New resuls on M8-H duality

M⁸-H duality (H=M⁴× CP₂) has taken a central role in TGD framework. M⁸-H duality allows to identify space-time regions as “roots” of octonionic polynomials in complexified M⁸ – M⁸_c. The polynomial is obtained from ordinary real polynomial P with rational coefficients by algebraic continuation. One obtains brane-like 6-surfaces as 6-spheres as universal solutions. They have M⁴ projection which is piece of hyper-surface for which Minkowski time as time coordinate of CD corresponds to a root t=r_n of P. For monic polynomials these time values are algebraic integers and Galois group permutes them.

Remark:O_c,O_c,C_c,R_c will be used in the sequel for complexifications of octonions, quaternions, etc.. number fields using commuting imaginary unit i appearing naturally via the roots of real polynomials.

M⁸-H duality allows to map space-time surfaces in M⁸ to H so that one has two equivalent descriptions for the space-time surfaces as algebraic surfaces in M⁸ and as minimal surfaces with 2-D singularities in H satisfying an infinite number of additional conditions stating vanishing of Noether charges for super-symplectic algebra actings as isometries for the “world of classical worlds” (WCW). Twistor lift allows variants of this duality. M⁸_H duality predicts that space-time surfaces form a hierarchy induced by the hierarchy of extensions of rationals defining an evolutionary hierarchy. This forms the basis for the number theoretical vision about TGD.

M⁸-H duality makes sense under 2 additional assumptions to be considered in the following more explicitly than in earlier discussions.

1. Space-time surface X⁴_c is identified as a 4-D root for a H_c-valued “imaginary” or “real” part of O_c valued polynomial obtained as an O_c continuation of a real polynomial P with rational coefficients, which can be chosen to be integers. These options correspond to complexified-quaternionic tangent – or normal spaces. For P(x)= xⁿ+.. ordinary roots are algebraic integers. The real 4-D space-time surface is projection of this surface from M⁸_c to M⁸. One could drop the subscripts “_c” but in the sequel they will be kept.

The tangent space of space-time surface and thus space-time surface itself contains a preferred M²_c⊂ M⁴_c or more generally, an integrable distribution of tangent spaces M²_c(x). The string world sheet like entity defined by this distribution is surface X²_c⊂ X⁴_c in R_c sense.

X²_c can be fixed by posing to the non-vanishing Q_c-valued part of octonionic polynomial condition that the C_c valued “real” or “imaginary” part in C_c sense for this polynomial vanishes. M²_c would be the simplest solution but also more general complex sub-manifolds X²_c⊂ M⁴_c are possible. In general one would obtain book like structures as collections of several string world sheets having real axis as back.

By assuming that R_c-valued “real” or “imaginary” part of the polynomial at this 2-surface vanishes. one obtains preferred M¹_c or E¹_c containing octonionic real and preferred imaginary unit or distribution of the imaginary unit having interpretation as complexified string. Together these kind 1-D surfaces in R_c sense would define local quantization axis of energy and spin. The outcome would be a realization of the hierarchy Rrightarrow C_c→ H_c→ O_c realized as surfaces.

Remark: Also M⁴_c appears as a special solution for any polynomial P. M⁴_c seems to be like a universal reference solution with which to compare other solutions. M⁴_c would intersect all other solutions along string world sheets X²_c. Also this would give rise to a book like structures with 2-D string world sheet representing the back of given book. The physical interpretation of these book like structures remains open in both cases.

Polynomial P(o) can be written at δ M⁸₊ as P(o)=P(r)e and its roots correspond to 6-spheres S⁶ represented as surfaces t_M=t= r_N, r_M= (_N²-r_E²)^1/2≤ r_N, r_E≤ r_N, where the value of Minkowski time t=r=r_N is a root of P(r) and r_M denotes radial Minkowski coordinate. The points with distance r_M from origin of t=r_N ball of M⁴ has as fiber 3-sphere with radius r =(_N²-r_E²)^1/2. At the boundary of S³ contracts to a point.

The basic assumption has been that particle vertices are 2-D partonic 2-surfaces and light-like 3-D surfaces – partonic orbits identified as boundaries between Minkowskian and Euclidian regions of space-time surface in the induced metric (at least at H level) – meet along their 2-D ends X² at these partonic 2-surfaces. This would generalize the vertices of ordinary Feynman diagrams. Obviously this would make the definition of the generalized vertices mathematically elegant and simple.

Note that this does not require that space-time surfaces X⁴ meet along 3-D surfaces at S⁶. The interpretation of the times t_n as moments of phase transition like phenomena is suggestive. ZEO based theory of consciousness suggests interpretation as moments for state function reductions analogous to weak measurements ad giving rise to the flow of experienced time.

I have also considered the possibility that 2-D string world sheets in M⁸ could correspond to intersections X⁴∩ S⁶? This is not possible since time coordinate t_M constant at the roots and varies at string world sheets.

Note that the compexification of M⁸ (or equivalently octonionic E⁸) allows to consider also different variants for the signature of the 6-D roots and hyperbolic spaces would appear for (ε₁, ε_i,..,ε₈), epsilon_i=+/- 1 signatures. Their physical interpretation – if any – remains open at this moment.

This interpretation gives a justification for the earlier proposal that the descriptions provided by the old-fashioned low energy hadron physics assuming SU(2)_L× SU(2)_R and acting acting as covering group for isometries SO(4) of E⁴ and by high energy hadron physics relying on color group SU(3) are dual to each other.

See the article About p-adic length scale hypothesis and dark matter hierarchy or the chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M⁸-H Duality, SUSY, and Twistors.

For a summary of earlier postings see Latest progress in TGD.

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