Quaternions, octonions, and TGD

Quaternions and octonions have been lurking around for decades in hope of getting deeper role in physics but as John Baez put it: "I would not have the courage to give octonions as a research topic for a graduate student". Quaternions are algebraically a 4-D structure and this strongly suggests that space-time could be analogous to complex plane.

Classical continuous number fields reals, complex numbers, quaternions, octonions have dimensions 1, 2, 4, 8 coming in powers of 2. In TGD imbedding space is 8-D structure and brings in mind octonions. Space-time surfaces are 4-D and bring in mind quaternions. String world sheets and partonic 2-surfaces are 2-D and bring in mind complex numbers. The boundaries of string world sheets are 1-D and carry fermions and of course bring in mind real numbers. These dimensions are indeed in key role in TGD and form one part of the number theoretic vision about TGD involving p-adic numbers, classical number fields, and infinite primes.

What quaternionicity could mean?

Quaternions are non-commutative: AB is not equal to BA. Octonions are even non-associative: A(BC) is not equal to (AB)C. This is problematic and in TGD problems is turned to a victory if space-time surfaces as 4-surface in 8-D M⁴× CP₂ are associative (or co-associative in which case normal space orthogonal to the tangent space is associative). This would be extremely attractive purely number theoretic formulation of classical dynamics.

What one means with quaternionicity of space-time is of course highly non-trivial questions. It seems however that this must be a local notion. The tangent space of space-time should have quaternionic structure in some sense.

1. It is known that 4-D manifolds allow so called almost quaternionic structure: to any point of space-time one can assign three quaternionic imaginary units. Since one is speaking about geometry, imaginary quaternionic units must be represented geometrically as antisymmetric tensors and obey quaternionic multiplication table. This gives a close connection with twistors: any orientable space-time indeed allows extension to twistor space which is a structure having as "fiber space" unit sphere representing the 3 quaternionic units.
   
   
2. A stronger notion is quaternionic Kähler manifold, which is also Kähler manifold - one of the quaternionic imaginary unit serves as global imaginary unit and is covariantly constant.CP₂ is example of this kind of manifold. The twistor spaces associated with quaternion-Kähler manifolds are known as Fano spaces and have very nice properties making them strong candidates for the Euclidian regions of space-time surfaces obtained as deformations of so called CP₂ type vacuum extremals represenging lines of generalized Feynman diagrams.
   
   
3. The obvious question is whether complex analysis including notions like analytic function, Riemann surface, residue integration crucial in twistor approach to scattering amplitudes, etc... generalises to quaternions. In particular, can one generalize the notion of analytic function as a power series in z to that for quaternions q. I have made attempts but was not happy about the outcome and had given up the idea that this could allow to define associative/ co-associative space-time surface in very practically manner. It was quite a surprise to find just month or so ago that quaternions allow differential calculus and that the notion of analytic function generalises elegantly but in a slightly more general manner than I had proposed. Also the conformal invariance of string models generalises to what one might call quaternion conformal invariance. What is amusing is that the notion of quaternion analyticity had been discovered for aeons ago (see this) and I had managed to not stumble with it earlier! See this.
   
   

Octonionicity and quaternionicity in TGD

In TGD framework one can consider further notions of quaternionicity and octonionicity relying on sub-manifold geometry and induction procedure. Since the signature of the imbedding space is Minkowskian, one must replace quaternions and octonions with their complexification called often split quaternions and split octonions. For instance, Minkowski space corresponds to 4-D subspace of complexified quaternions but not to an algebra. Its tangent space generates by multiplication complexified quaternions.

The tangent space of 8-D imbedding space allows octonionic structure and one can induced (one of the keywords of TGD) this structure to space-time surface. If the induced structure is quaternionic and thus associative (A(BC)= (AB)C), space-time surface has quaternionic structure. One can consider also the option of co-associativity: now the normal space of space-time surface in M⁴× CP₂ would be associative. Minkowskian regions of space-time surface would be associative and Euclidian regions representing elementary particles as lines of generalized Feynman diagrams would be co-associative.

Quaternionicity of space-time surface could provide purely number theoretic formulation of dynamics and the conjecture is that it gives preferred extremals of Kähler action. The reduction of classical dynamics to associativity would of course mean the deepest possible formulation of laws of classical physics that one can imagine. This notion of quaternionicity should be consistent with the quaternion-Kähler property for Euclidian space-time regions which represent lines of generalized Feynman graphs - that is elementary particles.

Also the quaternion analyticity could make sense in TGD framework in the framework provided by the 12-D twistor space of imbedding space, which is Cartesian product of twistor spaces of M⁴ and CP₂ which are the only twistor spaces with Kähler structure and for which the generalization of complex analysis is natural. Hence it seems that space-time in TGD sense might represent an intersection of various views about quaternionicity.

What about commutativity?: number theory in fermionic sector

Quaternions are not commutative (AB is not equal to AB in general) and one can ask could one define commutative and co-commutative surfaces of quaternionic space-time surface and their variants with Minkowski signature. This is possible.

There is also a physical motivation. The generalization of twistors to 8-D twistors starts from generalization in the tangent space M⁸ of CP₂. Ordinary twistors are defined in terms of sigma matrices identifiable as complexified quaternionic imaginary units. One should replaced the sigma matrices with 7 sigma matrices and the obvious guess is that they represent octonions. Massless irac operator and Dirac spinors should be replaced by their octonionic variant. A further condition is that this spinor structure is equivalent with the ordinary one. This requires that it is quaternionic so that one must restrict spinors to space-time surfaces.

This is however not enough - the associativity for spinor spinor dynamics forces them to 2-D string world sheets. The reason is that spinor connection consisting of sigma matrices replaced with octonion units brings in additional potential source of non-associativity. If induced gauge fields vanish, one has associativity but not quite: induce spinor connection is still non-associative. The stronger condition that induced spinor connection vanishes requires that the CP₂ projection of string world sheet is not only 1-D but geodesic circle. String world sheets would be possible only in Minkowskian regions of space-time surface and their orbit would contain naturally a light-like geodesic of imbedding space representing point-like particle.

Spinor modes would thus reside at 2-surfaces 2-D surfaces - string world sheets carrying spinors. String world sheets would in turn emerge as maximal commutative space-time regions: at which induced electroweak gauge fields producing problems with associativity vanish. The gamma matrices at string world sheets would be induced gamma matrices and super-conformal symmetry would require that string world sheets are determined by an action which is string world sheet area just as in string models. It would naturally be proportional to the inverse of Newton's constant (string tension) and the ratio hbar G/R² of Planck length and CP₂ radius squared would be fixed by quantum criticality fixing the values of all coupling strengths appearing in the action principle to be of order 10^-7. String world sheets would be fundamental rather than only emerging.

I have already earlier ended up to a weaker conjecture that spinors are localized to string world sheets from the condition that electromagnetic charge is well-defined quantum number for the induced spinor fields: this requires that induced W gauge fields and perhaps even potentials vanish and in the generic case string world sheets would be 2-D. Now one ends up with a stronger condition of commutativity implying that spinors at string world sheets behave like free particles. They do not act with induce gauge fields at string world sheets but just this avoidance behavior induces this interaction implicitly! Your behavior correlates with the behavior of the person whom you try to avoid! One must add that the TGD view about generalized Feynman graphs indeed allows to have non-trivial scattering matrix based on exchange of gauge bosons although the classical interaction vanishes.

Number theoretic dimensional hierarchy of dynamics

Number theoretical vision would imply a dimensional hierarchy of dynamics involving the dimensions of classical number fields. The classical dynamics for both space-times surface and spinors would simplify enormously but would be still consistent with standard model thanks to the topological view about interaction vertices as partonic 2-surfaces representing the ends of light-like 3-surface representing parton orbits and reducing the dynamics at fermion level to braid theory. Partonic 2-surfaces could be co-commutative in the sense that their normal space inside space-time surface is commutative at each point of the partonic 2-surface. The intersections of string world sheets and partonic 2-surfaces would consist of discrete points representing fermions. The light-like lines representing intersections of string world sheets with the light-like orbits of partonic 2-surfaces would correspond to orbits of point-like fermions (tangent vector of the light-like line would correspond to hypercomplex number with vanishing norm). The space-like boundary of string world sheet would correspond to real line. Therefore dimensional hierarchy would be realized.

The dimensional hierarchy would relate closely to both the generalization conformal invariance distinguish TGD from superstring models and to twistorialization. All "must be true" conjectures (physics geometry, physics as generalized number theory, M⁸-H duality, TGD as almost topological QFT, generalization of twistor approach to 8-D situation and induction of twistor structure, etc...) of TGD seems to converge to single coherent conceptual framework.

Have leptoquarks been observed in B meson decays?

Jester told in his blog "Resonaances" about an evidence for anomalies in the decays of B meson to K meson and lepton pair. There exist several anomalies.

1. The 3.7 sigma deviation from standard model predictions in the differential distribution of the B→ K^*μ⁺μ^- decay products.
   
   
2. The 2.6 sigma violation of lepton flavor universality in B⁺→ K⁺l⁺l^- decays.
   
   

The reported violation of lepton universality (, which need not be real) is especially interesting. The branching ratio B(B⁺→ K⁺e⁺e^-)/B(B⁺→ K⁺μ⁺μ^-)≈ .75 holds true. Standard model expectation is very near to unity.

Scalar lepto-quark has been proposed as an explanation of the anomaly. The lowest order diagram for lepton pair production in standard model is penguin diagram obtained from the self energy diagram for b quark involving tW^- intermediate in which W emits γ/Z decaying to lepton pair. Lepton universality is obvious. The penguin diagram involves 4 vertices and 4 propagators and the product of CKM matrix elements V_tbV^*_st. The diagram involving leptoquark is obtained from this diagram by a modification.

The diagram would induce an effective four-fermion coupling bbar_Lγ ^μs_L μ⁺_Lγ_μ μ^-_L representing neutral current breaking universality. Authors propose a heavy scalar boson exchanges with quantum numbers of lepto-quark and mass of order 10 TeV to explain why no anomalous weak interactions between leptons and quarks by lepto-quark exchange have not been observed. Scalar nature would suggest Higgs type coupling proportional to mass of the lepton and this could explain why the effect of exchange is smaller in the case of electron pair. The effective left-handed couplings would however suggest vector lepto-quarks with couplings analogous to W boson coupling. Note that the effect should reduce the rate: the measured rate for B_s → μ^-μ⁺ is .79+/- .20: reduction would be due to destructive interference of amplitudes.

General ideas

Some general ideas about TGD are needed in the model and are listed in order to avoid the impression that the model is just ad hoc construct.

1. In TGD all elementary particle can be regarded as pairs of wormhole contacts through which monopole magnetic flux flows: two wormhole contacts are necessary to get closed magnetic field lines. Monopole flux in turn guarantees the stability of the wormhole contact. In the case of weak bosons second wormhole contact carries fermion and antifermion at opposite throats giving rise to the net charges of the boson. The neutrino pair at the second wormhole contact neutralize the weak charges and guarantees short range of weak interactions.
   
   
2. The TGD inspired explanation of family replication phenomenon is in terms of the genus of the partonic 2-surfaces (wormhole throat) at the end of causal diamond. There is topological mixing of partonic topologies which depend on weak quantum numbers of the wormhole throat leading to CKM mixing. Lepton and quark families obvious correspond to each other: L(g)↔ q(g) and this is important in the model to be considered.
   

   The genera of the opposite wormhole throats are assumed to be identical for bosonic wormhole contacts. This can be assumed also for fermionic wormhole contacts for which only second throat carries fermion number. The universality of standard model couplings inspires the hypothesis that bosons are superpositions of the three lowest genera forming singlets with respect effective symmetry group SU(3)_g associated with the 3 lowest genera. Gauge bosons involve also superpositions of various fermion pairs with coefficients determined by the charge matrix.
   
   

3. p-Adic length scale hierarchy is one of the key predictions of TGD. p-Adic length scale hypothesis (to be used in the sequel) stating that p-adic primes are near powers of of 2: p≈ 2^k, k integer, relies on the success of p-adic mass calculations. p-Adic length scale hypothesis poses strong constraints on particle mass scales and one can readily estimate the mass of possible p-adically scaled up variants of masses of known elementary particles.
   

