It has become clear that M8-H duality generalizes and there is a connection with the twistorialization at the level of H.
Space-time surfaces as images of associative surfaces in M8
M8-H duality would provide an explicit construction of space-time surfaces as algebraic surfaces with an associative normal space (see this, this, and this). M8 picture codes space-time surface by a real polynomial with rational coefficients. One cannot exclude coefficients in an extension of rationals and also analytic functions with rational or algebraic coefficients can be considered as well as polynomials of infinite degree obtained by repeated iteration giving rise algebraic numbers as extension and continuum or roots as limits of roots.
M8-H duality maps these solutions to H and one can consider several forms of this map. The weak form of the duality relies on holography mapping only 3-D or even 2-D data to H and the strongest form maps entire space-time surfaces to H. The twistor lift of TGD allows to identify the space-time surfaces in H as base spaces of 6-D surfaces representing the twistor space of space-time surface as an S2 bundle in the product of twistor spaces of M4 and CP2. These twistor spaces must have Kähler structure and only the twistor spaces of M4 and CP2 have it so that TGD is unique also mathematically.
An interesting question relates to the possibility that also 6-D commutative space-time surfaces could be allowed. The normal space of the space-time surface would be a commutative subspace of M8c and therefore 2-D. Commutative space-time would be a 6-D surface X6 in M8.
This raises the following question: Could the inverse image of the 6-D twistor-space of 4-D space-time surface X4 so that X6 would be M8 analog of twistor lift? This requires that X6⊂ M8c has the structure of an S2 bundle and there exists a bundle projection X6→ X4.
The normal space of an associative space-time surface actually contains this kind of commutative normal space! Its existence guarantees that the normal space of X4 corresponds to a point of CP2. Could one obtain the M8c analog of the twistor space and the bundle bundle projection X6→ X4 just by dropping the condition of associativity. Space-time surface would be a 4-surface obtained by adding the associativity condition.
One can go even further and consider 7-D surfaces of M8 with real
and therefore well-ordered normal space. This would suggest
dimensional hierarchy: 7→ 6→ 4.
This leads to a possible interpretation of twistor lift of TGD at the level of M8 and also about generalization of M8-H correspondence to the level of twistor lift. Also the generalization of twistor space to a 7-D space is suggestive. The following arguments representa vision about "how it must be" that emerged during the writing of this article and there are a lot of details to be checked.
Commutative 6-surfaces and twistorial generalization of M8-H correspondence
Consider first the twistorial generalization of M8-H correspondence.
One can try to guess the form of the map of X6c to the product T(M4)× T(CP2).
- The complex 6-D surface X6c⊂ M8c has commutative normal space and thus corresponds to complexified octonionic complex numbers (z1+z2I). X6c has real dimension 12 just as the product T(M4)× T(CP2) of 6-D twistor spaces of M4 and CP2. It has a bundle structure with a complex 4-D base space which is mapped M4× CP2 by M8H duality. The fiber has complex dimension 2 and corresponds to the dimension for the product of twistor spheres of the twistor spaces of M4 and CP2.
- This suggests that M8-H duality generalizes so that it maps X6c ⊂ M8c to T(M4)× T(CP2) . It would map the point of X6c to its real projection identified as a point of T(M4). "Real" means here that the complex continuation of the number theoretical norm squared for octonions is real so that the components of M8 point are either real or imaginary with respect to the commuting imaginary unit i. The complex 6-D tangent space of X6c would be mapped to a point of T(CP2).
The beauty of this picture would be that the entire complex 6-D surface would carry physical information mapped directly to the twistor space.
The surfaces X6 have local normal space basis 1⊕ e7 . The problem is that this space is invariant under SU(3) for M8-H for CP2. Could one choose the 2-D normal space to be something else without losing the duality. If e7 and e1 are permuted, the tangent space basis vector transforms by a phase phase factor under U(1)× U(1). The 4-D sub-basis of normal space would be now (1,e1,e7,e2). This does not affect the M8-H-duality map to CP2. The 6-D space of normal spaces would be the flag manifold SU(2)/U(1)times U(1), which is nothing but the twistor space T(CP2).
What about the twistorial counterpart for the map of M4⊂ M8→ M4⊂ M8? One can consider several options.
What about the physical interpretation at the level of M8c.
- At the level of M8, M4 is replaced by M6 at least locally in the sense that one can use M6 coordinates for the point of X6. Can one identify the M6 image of this space as the projective space C4/C× obtained from C4 by dividing with complex scalings? This would give the twistor space CP3= SU(4)/U(3) of M4. This is not obvious since one has (complexified) octonions rather than C4 or its hypercomplex analog. This would be analogous to using several (4) coordinate charts glued together as in the case of sphere CP1.
