Schwarzschild deformed supergravity background: possible geometry origin of fermion generations and mass hierarchy

Abstract

TThe problem of fermion masses hierarchy in the Standard Model is considered on a toy model of a 10-dimensional space-time with a IIA supergravity type background. Dirac equation on this background, after compactification of extra 4- and 1-dimensional subspaces, gives the spectrum of Fermi fields which profiles in 5 dimensions and corresponding Higgs generated masses in 4 dimensions depend on the eigenvalues of Dirac operator on the named compact subspaces. Schwarzschild Euclidean deformation of the supergravity throat with the "apple-shaped" conical singularity permits to leave only three non-divergent angular modes interpreted as three generations of the down-type quarks. Calculated ratio $m_{d} / m_{s} = e^{-3}$ exactly coincides with its experimentally observed value for integer values of two free parameters of the 10-dimensional background. Equations for non-chiral modes coincide with the non-relativistic Schr\"odinger equation for an electron moving in a Coulomb field; the corresponding small fermion masses generated by the twisted boundary conditions are expressed through the degenerate hypergeometric functions.