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Attempts to resolve the long-standing Fermion mass hierarchy problem in frames of the AdS/CFT correspondence demand the knowledge of bulk fermion masses. The approach of the “old” conformal bootstrap in the anti-de Sitter (AdS) context permitted calculating bulk masses of scalar fields, as it was shown in three previous papers B. L. Altshuler [arXiv:1810.01105; Int. J. Mod. Phys. 2050001 (2020)10.1142/S0217751X20500013; J. High Energy Phys. 01 (2020) 13710.1007/JHEP01(2020)137]. In the present paper, this approach is extended to physically more interesting spin-1/2 bulk fields. Calculation of spinor-scalar vertices is performed in physical AdS space, and unexpectedly simple expressions for spinor-scalar bubbles (two-point one-loop self-energy Witten diagrams) are obtained. The “double-trace from UV to IR flow” subtraction of UV divergences used earlier in calculations of the UV-finite bulk tadpoles is applied to bubbles. This permitted us to write down in the frame of the old conformal bootstrap approach the nontrivial spectral equations for the bulk fermion masses; the SU(N) Yukawa model of spinor fields interacting with conformal invariant scalar field in case of four boundary dimensions is considered as a specific example.

The surprising simplicity of Barvinsky-Nesterov or equivalently Gelfand-Yaglom methods of calculation of quantum determinants permits to obtain compact expressions for UV-finite difference of one-loop quantum energies for two arbitrary values of parameter of the double-trace asymptotic boundary conditions. This result generalizes Gubser and Mitra calculation for particular case of difference of "regular" and "irregular" one-loop energies in one-brane RS-model. Approach developed in the paper also allows to get "in one line" the one-loop quantum energies in two-brane RS-model. The relationship between "one-loop" expressions corresponding to mixed Robin and to double-trace asymptotic boundary conditions is traced., Comment: 16 pages

Bi-tensor kernel in integral form of Einstein equations realizing Mach's idea of non-existence of empty space-times is taken as an inverse of differential operator ("Mach operator") defined conventionally as a second variation of Einstein's gravity Action over contravariant components of metric tensor. The choice of transverse gauge condition used in this definition does not influence results of the paper since only transverse and traceless tensor modes written on different background space-times are studied. Presence of ghosts among modes of Mach operator invalidates the integral formulation of Einstein equations. And the demand of absence of these ghosts proves to be a selection rule for dimensionality of the background space-time. In particular Mach operator written on De Sitter background or on the background of so called "Einstein Universe" does not possess tensor ghosts only in 4-dimensions. The similar demand gives non-trivial formula for dimensionalities of subspaces of the Freund-Rubin background.