In the following I give a brief summary about what has been done. I concentrate on M8-H duality since the most significant results are achieved here.
It is fair to say that the new view answers the following a long list of open questions.
What mathematical challenges one must meet?
- When M8-H correspondence is true (to be honest, this question emerged during this work!)? What are the explicit formulas expressing associativity of the tangent space or normal space of the 4-surface?
The key element is the formulation in terms of complexified M8 identified in terms of octonions and restriction to M8. One loses the number field property but for polynomials ring property is enough. The level surfaces for real and imaginary parts of octonionic polynomials with real coefficients define 4-D surfaces in the generic case.
Associativity condition is an additional condition reducing the dimension of the space-time surface unless some components of RE(P) or IM(P) are critical meaning that also their gradients vanish. This conforms with the quantum criticality of TGD and provides a concrete first principle realization for it.
- How this picture corresponds to twistor lift? The twistor lift of Kähler action (dimensionally reduced Kähler action in twistor space of space-time surface) one obtains two kinds of space-time regions. The regions, which are minimal surfaces and obey dynamics having no dependence on coupling constants, correspond naturally to the critical regions in M8 and H.
There are also regions in which one does not have extremal property for both Kähler action and volume term and the dynamics depends on coupling constant at the level of H. These regions are associative only at their 3-D ends at boundaries of CD and at partonic orbits, and the associativity conditions at these 3-surfaces force the initial values to satisfy the conditions guaranteeing preferred extremal property. The non-associative space-time regions are assigned with the interiors of CDs. . The particle orbit like space-time surfaces entering to CD are critical and correspond to external particles.
- The surprise was that M4⊂ M8 is naturally co-associative. If associativity holds true also at the level of H, M4 ⊂ H must be associative. This is possible if M8-H duality maps tangent space in M8 to normal space in H and vice versa.
- The connection to the realization of the preferred extremal property in terms of gauge conditions of subalgebra of SSA is highly suggestive. Octonionic polynomials critical at the boundaries of space-time surfaces would determine by M8-H correspondence the solution to the gauge conditions and thus initial values and by holography the space-time surfaces in H.
- A beautiful connection between algebraic geometry and particle physics emerges. Free many-particle states as disjoint critical 4-surfaces can be described by products of corresponding polynomials satisfying criticality conditions.
These particles enter into CD , and the non-associative and non-critical portions of the space-time surface inside CD describe the interactions. One can define the notion of interaction polynomial as a term added to the product
of polynomials. It can vanish at the boundary of CD and forces the 4-surface to be connected inside CD. It also spoils associativity: interactions are switched on. For bound states the coefficients of interaction polynomial are such that one obtains a bound state as associative space-time surface.
- This picture generalizes to the level of quaternions. One can speak about 2-surfaces of space-time surface
with commutative or co-commutative tangent space. Also these 2-surfaces would be critical. In the generic case commutativity/co-commutativity allows only 1-D curves.
At partonic orbits defining boundaries between Minkowskian and Euclidian space-time regions inside CD the string world sheets degenerate to the 1-D orbits of point like particles at their boundaries. This conforms with the twistorial description of scattering amplitudes in terms of point like fermions.
For critical space-time surfaces representing incoming states string world sheets are possible as commutative/co-commutative surfaces (as also partonic 2-surfaces) and serve as correlates for (long range) entaglement) assignable also to macroscopically quantum coherent system (heff/h=n hierarchy implied by adelic physics).
- The octonionic polynomials with real coefficients form a commutative and associative algebra allowing besides algebraic operations function composition. Space-time surfaces therefore form an algebra and WCW has algebra structure. This could be true for the entire hierarchy of Cayley-Dickson algebras, and one would have a highly non-trivial generalization of the conformal invariance and Cauchy-Riemann conditions to their n-linear counterparts at the n:th level of hierarchy with n=1,2,3,.. for complex numbers, quaternions, octonions,... One can even wonder whether TGD generalizes to this entire hierarchy!
What questions should be answered?
- One should prove more rigorously that criticality is possible without the reduction of dimension of the space-time surface.
- One must demonstrate that SSA conditions can be true for the images of the associative regions (with 3-D or 4-D). This would obviously pose strong conditions on the values of coupling constants at the level of H.
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the articles Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part I and Does M8-H duality reduce classical TGD to octonionic algebraic geometry?: part II.
- Does associativity hold true in H for minimal surface extremals obeying universal critical dynamics? As found, the study of the known extremals supports this view.
