations of the upper complex plane. Under a modular transformation, a modular form is multiplied by the kth power of a certain complex number; modular forms are classified by that exponent k, which is called the weight. In particle physics models employing modular symmetry, fields are defined by a modular weight as well as the usual assortment of quantum numbers and group representations.

Modular forms are a central topic in modern mathematics. Fermat's last theorem was proved as a corollary of the "modularity theorem" relating elliptic curves to modular forms. Their ultimate significance may be as part of the Langlands program connecting number theory and harmonic analysis, whereby e.g. the coefficients in a Taylor expansion of a harmonic function in some geometric space, are related to the number of solutions of polynomials over some corresponding number field.

Modular forms are also everywhere in string theory, initially because one of the symmetries of the string is modular invariance. This refers to the coordinate system one places on the two-dimensional (one space dimension, one time dimension) "worldsheet" that the string traces out in space-time (an analogue of the world-line of a particle). The internal coordinates one uses to describe the string should not have physical significance, and so the worldsheet has an internal diffeomorphism invariance like that of general relativity. But the physics must remain the same, also under "large" diffeomorphisms that discontinuously change the coordinate system, and these are described by the modular group. So probability amplitudes for strings tend to involve modular forms.

However, modular symmetry can also apply to extra space-time dimensions. If you have two extra space dimensions that form a torus, once again modular transformations of the coordinates along those two directions are possible. This kind of modular transformation underlies some of the dualities of string theory, and reaches all the way down to some electric-magnetic dualities of field theory.

(I should say that the truly general notion is that of an "automorphic form": basically a harmonic function on some space which respects a discrete symmetry of that space. Periodic functions on the real line are the same when shifted along by an integer number of wavelengths; modular forms transform neatly under the more intricate transformations of the modular group; and general automorphic forms do the same thing, but in yet higher dimensions. One expects automorphic forms to be part of brane amplitudes, for example.)

So once particle physicists started extending flavor symmetry to include modular weights, a few string theorists started looking for ways to obtain such models from modular transformations of the extra dimensions. One such paper has appeared today:

https://arxiv.org/abs/1901.03251
Unification of Flavor, CP, and Modular Symmetries
Alexander Baur, Hans Peter Nilles, Andreas Trautner, Patrick K.S. Vaudrevange

(For a while I confused the first author with @ohwilleke's favorite gravitational theorist, Alexandre Deur.)

I'm not trumpeting this paper as a clear breakthrough, though it is interesting that they have tied this modular renaissance to a separate line of thought involving C, P, and T symmetries. It's more that it provides the occasion to mention this new class of models, which I find intriguing because of their deep mathematical connections, covered e.g. in Edward Frenkel's book Love and Math, and which will probably show up on the physics blogs and other semi-popular science media, one day soon.

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Auto-Didact, arivero and ohwilleke

ohwilleke
Gold Member
One of the interesting new ideas in BSM model building began with this paper:

https://arxiv.org/abs/1706.08749
Are neutrino masses modular forms?
Ferruccio Feruglio

https://arxiv.org/abs/1901.03251
Unification of Flavor, CP, and Modular Symmetries
Alexander Baur, Hans Peter Nilles, Andreas Trautner, Patrick K.S. Vaudrevange

(For a while I confused the first author with @ohwilleke's favorite gravitational theorist, Alexandre Deur.)

Haha! (Totally outside any of his areas of research though. Aside from gravity, he does QCD, especially at the perturbative/non-perturbative threshold at JLabs with an experimental/phenomenological flavor, rather than HEP-theory.)

Interesting papers.

I think modular forms make a lot of sense as ways to understand and formulate the PMNS and CKM matrixes as dynamic processes. I've thought about the standing wave in two dimensions concept before but never articulated it as clearly, and never realized that there was a whole area of mathematics that went with it. I appreciate your explanation to connect those dots.

The abstract of the first article makes the following bold claim regarding its model (considering its somewhat dubious supersymmetric motivation):

The most economical model predicts neutrino mass ratios, lepton mixing angles, Dirac and Majorana phases uniquely in terms of the modulus vacuum expectation value, with all the parameters except one within the experimentally allowed range.

This snippet from the abstract is also intriguing and is very much in line with how I am inclined to interpret what we see at a deeper level:

Yukawa couplings are just modular forms and the only source of flavour symmetry breaking is the vacuum expectation value of a single complex field, the modulus.

I had been surprised that I hadn't seen it lately until I noticed that it was from the summer of 2017. With more reflection, I think I probably actually bookmarked this article when it came out and then didn't get back to writing about it. But, in late middle age (I just got bifocals a year ago and last week had them upgraded (downgraded?) to coke bottle thickness), I'm not surprised that an article a year and a half old isn't fresh in my memory anymore.

