Calabi-Yau manifolds appear in the expression for the energy emission rate in blackhole scattering

I received a like to a very interesting Nature article reporting the work of Driesse et al on calculation of gravitational scattering amplitudes of blackholes in a quantum field theory (QFT) model. The title of the article (see this) is "Emergence of Calabi Yau manifolds in high-precision black-hole scattering". There is also a popular article (see this) describing the findings. The theoretical motivation for using a QFT model is that blackholes are are elementary particle-like objects characterized byt only mass, spin, and charge.

Let us look first at the abstract of the article.

When two massive objects (black holes, neutron stars or stars) in our universe fly past each other, their gravitational interactions deflect their trajectories. The gravitational waves emitted in the related bound-orbit system-the binary inspiral-are now routinely detected by gravitational-wave observatories. Theoretical physics needs to provide high-precision templates to make use of unprecedented sensitivity and precision of the data from upcoming gravitational-wave observatories. Motivated by this challenge, several analytical and numerical techniques have been developed to approximately solve this gravitational two-body problem. Although numerical relativity is accurate it is too time-consuming to rapidly produce large numbers of gravitational-wave templates. For this, approximate analytical results are also required. Here we report on a new, highest-precision analytical result for the scattering angle, radiated energy and recoil of a black hole or neutron star scattering encounter at the fifth order in Newton's gravitational coupling G, assuming a hierarchy in the two masses. This is achieved by modifying state-of-the-art techniques for the scattering of elementary particles in colliders to this classical physics problem in our universe. Our results show that mathematical functions related to Calabi-Yau (CY) manifolds, 2n-dimensional generalizations of tori, appear in the solution to the radiated energy in these scatterings. We anticipate that our analytical results will allow the development of a new generation of gravitational-wave models, for which the transition to the bound-state problem through analytic continuation and strong-field resummation will need to be performed.

These findings look interesting from the TGD point of view. Calabi-Yau (CY) manifolds have an arbitrary complex dimension n. They generalize the notion of periodic orbit. In 1-D case orbit becomes a complex 2-D manifold, elliptic surface. But complex differential geometry allows a generalization to n-D real periodic orbits and their complex counterparts.

1. Torus is the simplest CY and 2-real-D elliptic doubly periodic surfaces appearing in complex analysis represent the basic example. I have discussed their representations at the level of space-time surfaces in the framework provided by holography= holomorphy vision. Weierstrass surfaces is one example (see this).

   The periods of planetary orbits in Coulomb force expressible in terms of elliptic integrals very probably led to the notion of elliptic Riemann surfaces by making the time variable complex. Elliptic Riemann surfaces are compact but define doubly periodic structures when represented in complex plane? Could the two periods define analogs of momenta?

2. The K-surface (see this) is a 4-(real)-dimensional CY manifold and a purely algebraic object having a unique topology. It appears in fourth order G⁴ in the calculation. K3 surface allows a Kähler metric. It is not clear to me how unique this metric is. The existence of the Kähler metric is important from the TGD point of view since induced metric codes for the Riemannian geometric aspects of TGD.

   Holography= holomorphy vision reduces TGD to algebraic geometry, which can be also regarded as Riemann geometry. Therefore an interesting question is whether a 4-real-D complex K3 surface could be represented in the TGD framework as a complex surface. Does Euclidean signature prevent this or does the K3 surface have a Minkowskian analog obtained by making the second complex coordinate hypercomplex?

   K3 surface can be represented as Fermat quartic surface x⁴+y⁴+z⁴+t⁴=0 in the twistor space CP₃ assigned M⁴. Twistor spaces of M⁴ and CP₂ appearing as factors of H=M⁴×CP₂ are unique in the sense that they are the only 4-D spaces allowing twistor bundles with a Kähler metric (see this).

   In TGD, CP₃ generalizes to its hypercomplex variant with one complex coordinate made hyperbolic and corresponds to SU(3,1)/SU(3)×U(1) (see this). This generalization allows to identify the base space of the twistor bundle as M⁴, rather than its compactified version. The hyperbolic counterpart of the quartic Fermat surface might serve as a one particular space-time surface in holography= holomorphy vision (see this, this , this, and this).

