Considerable progress in the understanding of M8-H duality

The idea of M⁸-H duality has progressed through frustratingly many several twists and turns. Consider first the development of the key ideas.

1. The first key idea was that one can interpret octonions O as M⁸ by using the number theoretic inner product defined by the real part of the octonion product. Later I gave up this assumption and considered complexified octonions, which do not form a number field, but finally found that the original option is the only sensible option.
2. The second key idea was that if either the tangent or normal space of the surface Y⁴ ⊂ M⁸ is quaternionic and therefore associative and if it also contains a commutative subspace, it can be parameterized by a point of CP₂ and mapped to H=M⁴× CP₂. This would be the first half or M⁸-H duality. How to map the M⁴ ⊂ M⁸ projection to M⁴× CP₂? This question did not have an obvious answer. The simplest map is direct identification whereas inversion is strongly suggested by Uncertainty Principle (UP) and the interpretation of M⁸ coordinates as components of 8-momentum. Note that one can considerably generalize the simplest view by replacing the fixed commutative subspace of quaternion space M⁴ with an integrable distribution of them in M⁸.
3. I considered first the option in which tangent space was assumed to be associative. The cold shower was that this option allows only trivial solutions (see this and this). Quaternionic normal space however works: any integrable distribution of quaternionic normal spaces defines an associative surface Y⁴.
4. If M⁸ is not complexified the surfaces Y⁴ in M⁸ are necessarily Euclidean. This is in sharp conflict with the original intuitive idea that they have a number theoretic Minkowski signature. It is the normal space, which must have a Minkowskian signature and this forces us to rethink how the M⁸-H duality is realized in Minkowskian degrees of freedom.
5. I have considered also the minimal option in which M⁸-H duality determines only the 3-D holographic data as 3-surfaces Y³⊂ M⁸ mapped by M⁸-H duality to H. The images of Y³ could define holographic data consistent with the holography = holomorphy (H-H) vision. Both M⁸ and H sides of the duality would be necessary.

   The physical interpretation for the space of 4-surfaces Y⁴ in M⁸ is as the analog of momentum space for particles identified as 3-D surfaces. In this interpretation the Y⁴ would be analog of time evolution with time replaced with energy. This is in conflict with physical intuition and suggests that maybe the minimal option is correct and indeed consistent with the fact that for point-like particles the momenta are at 3-D mass shells.

The motivation for reconsidering the M⁸-H duality came from the fact that the H-H hypothesis works extremely nicely for the space-time surfacex X⁴⊂ H. The roots of two generalized analytic functions f₁,f₂ of hypercomplex coordinate and 3 complex coordinates of H give as their roots space-time surfaces as minimal surfaces and the ansatz works for any action which is general coordinate invariant and expressible in terms of the induced geometry.

There is however the problem with the 3-D holographic data. How to fix them in such a way that they are consistent with functions f₁ and f₂ as analytic functions of H coordinates involving hypercomplex coordinate and 3 complex coordinates?

This led back to the original idea that the associative 4-surfaces Y⁴⊂ M⁸ might be definable in terms of real analytic functions f(o) of octonions as an octonionic generalization of the notion of holomorphy. The 3 alternative conditions f(o)=0 , f(o)=1 and the reality of f(o) are promising since they are invariant under octonionic automorphism group G₂. The argument goes as follows.

1. Since G₂ acts as automorphisms one has f(g₂(o))= g₂(f(o)) where g₂(0) is any local G₂ automorphism. If f(o)=0/1 is true then also f(g₂(o)) is true for any g₂⊂ G₂. One would have a huge dynamical spectrum generating symmetry analogous to the holomorphic symmetries of H-H vision. It would map the quaternionic normal spaces to quaternionic normal spaces and complex subspaces to complex subpaces.
2. Consider now the condition f(o)=0/1. The Taylor (or even Lauren -) expansion in powers of o gives only two terms. The first term is proportional to the octonionic real unit 1 of o and the second term to the octonionic imaginary part of Im(o)= o₇ of o.

