The emergence of twistor spaces from M8-H duality

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 M⁸_c 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⊂ M⁸_c 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 M⁸_c 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.

1. The complex 6-D surface X⁶_c⊂ M⁸_c has commutative normal space and thus corresponds to complexified octonionic complex numbers (z₁+z₂I). X⁶_c 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.
2. This suggests that M⁸-H duality generalizes so that it maps X⁶_c ⊂ M⁸_c to T(M⁴)× T(CP₂) . It would map the point of X⁶_c 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 X⁶_c 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.

One can try to guess the form of the map of X⁶_c to the product T(M⁴)× T(CP₂).

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.

1. 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.
2. 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_×.
3. 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?

What about the physical interpretation at the level of M⁸_c.

1. 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 M⁸_c.

2. Consider first the real part of X⁶_c⊂ M⁸_c. 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 S_z= +/- 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.

3. For the 6-D tangent space of X⁶_c 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.

4. The points of X⁶_c⊂ M⁸_c 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.

5. 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.

7-surfaces with real normal space and generalization of the notion of twistor space

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?

1. 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.
2. 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.
3. 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.

What could be the physical interpretation of the extended twistor space?

1. 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.

2. 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.

3. 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.

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.

1. 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).
2. 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.
3. At M⁸ level, one would have Kähler magnetic charges Q_K(M⁴), Q_K(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 Q_K,m(M⁴)/Q_K,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.

How could the Grassmannians of standard twistor approach emerge number theoretically?

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.

1. Projective space is FP_n-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 d_F=n-1. In the complex case, this space can be identified as U(n)/U(n-1)× U(1)= CP_n-1.
2. 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...d_k=n. Defining integers n_i= d_i-d_i-1, this space can in the complex case be expressed as U(n)/U(n₁)×.....U(n_k). The real dimension of this space is d_R=n²-∑_in_i².
3. For n=4 and F=C, one has the following important Grassmannians.
   1. 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 d_R=6.
   2. M=F₂ as the space of complex 2-flags corresponds to U(4)/U(2)× U(2) and has d_R=16-8= 8. This space is identified as a complexified Minkowski space with D_C= 4.
   3. The space F_1,2 consisting of 2-D complex flags carrying 1-D complex flags has representation U(4)/U(2)× U(1)× U(1) and has dimension D_R=10.

      F_1,2 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. F_1,2 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.

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.

1. The basic space is Q_c=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 Q_c as Q² and Q as C² corresponding to the decomposition of quaternion to 2 complex numbers. C in turn decomposes to R× R.
2. The interpretation C²= C⁴ gives the above described standard spaces. Note that the complexified and compactified Minkowski space is not same as Q_c=Q² and it seems that in TGD framework Q_c is more natural and the quark momenta in M⁴_c indeed are complex numbers as algebraic integers of the extension.

Number theoretic hierarchy R→ C→ Q brings in some new elements.

1. It is natural to define also the quaternionic projective space Q_c/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. M⁴_R 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.

2. 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 d_R=7. It has a natural projection to CP₃. The space F_1,2 is replaced with representation U(4)/U(2) and has dimension D_R=12.

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.

See the article Summary of TGD as it is towards end of 2021 or the chapter chapter with the same title.

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

Articles and other material related to TGD.