   One of the basic predictions is the possibility of p-adically scaled up variants of ordinary hadron physics and also of weak interaction physics. One such prediction is M₈₉ ~ hadron physics, which is scaled up variant of the ordinary M₁₀₇ hadron physics with mass scale which is by a factor 512 higher and corresponds to the energy scale relevant at LHC. Hence LHC might eventually demonstrate the feasibility of TGD.
   

   Quite generally, one can argue that one should speak about M₈₉ physics in which exotic variants of weak bosons and scaled up variants of hadrons appear. There would be no deep distinction between weak bosons and M₈₉ hadrons and elementary particles in general: all of them would correspond to string like objects involving both magnetic flux tubes carrying monopole flux between two wormhole throats and string world sheets connecting the light-like orbits of wormhole throats at which the signature of the induced metric changes.
   
   

4. TGD predicts dark matter hierarchy based on phases with non-standard value h_eff=n× h of Planck constant. The basic applications are to living matter but I have considered also particle physics applications.
   
   
   1. Dark matter in TGD sense provides a possible explanation for the experimental absence of super partners of ordinary particles: sparticles would be dark and would be characterized by the same p-adic mass scales as sparticles
      
      
   2. TGD predicts also colored leptons and there is evidence for meson like bound states of colored leptons. Light colored leptons are however excluded by the decay widths of weak bosons but also now darkness could save the situation.
      
      
   3. I have also proposed that RHIC anomaly observed in heavy ion collisions and its variant for proton heavy ion collisions at LHC suggesting string like structures can be interpreted in terms of low energy M₈₉ hadron physics but with large value of h_eff meaning that the M₈₉ p-adic length scale increases to M₁₀₇ p-adic length scale (ordinary hadronic length scale).
      
      
   One can consider also the adventurous possibility that vector lepto-quarks are dark in TGD sense.
5. TGD view about gauge bosons allows to consider also lepto-quark type states. These bosons would have quark and lepton at opposite wormhole throats. One can consider bosons which are SU(3)_g singlets defined by superpositions of L(g)q(g) or L(g)qbar(g). These states can be either M⁴ vectors or scalars (all bosons are vectors in 8-D sense in TGD by 8-D chiral symmetry guaranteeing separate conservation of B and L). Left handed couplings to quarks and leptons analogous to those of W bosons are suggested by the model for the anomalies. Vector lepto-quarks can be consistent with what is known about weak interactions only if they are dark in TGD sense. Scalar lepto-quarks could have ordinary value of Planck constant.
   
   

A TGD based model for the B anomaly in terms of lepto-quarks

It is natural to approach also the anomaly under discussion by assuming the basic framework just described. The anomaly in the decay amplitude of B→ Kμ^-μ⁺ could be due to an additional contribution based on a simple modification for the standard model amplitude.

1. In TGD framework, and very probably also in the model studied in the article, the starting point is the penguin diagram for lepton pair production in B→ Kμ^-μ⁺ decay involving only the decay b→ sl⁺l^- by virtual tW state emitting virtual γ/Z decaying to lepton pair and combining with t to form s.
   
   
   1. The diagram for lepton pair production involving virtual lepto-quark is obtained from the tW^- self-energy loop for b. One can go around the W^- branch of the loop to see what must happen. The loop starts with b→ tW^- followed by W^-→ l^-(g₁) νbar(g₁) producing on mass shell lepton l^-(g₁). This is followed by νbar(g₁)→ s X(Dbarνbar) producing on mass shell s. The genus of the virtual neutrino must ge g=1 unless
      leptonic CKM mixing is allow in the W decay vertex.
      

      After this one has X=∑Dbar(g)νbar(g) → Dbar(g₂)νbar(g₂). Any value of g₂ is possible. Finally, one has tDbar→ W⁺ and W⁺ νbar(g₂)) → l⁺(g₂). There are two loops involved and four lines contain a heavy particle (two W bosons, t, and X). The diagram contains 6 electroweak vertices whereas the standard model diagram has 4 vertices.
      
      

   2. All possible lepton pairs can be produced. The amplitude is proportional to the product V_tbV^*_tD(g2) implying breaking of lepton universality. The amplitude for production of e⁺μ^- pair is considerably smaller than that for μ⁺μ_- and τ⁺μ_- as the experimental findings suggest. If neutrino CKM mixing is taken into account, there is also a proportional to the matrix element V^L_l(g1)νg=1.
      

      In absence of leptonic CKM mixing (mixing explains the recently reported production of μ⁺e^- pairs in the decays of Higgs) only μ^-l⁺(g) pairs are produced. The possibility to have g≠ 1 is also a characteristic of lepton non-universality, which is however induced by the hadronic CKM mixing: lepto-quark couplings are universal.
      

      Note that the flavour universality of the gauge couplings means in the case of lepto-quarks that Lq pairs superpose to single SU(3)_g singlet as for ordinary gauge bosons. If L(g)q(g) would appear as separate particles, only μ⁺μ^- pairs would be produced in absence of leptonic CKM mixing.
      
      

2. A rough estimate for the ratio r of lepto-quark amplitude A(b→ sl^- (g₁)l⁺ (g₂) to the amplitude A(b→ sl^-(g)l⁺(g) involving virtual photon decaying to l⁺l^- pair is
   

   z =[X₁/X₂]× [F₁(x_X,x_t)/F₂(x_t)],
   

   X₁=V^L_l1ν(g=1) [∑_gV^L_l(g2)ν(g)
   V^*_D(g)t]V_tD(g2)g
   _X²g_W² ,
   

   X₂=V^*_dte² ,
   

   x_X= m²(X)/m²(W) , x_t= m²(t)/m²(W) .
   

   The functions F_i correspond come from the loop integral and depend on mass ratios appearing as the argument. The factors X_i collect various coupling parameters together.
   
   

3. The objection is that the model predicts a contribution to the scattering of leptons and quarks of the same family (L(g)-q(g) scattering) by the exchange of lepto-quark, which is of the same order of magnitude as for ordinary weak interactions. This should have been observed in high precision experiments testing standard model if the mass of the lepto-quark is of the same magnitude as weak boson mass. 10 TeV mass scale for lepto-quarks should guarantee that this is not the case and is probably the basic motivation for the estimate of the article. This requires that the ratio of the loop integrals appearing in z is of the order of unity. For a processional it should be easy to check this. Since the loop integral in the case of scalar lepto-quark studied in the article has the desired property and should not depend on the spin of the particles in the loops, one has good reasons to expect that the same holds true also for vector lepto-quarks.
   

   Without precise numerical calculation one cannot be sure that the loop integral ratio is not too large. In this case one could reduce the gauge coupling to lepto-quarks (expected to be rather near to weak coupling constant strength) but this looks like ad hoc trick. A more adventurous manner to overcome the problem would be to assume that lepto-quarks represent dark matter in TGD sense having effective Planck constant h_eff=n× h. Therefore they would not be visible in the experiments, which do not produce dark matter in elementary particle length scales.
   
   

4. The proposal of the article is that lepto-quark is scalar so that its coupling strength to leptons and quarks increases with mass scale. If I have understood correctly, the motivation for this assumption is that only in this manner the effect on the rate for e⁺e^- production is smaller than in the case of μ⁺μ^- pair. As found, the presence of CKM matrix elements in lepto-quark emission vertices at which quark charge changes, guarantees that both anomalous contributions to the amplitude are for electron pair considerably smaller than for muon pair.
   
   
5. Consider first a mass estimate for dark vector lepto-quark assumed to have weak boson mass scale. Even the estimate m(X)∼ m(W) is much higher than the very naive estimate as a sum of μ^- and s masses would suggest. Quite generally, if weak bosons, lepto-quarks, and M₈₉ hadrons are all basic entities of same M₈₉ physics, the mass scale is expected to be that of M₈₉ hadron physics and of the order of weak mass scale. A very naive scaling estimate for the mass would be by factor 512 and give an estimate around 50 GeV. If μ^- mass is scaled by the same factor 512, one obtains mass of order 100 GeV consistent with the estimate for the magnitude of the anomaly.
   

   Second p-adic mass scale estimate assumes vector or scalar lepto-quark with mass scale not far from 10 TeV. Ordinary μ^- corresponds to Gaussian Mersenne M_G,k, k=113. If p-adically scaled up variant of lepton physics is involved, the electron of the p-adically scaled up lepton physics could correspond to M₈₉. If muons correspond to Gaussian primes then the scaled up muon would correspond to the smallest Gaussian Mersenne prime below M₈₉, which is M_G,79. The mass of the scaled up muon would be obtained from muon mass by scaling by a factor 2^(113-79)/2 =2¹⁷=1.28× 10⁵ giving mass of order 10 TeV, which happens to be consistent with the conservative estimate of the article.
   

See the chapter New Particle Physics Predicted by TGD of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" or the article Have lepto-quarks been observed in the decays of B mesons?.

Permutations, braidings, and amplitudes

Nima Arkani-Hamed et al demonstrate that various twistorially represented on-mass-shell amplitudes (allowing light-like complex momenta) constructible by taking products of the 3-particle on mass-shell amplitude and its conjugate can be assigned with unique permutations of the incoming lines. The article describes the graphical representation of the amplitudes and its generalization. For 3-particle amplitudes, which correspond to ++- and +-- twistor amplitudes, the corresponding permutations are cyclic permutations, which are inverses of each other. One actually introduces double cover for the labels of the particles and speaks of decorated permutations meaning that permutation is always a right shift in the integer and in the range [1,2× n].

Amplitudes as representation of permutations

It is shown that for on mass shell twistor amplitudes the definition using on-mass-shell 3-vertices as building bricks is highly reducible: there are two moves for squares defining 4-particle sub-amplitudes allowing to reduce the graph to a simpler one. The first move is topologically like the s-t duality of the old-fashioned string models and second one corresponds to the transformation black ↔ white for a square sub-diagram with lines of same color at the ends of the two diagonals and built from 3-vertices.

One can define the permutation characterizing the general on mass shell amplitude by a simple rule. Start from an external particle a and go through the graph turning in in white (black) vertex to left (right). Eventually this leads to a vertex containing an external particle and identified as the image P(a) of the a in the permutation. If permutations are taken as right shifts, one ends up with double covering of permutation group with 2× n! elements - decorated permutations. In this manner one can assign to any any line of the diagram two lines. This brings in mind 2-D integrable theories where scattering reduces to braiding and also topological QFTs where braiding defines the unitary S-matrix. In TGD parton lines involve braidings of the fermion lines so that an assignment of permutation also to vertex would be rather nice.

BCFW bridge has an interpretation as a transposition of two neighboring vertices affecting the lines of the permutation defining the diagram. One can construct all permutations as products of transpositions and therefore by building BCFW bridges. BCFW bridge can be constructed also between disjoint diagrams as done in the BCFW recursion formula.

Can one generalize this picture in TGD framework? There are several questions to be answered.

1. What should one assume about the states at the light-like boundaries of string world sheets? What is the precise meaning of the supersymmetry: is it dynamical or gauge symmetry or both?
   
   
2. What does one mean with particle: partonic 2-surface or boundary line of string world sheet? How the fundamental vertices are identified: 4 incoming boundaries of string world sheets or 3 incoming partonic orbits or are both aspects involved?
   
   
3. How the 8-D generalization of twistors bringing in second helicity and doubling the M⁴ helicity states assignable to fermions does affect the situation?
   
   
4. Does the crucial right-left rule relying heavily on the possibility of only 2 3-particle vertices generalize? Does M⁴ massivation imply more than 2 3-particle vertices implying many-to-one correspondence between on-mass-shell diagrams and permutations? Or should one generalize the right-left rule in TGD framework?
   
   

Fermion lines for fermions massless in 8-D sense

What does one mean with particle line at the level of fermions?