   In TGD, a generalized complex manifold is obtained from a complex manifold by making one complex coordinate hypercomplex. H=M⁴×CP₂ and space-time surfaces Xsup>4 in H are generalized complex manifolds. Suppose that the double periodicity of the 2-dimensional case generalizes so that the hyperbolic variant of K3 surface could correspond to a lattice cell of a 4-D periodic structure. Could one assign the hyperbolic counterpart of K3 surface a 4-D variant of a plane wave? This would conform with the view that gravitational waves are involved with the scattering of blackholes. Could kind of representation generalize to all kinds of plane waves and could K3 be one of the simplest examples?

3. The 3-complex-dimensional CYs were not mentioned in the article. They appear in the spontaneous compactification of the string models. Now the topology is not unique and the famous number 10⁵⁰⁰ was introduced as a rough estimate for their number. This turned out to be an untestable and fatal production.

   What a 3-real-dimensional periodic "orbit" and its complex generalization could mean? By holography= holomorphy vision, space-time surfaces are representable as intersections of 2 3-D generalized complex manifolds X⁶ in H and could be seen as analogs of twistor spaces for M⁴ and CP₂. Also the twistor space CP₃ is a CY manifold.

   Could one of these 2 surfaces X⁶ be a generalized CY manifold with one hypercomplex coordinate in some cases? 6-D real periodicity would requires double periodicity in hyperbolic coordinate. 6-D real periodicity would require double periodicity also in hyperbolic coordinate, which looks unrealistic: could only one hypercomplex coordinate allow periodicity? 3-D (2-D) generalized complex homology would be non-trivial for these 6-surfaces (space-time surfaces).

   What a 3-real-dimensional periodic "orbit" and its complex generalization could mean? By holography= holomorphy vision (see this, this , this, and this), space-time surfaces are representable as intersections of 2 3-D generalized complex manifolds X⁶ and Y⁶ in H and could be seen as analogs of twistor spaces for M⁴ and CP₂. The twistor space CP₃ is a CY manifold. Also the SU(3)/U(1)\times U(1) as the twistor space of CP₂ is a Kähler manifold (see this): this makes TGD unique.

   Could it happen that X⁶ or Y⁶ is a generalized CY manifold with one hypercomplex coordinate? 6-D real periodicity would require double periodicity also in hyperbolic coordinate, which looks unrealistic since by hyper-complex analyticity only the second real hyperbolic coordinate of the pair (u,v) appears as argument in the function pair (f₁,f₂): H→ C² defining the space-time surface as its root. It would seem that only one hypercomplex coordinate can allow the periodicity? 3-D (2-D) generalized complex homology would be non-trivial for these 6-surfaces (space-time surfaces).

See the article Holography = holomorphy vision and elliptic functions and curves in TGD framework or the chapter Does M8 −H duality reduce classical TGD to octonionic algebraic geometry?: Part III.

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.

About the structure of Dirac propagator in TGD

The discussion of the notion of fermion propagator in TGD framework demonstrated that the construction is much more than a mere computational challenge. There are two alternative approaches. Fermionic propagation could correspond to a) a 4-D or lower-dimensional propagation at the space-time level for the induced spinor fields as analog of massless propagation or b) to 8-D propagation in H between points belonging to the space-time surface.

For the option a), the separate conservation of baryon and lepton numbers requires fixed H-chirality so that the spinor mode is sum of products of M⁴ and CP₂ spinors with fixed M⁴ and CP₂ chiralities whose product is +1 or -1. This suggests that M⁴ propagation is massless. It came as a total surprise that the propagation of color modes in the conventional sense is not possible in length scales above CP₂ scale. The M⁴ part of the propagator for virtual masses above the mass of the color partial wave is of the standard form but for virtual masses below it the progator is its conformal inversion. The connection with color confinement is highly suggestive.

