M^{8}H duality (H=M^{4}× CP_{2}) has taken a central role in TGD framework. M^{8}H duality allows to identify spacetime regions as "roots" of octonionic polynomials P in complexified M^{8}  M^{8}_{c}  or as minimal surfaces in H=M^{4}× CP_{2} having 2D singularities.
Remark:O_{c},H_{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.
Spacetime as algebraic surface in M^{8}_{c} regarded complexified octonions
The octonionic polynomial giving rise to spacetime surface as its "root" is obtained from ordinary real polynomial P with rational coefficients by algebraic continuation. The conjecture is that the identification in terms of roots of polynomials of even real analytic functions guarantees associativity and one can formulate this as rather convincing argument. Spacetime surface X^{4}_{c} is identified as a 4D 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 complexifiedquaternionic tangent or normal spaces. For P(x)= x^{n}+.. ordinary roots are algebraic integers. The real 4D spacetime 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.
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.
One obtains also branelike 6surfaces as 6spheres as universal solutions. They have M^{4} projection, which is a piece of hypersurface 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.
One cannot exclude rational functions or even real analytic functions in the sense that Taylor coefficients are octonionically real (proportional to octonionic real unit). Number theoretical vision  adelic physics suggests that polynomial coefficients are rational or perhaps in extensions of rationals. The real coefficients could in principle be replaced with complex numbers a+ib, where i commutes with the octonionic units and defines complexifiation of octonions. i appears also in the roots defining complex extensions of rationals.
Branelike solutions
One obtains also 6D branelike solutions to the equations.
 In general the zero loci for imaginary or real part are 4D but the 7D lightcone δ M^{8}_{+} of M^{8} with tip at the origin of coordinates is an exception. At δ M^{8}_{+} the octonionic coordinate o is lightlike and one can write o= re, where 8D time coordinate and radial coordinate are related by t=r and one has e=(1+e_{r})/\sqrt2 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 6spheres S^{6} represented as surfaces t_{M}=t= r_{N}, r_{M}= \sqrtr_{N}^{2}r_{E}^{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 3sphere with radius r =\sqrtr_{N}^{2}r_{E}^{2}. At the boundary of S^{3} contracts to a point.
 These 6spheres are analogous to 6D branes in that the 4D solutions would intersect them in the generic case along 2D surfaces X^{2}. The boundaries r_{M}=r_{N} of balls belong to the boundary of M^{4} lightcone. In this case the intersection would be that of 4D and 3D 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 6spheres t_{M}=r_{N} would be very special. At these 6spheres the 4D spacetime surfaces X^{4} as usual roots of P(o) could meet. Brane picture suggests that the 4D solutions connect the 6D branes with different values of r_{n}.
The basic assumption has been that particle vertices are 2D partonic 2surfaces and lightlike 3D surfaces  partonic orbits identified as boundaries between Minkowskian and Euclidian regions of spacetime surface in the induced metric (at least at H level)  meet along their 2D ends X^{2} at these partonic 2surfaces. 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 spacetime surfaces X^{4} meet along 3D 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 2D partonic surfaces at the 6D roots as initial data fixing the 4D 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 3D holography coded by ZEO alone in the sense that pairs of 3surfaces at boundaries of CD define unique preferred extremals. The reduction to 2D 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 2D 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 6D 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 6D branelike solutions S^{6}_{c} have also lowerD counterparts. The condition determining X^{2} states that the C_{c}valued "real" or "imaginary" for the nonvanishing Q_{c}valued "real" or "imaginary" for P vanishes. This condition allows universal branelike 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 hypersurfaces in M^{2}_{c}.
Explicit realization of M^{8}H duality
M^{8}H duality allows to map spacetime surfaces in M^{8} to H so that one has two equivalent descriptions for the spacetime surfaces as algebraic surfaces in M^{8} and as minimal surfaces with 2D singularities in H satisfying an infinite number of additional conditions stating vanishing of Noether charges for supersymplectic algebra actings as isometries for the "world of classical worlds" (WCW). Twistor lift allows variants of this duality. M^{8}_{H} duality predicts that spacetime 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.
 Associativity condition for tangent/normal space is the first essential condition for the existence of M^{8}H duality and means that tangent  or normal space is quaternionic.
 Also second condition must be satisfied. The tangent space of spacetime surface and thus spacetime surface itself must contain a preferred M^{2}_{c}⊂ M^{4}_{c} or more generally, an integrable distribution of tangent spaces M^{2}_{c}(x) and similar distribution of their complements E^{2}c(x). The string world sheet like entity defined by this distribution is 2D surface X^{2}_{c}⊂ X^{4}_{c} in R_{c} sense. E^{2}_{c}(x) would correspond to partonic 2surface.
One can imagine two realizations for this condition.
Option I: Global option states that the distributions M^{2}_{c}(x) and E^{2}_{c}(x) define slicing of X^{4}_{c}.
Option II: Only discrete set of 2surfaces satisfying the conditions exist, they are mapped to H, and strong form of holography (SH) applied in H allows to deduce spacetime surfaces in H. This would be the minimal option.
How these conditions would be realized?
 The basic observation is that X^{2}c can be fixed by posing to the nonvanishing H_{c}valued part of octonionic polynomial P condition that the C_{c} valued "real" or "imaginary" part in C_{c} sense for P vanishes. M^{2}_{c} would be the simplest solution but also more general complex submanifolds X^{2}_{c}⊂ M^{4}_{c} are possible. This condition allows only a discrete set of 2surfaces as its solutions so that it works only for Option II.
These surfaces would be like the families of curves in complex plane defined by u=0 an v= 0 curves of analytic function f(z)= u+iv. One should have family of polynomials differing by a constant term, which should be real so that v=0 surfaces would form a discrete set.
 One can generalize this condition so that it selects 1D surface in X^{2}_{c}. By assuming that R_{c}valued "real" or "imaginary" part of quaternionic part of P at this 2surface 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 1D surfaces in R_{c} sense would define local quantization axis of energy and spin. The outcome would be a realization of the hierarchy R_{c}→ C_{c}→ H_{c}→ O_{c} realized as surfaces.
This option could be made possible by SH. SH states that preferred extremals are determined by data at 2D surfaces of X^{4}. Even if the conditions defining X^{2}_{c} have only a discrete set of solutions, SH at the level of H could allow to deduce the preferred extremals from the data provided by the images of these 2surfaces under M^{8}H duality. Associativity and existence of M^{2}(x) would be required only at the 2D surfaces.
 I have proposed that physical string world sheets and partonic 2surfaces appear as singularities and correspond to 2D folds of spacetime surfaces at which the dimension of the quaternionic tangent space degenerates from 4 to 2. This interpretation is consistent with a book like structure with 2pages. Also 1D real and imaginary manifolds could be interpreted as folds or equivalently books with 2 pages.
For the singular surfaces the dimension quaternionic tangent or normal space would reduce from 4 to 2 and it is not possible to assign CP_{2} point to the tangent space. This does not of course preclude the singular surfaces and they could be analogous to poles of analytic function. Lightlike orbits of partonic 2surfaces would in turn correspond to cuts.
Does M^{8}H duality relate hadron physics at high and low energies?
During the writing of this article I realized that M^{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 subgroup 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 oldfashioned 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 chapter TGD view about McKay Correspondence, ADE Hierarchy, Inclusions of Hyperfinite Factors, M^{8}H Duality, SUSY, and Twistors or the article About padic length scale hypothesis and dark matter hierarchy.