1. How the addition of CP₂ helicity and complete correlation between M⁴ and CP₂ chiralities does affect the rules of N=4 SUSY? Chiral invariance in 8-D sense guarantees fermion number conservation for quarks and leptons separately and means conservation of the product of M⁴ and CP₂ chiralities for 2-fermion vertices. Hence only M⁴ chirality need to be considered. M⁴ massivation allows more 4-fermion vertices than N=4 SUSY.
   
   
2. One can assign to a given partonic orbit several lines as boundaries of string world sheets connecting the orbit to other partonic orbits. Supersymmetry could be understoond in two manners.
   
   
   1. The fermions generating the state of super-multiplet correspond to boundaries of different string world sheets which need not connect the string world sheet to same partonic orbit. This SUSY is dynamical and broken. The breaking is mildest breaking for line groups connected by string world sheets to same partonic orbit. Right handed neutrinos generated the least broken N=2 SUSY.
      
      
   2. Also single line carrying several fermions would provide realization of generalized SUSY since the multi-fermion state would be characterized by single 8-momentum and helicity. One would have N =4 SUSY for quarks and leptons separately and N =8 if both quarks and leptons are allowed. Conserved total for quark and antiquarks and leptons and antileptons characterize the lines as well.

      What would be the propagator associated with many-fermion line? The first guess is that it is just a tensor power of single fermion propagator applied to the tensor power of single fermion states at the end of the line. This gives power of 1/p²ⁿ to the denominator, which suggests that residue integral in momentum space gives zero unless one as just single fermion state unless the vertices give compensating powers of p. The reduction of fermion number to 0 or 1 would simplify the diagrammatics enormously and one would have only 0 or 1 fermions per given string boundary line. Multi-fermion lines would represent gauge degrees of freedom and SUSY would be realized as gauge invariance. This view about SUSY clearly gives the simplest picture, which is also consistent with the earlier one, and will be assumed in the sequel
      
      

3. The multiline containing n fermion oscillator operators can transform by chirality mixing in 2ⁿ manners at 4-fermion vertex so that there is quite a large number of options for incoming lines with n_i fermions.
   
   
   
4. In 4-D Dirac equation light-likeness implies a complete correlation between fermion number and chirality. In 8-D case light-likeness should imply the same: now chirality correspond to fermion number. Does this mean that one must assume just superposition of different M⁴ chiralities at the fermion lines as 8-D Dirac equation requires. Or should one assume that virtual fermions at the end of the line have wrong chirality so that massless Dirac operator does not annihilate them?
   
   

Fundamental vertices

One can consider two candidates for fundamental vertices depending on whether one identifies the lines of Feynman diagram as fermion lines or as light-like orbits of partonic 2-surfaces. The latter vertices reduces microscopically to the fermionic 4-vertices.

1. If many-fermion lines are identified as fundamental lines, 4-fermion vertex is the fundamental vertex assignable to single wormhole contact in the topological vertex defined by common partonic 2-surface at the ends of incoming light-like 3-surfaces. The discontinuity is what makes the vertex non-trivial.
   
   
2. In the vertices generalization of OZI rule applies for many-fermion lines since there are no higher vertices at this level and interactions are mediated by classical induced gauge fields and chirality mixing. Classical induced gauge fields vanish if CP₂ projection is 1-dimensional for string world sheets and even gauge potentials vanish if the projection is to geodesic circle. Hence only the chirality mixing due to the mixing of M⁴ and CP₂ gamma matrices is possible and changes the fermionic M⁴ chiralities. This would dictate what vertices are possible.
   
   
3. The possibility of two helicity states for fermions suggests that the number of amplitudes is considerably larger than in N=4 SUSY. One would have 5 independent fermion amplitudes and at each 4-fermion vertex one should be able to choose between 3 options if the right-left rule generalizes. Hence the number of amplitudes is larger than the number of permutations possibly obtained using a generalization of right-left rule to right-middle-left rule.
   
   
4. Note however that for massless particles in M⁴ sense the reduction of helicity combinations for the fermion and antifermion making virtual gauge boson happens. The fermion and antifermion at the opposite wormhole throats have parallel four-momenta in good approximation. In M⁴ they would have opposite chiralities and opposite helicities so that the boson would be M⁴ scalar. No vector bosons would be obtained in this manner.

   In 8-D context it is possible to have also vector bosons since the M⁴ chiralities can be same for fermion and anti-fermion. The bosons are however massive, and even photon is predicted to have small mass given by p-adic thermodynamics. Massivation brings in also the M⁴ helicity 0 state. Only if zero helicity state is absent, the fundamental four-fermion vertex vanishes for ++++ and ---- combinations and one extend the right-left rule to right-middle-left rule. There is however no good reason for he reduction in the number of 4-fermion amplitudes to take place.
   
   

Partonic surfaces as 3-vertices

At space-time level one could identify vertices as partonic 2-surfaces.

1. At space-time level the fundamental vertices are 3-particle vertices with particle identified as wormhole contact carrying many-fermion states at both wormhole throats. Each line of BCFW diagram would be doubled. This brings in mind the representation of permutations and leads to ask whether this representation could be re-interpreted in TGD framework. For this option the generalization of the decomposition of diagram to 3-particle vertices is very natural. If the states at throats consist of bound states of fermions as SUSY suggests, one could characterize them by total 8-momentum and helicity in good approximation. Both helicities would be however possible also for fermions by chirality mixing.
   
   
2. A genuine decomposition to 3-vertices and lines connecting them takes place if two of the fermions reside at opposite throats of wormhole contact identified as fundamental gauge boson (physical elementary particles involve two wormhole contacts).

   The 3-vertex can be seen as fundamental and 4-fermion vertex becomes its microscopic representation. Since the 3-vertices are at fermion level 4-vertices their number is greater than two and there is no hope about the generalization of right-left rule.
   
   

OZI rule implies correspondence between permutations and amplitudes

The realization of the permutation in the same manner as for N=4 amplitudes does not work in TGD. OZI rule following from the absence of 4-fermion vertices however implies much simpler and physically quite a concrete manner to define the permutation for external fermion lines and also generalizes it to include braidings along partonic orbits.

1. Already N=4 approach assumes decorated permutations meaning that each external fermion has effectively two states corresponding to labels k and k+n (permutations are shifts to the right). For decorated permutations the number of external states is effectively 2ⁿ and the number of decorated permutations is 2× n!. The number of different helicity configurations in TGD framework is 2ⁿ for incoming fermions at the vertex defined by the partonic 2-surface. By looking the values of these numbers for lowest integers one finds 2n≥ 2ⁿ: for n=2 the equation is saturated. The inequality log(n!) >nlog(n)/e)+1 (see Wikipedia) gives

   log(2n!)/log(2ⁿ)geq; log(2)+ 1+nlog(n/e)/nlog(2)= log(n/e)/log(2) +O(1/n)

   so that the desired inequality holds for all interesting values of n.
   
   

2. If OZI rule holds true, the permutation has very natural physical definition. One just follows the fermion line which must eventually end up to some external fermion since the only fermion vertex is 2-fermion vertex. The helicity flip would map k→ k+n or vice versa.
   
   
3. The labelling of diagrams by permutations generalizes to the case of diagrams involving partonic surfaces at the boundaries of causal diamond containing the external fermions and the partonic 2-surfaces in the interior of CD identified as vertices. Permutations generalize to braidings since also the braidings along the light-like partonic 2-surfaces are allowed. A quite concrete generalization of the analogs of braid diagrams in integrable 2-D theories emerges.
   
   
4. BCFW bridge would be completely analogous to the fundamental braiding operation permuting two neighboring braid strands. The almost reduction to braid theory - apart from the presence of vertices conforms with the vision about reduction of TGD to almost topological QFT.
   
   

To sum up, the simplest option assumes SUSY as both gauge symmetry and broken dynamical symmetry. The gauge symmetry relates string boundaries with different fermion numbers and only fermion number 0 or 1 gives rise to a non-vanishing outcome in the residue integration and one obtains the picture used hitherto. If OZI rule applies, the decorated permutation symmetry generalizes to include braidings at the parton orbits and k→ k +/- n corresponds to a helicity flip for a fermion going through the 4-vertex. OZI rule follows from the absence of non-linearities in Dirac action and means that 4-fermion vertices in the usual sense making theory non-renormalizable are absent. Theory is essentially free field theory in fermionic degrees of freedom and interactions in the sense of QFT are transformed to non-trivial topology of space-time surfaces.

See the chapter Classical part of the twistor story of "Towards M-matrix" or the article Classical part of the twistor story.

What about non-planar Feynman diagrams in twistor approach

Non-planar Feynman diagrams have remained a challenge for the twistor approach. The problem is simple: there is
no canonical ordering of the extrenal particles and the loop integrand involving tricky shifts in integrations to get finite outcome is not unique and well-defined so that twistor Grasmann approach encounters difficulties.

Recently Nima Arkani-Hamed et al have have also considered non-planar MHV diagrams (having minimal number of "wrong" helicities) of N=4 SUSY, and shown that they can be reduced to non-planar diagrams for different permutations of vertices of planar diagrams ordered naturally. There are several integration regions identified as positive Grassmannians corresponding to different orderings of the external lines inducing non-planarity. This does not however hold true generally.

At the QFT limit the crossings of lines emerges purely combinatorially since Feynman diagrams are purely combinatorial objects with the ordering of vertices determining the topological properties of the diagram. Non-planar diagrams correspond to diagrams, which do not allow crossing-free imbedding to plane but require higher genus surface to get rid of crossings.

1. The number of the vertices of the diagram and identification of lines connecting them determines the diagram as a graph. This defines also in TGD framework Feynman diagram like structure as a graph for the fermion lines and should be behind non-planarity in QFT sense.
   
   
2. Could 2-D Feynman graphs exists also at geometric rather than only combinatorial level? Octonionization at imbedding space level requires identification of preferred M²⊂ M⁴ defining a preferred hyper-complex sub-space. Could the projection of the Fermion lines defined concrete geometric representation of Feynman diagrams?
   
   
3. Despite their purely combinatorial character Feynman diagrams are analogous to knots and braids. For years ago I proposed the generalization of the construction of knot invariants in which one gradually eliminates the crossings of the knot projection to end up with a trivial knot is highly suggestive as a procedure for constructing the amplitudes associated with the non-planar diagrams. The outcome should be a collection of planar diagrams calculable using twistor Grassmannian methods. Scattering amplitudes could be seen as analogs of knot invariants. The reduction of MHV diagrams to planar diagrams could be an example of this procedure.
   
   

One can imagine also analogs of non-planarity, which are geometric and topological rather than combinatorial and not visible at the QFT limit of TGD.

1. The fermion lines representing boundaries of string world sheets at the light-like orbits of partonic 2-surfaces can get braided. The same can happen also for the string boundaries at space-like 3-surfaces at the ends of the space-time surface. The projections of these braids to partonic 2-surfaces are analogs of non-planar diagrams. If the fermion lines at single wormhole throat are regarded effectively as a line representing one member of super-multiplet, this kind of braiding remains below the resolution used and cannot correspond to the braiding at QFT limit.
   
   
2. 2-knotting and 2-braiding are possible for partonic 2-surfaces and string world sheets as 2-surfaces in 4-D space-time surfaces and have no counterpart at QFT limit.
   
   
3. If one can approximate space-time sheets by maps from M⁴ to CP₂, one expects General Relativity and QFT description to be good approximations. GRT space-time is obtained by replacing space-time sheets with single sheet - a piece of slightly deformed Minkowski space but without assupmtion about imbedding to H. Induced classical gravitational field and gauge fields are sums of those associated with the sheets. The generalized Feynman diagrams with lines at various sheets and going also between sheets are projected to single piece of M⁴. Many-sheetedness makes 1-homology non-trivial and implies analog of braiding, which should be however invisible at QFT limit.
   
   

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

About the twistorial description of light-likeness in 8-D sense using octonionic spinors

The twistor approach to TGD require that the expression of light-likeness of M⁴ momenta in terms of twistors generalizes to 8-D case. The light-likeness condition for twistors states that the 2× 2 matrix representing M⁴ momentum annihilates a 2-spinor defining the second half of the twistor. The determinant of the matrix reduces to momentum squared and its vanishing implies the light-likeness. This should be generalized to a situation in one has M⁴ and CP₂ twistor, which are not light-like separately but light-likeness in 8-D sense holds true (allowing massive particles in M⁴ sense and thus generalization of twistor approach for massive particles).