For light-like fermion lines at light-like partonic orbits, there are good reasons to expect that the condition s₁=s₂ is satisfied and implies that the propagation from s₁ is possible to only a discrete set of points s₂. Also this has direct relevance for the understanding of color confinement and more or less implies the intuitively deduced TGD based model for elementary fermions as 1-dimensional geometric objects.

Although the option b) need not provide a realistic propagator, it could provide a very useful semiclassical picture. If the condition s₁=s₂ is assumed, fermionic propagation along light-like geodesics of H is favored and in accordance with the model for elementary particles. This allows a classical space-time picture of particle massivation by p-adic thermodynamics and color confinement.

Also the interpretational and technical problems related to the construction of 4-D variants of super-conformal representations having spinor modes as ground states and to the p-adic thermodynamics are discussed.

See the article About the structure of Dirac propagator in TGD or the chapter with the same title.

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.

Comments about the unified theory of gravitation by Mikko Partanen and Jukka Tulkki

Esa Sakkinen set to me link to an article by Mikko Partanen and Jukka Tulkki proposing a unified theory of gravity (see this). The amusing coincidence is that Jukka Tulkki was an assistant in the Physics Laboratory of the Technological University where I made my thesis more than 4 decades ago. We had nice discussions.

In contrast to the most "breakthrough papers" in this field appearing with rate one per week, this article was coherent and well-written and did not contain the fatal error at the first page so that I decided to try to understand what is said in the article.

The key concepts of the unified theory of gravitation are "8-spinor", "space-time dimension field", and "gravity field" as analog of U(2) gauge potential with the labels of components corresponding to space-time coordinates. The claim was that this gives a renormalizable theory. This might be the case, but I dare to be skeptical about the claim that it is a theory of gravitation.

1. Basic notions

1.1 The notion of 8-spinor

1. At the first look, the 8-spinor has nothing to do with spinors as they are usually defined. The components can be, for example, of electromagnetic (em) field or vector potential or even space-time coordinates and some components are identically zero. Normally spinors have definite transformation properties in coordinate transformations and symmetries. Therefore spinors in the standard sense of the word are not in question.

   I would be cautious and speak just of an array of 8 numbers with a physical interpretation. What is non-trivial is that Maxwell action can be represented as a bilinear ΨΨ of this 8-spinor.

   In physics the spinor is much more than an array of numbers and the notion of spinor structure is a delicate concept: already in relativity. For instance, the spinor structure does not exist for all space-time topologies. This has a key role in TGD.

   In the theoretical vision of TGD, octonions and quaternions are central and octospinors emerge (see this). In this framework, the notion of octovector and octospinor could make sense as will be found later. Octovectors, octospinors and their conjugates as representations of SO(1,7) form a kind of holy trinity by triality symmetry.

2. The U(1) invariance as gauge invariance of Maxwell's em field holds true for each component of the 8-spinor formed from the em field. The components of 8-spinor formed from a vector potential suffer a non-trivial gauge transformation. This is not the case in non-abelian gauge theory and the 8-spinor components are Lie algebra-valued. This U(1)⁸ symmetry for spinors looks strange to me. In octonionic interpretation the multiplication with quaternions and octonions would define analogs of U(2) gauge transformations.

1.2 The notion of space-time dimension field

The notion of space-time dimension field is new.

1. It is stated that it is not a physical field. The defining equations say that it is a covariant constant. It has a spacetime index "a" labelling space-time dimensions. Its 4 components are exponentiation for each kernel matrix t^a with coefficient X_a depending on the location.
2. The kernel generators t^a are 8x8 matrices, which satisfy commutation relations of quaternionic units and realize U(2) algebra, which is the same as electroweak algebra. The connection with quaternions was not noticed. The exponent of exp(iX_at^a) could be interpreted as a local U(1) gauge transformation generated by t^a. For some reason, more general exponents defining U(2) gauge transformation were not considered. It is claimed that the t^a's correspond to the generators of the electroweak gauge group.

