### M Theory Lesson 274

Tony Smith brings our attention to a recent set of slides by C. Vafa of F theory fame. Vafa is discussing some simple $3 \times 3$ matrices for neutrino and charged lepton masses and mixings. The final slide indicates a mass hierarchy of the form

$m_1 : m_2 : m_3 = c^4 : c^2 : 1$

where $c$ is a characteristic GUT coupling. This expression should be familiar to AF readers, at least in the context of MUB operations.

Recall that a $3 \times 3$ symmetric matrix is the sum of a symmetric 1-circulant and a 2-circulant, which is automatically symmetric. Thus a symmetric matrix $M$ with first row $(c^2 , c, 1)$ may be expressed as where the second factor is itself a sum of circulants. This makes explicit the contribution from dimension $2$ that we see in the Fourier decompositions for the mixing matrices. Vafa's (neutrino) Yukawa matrix is just of this form for $a=0$ and $c$ a cubed root of unity.

$m_1 : m_2 : m_3 = c^4 : c^2 : 1$

where $c$ is a characteristic GUT coupling. This expression should be familiar to AF readers, at least in the context of MUB operations.

Recall that a $3 \times 3$ symmetric matrix is the sum of a symmetric 1-circulant and a 2-circulant, which is automatically symmetric. Thus a symmetric matrix $M$ with first row $(c^2 , c, 1)$ may be expressed as where the second factor is itself a sum of circulants. This makes explicit the contribution from dimension $2$ that we see in the Fourier decompositions for the mixing matrices. Vafa's (neutrino) Yukawa matrix is just of this form for $a=0$ and $c$ a cubed root of unity.

## 5 Comments:

Hi Marni

I hear via MyFace that you have a visa extension. Yay... clapping...whistling...wooohoooo......

Kerie

In Lubos's blog I learned that F theorists are finally realizing that complex projective spaces might be physically and mathematically interesting. Maybe they will some day discover CP2 after realizing that the standard explanations for family replication might be wrong, that color might not be what is believed to be, that space-time might be 4-D after all, and so on.

The question is whether they willing to learn these lessons if someone has already done it? Isn't it disappointing to discover something wonderful and find that some idiot did it for decades ago?

What has led to astray is the lack of any kind of physical principles. The basic flaw was the assumption that GUTs are appropriate QFT limit of any unification. This is certainly the dullest possible working hypothesis involving absolutely minimum amount of imagination. As a consequence F-theorists are filling hep-th with incredibly baroque constructions in order just to reproduce SU(5) GUT! The saying about mountains giving birth to mice fits here very well.

That Einstein with purely conceptual thinking was able to develop a theory whose supersymmetric version might be UV finite shows how incredibly important it is to base the theoretical activities to some physical principles.

Yes, I know that it is useless to preach for an audience that is sleeping. The intellectual activity in every field of life comes in bursts followed by long periods of stagnation. Consciousness theorist inside me says that this is kind of fractal counterpart of sleep-awake cycle so that one must just tolerate the snoring.

Isn't it disappointing to discover something wonderful and find that some idiot did it for decades ago?LOL, Matti. But I am sure that in the time honoured tradition of string theory, they will simply completely ignore the 20 or 30 years of work that went before them and tell everyone that it was their idea.Hi Kerie! Yes, hard to believe that after 4 or 5 months that I finally have a real visa!

Maybe they will some day discover CP2 after realizing that the standard explanations for family replication might be wrong...CP^2 also arises in supergravity, as the projective charge space of a class of N=2 extremal black holes. The symmetries of the black hole are just the symmetries of CP^2.However, for some reason in the literature, CP^2 is not mentioned explicitly. Instead, only the black hole orbits are given explicitly, in terms of quotient groups. Perhaps Vafa's work will be trend changing and people will become more comfortable discussing the projective space first and its symmetries later.

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