The case of M⁸=M⁴× E⁴

M⁸-H duality suggests that it might be useful to consider first the twistorialiation of 8-D light-likeness first the simpler case of M⁸ for which CP₂ corresponds to E⁴. It turns out that octonionic representation of gamma matrices provide the most promising formulation.

In order to obtain quadratic dispersion relation, one must have 2× 2 matrix unless the determinant for the 4× 4 matrix reduces to the square of the generalized light-likeness condition.

1. The first approach relies on the observation that the 2× 2 matrices characterizing four-momenta can be regarded as hyper-quaternions with imaginary units multiplied by a commuting imaginary unit. Why not identify space-like sigma matrices with hyper-octonion units?
   
   
2. The square of hyper-octonionic norm would be defined as the determinant of 4× 4 matrix and reduce to the square of hyper-octonionic momentum. The light-likeness for pairs formed by M⁴ and E⁴ momenta would make sense.
   
   
3. One can generalize the sigma matrices representing hyper-quaternion units so that they become the 8 hyper-octonion units. Hyper-octonionic representation of gamma matrices exists (γ₀=σ_z× 1, γ_k= σ_y× I_k) but the octonionic sigma matrices represented by octonions span the Lie algebra of G₂ rather than that of SO(1, 7). This dramatically modifies the physical picture and brings in also an additional source of non-associativity. Fortunately, the flatness of M⁸ saves the situation.
   
   
4. One obtains the square of p²=0 condition from the massless octonionic Dirac equation as vanishing of the determinant much like in the 4-D case. Since the spinor connection is flat for M⁸ the hyper-octonionic generalization indeed works.
   
   

This is not the only possibility that I have considered.

1. Is it enough to allow the four-momentum to be complex? One would still have 2× 2 matrix and vanishing of complex momentum squared meaning that the squares of real and imaginary parts are same (light-likeness in 8-D sense) and that real and imaginary parts are orthogonal to each other. Could E⁴ momentum correspond to the imaginary part of four-momentum?
   
   
2. The signature causes the first problem: M⁸ must be replaced with complexified Minkowski space M_c⁴ for to make sense but this is not an attractive idea although M_c⁴ appears as sub-space of complexified octonions.
   
   
3. For the extremals of Kähler action Euclidian regions (wormhole contacts identifiable as deformations of CP₂ type vacuum extremals) give imaginary contribution to the four-momentum. Massless complex momenta and also color quantum numbers appear also in the standard twistor approach. Also this suggest that complexification occurs also in 8-D situation and is not the solution of the problem.
   
   

The case of M⁸=M⁴× CP₂

What about twistorialization in the case of M⁴× CP₂? The introduction of wave functions in the twistor space of CP₂ seems to be enough to generalize Witten's construction to TGD framework and that algebraic variant of twistors might be needed only to realize quantum classical correspondence. It should correspond to tangent space counterpart of the induced twistor structure of space-time surface, which should reduce effectively to 4-D one by quaternionicity of the space-time surface.

1. For H=M⁴× CP₂ the spinor connection of CP₂ is not trivial and the G₂ sigma matrices are proportional to M⁴ sigma matrices and act in the normal space of CP₂ and to M⁴ parts of octonionic imbedding space spinors, which brings in mind co-associativity. The octonionic charge matrices are also an additional potential source of non-associativity even when one has associativity for gamma matrices.
   

   Therefore the octonionic representation of gamma matrices in entire H cannot be physical. It is however equivalent with ordinary one at the boundaries of string world sheets, where induced gauge fields vanish. Gauge potentials are in general non-vanishing but can be gauge transformed away. Here one must be of course cautious since it can happen that gauge fields vanish but gauge potentials cannot be gauge transformed to zero globally: topological quantum field theories represent basic example of this.
   
   

2. Clearly, the vanishing of the induced gauge fields is needed to obtain equivalence with ordinary induced Dirac equation. Therefore also string world sheets in Minkowskian regions should have 1-D CP₂ projection rather than only having vanishing W fields if one requires that octonionic representation is equivalent with the ordinary one. For CP₂ type vacuum extremals electroweak charge matrices correspond to quaternions, and one might hope that one can avoid problems due to non-associativity in the octonionic Dirac equation. Unless this is the case, one must assume that string world sheets are restricted to Minkowskian regions. Also imbedding space spinors can be regarded as octonionic (possibly quaternionic or co-quaternionic at space-time surfaces): this might force vanishing 1-D CP₂ projection.
   
   
   1. Induced spinor fields would be localized at 2-surfaces at which they have no interaction with weak gauge fields: of course, also this is an interaction albeit very implicit one! This would not prevent the construction of non-trivial electroweak scattering amplitudes since boson emission vertices are essentially due to re-groupings of fermions and based on topology change.
      
      
   2. One could even consider the possibility that the projection of string world sheet to CP₂ corresponds to CP₂ geodesic circle at which also the induced gauge potentials vanish so that one could assign light-like 8-momentum to entire string world sheet, which would be minimal surface in M⁴× S¹. This would mean enormous technical simplification in the structure of the theory. Whether the spinor harmonics of imbedding space with well-defined M⁴ and color quantum numbers can co-incide with the solutions of the induced Dirac operator at string world sheets defined by minimal surfaces remains an open problem.
      
      
   3. String world sheets cannot be present inside wormhole contacts, which have 4-D CP₂ projection so that string world sheets cannot carry vanishing induced gauge fields. Therefore the strings in TGD are open.
      
      

Summarizing

To sum up, the generalization of the notion of twistor to 8-D context allows description of massive particles using twistors but requires that octonionic Dirac equation is introduced. If one requires that octonionic and ordinary description of Dirac equation are equivalent, the description is possible only at surfaces having at most 1-D CP₂ projection - geodesic circle for the most stringent option. The boundaries of string world sheets are such surfaces and also string world sheets themselves if they have 1-D CP₂ projection, which must be geodesic circle if also induce gauge potentials are required to vanish. In spirit with M⁸-H duality, string boundaries give rise to classical M⁸ twistorizalization analogous to the standard M⁴ twistorialization and generalize 4-momentum to massless 8-momentum whereas imbedding space spinor harmonics give description in terms of four-momentum and color quantum numbers. One has SO(4)-SU(3) duality: a wave function in the space of 8-momenta corresponds to SO(4) description of hadrons at low energies as opposed to that for quarks at high energies in terms of color. The M⁴ projection of the 8-D M⁸ momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle (EP) in the most concrete form: gravitational four-momentum equals to inertial four-momentum.
EP for internal quantum numbers is clearly more delicate. In twistorialization also helicity is brought and for CP₂ degrees of freedom M⁸ helicity means that electroweak spin is described in terms of helicity.

Biologists have a principle known as "ontogeny recapitulates phylogeny" (ORP) stating that the morphogenesis of the individual reflects evolution of the species. The principle seems to be realized also in theoretical physics - at least in TGD Universe. ORP would now say that the evolution of theoretical physics via the emergence of increasingly complex notion of particle reflects the structure physics itself. Point like particles are really there as points at partonic 2-surfaces carrying fermion number: their 1-D orbits correspond to the boundaries of string world sheets; 2-D hyper-complex string world sheets in flat space (M⁴× S¹) are there and carry induced spinors; also complex (or co-complex) partonic 2-surfaces (Euclidian string world sheets) and carry particle numbers; 3-D space-like surfaces at the ends of causal diamonds (CDs) and the 3-D light-like orbits of partonic 2-surfaces are there; 4-D space-time surfaces are there as quaternionic or co-quaternionic sub-manifolds of 8-D octonionic imbedding space: there the hierarchy ends since there are no higher-dimensional classical number fields. ORP would thus also realize evolution of mathematics at the level of physics.

The M⁴ projection of the 8-D M⁸ momentum must by quantum classical correspondence be equal to the four-momentum assignable to imbedding space-spinor harmonics serving as building bricks for various super-conformal representations. This is nothing but Equivalence Principle in the most concrete form: gravitational four-momentum equals to inertial four-momentum.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

The only game in the town: again!

I had thought that after its impressive failures super string theory as a physical theory would have been dead and buried as a theory of everything and would be allowed to lay in peace. I also thought that "string theory as the only in the game" nightmare would belong to horrors of past like the concentration camps of nazis. I was wrong.

There seems to be a media campaign to revive the beliefs in this thinking. As superstrings have turned out to be unable to say anything interesting about physics, some super string gurus have decided to "prove" mathematically that superstring theory is the only possible theory of quantum gravitation and blog audience is expected to believe the "proofs". Superstrings have been the only possible quantum theory of gravitation but for reasons having very little to do with science.

There is a popular article about these attempts in Quanta Magazine. The opinions of various academic authorities are asked about about this claim.

The arguments in favor of this claim - certainly not proofs in any imaginable sense of the word - represent some pieces of evidence that certain conformal field theories in 2-D Minkowski defining boundary of AdS₃ have a mass spectrum with exponentially growing density of states characterizing strings like objects so that AdS₃/CFT is plausible and theories could be dual to a string theory in 3-dimensional AdS₃. From this it is concluded by some magic leap of lawyer logic completely incomprehensible to me that super string theory is the only possible theory of quantum gravity. An additional conclusion of Lubos is that all those who do not believe this are idiots and enemies of science.

Peter Woit points out some of the fallacies of the argument. First of all, 3-D quantum gravity is topological quantum field theory so that it cannot have anything to do with super string model and gravitation since there are no gravitons: here one could however argue that the corresponding string theory is higher dimensional and involve S⁷ factor. Secondly the dimension is wrong: it should be 4 rather than 3.

These articles are excellent examples of the sloppy thinking prevailing in recent day theoretical physic - to say nothing about the catastrophic situation in blogs! One should first of all precisely define what string theory means. There are many theories involving string like objects as basic dynamical entities. That some conformal field theory leads to density of states of form assignable to strings like objects tells absolutely nothing about the status of super strings as theory of quantum gravitation.

Below some attempts to encourage more precise thinking based by posing three questions: What does one really mean with a theory of gravity and what does one believe to be true about gravity? What does one really mean with super-conformal invariance? What does one really mean with string like objects?

What does one mean with a theory of gravity?

Can one just start from the quantization attempts of general relativity and require generalization of the lowest order Feynman diagrammatics in Minkowski space background? Is gravitation really what we believe it to be. There are anomalies even at the level of solar system: Pioneer and Flyby anomalies. Dark matter is still mystery and all the proposed models have failed. Galactic dark matter does not seem to be a spherical halo as has been believed.

TGD has led to rather detailed view about new quantum physics realized in astrophysical scales via the hierarchy of Planck constants and explaining various anomalies (see for instance this and this). This view about quantum gravitation is based on experimental findings and differs dramatically from the ultra-naive assumption behind super string models assuming that everything is understood down to Planck scale.

What does one mean with super-conformal symmetry?

When one speaks of AdS/CFT, one should not forget that it is based on one particular definition of super-conformal invariance: the conformal invariance associated with 2-D surfaces. In TGD framework 2-D surfaces are replaced by 3-D surfaces with one light-like direction so that one still has metric 2-dimensionality. This implies a gigantic extension of super-conformal symmetries and replaces AdS/CFT duality involving 10-D space-times AdS_n ×S^10-n with a duality which realizes the counterpart of ordinary holography stating that 2-D partonic surfaces and their tangent space data plus possibly string world sheets code for physics.

This of course shows that superstrings are certainly not the only game in the town: the super-conformal symmetries of super-string models are infinitesimal as compared to those of TGD. TGD also predicts space-time dimension correctly, is unique since 4-D Minkowski space M⁴ and CP₂ are twistorially completely unique (their twistor space allows Kähler structure). And of course, the symmetries of TGD are those of standard model and TGD leads to a flow of non-trivial predictions in all scales: applications to quantum biology, neuroscience, and consciousness the most fascinating examples.

I do not want that TGD would be the only game in the town although it is very regrettable that I am still the only developer of this marvellous theory. The sociology of science is profoundly irrational.

What does one mean with super-symmetry?

Superstrings assume a very specific form of space-time super symmetry. The realization of space-time SUSY relies typically on the assumption of Majorana spinors leading to the non-conservation of fermion numbers. The failure of LHC to find N=1 SUSY has finally activated the attempts to build more realistic SUSY scenarios with Dirac fermions.

In TGD framework a different kind of SUSY indeed emerges as a dynamical symmetry and is unavoidably broken at the level of quantum state. The fermionic oscillator operators at partonic surfaces labelled by fermion number, helicity, weak quantum numbers, and what might be called classical 8-momentum generate the SUSY. There is a hierarchy of SUSY breakings. Right-handed neutrino generates N=2 SUSY with smallest breaking. The other H-spinor components generate N=4 SUSYs for both quarks and leptons combining to form N=8 SUSY of N=8 supergravity. These states can still have relatively low mass since statistics allows same wave function at the partonic 2-surface for the spinor modes involved. For higher modes there is "wiggling" in CP₂ scale and masses are measured using CP₂ mass as a natural unit. Hence a theory resembling N=4 SUSY should be a good approximation to TGD. By adding conformal weight one obtains super-conformal algebra.

What does one mean with strings?

First of all, I hate neither strings nor physics. I hate only attempts to build hegemonies based on "The only game in the town" thinking: intelligence and arrogance are mutually exclusive!;-). I just want want to tell that superstrings were the first and - sad to say - wrong guess and a much more feasible option emerged 7 years befor the first super string revolution (1984).

Strings in TGD

Consider first the notion of string in TGD. TGD predicts 4-D string like objects with 2-D string world sheet as projection to dominate the cosmology before transition to a radiation dominated phase by the counterpart of inflationary period. String like objects which I call magnetic flux tubes dominate physics of later times in all scales.

Also genuine 2-D string world sheets emerge also from TGD from a very general argument forcing the induced spinor fields to 2-D string world sheets: the electromagnetic charge for the modes of induced spinor fields must be well-defined and this requires that classical W boson fields vanish implying restriction to string world-sheets.

Recently a stronger variant of this condition has emerged. The octonionic variant of Dirac operator allows to define generalization of the basic formulas for 4-D twistors. The octonionic Dirac operator in its algebraic version must be equivalent with the ordinary one: associativity condition requires that all induced weak gauge fields must vanish at string world sheets so that they must have 1-D CP₂ projection and can appear only in regions with Minkowskian signature of the induced metric.

Furthermore, the gamma matrices associated with string world sheets are most naturally induced gamma matrices and SUSY requires that world sheet area appear besides Kähler action in Minkowskian regions. Gravitational constant would appear at the level of basic definition of theory and quantum criticality condition would fix its ratio to CP₂ length squared. The view about the relationship between inertial and gravitational four-momentum becomes more precise. This suggests that string world sheets do somewhat more than just "emerge" in TGD framework.

Twistor strings and their TGD counterparts

Strings could also mean twistor strings of Witten, which he proposed as a manner to understand the scattering amplitudes of super-symmetric Yang-Mills theories (SYMs). This construction generalizes in a natural manner to TGD by replacing Witten's string world sheets (2-D surfaces) with unions of partonic 2-surfaces and string world sheets assumed to be surface of space-time surface which is base space of its twistor space having imbedding as a 6-D surface in the production of twistor spaces of M⁴ and CP₂.

I have been just working with the twistorial construction of the generalization of S-matrix in TGD framework inspired by several breakthroughs in the mathematical understanding of TGD.

1. TGD as a physics based of 4-D space-time surface in 8-D imbedding space can be lifted to that for their 6-D twistor spaces representable as 6-surfaces in 12-D Cartesian product of twistor spaces of M⁴ and CP₂ (see this).
   
   
2. Extended conformal invariance can be realized as conformal gauge condition and classically these conditions state that all conformal charges of space-time surfaces vanish. This realized the notion of preferred extremal precisely (see this). As a matter fact, one has infinite hierarchy of conformal symmetry breakings defined by the vanishing conditions for the sub-algebras of conformal algebras with conformal weights coming as multiples of integer n. This gives rise to the hierarchy of Planck constants and dark matters.
   
   
3. Preferred extremal property fixes the space-time regions to a very high degree and there are arguments suggesting that in Euclidian regions one as quaternion-Kähler manifolds having twistor spaces which are so called Fano spaces (see this).
   
   
4. The generalization of ordinary complex analyticity to quaternion analyticity seems to make sense after all and extends the 3-D light-like conformal symmetry to its quaternionic analog in the interior of 4-D space-time surfaces (see this).
   
   
5. Witten's twistor string theory generalizes in natural manner to TGD (see this and this) framework. 2-D objects appear as partonic 2-surfaces and string world sheets carrying induced fermion fields and both are needed.
   
   
6. Also new insights about how projective sub-manifolds of twistor Grassmannians appearing as surfaces over which twistor amplitudes are obtained as residue integrals emerge. One can identify the complex moduli characterizing polynomials defining partonic 2-surfaces in the twistor space of imbedding space as complex coefficients of these polynomials and defined modulo overall complex scaling so that the moduli space
   is projective sub-manifold of a projective space or more generally, Grassmannian.
   

   One can also generalize also the notion of positive Grassmannian to complex situation. In the real case positivity follows from the condition that the integrand of the twistor integral is projectively well-defined and therefore non-negative. In complex case the positivity can be generalized to the assumption that various complex coordinates are positivity in the sense that they have values in hyperbolic space realized as upper half-plane of complex plane. Hyperbolic space defining various 2-D hyperbolic geometries. Positivity and its generalization have deep number theoretic meaning in TGD framework since they allow the algebraic continuation of the complex amplitudes to p-adic number fields (see this and this) .
   

Twistorial questions to ponder

For a non-specialist lacking the technical skills, the work related to twistors (for TGD view and references see
this) is a garden of mysteries and there are a lot of questions to be answered: most of them of course trivial for the specialist. Here a just few of them.

1. How the twistor string approach of Witten and its possible TGD generalization relate to the approach involving residue integration over projective sub-manifolds of Grassmannians G(k,n). Nima et al argue that one can transform Grassmannian representation to twistor string representation for tree amplitudes. The integration over G(k,n) translates to integration over the moduli space of complex curves of degree d= k-1+l, l≥ g is the number of loops. The moduli correspond to complex coefficients of the polynomial of degree d and they form naturally a projective space since an overall scaling of coefficients does not change the surfaces. One can expect also in the general case that moduli space of partonic 2-surfaces is projective sub-manifold of a projective space. What is so nice that loop corrections would correspond to the inclusion of higher degree surfaces.
   

   This connection gives hopes for understanding the integration contours in G(k,n) at deeper level in terms of the moduli spaces of partonic 2-surfaces possibly restricted by conformal gauge conditions.
   
   

2. The notion of positive Grassmannian is one of the central notions introduced by Nima et al. The claim is that the sub-spaces of the real Grassmannian G(k,n) contributing to the amplitudes for ++-- signature are such that the determinants of the k× k minors associated with ordered columns of the k× n matrix C representing point of G(k,n) are positive. To be precise, the signs of all minors are positive or negative simultaneously: only the ratios of the determinants defining projective invariants are positive.
   

   At the boundaries of positive regions some of the determinants vanish. What happens that some k-volumes degenerate to a lower-dimensional volume. Boundaries are responsible for the leading singularities of the scattering amplitudes and the integration measure associated with G(k,n) has logarithmic singularity at the boundaries. These boundaries obviously correspond to the boundaries of the moduli space for the partonic 2-surfaces.
   

   This condition has a partial generalization to the complex case: the determinants are non-vanishing. A possible further manner to generalize this condition would be that the determinants have positive real part so that apart from rotation by π/2 they would be in the upper half plane of complex plane - the hyperbolic plane playing key role in the theory of hyperbolic 2-manifolds for which it serves as universal covering space by a finite discrete subgroup of Lorentz group SL(2,C). The upper half-plane has therefore a deep meaning in the theory of Riemann surfaces and might have counterpart at the moduli space of partonic 2-surfaces. The projective space would be based - not on projectivization of Cⁿ but that of Hⁿ, H the upper half plane.
   
   

3. Could positivity have some even deeper meaning? Why positivity? In TGD framework the number theoretical universality of amplitudes suggests this. Canonical identification maps ∑ x_npⁿ→ ∑ x_np^-n p-adic number to non-negative reals. p-Adicization is possible for angle variables by replacing them by discrete phases, which are roots of unity. For non-angle like variables, which are non-negative by using canonical identification or its variant. The positivity should hold true for all structures involved, the G(2,n) points defined by the twistors characterizing momenta and helicities of particles (actually pairs of orthogonal planes defined by twistors and their conjugates), the moduli space of partonic 2-surfaces, etc...
   
   

For a summary about recent TGD view about construction of twistor amplitudes and TGD counterpart of N= 4 SUSY as dynamical and broken supersymmetry with Dirac spinors see the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Witten's twistor string approach and TGD

The twistor Grassmann approach has led to a phenomenal progress in the understanding of the scattering amplitudes of gauge theories, in particular the N=4 SUSY.

As a non-specialist I have been frustrated about the lack of concrete picture, which would help to see how twistorial amplitudes might generalize to TGD framework. A pleasant surprise in this respect was the proposal of a particle interpretation for the twistor amplitudes by Nima Arkani Hamed et al in the article "Unification of Residues and Grassmannian Dualities" .

In this interpretation incoming particles correspond to spheres CP₁ so that n-particle state corresponds to (CP₁)ⁿ/Gl(2) (the modding by Gl(2) might be seen as a kind of formal generalization of particle identity by replacing permutation group S₂ with Gl(2) of 2× 2 matrices). If the number of "wrong" helicities in twistor diagram is k, this space is imbedded to CP_k-1ⁿ/Gl(k) as a surface having degree k-1 using Veronese map to achieve the imbedding. The imbedding space can be identified as Grassmannian G(k,n) . This surface defines the locus of the multiple residue integral defining the twistorial amplitude.

The particle interpretation brings in mind the extension of single particle configuration space E³ to its Cartesian power E³ⁿ/S_n for n-particle system in wave mechanics. This description could make sense when point-like particle is replaced with 3-surface or partonic 2-surface: one would have Cartesian product of WCWs divided my S_n. The generalization might be an excellent idea as far calculations are considered but is not in spirit with the very idea of string models and TGD that many-particle states correspond to unions of 3-surfaces in H (or light-like boundaries of causal diamond (CD) in Zero Energy Ontology (ZEO).

Witten's twistor string theory

Witten's twistor string theory is more in spirit with TGD at fundamental level since it is based on the identification of generalization of vertices as 2-surfaces in twistor space.

1. There are several kinds of twistors involved. For massless external particles in eigenstates of momentum and helicity null twistors code the momentum and helicity and are pairs of 2-spinor and its conjugate. More general momenta correspond to two independent 2-spinors.
   

   One can perform twistor Fourier transform for the conjugate 2-spinor to obtain twistors coding for the points of compactified Minkowski space. Wave functions in this twistor space characterized by massless momentum and helicity appear in the construction of twistor amplitudes. BCFW recursion relation allows to construct more complex amplitudes assuming that intermediate states are on mass shells massless states with complex momenta.
   

   One can perform twistor Fourier transformation (there are some technical problems in Minkowski signature) also for the second 2-spinor to get what are called momentum twistors providing in some aspects simpler description of twistor amplitudes. These code for the four-momenta propagating between vertices at which the incoming particles arrive and the differences if two subsequent momenta are equal to massless external momenta.
   
   

2. In Witten's theory the interactions of incoming particles correspond to amplitudes in which the twistors appearing as arguments of the twistor space wave functions characterized by momentum and helicity are localized to complex curves X² of twistor space CP₃ or its Minkowskian counterpart. This can be seen as a non-local twistor space variant of local interactions in Minkowski space.
   