- If M8-H duality generalizes as such, the points of M6 could be mapped to the 6-D analog of cd4 such that the image point is defined as the intersection of a geodesic line with direction given by the 6-D momentum with the 5-D light-like boundary of 6-D counterpart cd6 of cd? Does the slicing of M6 by 5-D light-boundaries of cd6 for various values of 6-D mass squared have interpretation as CP3? Note that the boundary of cd6 does not contain origin and the same applies to CP3= C4/C×.
- Or could one identify the octonionic analog of the projective space CP3=C4/C×? Could the octonionic M8 momenta be scaled down by dividing with the momentum projection in the commutative normal space so that one obtains an analog of projective space? Could one use these as coordinates for M6?
The scaled 8-momenta would correspond to the points of the octonionic analog of CP3. The scaled down 8-D mass squared would have a constant value.
A possible problem is that one must divide either from left or right and results are different in the general case. Could one require that the physical states are invariant under the automorphisms generated o→ gog-1, where g is an element of the commutative subalgebra in question?
7-surfaces with real normal space and generalization of the notion of twistor space
- The first thing to notice is that in the twistor Grassmann approach twistor space provides an elegant description of spin. Partial waves in the fiber S2 of twistor space representation of spin as a partial wave. All spin values allow a unified treatment.
The problem is that this requires massless particles. In the TGD framework 4-D masslessness is replaced with its 8-D variant so that this difficulty is circumvented. This kind of description in terms of partial waves is expected to have a counterpart at the level of the twistor space T(M(4)× T(CP2). At level of M8 the description is expected to be in terms of discrete points of M8c.
- Consider first the real part of X6c⊂ M8c. At the level of M8 the points of X4 correspond to points. The same must be true also at the level of X6. Single point in the fiber space S2 would be selected. The interpretation could be in terms of the selection of the spin quantization axis.
Spin quantization axis corresponds to 2 diametrically opposite points of S2. Could the choice of the point also fix the spin direction? There would be two spin directions and in the general case of a massive particle they must correspond to the values Sz= +/- 1/2 of fermion spin. For massless particles in the 4-D sense two helicities are possible and higher spins cannot be excluded. The allowance of only spin 1/2 particles conforms with the idea that all elementary particles are constructed from quarks and antiquarks. Fermionic statistics would mean that for fixed momentum one or both of the diametrically opposite points of S2 defining the same and therefore unique spin quantization axis can be populated by quarks having opposite spins.
- For the 6-D tangent space of X6c or rather, its real projection, an analogous argument applies. The tangent space would be parametrized by a point of T(CP2) and mapped to this point. The selection of a point in the fiber S2 of T(CP2) would correspond to the choice of the quantization axis of electroweak spin and diametrically opposite points would correspond to opposite values of electroweak spin 1/2 and unique quantization axis allows only single point or pair of diametrically opposite points to be populated.
Spin 1/2 property would hold true for both ordinary and electroweak spins and this conforms with the properties of M4× CP2 spinors.
- The points of X6c⊂ M8c would represent geometrically the modes of H-spinor fields with fixed momentum. What about the orbital degrees of freedom associated with CP2?
M4 momenta represent orbital degrees of M4 spinors so that E4 parts of E8 momenta should represent the CP2 momenta. The eigenvalue of CP2 Laplacian defining mass squared eigenvalue in H should correspond to the mass squared value in E4 and to the square of the radius of sphere S3 ⊂ E4.
This would be a concrete realization for the SO(4)=SU(2)L× SU(2)R↔ SU(3) duality between hadronic and quark descriptions of strong interaction physics. Proton as skyrmion would correspond to a map S3 with radius identified as proton mass. The skyrmion picture would generalize to the level of quarks and also to the level of bound states of quarks allowed by the number theoretical hierarchy with Galois confinement. This also includes bosons as Galois confined many quark states.
- The bound states with higher spin formed by Galois confinement should have the same quantization axis in order that one can say that the spin in the direction of the quantization axis is well-defined. This freezes the S2 degrees of freedom for the quarks of the composite.
It would seem that twistorialization could correspond to the introduction
of 6-surfaces of M8, which have commutative normal space. The next step is to ask whether it makes sense to consider 7-surfaces with a real norma space allowing well-ordering? This would give a hierarchy of surfaces of M8 with dimensions 7, 6, and 4. The 7-D space would have bundle projection to 6-D space having bundle projection to 4-D space.
What could be the physical interpretation of 7-D surfaces of M8 with real normal space in the octonionic sense and of their H images?
What could be the physical interpretation of the extended twistor space?