For o² one obtains o²= o₀²-o₇• o₇ + 2o₀ o₇. The coefficients of these parts depend on the real part o₀ of o and the length r₇ of the imaginary Im(o). The higher powers of o involve products of two octonions of form o₁= α₁+ β₁o₇ and o₂= α₂+ β₂0₇ and the product is of form o₁o₂= (α₁α₂ -β₁β₂) +(α₁β₂+ α₃β₁)o₇. By induction one finds that the coefficients for any power depend only on o₀ and the radius r₇ of 6-sphere only. In particular, the function f(o) is expressible has the general form

f(o) =f₁(o₀,r₇)+f₂(o₀,r₇) .

The detailed forms of these functions have been discussed in the earlier articles (see this, this and this but are not relevant for what follows. One can consider 3 G₂ invariant options.

1. The condition f(o)=0 or f(o)=1 gives the roots of f₁ and f₂ as o₀= h₁(r₇) and o₀= h₂(r₇). Together these conditions give a discrete set of roots (o₀,r₇)_n.

These roots define a set of 6-spheres S⁶ with r₇= constant. Can one assign an associative 4-surface Y⁴⊂ M⁸ to a given S⁶? The condition that the normal space is quaternionic is satisfied if one fixes complement E⁴ of quaternionic sub-space M⁴ and restricts the points of S⁶ to the intersection of E⁴ and S⁶, which is 2-sphere S². The normal space M⁴ of E⁴ would define the quaternionic subspace M⁴, and it should be the same for all points of Y⁴. The problem is how to assign Y⁴ to S² taking the role of holographic data.

One can also consider a less general G₂ invariant option for which only f₂=0 is satisfied so that f(o) is real in the octonionic sense. Also this set of solutions is invariant under G₂ since automorphisms map real octonions to real octonions. Now the solutions are 7-D surfaces m⁰= h₂(r₇). The roots define a set of orbits of 6-spheres S⁶ such that one has r₀= h₂^-1(m⁰).

Can one associate an associative 4-surface Y⁴⊂ M⁸ to a given orbit of S⁶? The condition that the normal space is quaternionic is satisfied if one fixes the complement E⁴ of a fixed quaternionic sub-space M⁴ ⊂ M⁸ and restricts the points of given S⁶(m⁰) to the intersection of E⁴ and S⁶(m⁰), which is a 2-sphere S²(m₀). The normal space M⁴ of E⁴ should define the quaternionic subspace, and it would be the same for all points of Y⁴. The union Y³ of the 2-surfaces S²(m₀) is a 3-surface and strongly brings in mind the orbit of a partonic 2-surface essential for holography in H.

Could Y³ define holographic data for Y⁴ ⊂ M⁸ with constant quaternionic normal space M⁴? Can one increase the dimension of M⁴ to M⁵ so that one would have 3-D intersection of E³ and S⁶(m⁰) and would obtain Y⁴. The problem is that the normal space is not quaternionic without additional conditions. This looks artificial. It seems that something is not understood.

1. The radical option would be to give up the condition that one has 4-D surfaces in M⁸. It is enough that the M⁸-H duality maps to H only the 3-D surfaces Y³ as pre-images of partonic orbits X³ serving as holographic data in H identifiable as partonic orbits. One can hope that M⁸-H duality is needed to fix the holographic data for H-H in an internally consistent manner. This picture would also conform with the fact that momentum space for point-like particles consists of 3-D mass shells.
2. What about M⁸-H duality for M⁴ coordinates? Could the M⁴⊂ H point correspond to the projection of the E⁴× S⁶ point to M⁴⊂ M⁸ as such or is an inversion suggested by Uncertainty Principle involved?
3. If the minimal proposal works, one can construct more general solutions for Y³ by applying local G₂ automorphisms to the basic solutions. The CP₂ points for the image of Y³ in H would not be constant anymore. The case in which the sphere S²⊂ M⁸ is mapped to a homologically non-trivial sphere of CP₂ is of special physical interest.
4. What can one say about the elements g₂(o) of the local G₂? The action of G₂ on octonions allows a matrix representation but the matrix elements are octonions so that the rules of multiplication are not standard and non-associative. Associativity is obtained if one considers only elements of G₂ belonging to a local SU(3) subgroup having physical interpretations as a color group.
5. Holomorphy= holography vision inspires the question whether g₂(o) is an analytic function of the generalized complex coordinates of M⁸. Could this imply that the holographic data mapped to H by M⁸-H duality are consistent with the holomorphy in H?

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

Source: https://matpitka.blogspot.com/2025/10/considerable-progress-in-understanding.html