   The surfaces X² are characterized by their degree d (of the polynomial of complex coordinates defining the algebraic 2-surface) the genus g of the algebraic surface, by the number k of "wrong" (helicity violating) helicities, and by the number of loops of corresponding diagram of SUSY amplitude: one has d= k-1+ l, g≤ l. The interaction vertex in twistor space is not anymore completely local but the n particles are at points of the twistorial surface X².
   
   

Generalization of the notion of twistor to 8-D context

In the addition to the article Classical part of the twistor story a proposal generalizing Witten's approach to TGD is discussed. The following gives a very concise summary of the article.

Consider first various aspects of twistorialization.

1. The fundamental challenge is the generalization of the notion of twistor associated with massless particle to 8-D context, first for M⁴=M⁴× E⁴ and then for H= M⁴ × CP₂. The notion of twistor space solves this question at geometric level. As far as construction of the TGD variant of Witten's twistor string is considered, this might be quite enough.
   

   What is especially nice, that twistorialization extends to from spin to include also electroweak spin. These two spins correspond correspond to M⁴ and CP₂helicities for the twistor space amplitude, and are non-local properties of these amplitudes. In TGD framework only twistor amplitudes for which helicities correspond to that for massless fermion and antifermion are possible and by fermion number conservation the numbers of positive and negative helicities are identical and equal to the fermion number (or antifermion number). Separate lepton and baryon number conservation realizing 8-D chiral symmetry implies that M⁴ and CP₂ helicities are completely correlated. At twistorial level this means restriction to amplitudes with k=2(n(F)-n(Fbar)) if
   no mixing of M⁴and CP₂ occurs (massivation in M4 sense) . This in quark and lepton sector separately.
   
   

2. M⁸-H duality and quantum-classical correspondence however suggest that M⁸ twistors might allow tangent space description of four-momentum, spin, color quantum numbers and electroweak numbers and that this is needed. What comes in mind is the identification of fermion lines as light-like geodesics possessing M⁸ valued 8-momentum, which would define the long sought gravitational counterparts of four-momentum and color quantum numbers at classical point-particle level. The M⁸ part of this four-momentum would be equal to that associated with imbedding space spinor mode characterizing the ground state of super-conformal representation for fundamental fermion.
   

   Hence one might also think of starting from the 4-D condition relating Minkowski coordinates to twistors and looking what it could mean in the case of M⁸. The generalization is indeed possible in M⁸ =M⁴× E⁴ by its flatness if one replaces gamma matrices with octonionic gamma matrices.
   

   In the case of M⁴× CP₂ situation is different since for octonionic gamma matrices SO(1,7) is replaced with G₂, and the induced gauge fields have different holonomy structure than for ordinary gamma matrices and octonionic sigma matrides appearing as charge matrices bring in also an additional source of non-associativity. Certainly the notion of the twistor Fourier transform fails since CP₂ Dirac operator cannot be algebraized.
   

   Algebraic twistorialization however works for the light-like fermion lines at which the ordinary and octonionic representations for the induced Dirac operator are equivalent. One can indeed assign 8-D counterpart of twistor to the particle classically as a representation of light-like hyper-octonionic four-momentum having massive M⁴ and CP₂ projections and CP₂ part perhaps having interpretation in terms of classical tangent space representation for color and electroweak quantum numbers at fermionic lines.
   

   If all induced electroweak gauge fields - rather than only charged ones as assumed hitherto - vanish at string world sheets, the octonionic representation is equivalent with the ordinary one. The CP₂ projection of string world sheet should be 1-dimensional: inside CP₂ type vacuum extremals this is impossible, and one could even consider the possibility that the projection corresponds to CP₂ geodesic circle. This would be enormous technical simplification. What is important that this would not prevent obtaining non-trivial scattering amplitudes at elementary particle level since vertices would correspond to re-arrangement of fermion lines between the generalized lines of Feynman diagram meeting at the vertices (partonic 2-surfaces).
   
   

3. In the fermionic sector one is forced to reconsider the notion of the induced spinor field. The modes of the imbedding space spinor field should co-incide in some region of the space-time surface with those of the induced spinor fields. The light-like fermionic lines defined by the boundaries of string world sheets or their ends are the obvious candidates in this respect. String world sheets is perhaps too much to require. Number theoretically string world sheets would be commutative surfaces analogous to space-time surfaces as quaternionic surfaces of octonionic imbedding space.
   

   The only reasonable identification of string world sheet gamma matrices is as induced gamma matrices and super-conformal symmetry requires that the action contains string world sheet area as an additional term just as in string models. String tension would correspond to gravitational constant and its value - that is ratio to the CP₂ radius squared, would be fixed by quantum criticality.
   
   

Generalization of Witten's construction in TGD framework

The generalization of the Witten's geometric construction of scattering amplitudes relying on the induction of the twistor structure of the imbedding space to that associated with space-time surface looks surprisingly straight-forward and would provide more precise formulation of the notion of generalized Feynman diagrams forcing to correct some wrong details.

1. All generalized Feynman graphs defined in terms of Euclidian regions of space-time surface are lifted to twistor spaces. Incoming particles correspond quantum mechanically to twistor space amplitudes defined by their momenta and helicities and and classically to the entire twistor space of space-time surface as a surface in the twistor space of H. Of course, also the Minkowskian regions have this lift. The vertices of Feynman diagrams correspond to regions of twistor space in which the incoming twistor spaces meet along their 5-D ends having also S² bundle structure over space-like 3-surfaces. These space-like 3-surfaces correspond to ends of Euclidian and Minkowskian space-time regions separated from each other by light-like 3-surfaces at which the signature of the metric changes from Minkowskian to Euclidian. These "partonic orbits" have as their ends 2-D partonic surfaces. By strong form of General Coordinate Invariance implying strong of holography, these 2-D partonic surfaces and their 4-D tangent space data should code for quantum physics. Their lifts to twistor space are 4-D S² bundles having partonic 2-surface X² as base.
   

   One of the nice outcomes is that the genus appearing in Witten's formulation naturally corresponds to family replication in TGD framework.
   
   

2. All elementary particles are many particle bound states of massless fundamental fermions: the non-collinearity of massless momenta explains massivation. The fundamental fermions are localized at wormhole throats defining the light-like orbits of partonic 2-surfaces. Throats are associated with wormhole contacts connecting two space-time sheets. Stability of the contact is guaranteed by non-vanishing monopole magnetic flux through it and this requires the presence of second wormhole contact so that a closed magnetic flux tube carrying monopole flux and involving the two space-time sheets is formed. The net fermionic quantum numbers of the second throat correspond to particle's quantum numbers and above weak scale the weak isospins of the throats sum up to zero.
   
   
3. Fermionic 2-vertex is the only local many-fermion vertex being analogous to a mass insertion. The non-triviality of the theory could be seen to follow from the analog of OZI mechanism: the fermionic lines inside partonic orbits are redistributed in vertices. Lines can also turn around in time direction which corresponds to creation or annihilation of a pair. 3-particle vertices are obtained only in topological sense as 3 space-time surfaces are glued together at their ends. The interaction between fermions at different wormhole throats is described in terms of string world sheets.
   

   
   

4. The earlier proposal was that the fermions in the internal fermion lines are massless in M⁴ sense but have non-physical helicity so that the algebraic M⁴ Dirac operator emerging from the residue integration over internal four-momentum does not annihilate the state at the end of the propagator line. Now the algebraic induced Dirac operator defines the propagator at fermion lines. Should one assume generalization of non-physical helicity also now?
   
   
5. All this stuff must be lifted to twistorial level and one expects that the lift to S² bundle allows an alternative description of fermions and spinor structure so that one can speak of induced twistor structure instead of induced spinor structure. This approach allows also a realization of M⁴ conformal symmetries in terms of globally well-defined linear transformations so that it might be that twistorialization is not a mere reformulation but provides a profound unification of bosonic and fermionic degrees of freedom.
   

The emergence of fundamental 4-fermion vertex and of boson exchanges

The emergence of the fundamental 4-fermion vertex and of boson exchanges deserves a more detailed discussion.

1. I have proposed that the discontinuity of the Dirac operator at partonic two-surface (corner of fermion line) defines both the fermion boson vertex and TGD analog of mass insertion (not scalar but imbedding space vector) giving rise to mass parameter having interpretation as Higgs vacuum expectation and various fermionic mixing parameters at QFT limit of TGD obtained by approximating many-sheeted space-time of TGD with the single sheeted region of M⁴ such that gravitational field and gauge potentials are obtained as sums of those associated with the sheets.
   
   
2. Non-trivial scattering requires also correlations between fermions at different partonic 2-surfaces. Both partonic 2-surfaces and string world sheets are needed to describe these correlations. Therefore the string world sheets and partonic 2-surfaces cannot be dual: both are needed and this means deviation from Witten's theory. Fermion vertex corresponds to a "corner" of a fermion line at partonic 2-surface at which generalized 4-D lines of Feynman diagram meet and light-like fermion line changes to space-like one. String world sheet with its corners at partonic 2-surfaces (wormhole throats) describes the momentum exchange between fermions. The space-like string curve connecting two wormhole throats serves as the analog of the exchanged gauge boson.
   
   
3. Two kinds of 4-fermion amplitudes can be considered depending on whether the string connects throats of single wormhole contact (CP₂ scale) or of two wormhole contacts (p-adic length scale - typically of order elementary particle Compton length). If string worlds sheets have 1-D CP₂ projection, only Minkowskian string world sheets are possible. The exchange in Compton scale should be assignable to the TGD counterpart of gauge boson exchange and the fundamental 4-fermion amplitude should correspond to single wormhole contact: string need not to be involved now. Interaction is basically classical interaction assignable to single wormhole contact generalizing the point like vertex.
   
   
4. The possible TGD counterparts of BCFW recursion relations should use the twistorial representations of fundamental 4-fermion scattering amplitude as seeds. Yangian invariance poses very strong conditions on the form of these amplitudes and the exchange of massless bosons is suggestive for the general form of amplitude.
   

   The 4-fermion amplitude assignable to two wormhole throats defines the analog of gauge boson exchange and is expressible as fusion of two fundamental 4-fermion amplitudes such that the 8-momenta assignable to the fermion and anti-fermion at the opposite throats of exchanged wormhole contact are complex by BCFW shift acting on them to make the exchanged momenta massless but complex. This entity could be called fundamental boson (not elementary particle).
   
   

5. Can one assume that the fundamental 4-fermion amplitude allows a purely formal composition to a product of BFF amplitudes? Two 8-momenta at both FFB vertices must be complex so that at least two external fermionic momenta must be complex. These external momenta are naturally associated with the throats of the a wormhole contact defining virtual fundamental boson. Rather remarkably, without the assumption about product representation one would have general four-fermion vertex rather than boson exchange. Hence gauge theory structure is not put in by hand but emerges.
   
   

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

Could quaternion analyticity make sense for the preferred extremals?

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. The obvious ideas coming in mind are appropriately defined quaternionic and octonion analyticity. I have used a considerable amount of time to consider these possibilities but had to give up the idea about octonion analyticity could somehow allow to preferred extemals.

Basic idea

One can argue that quaternion analyticity is the more natural option in the sense that the local octonionic imbedding space coordinate (or at least M⁸ or E⁸ coordinate, which is enough if M⁸-H duality holds true) would for preferred extremals be expressible in the form

o(q)= u(q) + v(q)× I .

Here q is quaternion serving as a coordinate of a quaternionic sub-space of octonions, and I is octonion unit belonging to the complement of the quaternionic sub-space, and multiplies v(q) from right so that quaternions and qiaternionic differential operators acting from left do not notice these coefficients at all. A stronger condition would be that the coefficients are real. u(q) and v(q) would be quaternionic Taylor- of even Laurent series with coefficients multiplying powers of q from right for the same reason.

The 4-D generalization of conformal invariance suggests strongly that the notion of analytic function generalizes somehow. The obvious ideas coming in mind are appropriately defined quaternionic and octonion analyticity. I have used a considerable amount of time to consider these possibilities but had to give up the idea about octonion analyticity could somehow allow to preferred extemals.