- The first guess is that the images in H correspond to 7-D surfaces as generalizations of 6-D twistor space in the product of similar 7-D generalization of twistor spaces of M4 and CP2. One would have a bundle projection to the twistor space and to the 4-D space-time.
- SU(3)/U(1)× U(1) is the twistor space of CP2. SU(3)/SU(2)× U(1) is the twistor space of M4? Could 7-D SU(3)/U(1) resp. SU(4)/SU(3) correspond to a generalization of the twistor spaces of M4 resp. CP2? What could be the interpretation of the fiber added to the twistor spaces of M4, CP2 and X4? S3 isomorphic to SU(2) and having SO(4) as isometries is the obvious candidate.
- The analog of M8-H duality in Minkowskian sector in this case could be to use coordinates for M7 obtained by dividing M8 coordinates by the real part of the octonion. Is it possible to identify RP7= M8/R× with SU(4)/SU(3) or at least relate these spaces in a natural manner. It should be easy to answer these questions with some knowhow in practical topology.
A possible source of problems or of understanding is the presence of a commuting imaginary unit implying that complexification is involved in Minkowskian degrees of freedom whereas in CP2 degrees of freedom it has no effect. RP7 is complexified to CP7 and the octonionic analog of CP3 is replaced with its complexification.
One expects that the action in the 7-D situation is analogous to Chern-Simons action associated with 8-D Kahler action, perhaps identifiable as a complexified 4-D Kähler action.
- Twistorialization takes care of spin and electroweak spin. The remaining standard model quantum numbers are Kähler magnetic charges for M4 and CP2 and quark number. Could the additional dimension allow their geometrization as partial waves in the 3-D fiber?
The first thing to notice is that it is not possible to speak about the choice of quantization axis for U(1) charge. It is however
possible to generalize the momentum space picture also to the 7-D branes X7 of M8 with real normal space and select only discrete points of cognitive representation carrying quarks. The coordinate of 7-D generalized momentum in the 1-D fiber would correspond to some charge interpreted as a U(1) momentum in the fiber of 7-D generalization of the twistor space.
- One can start from the level of the 7-D surface with a real normal space. For both M4 and CP2, a plausible guess for the identification of 3-D fiber space is as 3-sphere S3 having Hopf fibration S3→ S2 with U(1) as a fiber.
At H side one would have a wave exp(iQ φ/2π) in U(1) with charge Q and at M8 side a point of X7 representing Q as 7:th component of 7-D momentum.
Note that for X6 as a counterpart of twistor space the 5:th and 6:th components of the generalized momentum would represent spin quantization axis and sign of quark spin as a point of S2. Even the length of angular momentum might allow this kind representation.
- Since both M4 and CP2 allow induced Kähler field, a possible identification of Q would be as a Kähler magnetic charge. These charges are not conserved but in ZEO the non-conservation allows a description in terms of different values of the magnetic charge at opposite halfs of the light-cone of M8 or CD.
Instanton number representing a change of magnetic charge would not be a charge in strict sense and drops from consideration.
How could the Grassmannians of standard twistor approach emerge number theoretically?
- At M4 side, the 7-D bundle would be SU(4)/SU(3)→ SU(4)/SU(3)× U(1). At CP2 side the bundle would be SU(3)/U(1)→ SU(3)/U(1)× U(1).
- For the induced bundle as 7-D surface in the SU(4)/SU(3)× SU(3)/U(1), the two U(1):s are identified. This would correspond to an identification φ(M4)= φ(CP2) but also a more general correspondence φ(M4)= (n/m)φ(CP2) can be considered. m/n can be seen as a fractional U(1) winding number or as a pair of winding numbers characterizing a closed curve on torus.
- At M8 level, one would have Kähler magnetic charges QK(M4), QK(CP2) represented associated with U(1) waves at twistor space level and as points of X7 at M8 level involving quark. The same wave would represent both M4 and CP2 waves that would correlate the values of Kähler magnetic charges by QK,m(M4)/QK,m(CP2)= m/n if both are non-vanishing. The value of the ratio m/n affects the dynamics of the 4-surfaces in M8 and via twistor lift the space-time surfaces in H.
One can identify the TGD counterparts for various Grassmann manifolds appearing in the standard twistor approach.
Consider first, the various Grassmannians involved with the standard twistor approach (this) can be regarded as flag-manifolds of 4-complex dimensional space T.
This picture generalizes to TGD and actually generalizes so that also the real Minkowski space is obtained naturally. Also the complexified Minkowski space has a natural interpretation in terms of extensions of rationals forcing complex algebraic integers as momenta. Galois confinement would guarantee that physical states as bound states have real momenta.