The second paper goes too far into speculative string theory concepts to capture my interest. While automorphism are an elegant extension, so to speak, of modular forms, I have a pretty dim view of string theories reliance on dimensions N>>4, let alone torus topologies in space-time. While a torus topology of space-time is mathematically interesting and conceptually possible to describe, its seems extremely unlikely to be physical and has no observational motivation. The abstract of the second paper states (with weasel words in bold and hard truths underlined):

Flavor symmetry plays a crucial role in the standard model of particle physics but its origin is still unknown. We develop a new method (based on outer automorphisms of the Narain space group) to determine flavor symmetries within compactified string theory. A picture emerges where traditional (discrete) flavor symmetries, CP-like symmetries and modular symmetries (like T-duality) of string theory combine to unified flavor symmetries. The groups depend on the geometry of compact space and the geographical location of fields in the extra dimensions. We observe a phenomenon of "local flavor groups" with potentially different flavor symmetries for the various sectors of quarks and leptons. This should allow interesting connections to existing bottom-up attempts in flavor model building.

Basically, it comes across as utter bullshit from people who have no clue what is going on. I know, that's harsh. And, finding a mathematical tool that could connect flavor, CP and the T-duality in a unified fashion is pretty cool. Still, my rule of thumb is that if you can't explain what you are doing plainly and without weasel words, you are probably not thinking about what you are doing very clearly either.

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Auto-Didact
Klystron
Gold Member
Edward Frenkel's book Love and Math,

Thanks for the reference. Recently re-read Amir Aczel's Fermat's Last Theorem. Frenkel's book sounds like a good source before Stewart's Algebraic Number Theory and Fermat's Last Theorem. Not so much a digression as an extension of the use of elliptic functions and modular forms in my reading list.

@mitchell porter damn, I was just about to do a thread on modular forms and another one on the Langlands program more or less in the mathematical physics spirit of my PDE thread here... seems like you beat me to it this time :) Have you read Frenkel's book? It is still on my to read list.

To make things clearer, modular forms are periodic holomorphic functions in the upper half complex plane which can be conformally mapped onto Riemann surfaces, such as the Riemann sphere; an example of such a conformal mapping is the Möbius transformation onto the Poincaré disc.

Moreover, modular forms can also be approached from an (algebraic) geometry perspective as sections of line bundles on the moduli space of elliptic curves (compare this with sections of the Möbius strip being spinors). This essentially is all intimately related to the conceptual basis of twistor theory.

An interesting jumping off point for learning more about modular forms is through studying Ramanujan's ##\tau##-function, i.e. engaging in "numerology" just as Ian McDonald did, which turned out to reveal to be a deep connection between modular forms and the properties of affine root systems of the classical Lie algebras.

arivero
https://arxiv.org/abs/1908.11032Shotaro Shiba Funai, Hirotaka Sugawara
(Submitted on 29 Aug 2019)
The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs and gauge particles -- more precisely, their mass ratios) are expressed as coefficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our intention here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (Kähler potential and the metric) of the moduli space of the Calabi-Yau manifold, thus providing a way to calculate the metric of Calabi-Yau manifold itself directly from the experimental data.
Here we have some physics numerology using modular forms. The authors just take matrices appearing in the theory of modular forms, and use them to approximate the standard model mass matrices, without having a definite physical framework. I don't believe their specific proposals because Koide gets the charged leptons exactly, whereas they only approximate them; but it is a step forward in terms of ambition, for this research program of masses from modular symmetry.

ohwilleke, arivero and Auto-Didact
I really want to learn the use of modular forms in mass models in particle physics. Can you please suggest me where to start ? I have worked in models based on discrete symmetry before but it is just hard for me to understand the modular forms. Please help.

ohwilleke
Gold Member
Re-reading the Moonshine paper. Is it really true, as asserted, that?

The mass ratios are scale independent quantities [4] and do not vary with energy scale.

I could see that relative order of magnitude might be preserved, but it isn't at all obvious to me that the ratios of the SM mass ratios are the same at all energy levels.

They cite to K. G. Chetyrkin and A. R´etey, Nucl. Phys. B583 (2000) 3 [arXiv:hep-ph/9910332]. for this proposition.

I was reading this in response to Kishan M's comment on the theory that one of the more practical research strategies to get right to the part of the theory of modular forms that you care about might be to trace back footnotes in a couple of the papers until you get to what you want. It might not be quite as straightforward as a textbook, but likely has the benefit of focusing on what is most pertinent.

Kishan M
Re-reading the Moonshine paper. Is it really true, as asserted, that?

"The mass ratios are scale independent quantities [4] and do not vary with energy scale."

I could see that relative order of magnitude might be preserved, but it isn't at all obvious to me that the ratios of the SM mass ratios are the same at all energy levels.
It's not literally true, because of RG flow, as critics of the Koide formula like to remind us... They must mean is that the change between IR and UV is small enough, that mass ratios in the IR can still be a guide to the UV physics. (I'm not sure I'd agree, not for all masses, but it's the only reasonable interpretation I can make.)

I really want to learn the use of modular forms in mass models in particle physics. Can you please suggest me where to start ? I have worked in models based on discrete symmetry before but it is just hard for me to understand the modular forms. Please help.
Unfortunately I don't know an ideal introduction to modular forms. The Fields medalist Richard Borcherds has many excellent video tutorials on complex analysis and related topics, and one of them is an introduction to modular forms, but even he says they're weird, connecting unexpected parts of math. I think an ideal introduction (especially for physicists) would emphasize modular transformations both as hyperbolic tilings of the complex plane, and as twists of a torus... This is the key to the math of Ramanujan and Wiles, too.

Kishan M and ohwilleke