Basic idea

One can argue that quaternion analyticity is the more natural option in the sense that the local octonionic imbedding space coordinate (or at least M⁸ or E⁸ coordinate, which is enough if M⁸-H duality holds true) would for preferred extremals be expressible in the form

o(q)= u(q) + v(q)× I .

Here q is quaternion serving as a coordinate of a quaternionic sub-space of octonions, and I is octonion unit belonging to the complement of the quaternionic sub-space, and multiplies v(q) from right so that quaternions and qiaternionic differential operators acting from left do not notice these coefficients at all. A stronger condition would be that the coefficients are real. u(q) and v(q) would be quaternionic Taylor- of even Laurent series with coefficients multiplying powers of q from right for the same reason.

I ended up to this idea after finding two very interesting articles discussing the generalization of Cauchy-Riemann equations. The first article was about so called triholomorphic maps between 4-D almost quaternionic manifolds. The article gave as a reference an article about quaternionic analogs of Cauchy-Riemann conditions discussed by Fueter long ago (somehow I have managed to miss Fueter's work just like I missed Hitchin's work about twistorial uniqueness of M⁴ and CP₂), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The signature of M⁴ metric is a problem. I have proposed complexification of M⁸ and M⁴ to get rid of the problem by assuming that the imbedding space corresponds to surfaces in the space M⁸ identified as octonions of form o₈= Re(o)+i Im(o), where o is imaginary part of ordinary octonion and i is commuting imaginary unit. M⁴ would correspond to quaternions of form q₄= Re(q)+i Im(q). What is important is that powers of q₄ and o₈ belong to this sub-space (as follows from the vanishing of cross product term in the square of octonion/quaternion) so that powers of q₄ (o₈) has imaginary part proportional to Im(q) (Im(o)).

I ended up to reconsider the idea of quaternion analyticity after having found two very interesting articles discussing the generalization of Cauchy-Riemann equations. The first article was about so called triholomorphic maps between 4-D almost
quaternionic manifolds. The article gave as a reference an article about quaternionic analogs of Cauchy-Riemann conditions discussed by Fueter long ago (somehow I have managed to miss Fueter's work just like I missed Hitchin's work about twistorial uniqueness of M⁴ and CP₂), and also a new linear variant of these conditions, which seems especially interesting from TGD point of view as will be found.

The first form of Cauchy-Rieman-Fueter conditions

Cauhy-Riemann-Fueter (CRF) conditions generalize Cauchy-Riemann conditions. These conditions are however not unique. Consider first the translationally invariant form of CRF conditions.

1. The translationally invariant form of CRF conditions is ∂_q*f=0 or explicitly
   

   ∂_qf=(∂_t- ∂_x I-∂_y J -∂_zK)f=0 .
   

   This form does not allow quaternionic Taylor series. Note that the Taylor coefficients multiplying powers of the coordinate from right are arbitrary quaternions. What looks pathological is that even linear functions of q fail be solve
   this condition. What is however interesting that in flat space the equation is equivalent with Dirac equation for a pair of Majorana spinors.
   
   

2. The condition allows functions depending on complex coordinate z of some complex-plane only. It also allows functions satisfying two separate analyticity conditions, say
   

   ∂u*f= (∂_t- ∂_x I)f=0 ,
   

   ∂_v*f=-(∂_y J+∂_z K)f=-J(∂_y -∂ _zI) f=0 .
   

   In the latter formula J multiplies from left! One has good hopes of obtaining holomorphic functions of two complex coordinates. This might be enough to understand the preferred extremals of Kähler action as quaternion analytic mops.
   

   
   There are potential problems due to non-commutativity of u=t+/- xI and v=yJ+/- zK= (y+/- zI) J (note that J ~ multiplies from right!) and ∂_u and ∂_v. A prescription for the ordering of the powers u and v in the polynomials of u and v appearing in the double Taylor series seems to be needed. For instance, powers of u can be taken to be at left and v or of a related variable at right.
   

   By the linearity of ∂_v* one can leave J to the left and commute only (∂_y -∂ _zI) through the u-dependent part of the series: this operation is trivial. The condition ∂_vf=0 is satisfied if the polynomials of y and z are polynomials of y+iz multiplied by J from right. The solution ansatz is thus product of Taylor series of monomials f_mn= (x+iy)^m
   (y+iz)ⁿJ with Taylor coefficients a_mn, which multiply the monomials from right
   and are arbitrary quaternions. Note that the monomials (y+iz)ⁿ do not reduce to polynomials of v and that the ordering of these powers is arbitrary. If the coefficients a_mn are real f maps 4-D quaternionic region to 2-D region spanned by J and K. Otherwise the image is 4-D.
   
   

3. By linearity the solutions obey linear superposition. They can be also multiplied if product is defined as ordered product in such a manner that only the powers t+ix and y+iz are multiplied together at left and coefficients a_mn are multiplied together at right. The analogy with quantum non-commutativity is obvious.
   

   
   

4. In Minkowskian signature one must multiply imaginary units I,J,K with an additional commuting imaginary unit i. This would give solutions as powers of (say) t+ex, e=iI with e²=1 representing imaginary unit of hyper-complex numbers. The natural interpretation would be as algebraic extension which is analogous to the extension of rational number by adding algebraic number, say 2^1/2 to get algebraically 2-dimensional structure but as real numbers 1-D structure. Only the non-commutativity with J and K distinguishes e from e=+/- 1 and if J and K do not appear in the function, one can replace e by +/- 1 in t+ex to get just t+/- x appearing as argument for waves propagating with light velocity.
   
   

Second form of CRF conditions

Second form of CRF conditions proposed in the second reference is tailored in order to realize the almost obvious manner to realize quaternion analyticity.

1. The ingenious idea is to replace preferred quaternionic imaginary unit by a imaginary unit which is in radial direction: e_r= (xI+yJ+zK)/r and require analyticity with respect to the coordinate t+e r. The solution to the condition is power series in t+re_r= q so that one obtains quaternion analyticity.
   
   
2. The excplicit form of the conditions is
   

   l (∂_t- e_r ∂_r)f= (∂_t-e_r/r r∂_r)f=0 .
   

   This form allows both the desired quaternionic Taylor series and ordinary holomorphic functions of complex variable in one of the 3 complex coordinate planes as general solutions.
   
   

3. This form of CRF is neither Lorentz invariant nor translationally invariant but remains invariant under simultaneous scalings of t and r and under time translations. Under rotations of either coordinates or of imaginary units the spatial part transforms like vector so that quaternionic automorphism group SO(3) serves as a moduli space for these operators.
   
   
4. The interpretation of the latter solutions inspired by ZEO would be that in Minkowskian regions r corresponds to the light-like radial coordinate of the either boundary of CD, which is part of δ M⁴_+/-. The radial scaling operator is that assigned with the light-like radial coordinate of the light-cone boundary. A slicing of CD by surfaces parallel to the δ M⁴_+/- is assumed and implies that the line r=0 connecting the tips of CD is in a special role. The line connecting the tips of CD defines coordinate line of time coordinate. The breaking of rotational invariance corresponds to the selection of a preferred quaternion unit defining the twistor structure and preferred complex sub-space.
   

   In regions of Euclidian signature r could correspond to the radial Eguchi-Hanson coordinate of CP₂ and r=0 corresponds to a fixed point of U(2) subgroup under which CP₂ complex coordinates transform linearly.
   
   

5. Also in this case one can ask whether solutions depending on two complex local coordinates analogous to those for translationally invariant CRF condition are possible. The remain imaginary units would be associated with the surface of sphere allowing complex structure.
   
   

Generalization of CRF conditions?

Could the proposed forms of CRF conditions be special cases of much more general CRF conditions as CR conditions are?

1. Ordinary complex analysis suggests that there is an infinite number of choices of the quaternionic coordinates related by the above described quaternion-analytic maps with 4-D images. The form of of the CRF conditions would be different in each of these coordinate systems and would be obtained in a straightforward manner by chain rule.
   
   
2. One expects the existence of large number of different quaternion-conformal structures not related by quaternion-analytic transformations analogous to those allowed by higher genus Riemann surfaces and that these conformal equivalence classes of four-manifolds are characterized by a moduli space and the analogs of Teichmueller
   parameters depending on 3-topology. In TGD framework strong form of holography suggests that these conformal equivalence classes for preferred extremals could reduce to ordinary conformal classes for the partonic 2-surfaces. An attractive possibility is that by conformal gauge symmetries the functional integral over WCW reduces to the integral over the conformal equivalence classes.
   
   
3. The quaternion-conformal structures could be characterized by a standard choice of quaternionic coordinates reducing to the choice of a pair of complex coordinates. In these coordinates the general solution to quaternion-analyticity conditions would be of form described for the linear ansatz. The moduli space corresponds to that for complex or hyper-complex structures defined in the space-time region.
   
   

Geometric formulation of the CRF conditions

The previous naive generalization of CRF conditions treats imaginary units without trying to understand their geometric content. This leads to difficulties when when tries to formulate these conditions for maps between quaternionic and hyper-quaternionic spaces using purely algebraic representation of imaginary units since it is not clear how these units relate to each other.

In the first article the CRF conditions are formulated in terms of the antisymmetric (1,1) type tensors representing the imaginary units: they exist for almost quaternionic structure and presumably also for almost hyper-quaternionic structure needed in Minkowskian signature.

The generalization of CRF conditions is proposed in terms of the Jacobian J of the map mapping tangent space TM to TN and antisymmetric tensors J_u and J_u representing the quaternionic imaginary units of N and M. The generalization of CRF conditions reads as

J- ∑_u J_u J j_u=0 .

For N=M it reduces to the translationally invariant algebraic form of the conditions discussed above. These conditions seem to be well-defined also when one maps quaternionic to hyper-quaternionic space or vice versa. These conditions are not unique. One can perform an SO(3) rotation (quaternion automorphism) of the imaginary units mediated by matrix Λ^uv to obtain

J- Λ^uv J_uJ j_v=0 .

The matrix Λ can depend on point so that one has a kind of gauge symmetry. The most general triholomorphic map allows the presence of Λ Note that these conditions make sense on any coordinates and complex analytic maps generate new forms of these conditions.

Covariant forms of structure constant tensors reduce to octonionic structure constants and this allows to write the conditions explicitly. The index raising of the second index of the structure constants is however needed using the metrics of M and N. This complicates the situation and spoils linearity: in particular, for surfaces induced metric is needed. Whether local SO(3) rotation can eliminate the dependence on induced metric is an interesting question. Minkowskian imaginary units differ only by multiplication by i from Euclidian so that Minkowskian structure constants differ only by sign from those for Euclidian ones.

In the octonionic case the geometric generalization of CRF conditions does not seem to make sense. By non-associativity of octonion product it is not possible to have a matrix representation for the matrices so that a faithful representation of octonionic imaginary units as antisymmetric 1-1 forms does not make sense. If this representation exists it it must map octonionic associators to zero. Note however that CRF conditions do not involve products of three octonion units so that they make sense as algebraic conditions at least.

Does residue calculus generalize?

CRF conditions allow to generalize Cauchy formula allowing to express value of analytic function in terms of its boundary values. This would give a concrete realization of the holography in the sense that the physical variables in the interior could be expressed in terms of the data at the light-like partonic orbits and and the ends of the space-time surface. Triholomorphic function satisfies d'Alembert/Laplace equations - in induced metric in TGD framework- so that the maximum modulus principle holds true. The general ansatz for a preferred extremals involving Hamilton-Jacobi structure leads to d'Alembert type equations for preferred extremals.

Could the analog of residue calculus exist? Line integral would become 3-D integral reducing to a sum over poles and possible cuts inside the 3-D contour. The space-like 3-surfaces at the ends of CDs could define natural integration contours, and the freedom to choose contour rather freely would reflect General Coordinate Invariance. A possible choice for the integration contour would be the closed 3-surface defined by the union of space-like surfaces at the ends of CD and by the light-like partonic orbits.

