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 P in complexified M8 - M8c - or as minimal surfaces in H=M4× CP2 having 2-D singularities.
Remark:Oc,Hc,Cc,Rc 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.
Space-time as algebraic surface in M8c regarded complexified octonions
The octonionic polynomial giving rise to space-time 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. Space-time surface X4c is identified as a 4-D root for a Hc-valued "imaginary" or "real" part of Oc valued polynomial obtained as an Oc 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 M8c to M8. One could drop the subscripts "c" but in the sequel they will be kept.
M4c appears as a special solution for any polynomial P. M4c seems to be like a universal reference solution with which to compare other solutions.
One obtains also brane-like 6-surfaces as 6-spheres as universal solutions. They have M4 projection, which is a piece of hyper-surface for which Minkowski time as time coordinate of CD corresponds to a root t=rn 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.
One obtains also 6-D brane-like solutions to the equations.
Explicit realization of M8-H duality
- In general the zero loci for imaginary or real part are 4-D but the 7-D light-cone δ M8+ of M8 with tip at the origin of coordinates is an exception. At δ M8+ 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+er)/\sqrt2 such that one as e2=e.
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 tM=t= rN, rM= \sqrtrN2-rE2≤ rN, rE≤ rN, where the value of Minkowski time t=r=rN is a root of P(r) and rM denotes radial Minkowski coordinate. The points with distance rM from origin of t=rN ball of M4 has as fiber 3-sphere with radius r =\sqrtrN2-rE2. At the boundary of S3 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 X2. The boundaries rM=rN of balls belong to the boundary of M4 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 tM=rN would be very special. At these 6-spheres the 4-D space-time surfaces X4 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 rn.
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 tn 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 M8-H duality possible as also classical twistor lift.
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 tM 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, εi,..,ε8), epsiloni=+/- 1 signatures. Their physical interpretation - if any - remains open at this moment.
- The universal 6-D brane-like solutions S6c have also lower-D counterparts. The condition determining X2 states that the Cc-valued "real" or "imaginary" for the non-vanishing Qc-valued "real" or "imaginary" for P vanishes. This condition allows universal brane-like solution as a restriction of Oc to M4c (that is CDc) and corresponds to the complexified time=constant hyperplanes defined by the roots t=rn of P defining "special moments in the life of self" assignable to CD. The condition for reality in Rc sense in turn gives roots of t=rn a hyper-surfaces in M2c.
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. M8H 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.
- Associativity condition for tangent-/normal space is the first essential condition for the existence of M8-H duality and means that tangent - or normal space is quaternionic.
- Also second condition must be satisfied. The tangent space of space-time surface and thus space-time surface itself must contain a preferred M2c⊂ M4c or more generally, an integrable distribution of tangent spaces M2c(x) and similar distribution of their complements E2c(x). The string world sheet like entity defined by this distribution is 2-D surface X2c⊂ X4c in Rc sense. E2c(x) would correspond to partonic 2-surface.
One can imagine two realizations for this condition.
Option I: Global option states that the distributions M2c(x) and E2c(x) define slicing of X4c.
Option II: Only discrete set of 2-surfaces satisfying the conditions exist, they are mapped to H, and strong form of holography (SH) applied in H allows to deduce space-time surfaces in H. This would be the minimal option.
How these conditions would be realized?
Does M8-H duality relate hadron physics at high and low energies?
- The basic observation is that X2c can be fixed by posing to the non-vanishing Hc-valued part of octonionic polynomial P condition that the Cc valued "real" or "imaginary" part in Cc sense for P vanishes. M2c would be the simplest solution but also more general complex sub-manifolds X2c⊂ M4c are possible. This condition allows only a discrete set of 2-surfaces 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 1-D surface in X2c. By assuming that Rc-valued "real" or "imaginary" part of quaternionic part of P at this 2-surface vanishes. one obtains preferred M1c or E1c 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 Rc sense would define local quantization axis of energy and spin. The outcome would be a realization of the hierarchy Rc→ Cc→ Hc→ Oc realized as surfaces.
This option could be made possible by SH. SH states that preferred extremals are determined by data at 2-D surfaces of X4. Even if the conditions defining X2c 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 2-surfaces under M8-H duality. Associativity and existence of M2(x) would be required only at the 2-D surfaces.
- I have proposed that physical string world sheets and partonic 2-surfaces appear as singularities and correspond to 2-D folds of space-time 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 2-pages. Also 1-D 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 CP2 point to the tangent space. This does not of course preclude the singular surfaces and they could be analogous to poles of analytic function. Light-like orbits of partonic 2-surfaces would in turn correspond to cuts.
During the writing of this article I realized that M8-H duality has very nice interpretation in terms of symmetries. For H=M4× CP2 the isometries correspond to Poincare symmetries and color SU(3) plus electroweak symmetries as holonomies of CP2. For octonionic M8 the subgroup SU(3) ⊂ G2 is the sub-group of octonionic automorphisms leaving fixed octonionic imaginary unit invariant - this is essential for M8-H duality. SU(3) is also subgroup of SO(6)== SU(4) acting as rotation on M8= M2× E6. The subgroup of the holonomy group of SO(4) for E4 factor of M8= M4× E4 is SU(2)× U(1) and corresponds to electroweak symmetries. One can say that at the level of M8 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 E4 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, M8-H Duality, SUSY, and Twistors or the article About p-adic length scale hypothesis and dark matter hierarchy.