- Projective space is FPn-1 the Grasmannian F1(Fn) formed by the k-D planes of Vn where F corresponds to the field of real, complex or quaternionic numbers, are the simplest spaces of this kind. The F-dimension is dF=n-1. In the complex case, this space can be identified as U(n)/U(n-1)× U(1)= CPn-1.
- More general flag manifolds carry at each point a flag, which carries a flag which carries ... so that one has a hierarchy of flag dimensions d0=0<d1<d2...dk=n. Defining integers ni= di-di-1, this space can in the complex case be expressed as U(n)/U(n1)×.....U(nk). The real dimension of this space is dR=n2-∑ini2.
- For n=4 and F=C, one has the following important Grassmannians.
- The twistor space CP3 is projective is of complex planes in T=C4 and given by CP3=U(4)/U(3)× U(1) and has real dimension dR=6.
- M=F2 as the space of complex 2-flags corresponds to U(4)/U(2)× U(2) and has dR=16-8= 8. This space is identified as a complexified Minkowski space with DC= 4.
- The space F1,2 consisting of 2-D complex flags carrying 1-D complex flags has representation U(4)/U(2)× U(1)× U(1) and has dimension DR=10.
F1,2 has natural projection ν to the twistor space CP3 resulting from the symmetry breaking U(3)→ U(2)× U(1) when one assigns to 2-flag a 1-flag defining a preferred direction. F1,2 also has a natural projection μ to the complexified and compactified Minkowski space M=F2 resulting in the similar manner and is assignable to the symmetry breaking U(2)× U(2)→ U(1)× U(1) caused by the selection of 1-flag.
These projections give rise to two correspondences known as Penrose transform. The correspondence μ ∘ ν-1 assigns to a point of twistor space CP3 a point of complexified Minkowski space. The correspondence ν ∘ μ-1 assigns to the point of complexified Minkowski space a point of twistor space CP3. These maps are obviously not unique without further conditions.
Number theoretic hierarchy R→ C→ Q brings in some new elements.
- The basic space is Qc=Q2 identifiable as a complexified Minkowski space. The idea is that number theoretically preferred flags correspond to fields R,C,Q with real dimensions 1,2,4. One can interpret Qc as Q2 and Q as C2 corresponding to the decomposition of quaternion to 2 complex numbers. C in turn decomposes to R× R.
- The interpretation C2= C4 gives the above described standard spaces. Note that the complexified and compactified Minkowski space is not same as Qc=Q2 and it seems that in TGD framework Qc is more natural and the quark momenta in M4c indeed are complex numbers as algebraic integers of the extension.
To sum up, the Grassmannians associated with M4 as 6-D twistor space and its 7-D extension correspond to a complexification by a commutative imaginary unit i - that is "vertical direction". The Grassmannians associated with CP2 correspond to "horizontal ", octonionic directions and to associative, commutative and well-ordered normal spaces of the space-time surface and its 6-D and 7-D extensions. Geometrization of the basic quantum states/numbers - not only momentum - representing them as points of these spaces is in question.
- It is natural to define also the quaternionic projective space Qc/Q=Q2/Q (see this), which corresponds to real Minkowski space. By non-commutativity this space has two variants corresponding to left and right division by quaternionic scales factor. A natural condition is that the physical states are invariant under automorphisms q→ hqh-1 and depend only on the class of the group element. For the rotation group this space is characterized by the direction of the rotation axis and by the rotation angle around it and is therefore 2-D.
This space is projective space QP1, quaternionic analog of Riemann sphere CP1 and also the quaternionic analog of twistor space CP3 as projective space. Therefore the analog of real Minkowski space emerges naturally in this framework. More generally, quaternionic projective spaces Qn have dimension d=4n and are representable as coset spaces of symplectic groups defining the analogs of unitary/orthogonal groups for quaternions as Sp(n+1)/Sp(n)× Sp(1) as one can guess on basis of complex and real cases. M4R would therefore correspond to Sp(2)/Sp(1)× SP(1).
QP1 is homeomorphic to 4-sphere S4 appearing in the construction of instanton solutions in E4 effectively compactified to S4 by the boundary conditions at infinity. An interesting question is whether the self-dual Kähler forms in E4 could give rise to M4 Kähler structure and could correspond to this kind of self-dual instantons and therefore what I have called Hamilton-Jacobi structures.
- The complex flags can also contain real flags. For the counterparts of twistor spaces this means the replacement of U(1) with a trivial group in the decompositions.
The twistor space CP3 would be replaced U(4)/U(3) and has real dimension dR=7. It has a natural projection to CP3. The space F1,2 is replaced with representation U(4)/U(2) and has dimension DR=12.
See the the chapter Summary of TGD as it is towards end of 2021 or the article
with the same title.