Is M8 -H Duality Consistent with Fourier Analysis at the Level of M4 x CP2?

Abstract

M⁸-H duality predicts that space-time surfaces as algebraic surfaces in complexified M⁸ (complexified octonions) determined by polynomials can be mapped to H = M⁴ x CP₂. The polynomials do not involve periodic functions typically associated with the minimal space-time surfaces in H. Since M⁸ is analogous to momentum space, the periodicity is not needed. However, the representation of the space-time surfaces in H obey dynamics and the H-images of X⁴ ⊂ M⁸ should involve periodic functions and Fourier analysis for CP₂ coordinates as functions of M⁴ coordinates. Neper number, and therefore trigonometric and exponential functions are p-adically very special. As a consequence, Fourier analysis extended to allow exponential functions in the case of hyperbolic signature is a number theoretically universal concept making sense also for p-adic number fields. The proposal is that by its non-locality, the map of the tangent space of the space-time surface X⁴ ⊂ M⁸ to CP₂ point brings in dynamics and therefore requires the expansion of CP₂ coordinates as exponential and trigonometric functions of M⁴ coordinates that is Fourier analysis. The connection with hierarchy of effective Planck constants and the implications for the  quantum model of cognitive process are considered.