- #1

- 1,235

- 405

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

https://arxiv.org/abs/1706.08749

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

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

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.
Last edited: