^{8}-H duality (H=M

^{4}× CP

_{2}) has taken a central role in TGD framework. M

^{8}-H duality allows to identify space-time regions as "roots" of octonionic polynomials in complexified M

^{8}- M

^{8}

_{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 M

^{4}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.

M^{8}-H duality allows to map space-time surfaces in M^{8} to H so that one has two equivalent descriptions for the space-time surfaces as algebraic surfaces in M^{8} 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. M^{8}_{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.

M^{8}-H duality makes sense under 2 additional assumptions to be considered in the following more explicitly than in earlier discussions.

- Space-time surface X
^{4}_{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)= x^{n}+.. ordinary roots are algebraic integers. The real 4-D space-time surface is projection of this surface from M^{8}_{c}to M^{8}. 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 M

^{2}_{c}⊂ M^{4}_{c}or more generally, an integrable distribution of tangent spaces M^{2}_{c}(x). The string world sheet like entity defined by this distribution is surface X^{2}_{c}⊂ X^{4}_{c}in R_{c}sense.X

^{2}_{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. M^{2}_{c}would be the simplest solution but also more general complex sub-manifolds X^{2}_{c}⊂ M^{4}_{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 M^{1}_{c}or E^{1}_{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 R_{\}rightarrow C_{c}→ H_{c}→ O_{c}realized as surfaces.**Remark:**Also M^{4}_{c}appears as a special solution for any polynomial P. M^{4}_{c}seems to be like a universal reference solution with which to compare other solutions. M^{4}_{c}would intersect all other solutions along string world sheets X^{2}_{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. - Associativity condition for tangent-/normal space is second essential condition and means that tangent - or normal space is quaternionic. The conjecture is that the identification in terms of roots of polynomials guarantees this and one can formulate this as rather convincing argument.

- In general the zero loci for imaginary or real part are 4-D but the 7-D light-cone δM
^{8}_{+}of M^{8}with tip at the origin of coordinates is an exception . At δM^{8}_{+}the octonionic coordinate o is light-like and one can write o= re, where 8-D time coordinate and radial coordinate are related by t=r and one has e=(1+e_{r})/2^{1/2}such that one as e^{2}=e.Polynomial P(o) can be written at δ M

^{8}_{+}as P(o)=P(r)e and its roots correspond to 6-spheres S^{6}represented as surfaces t_{M}=t= r_{N}, r_{M}= (_{N}^{2}-r_{E}^{2})^{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 M^{4}has as fiber 3-sphere with radius r =(_{N}^{2}-r_{E}^{2})^{1/2}. At the boundary of S^{3}contracts to a point. - These 6-spheres are analogous to 6-D branes in that the 4-D solutions would intersect them in the generic case along 2-D surfaces X
^{2}. The boundaries r_{M}=r_{N}of balls belong to the boundary of M^{4}light-cone. In this case the intersection would be that of 4-D and 3-D surface, and empty in the generic case (it is however quite not clear whether topological notion of "genericity" applies to octonionic polynomials with very special symmetry properties). - The 6-spheres t
_{M}=r_{N}would be very special. At these 6-spheres the 4-D space-time surfaces X^{4}as usual roots of P(o) could meet. Brane picture suggests that the 4-D solutions connect the 6-D branes with different values of r_{n}.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 X

^{2}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 X

^{4}meet along 3-D surfaces at S^{6}. 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. - One could perhaps interpret the free selection of 2-D partonic surfaces at the 6-D roots as initial data fixing the 4-D roots of polynomials. This would give precise content to strong form of holography (SH), which is one of the central ideas of TGD and strengthens the 3-D holography coded by ZEO alone in the sense that pairs of 3-surfaces at boundaries of CD define unique preferred extremals. The reduction to 2-D holography would be due to preferred extremal property realizing the huge symplectic symmetries and making M
^{8}-H duality possible as also classical twistor lift.I have also considered the possibility that 2-D string world sheets in M

^{8}could correspond to intersections X^{4}∩ S^{6}? This is not possible since time coordinate t_{M}constant at the roots and varies at string world sheets.Note that the compexification of M

^{8}(or equivalently octonionic E^{8}) allows to consider also different variants for the signature of the 6-D roots and hyperbolic spaces would appear for (ε_{1}, ε_{i},..,ε_{8}), epsilon_{i}=+/- 1 signatures. Their physical interpretation - if any - remains open at this moment. - The universal 6-D brane-like solutions S
^{6}_{c}have also lower-D counterparts. The condition determining X^{2}states that the C_{c}-valued "real" or "imaginary" for the non-vanishing Q_{c}-valued "real" or "imaginary" for P vanishes. This condition allows universal brane-like solution as a restriction of O_{c}to M^{4}_{c}(that is CD_{c}) and corresponds to the complexified time=constant hyperplanes defined by the roots t=r_{n}of P defining "special moments in the life of self" assignable to CD. The condition for reality in R_{c}sense in turn gives roots of t=r_{n}a hyper-surfaces in M^{2}_{c}.

^{8}-H duality has very nice interpretation in terms of symmetries. For H=M

^{4}× CP

_{2}the isometries correspond to Poincare symmetries and color SU(3) plus electroweak symmetries as holonomies of CP

_{2}. For octonionic M

^{8}the subgroup SU(3) ⊂ G

_{2}is the sub-group of octonionic automorphisms leaving fixed octonionic imaginary unit invariant - this is essential for M

^{8}-H duality. SU(3) is also subgroup of SO(6)== SU(4) acting as rotation on M

^{8}= M

^{2}× E

^{6}. The subgroup of the holonomy group of SO(4) for E

^{4}factor of M

^{8}= M

^{4}× E

^{4}is SU(2)× U(1) and corresponds to electroweak symmetries. One can say that at the level of M

^{8}one has symmetry breaking from SO(6) to SU(3) and from SO(4)= SU(2)× SO(3) to U(2).

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 E^{4} 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, M^{8}-H Duality, SUSY, and Twistors.

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