In TGD, quaternionic and octonionic structures are central and quaternionization for both M⁴ and CP₂ take place. t^a would correspond to quaternion units and their action on octonions is by multiplication. 8-spinors could correspond to octospinors or octovectors.

2. What calculations were done?

QED and gravitation in lowest order was considered but I do not think that this had much to do with the claimed unified theory of gravitation. I understand that this was essentially QED with coupling to the gravitational field described in standard way in the lowest order. Gravitation was brought in via a connection and claimed to be consistent with general relativity.

3. What unified theory of gravity could mean?

To my view, the second part of the article was the new thing.

1. The non-trivial claim was that quantized gravity is describable as a gauge theory using a compact, finite-dimensional gauge group U(2). Gravitational theories in which the gauge group is the Lorentz group SO(1,3) have been proposed. SO(1,3) finite-dimensional but not compact and this leads to problems. It has been also proposed that gravity is a gauge theory of translation group: also now the gauge group would be non-compact.

   This creates questions: How would the quaternionic automorphism group U(2) of quaternions act as a gauge group and produce a theory of gravitation? What happens to the general coordinate invariance? If GCI is present, what happens to the Poincare symmetry? The group U(2) is not the Lorentz group. Does one really obtain the metric theory of gravitation? It was claimed that this is the case.

2. Consider now a possible number theoretic interpretation in terms of octonions and quaternions.
   1. There are two indices: the indices a for the t^a and the indices μ for the space-time coordinates. "a" would correspond to the vierbein indices in general relativity. The four quaternionic units as labels of gauge Lie algebra generators would correspond to the labels vielbein vectors in spacetime.
   2. One way to proceed is to assume that space-time allows quaternionic structure and quaternion units allow to define the analog of vielbein. The U(2) gauge group would act as quaternionic multiplication on octonionic 8-vectors and 8-spinos: they were proposed to be 8-spinors but in the very weird sense in which they actually are not spinors.
   3. Could electric and magnetic fields allow an interpretation as octo-vectors? This would explain why the octospinors have identically vanishing components: they would correspond to octonionic and quaternionic units vanishing for 3-vectors.
   4. It is argued that this gives a gauge theory of gravity in which the kernel generators t^a, identifiable as quaternion units in TGD, correspond to the generators of the gauge group U(2). This would guarantee renormalizability. It can do that, but it is difficult to see how the emerging theory could describe gravitation.

   4. Some critical comments

   At least following critical comments can be raised.

   1. Is the space-time dimension field really needed?
   2. The gravitational field was identified as a gauge potential in non-Abelian U(2). However, the gauge transformations and covariant derivatives were defined as if t^a would generate the Abelian Lie algebra U(1)⁴. This I fail to understand. I would allow the quaternionic action as multiplication as symmetries.
   3. An action density for the gravitational field was introduced. The proposed action was instanton density E*B rather than E²-B². The inner product E*B is a total divergence and it cannot serve as an action.
   To sum up, octonions and quaternions could make it possible to formulate the notion of 8-spinor mathematically. Octovectors, octospinors and their conjugates would form a triad. If one transforms tensor indices to vielbein indices, one can associate octovectors with various vectors and antisymmetric tensors having only spatial indices.

   5. Analogies with number theoretic vision of TGD

   In the number theoretic vision of TGD, quaternionic and octonionic structures play a central role.