- Could one construct the scattering amplitudes at the level of M8? Here the possible problems are caused by the exponents of action (Kähler action and volume term) at H side. Twistorial construction however leads to a proposal that the exponents actually cancel. This happens if the scattering amplitude can be thought as an analog of Gaussian path integral around single extremum of action and conforms with the integrability of the theory. In fact, nothing prevents from defining zero energy states in this manner! If this holds true then it might be possible to construct scattering amplitudes at the level of M8.
- What about coupling constants? Coupling constants make themselves visible at H side both via the vanishing conditions for Noether charges in sub-algebra of SSA and via the values of the non-vanishing Noether charges. M8-H correspondence determining the 3-D boundaries of interaction regions within CDs suggests that these couplings must emerge from the level M8 via the criticality conditions posing conditions on the coefficients of the octonionic polynomials coding for interactions.
Could all coupling constant emerge from the criticality conditions at the level of M8? The ratio of R2/lP2 of CP2 scale and Planck length appears at H level. Also this parameter should emerge from M8-H correspondence and thus from criticality at M8 level. Physics would reduce to a generalization of the catastrophe theory of Rene Thom!
- There are questions related to ZEO. Is the notion of CD general enough or should one generalize it to the analog of the polygonal diagram with light-like geodesic lines as its edges appearing in the twistor Grassmannian approach to scattering diagrams? Octonionic approach gives naturally the light-like boundaries assignable to CDs but leaves open the question whether more complex structures with light-like boundaries are possible. How the space-time surfaces associated with different quaternionic structures of M8 and with different positions of tips of CD interact?
- Real analyticity requires that the octonionic polynomials have real coefficients. This forces the origin of octonionic coordinates to be at real line (time axis) in the octonionic sense. All CDs cannot be located along this line. How do the varieties associated with octonionic polynomials with different origins interact? The polynomials with different origins neither commute nor are associative. How could one avoid losing the extremely beautiful associative and commutative algebra? It seems that one cannot form their products and sums and must form the Cartesian product of M8:s with different origins and formulate the interaction in this framework. Could Cayley-Dickson hierarchy be necessary to describe the interactions between different CDs (note however that the dimension are powers of 2) than multiples of 8?
Is the interaction nr well-defined only at the level of H inside CD to which these 4-D varieties arrive through the boundary of this CD? All CDs, whose tips are along light-like ray of CD boundary, share this ray. There is a common M20 shared by these CDs. Could M20 make possible the interaction. The CDs able to interact with given CD would have tips at the 3-D boundary of this CD and share common M20. These M20:s are labelled by the points of twistor sphere so that twistoriality seems to enter into the game in non-trivial manner also at the level of M8!
This however allows interactions only between varieties with same M40. What about a more general picture in which unit octonions define 6-D sphere S6 of directions of 8-D light-rays and parameterize different quaternionic structures with fixed M20. Could the condition for interaction be that S6 coordinates are same so that M20 is shared. In this case light rays in 7-D light-cone would parameterize the allowed origins for octonionic polynomials for which the interaction of zero loci is possible. The images of these light-rays in H would be more complex. This could allow varieties with different M40 but common M20 to interact via the common light ray/M20. Somewhat similar picture involving preferred M20 for given connected part of twistor graph emerges from the construction of twistor amplitudes.
What would be the interaction at the intersection? Light-like ray naturally defines a string like object with fermions at its ends. Could this fermionic string be transferred between space-time surfaces in the intersecting CDs. Also a branching of a string like object between space-time varieties in different CDs intersecting along this ray would be possible. This would describe stringy reaction vertex with incoming strings in different CDs. It must be admitted that this unavoidably brings in mind branes and all the nasty things that I have said about them during years!
Or could the strange singularity behavior in Minkowski signature play a role in the interactions? In the generic situation the intersection consists of discrete points but as the study of o2 shows the surface RE(o2)=0 and IM(o2)=0 can have dimensions 5 and 6 and their intersection naively expected to consist of discrete points can be interior or exterior of light-cone. Could the zero loci of singular polynomials play a key role in the interaction and allow 4-varieties in M8 to interact by being glued to this higher than 4-D objects. 4-D space-time variety could have 2-D intersection with 6-D variety and the 6-D variety could allow the interactions to between two 4-D varieties by correlating them. Also this strongly brings in mind branes but looks less elegant that above proposal.
- What is the connection with Yangian symmetry, whose generalization in TGD framework is highly suggestive?