Koide from S-duality

1) Crackpot idea of the day: "the bottom quark is S-dual to the rho meson".

Gorsky et al conjecture that holographic QCD has a "flavor S-duality" in which vector mesons are dual to baryons. This is to be realized in string theory by a web of 5-branes.

Quark-hadron duality shows a kind of continuity between properties of quarks and properties of hadrons.

And the mass of the rho meson has been estimated at sqrt(6) times the constituent quark mass; while in the simplest version of Rivero's waterfall, the bottom quark mass comes out as 2 . sqrt(6) . sqrt(6) times the constituent quark mass. Also, the Brannen mass scale of the Koide triple containing the bottom quark equals the mass of the proton, the prototypical baryon.

2) Vague bonus idea: Koide relations are an echo of this S-duality.

This just comes from Brannen and Sheppeard's discussion of the discrete Fourier transform, in the context of circulant matrices (they obtain Koide masses as eigenvalues of a circulant). One might start with the realization of geometric Langlands via S-duality, and then look for analogues over finite fields. Sheppeard has occasionally hinted at something like this.