   1. M⁸-H, where one has H=M⁴×CP₂ duality as analog of momentum-position duality means that M⁸ has interpretation in terms of octonions with Minkowski scalar product defined as Re(o₁o₂). At the level of M⁸ 4-D associative surfaces are the counterparts of space-time surfaces in H.
   2. Complexified octonionic spinors are 8-component spinors and quaternionic spinors are an associative 4-D subspace of them. The octo-spinor property gives rise to spin and electroweak spin. These have a central role in the twistorialization of TGD. The non-associativity of course brings some delicacies and Associativity is proposed to be the dynamic principle of number theoretic TGD defined in octonionic M⁸. Octonionic automorphism group G₂ and its subgroup SU(3) identified as color group in TGD play a key role in TGD and quaternion multiplication corresponds to the action of electroweak U(2).
   3. What happens to the Lorentz symmetry violated by the replacement of the Lorentz group with U(2)? This was also a longstanding problem of M⁸-H duality but is now solved. The first wrong guess was that the surface Y⁴ in M⁸ is quaternionic and has Minkowskian signature. The guess was wrong. The surfaces Y² in M⁸ have Euclidean number theoretic induced metric and their normal spaces are quaternionic and Minkowskian. This is also required by the associativity as a dynamic principle

   For the almost most recent summary of TGD see this and this.

   For the holography= holomorphy vision and its relationship to an analog of Langlands duality relating geometric and number theoretic visions of TGD see this and this.

   The most recent summary of twistorialization of TGD in both H and M⁸ can be found at here.

   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.

Twistorialization at the level of M8

M⁸-H duality as analog of momentum-position duality for 3-surfaces as particles (see this, this, this and this) is a central part of TGD. I have already earlier considered several variants of what the twistor lift at the level of M⁸ could mean. There are several questions to be answered.

Identification of the twistor spaces

What are the twistor spaces of T(M⁸) and T(Y⁴) for the M⁸-H dual Y⁴ of the space-time surfaces X⁴⊂ H?

1. The 12-D space of light-like geodesics in M(1,7) would be the naive guess for the twistor space of M⁸. Now however the Minkowski metric of M⁸ is number theoretic and given by real part of octonionic product and 14-D G₂, is the number theoretic symmetry group so that the 12-D G₂/U(1)× U(1) is the natural candidate for the octonionic twistor space of M⁸. U(1)× U(1) has an interpretation as color Cartan algebra.
2. Without further conditions, the twistor sphere defined by light-like rays at a given point of M⁸ is a 6-D and the space S⁶=G₂/SU(3) is the natural identification for it. With this identification, the dimension of the total twistor space T(M⁸) would be 8+6=14, the dimension of G₂. This does not conform with the identification as T(M⁸)=G₂. It is also an open question whether S⁶ possesses the twistorially highly desirable Kähler structure.
3. How could one reduce the dimension of the space of light-like rays of M⁸ from 6 to 4? Could the condition that the light-rays are associated with a point of M⁸-H dual Y⁴ ⊂ M⁸ are quaternionic, allow to achieve this. M⁸-H duality in its recent form indeed requires that the normal space for a given point of Y⁴⊂ M⁸ as M⁸-H dual of X⁴⊂ H is quaternionic and Minkowskian in number theoretic sense (see this). This suggests a direct connection between twistorialization and M⁸-H duality.
   1. Could one require that the light-like 8-momentum has vanishing tangential component to Y⁴ and is therefore quaternionic? This would replace the twistor sphere with a union of twistor spheres associated with Minkowskian mass shells p²=m². The space of light rays would be 3- rather than 4-D and the wistor space of M⁸ would be 11-D rather than 12-D. One dimension is missing.
   2. The physical intuition suggests that the light rays do not have a momentum component in the direction of the tangent space of Y³ defining the 3-D holographic data but that they have a component tangential to Y⁴ in a direction normal to Y³. This would conform with non-point-likeness: by general coordinate invariance, the momentum component tangential to Y³ would not correspond to anything physical.

      The additional condition would be that these light-like vectors are quaternionic. The space of allowed 8-D light-like vectors would be 4-D and the twistor space could be G₂. The associativity of the dynamics at the level of M⁸ requires that the normal space is quaternionic and thus Minkowskian and also contains a commutative subspace. Can these two quaternionicity conditions be consistent with each other? If so, 8-D associative light-likeness respecting the 3-dimensality of holographic data implies the desired 4-dimensionality of the analog of the twistor sphere.

4. The section of the twistor bundle assigned to Y⁴ assigns to each point of Y⁴ a light-like vector. If also quaternionic units are chosen in an integrable way, this would define the M⁸ counterpart of the H-J structure which, when mapped to H by M⁸-H duality, would provide the H-J structure of H.

   If the selected light-like vectors have a vanishing tangential component in Y⁴, the light-like vectors in H are in M⁴. If this is not the case, the light-like vectors in M⁴ have also CP₂ component. For instance, light-like geodesics in M⁴× S¹, S¹⊂ CP₂ are possible. It therefore seems that the TGD view of twistorialization indeed makes possible the twistor description of massive particles.

Spinorial aspects of M⁸ twistorialization

What about the spinorial aspects of M⁸ twistorialization? One should generalize a) the map of the points of sphere S² to the 2× 2 matrices defined by a bi-spinor and its dual, b) the masslessness condition as vanishing of a determinant of the analog of the quaternionic matrix and c) the coincidence relation. One should also understand how the counterparts of the electroweak couplings are represented and solve the Dirac equation in M⁸.

1. In the case of M⁴, the map of massless momenta are mapped to the bispinors providing a matrix representation of quaternions in terms of Pauli sigma matrices. A possible way to achieve this is to introduce octonionic spinor structure (see this, this and this) in which massless 8-D momenta correspond to octonions, which should be associative and therefore quaternionic. This would conform with the above identification of light rays.
2. Octonionic spinors, presumably complexified with i=(-1)^1/2 commuting with the octonionic units, should be also defined. The map of quaternionic massless 8-momenta to the octonionic counterparts of the Pauli spin matrices representing quaternionic basis would define octonionic spinors satisfying the quaternionicity condition.
   1. The condition that the twistor space allows Käahler structure and has S²× S² as fiber might leave only the product T(M⁸)=T(M⁴)× T(E⁴), which is consistent with M⁸= M⁴× E⁴. The mechanism of the dimensional reduction would be the same as in the case of H. Whether one can identify T(E⁴) as CP₃ is quite not clear.
   2. Very naively, in the spinorial approach the extended twistor space C⁴ is replaced with C⁸. Division with 2-complex-dimensional planes CP₂ would give Grassmannian Gr_c(2,8) with dimension 2× (8-2)=12, which is a complex manifold having the representation U(8)/U(2)× U(6). The fiber would be CP₂. Minkowskian signature would suggest that U(6) is replaced with U(5,1) and U(8) with U(7,1).
   3. The number theoretic G₂/U(1)× U(1) is the third possible identification but it is not clear whether it is consistent with the number theoretic M⁴ signature and CP₂ fiber.
3. The matrices defined by bi-spinor pairs associated with M⁴ twistors can be regarded as quaternions. The quaternionicity condition means that the octonionic spinor pairs actually reduce to M⁴ bi-spinor pairs on a suitable basis, which however depends on the point of Y⁴?

   If commutative i is introduced and quaternions are not replaced with their 2× 2 matrix representations involving commuting imaginary units, a doubling of degrees of freedom takes place. Does this mean that both M⁴ chiralities are obtained? Could this solve the googly problem in M⁴?

Also in the case of octonionic spinors complexification would double the degrees of freedom. Does one obtain in this way both spin and electroweak spin?

1. What happens to M⁸ spinors as tensor products of Minkowski spinors and electroweak spinors when the octonionic Dirac operator acts on a quaternionic subspace. The electroweak degrees of freedom do not disappear but become passive. One has 8-D complex spinors, which are enough to represent a single H-chirality if the octonionic gamma matrices, which are quaternionic at Y⁴, are not represented in terms of Pauli sigma matrices and i is introduced.
2. The electroweak gauge potentials as induced spinor connection represent the geometric view of physics realized at the level of H. Number theoretical vision suggests that the M⁸ spinor connection cannot involve sigma matrices, which would be defined as commutators of octonionic units and be octonionic units themselves. Kähler coupling is however possible.

