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

#### Abstract

*M ^{8}*-

*H*duality predicts that space-time surfaces as algebraic surfaces in complexified

*M*(complexified octonions) determined by polynomials can be mapped to

^{8}*H*=

*M*x

^{4 }*CP*. The polynomials do

_{2}*not*involve periodic functions typically associated with the minimal space-time surfaces in

*H*. Since

*M*is analogous to momentum space, the periodicity is not needed. However, the representation of the space-time surfaces in

^{8}*H*obey dynamics and the

*H*-images of

*X*⊂

^{4}*M*should involve periodic functions and Fourier analysis for

^{8}*CP*coordinates as functions of

_{2}*M*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

^{4}*X*⊂

^{4}*M*to

^{8}*CP*point brings in dynamics and therefore requires the expansion of

_{2}*CP*coordinates as exponential and trigonometric functions of

_{2 }*M*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.

^{4}