On M8-H-duality, p-adic Length Scale Hypothesis & Dark Matter Hierarchy

Abstract

M⁸-H duality, p-adic length scale hypothesis and dark matter hierarchy as phases of ordinary matter with effective Planck constant h_eff  = nh₀ are basic assumptions of TGD, which all reduce to number theoretic vision. In the sequel M⁸-H duality, p-adic length scale hypothesis and dark matter hierarchy are discussed from number theoretic perspective. Several new results emerge. Strong form of holography (SH) allows to weaken strong form of M⁸-H duality mapping space-time surfaces X⁴ ⊂ M⁸ to H=M⁴ X CP₂ that it allows to map only certain complex 2-D sub-manifolds of quaternionic space-time surface to H: SH allows to determine X⁴ ⊂ M⁸ from this 2-D data. Complex sub-manifolds are determined by conditions completely analogous to those determined space-time surface as quaternionic sub-manifold and only discrete set of them is obtained. M⁸ duality allows to relate p-adic length scales L_p to differences for the roots of the polynomial defining the extension defining "special moments in the life of self" assignable causal diamond (CD) central in zero energy ontology (ZEO). Hence p-adic length scale hypothesis emerges both from p-adic mass calculations and M⁸-H duality. It is proposed that the size scale of CD correspond to the largest dark scale nL_p for the extension and that the sub-extensions of extensions could define hierarchy of sub-CDs. Skyrmions are an important notion if nuclear and hadron physics. M⁸-H duality suggests an interpretation of skyrmion number as winding number as that for a map defined by complex polynomial.