   What could the form of the Kähler gauge potential be? The Kähler form should be apart from a multiplicative imaginary unit i equal to the theoretical flat metric of M⁸ so that the Kähler function would represent harmonic oscillator potential. The octonionic Dirac equation would have a unique coupling to the Kähler gauge potential with Kähler coupling constant absorbed to it. This would guarantee that the solutions of the modified Dirac equation in M⁸ have a finite norm. Presumably the solutions can be found by generalizing the procedure to solve Dirac equation in harmonic oscillator potential.

3. The octonionic Dirac operator, which reduces to the quaternionic M⁴ Dirac operator and for the local quaternionic M⁴ identified as a normal space, the fermions are massless. How to solve this problem? As found, the non-vanishing M⁴ mass requires that the light-like M⁸ momentum has a component in the direction of Y⁴ having a natural interpretation as the analog of the square root of the Higgs field.
4. Complexified octonionic spinors form a complex 8-D space, which corresponds to a single fermion chirality. Do different H chiralities emerge from the mere octonionic picture or must one introduce them in the same way in the case of H? The couplings of quarks and leptons to the induced Kähler form are different and this should be true also at the level of M⁸: it seems that both quarks and leptons should be introduced unless on is read consider either leptons or quarks as fundamental fermions.
5. Color SU(3) acts as a number theoretic symmetry of octonions. SU(3) as an automorphism group transforms to each other different quaternionic normal spaces represented as points of CP₂. This representation is realized at the level of H in terms of spinor harmonics. The idea that the low energy and higher energy models for hadron in terms of SO(4) and SU(3) symmetries are dual suggest that fermionic SO(4) harmonics in M⁸ could be analogous to the representation of color as spinor harmonics in CP₂.

This picture suggests that 6-D Kähler action as the Kähler function of the twistor space of M⁸ could determined surfaces Y⁴ as its preferred extremals and that holography= holomorphy principle determines the extremals also now. The 12-D twistor bundle with 4-D fiber should have Kähler structure. This gives very strong condition. One possibility is that it is just the Cartesian product of twistor spaces for M⁴ and E⁴.

See the article Twistors and holography= holomorphy vision or the chapter with the same title.

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.

Twistors and holography= holomorphy vision

Twistorialization involves several problems. Mention only the identification of the twistor space, the googly problem meaning that only second massless M⁴ chirality allows geometrization in this way, the problem that massive fields do not allow twistorialization, and the problem that in general relativity only space-times with vanishing Weyl tensor allow twistor structure.

In the TGD framework, twistorialization should be performed for H=M⁴× CP₂. Now there are no primary bosonic fields since they are represented in terms of the induced spinor connection and metric and also classical color fields are obtained by induction. Twistor lift was based on the replacement of space-time surfaces in H=M⁴× CP₂ with the analogs of their 6-D twistor spaces X⁶ as sphere bundles as a surfaces in the twistor space T(H) of H identified as the product T(M⁴)× T(CP₂) of twistor spaces H. In TGD, the replacement of T(M⁴)=CP₃ with CP_{2,1} having one hypercomplex coordinate is natural. Dimensional reduction for the extremals of 6-D Kähler action and the identification of the fiber spheres CP₁ of T(M⁴) and T(CP₂) was needed to product to produce the X⁶ as a sphere bundle over X⁴.

Holography= holomorphy (H-H) vision in turn allows to solve the field equations for any general coordinate invariant action expressible in terms of the induced geometry allows to solve the field equations, which are extremely nonlinear partial differential equations, exactly by reducing them to purely algebraic local equations. The independence of action means universality. H-H vision conforms with T(H) view but one can ask whether one could twist TGD without the introduction of T(H) by representing the twistor spheres of T(M⁴) and T(CP₂) as homologically non-trivial spheres of the causal diamond CD (missing the line connecting its tips) and CP₂. The second condition involved with the H-H principle would represent the identification of the twistor spheres.

In this article various problems of the twistorialization are discussed in the TGD framework and the question whether the H-H principle is enough for twistorialization is discussed.

See the article Twistors and holography= holomorphy vision or the chapter with the same title.

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.