Poles and cuts would be in the interior of the space-time surface. Poles have co-dimension 2 and cuts co-dimension 1. Strong form of holography suggests that partonic 2-surfaces and perhaps also string world sheets serve as candidates for poles. Light-like 3-surfaces (partonic orbits) defining the boundaries between Euclidian and Minkowskian regions are singular objects and could serve as cuts. The discontinuity would be due to the change of the signature of the induced metric. There are CDs inside CDs and one can also consider the possibility that sub-CDs define cuts, which in turn reduce to cuts associated with sub-CDs.

Could one understand the preferred extremals in terms of quaternion-analyticity?

Could one understand the preferred extremals in terms of quaternion-analyticity or its possible generalization to an analytic representation for co-quaternionicity expected in space-time regions with Euclidian signature? What is the generalization of the CRF conditions for the counterparts of quaternion-analytic maps from hyper-quaternionic X⁴ to quaternionic CP₂ and from quaternionic X⁴ to hyper-quaternionic M⁴? It has already become clear that this problem can be probably solved by using the the geometric representation for quaternionic imaginary units.

The best thing to do is to look whether this is possible for the known extremals: CP₂ type vacuum extremals, vacuum extremals expressible as graph of map from M⁴ to a Lagrangian sub-manifold of CP₂, cosmic strings of form X²× Y²⊂ M⁴ × CP₂ such that X² is string world sheet (minimal surface) and Y² complex sub-manifold of CP₂. One can also check whether Hamilton-Jacobi structure of M⁴ and of Minkowskian space-time regions and complex structure of CP₂ have natural counterparts in the quaternion-analytic framework.

1. Consider first cosmic strings. In this case the quaternionic-analytic map from X⁴ = X²× Y² to M⁴× CP₂ with octonion structure would be map X⁴ to 2-D string world sheet in M² and Y² to 2-D complex manifold of CP₂. This could be achieved by using the linear variant of CRF condition. The map from X⁴ to M⁴ would reduce to ordinary hyper-analytic map from X² with hyper-complex coordinate to M⁴ with hyper-complex coordinates just as in string models. The map from X⁴ to CP₂ would reduce to an ordinary analytic map from X² with complex coordinates. One would not leave the realm of string models.
   
   
2. For the simplest massless extremals (MEs) CP₂ coordinates are arbitrary functions of light-like coordinate u=k•m, k constant light-like vector, and of v=ε • m, ε a polarization vector orthogonal to k. The interpretation as classical counterpart of photon or Bose-Einstein condense of photons is obvious. There are good reasons to expect that this ansatz generalizes by replacing the variables u and v with coordinate along the light-like and space-like coordinate lines of Hamilton-Jacobi structure. The non-geodesic motion of photons with light-velocity and variation of the polarization direction would be due to interactions with the space-time sheet to which it is topologically condensed. Note that light-likeness condition for the coordinate curve gives rise to Virasoro conditions. This observation led long time ago to the idea that 2-D conformal invariance must have a non-trivial generalization to 4-D case.
   

   Now space-time surface would have naturally M⁴ coordinates and the map M⁴→ M⁴ would be just identity map satisfying the radial CRF condition. Can one understand CP₂ coordinates in terms of quaternion- analyticity? The dependence of CP₂ coordinates on u=t-x only can be formulated as CFR condition ∂_u* s^k=0 and this could is expected to generalize in the formulation using the geometric representation of quaternionic imaginary units at both sides. The dependence on light-light coordinate u follows from the translationally invariant CRF condition.
   

   The dependence on the real coordinate v is however problematic since the dependence is naturally on complex coordinate w assignable to the polarization plane of form z= f(w). This would give dependence on 2 transversal coordinates and CP₂ projection would be 3-D rather than 2-D. One can of course ask whether this dependence is actually present for preferred extremals? Could the polarization vector be complex local polarization vector orthogonal to the light-like vector? In quantum theory complex polarization vectors are used of routinely and become oscillator operators in second quantization and in TGD Universe MEs indeed serve as space-time correlates for photons or their BE condensates.
   
   

3. Vacuum extremals with Lagrangian manifold as (in the generic case 2-D) CP₂ projection are not expected to be preferred extremals for obvious reasons. One one can however try similar approach. Hyper-quaternionic structure for space-time surface using Hamilton-Jacobi structure is the first guess. CP₂ should allow a quaternionic coordinate decomposing to a pair of complex coordinates such that second complex coordinate is constant for 2-D Lagrangian manifold and second parameterizes it. Any 2-D surface allows complex structure defined by the induced metric so that there are good hopes that these coordinates exist. The quaternion-analytic map would map in the most general case is trivial for both hypercomplex and complex coordinate of M⁴ but the quaternionic Taylor coefficients reduce to real numbers to that the image is 2-D.
   
   
4. For CP₂ type vacuum extremals the M⁴ projection is random light-like curve. Now one expects co-quaternionicity and that quaternion-analyticity is not the correct manner to formulate the situation. "Co-" suggests that instead of expressing surface as graph one should perhaps express it in terms of conditions stating that some quaternionic analytic functions in H are vanish.
   

   One can fix the coordinates of X⁴ to be complex coordinates of CP₂ so that one gets rid of the degeneracy due to the choice of coordinates. M⁴ allows hyper-quaternionic coordinates and Hamilton-Jacobi structures define different choices of hyper-quaternionic coordinates. Now the second light- like coordinate would vary along random light-like curves providing slicing of M⁴ by 3-D surfaces.
   Hamilton-Jacobi structure defines at each point a plane M²(x) fixed by the light-like vector at the point and the 2-D orthogonal plane. In fact 4-D coordinate grid is defined. This local choice must be integrable, which means that one has slicing by 2-D string world sheets and polarization planes orthogonal to them.
   

   The problem is that the mapping of quaternionic CP₂ coordinate to hyper-quaternionic coordinates of M⁴ (say v=0, w=0) in terms of quaternionic analyticity is not easy. "Co-" suggets that , one could formulate light-likeness condition using Hamilton-Jacobi structure as conditions w*-constant=0 and v-constant=0. Note that one has u*=v.
   
   

5. In the naive generalization CRF conditions are linear. Whether this is the case in the formulation using the geometric representation of the imaginary units is not clear since the quaternionic imaginary units depend on the vielbein of the induced 3-metric (note however that the SO(3) gauge rotation appearing in the conditions could allow to compensate for the change of the tensors in small deformations of the spaced-time surface). If linearity is real and not true only for small perturbations, one could have linear superpositions of different types of solutions, which looks strange. Could the superpositions describe perturbations of say cosmic strings and massless extremals?
   
   
6. Both forms of algebraic C-R-F conditions generalize to the octonionic situation and right multiplication of powers of octonion by Taylor coefficients plus linearity imply that there are no problems with associativity. This inspires several questions.
   

   Could octonion analytic maps of imbedding space allow to construct new solutions from the existing ones? Could quaternion analytic maps applied at space-time level act as analogs of holomorphic maps and generalize conformal gauge invariance to 4-D context?
   

Conclusions

To sum up, connections between different conjectures related to the preferred extremals - M⁸-H duality, Hamilton-Jacobi structure, induced twistor space structure, quaternion-Kähler property and its Minkowskian counterpart, and even quaternion analyticity, are clearly emerging. The underlying reason is strong form of GCI forced by the construction of WCW geometry and implying strong from of holography posing extremely powerful quantization conditions on the extremals of Kähler action in ZEO. Without the conformal gauge conditions the mutual inconsistency of these conjectures looks rather infeasible.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.

The first form of Cauchy-Rieman-Fueter conditions

Cauhy-Riemann-Fueter (CRF) conditions generalize Cauchy-Riemann conditions. These conditions are however not unique. Consider first the translationally invariant form of CRF conditions.

1. The translationally invariant form of CRF conditions is ∂_q*f=0 or explicitly
   

   ∂_qf=(∂_t- ∂_x I-∂_y J -∂_zK)f=0 .
   

   This form does not allow quaternionic Taylor series. Note that the Taylor coefficients multiplying powers of the coordinate from right are arbitrary quaternions. What looks pathological is that even linear functions of q fail be solve
   this condition. What is however interesting that in flat space the equation is equivalent with Dirac equation for a pair of Majorana spinors.
   
   

2. The condition allows functions depending on complex coordinate z of some complex-plane only. It also allows functions satisfying two separate analyticity conditions, say
   

   ∂u*f= (∂_t- ∂_x I)f=0 ,
   

   ∂_v*f=-(∂_y J+∂_z K)f=-J(∂_y -∂ _zI) f=0 .
   

   In the latter formula J multiplies from left! One has good hopes of obtaining holomorphic functions of two complex coordinates. This might be enough to understand the preferred extremals of Kähler action as quaternion analytic mops.
   

   
   There are potential problems due to non-commutativity of u=t+/- xI and v=yJ+/- zK= (y+/- zI) J (note that J ~ multiplies from right!) and ∂_u and ∂_v. A prescription for the ordering of the powers u and v in the polynomials of u and v appearing in the double Taylor series seems to be needed. For instance, powers of u can be taken to be at left and v or of a related variable at right.
   

   By the linearity of ∂_v* one can leave J to the left and commute only (∂_y -∂ _zI) through the u-dependent part of the series: this operation is trivial. The condition ∂_vf=0 is satisfied if the polynomials of y and z are polynomials of y+iz multiplied by J from right. The solution ansatz is thus product of Taylor series of monomials f_mn= (x+iy)^m
   (y+iz)ⁿJ with Taylor coefficients a_mn, which multiply the monomials from right
   and are arbitrary quaternions. Note that the monomials (y+iz)ⁿ do not reduce to polynomials of v and that the ordering of these powers is arbitrary. If the coefficients a_mn are real f maps 4-D quaternionic region to 2-D region spanned by J and K. Otherwise the image is 4-D.
   
   

3. By linearity the solutions obey linear superposition. They can be also multiplied if product is defined as ordered product in such a manner that only the powers t+ix and y+iz are multiplied together at left and coefficients a_mn are multiplied together at right. The analogy with quantum non-commutativity is obvious.
   

   
   

4. In Minkowskian signature one must multiply imaginary units I,J,K with an additional commuting imaginary unit i. This would give solutions as powers of (say) t+ex, e=iI with e²=1 representing imaginary unit of hyper-complex numbers. The natural interpretation would be as algebraic extension which is analogous to the extension of rational number by adding algebraic number, say 2^1/2 to get algebraically 2-dimensional structure but as real numbers 1-D structure. Only the non-commutativity with J and K distinguishes e from e=+/- 1 and if J and K do not appear in the function, one can replace e by +/- 1 in t+ex to get just t+/- x appearing as argument for waves propagating with light velocity.
   
   

Second form of CRF conditions

Second form of CRF conditions proposed in the second reference is tailored in order to realize the almost obvious manner to realize quaternion analyticity.

1. The ingenious idea is to replace preferred quaternionic imaginary unit by a imaginary unit which is in radial direction: e_r= (xI+yJ+zK)/r and require analyticity with respect to the coordinate t+e r. The solution to the condition is power series in t+re_r= q so that one obtains quaternion analyticity.
   
   
2. The excplicit form of the conditions is
   

   l (∂_t- e_r ∂_r)f= (∂_t-e_r/r r∂_r)f=0 .
   

   This form allows both the desired quaternionic Taylor series and ordinary holomorphic functions of complex variable in one of the 3 complex coordinate planes as general solutions.
   
   

3. This form of CRF is neither Lorentz invariant nor translationally invariant but remains invariant under simultaneous scalings of t and r and under time translations. Under rotations of either coordinates or of imaginary units the spatial part transforms like vector so that quaternionic automorphism group SO(3) serves as a moduli space for these operators.
   
   
4. The interpretation of the latter solutions inspired by ZEO would be that in Minkowskian regions r corresponds to the light-like radial coordinate of the either boundary of CD, which is part of δ M⁴_+/-. The radial scaling operator is that assigned with the light-like radial coordinate of the light-cone boundary. A slicing of CD by surfaces parallel to the δ M⁴_+/- is assumed and implies that the line r=0 connecting the tips of CD is in a special role. The line connecting the tips of CD defines coordinate line of time coordinate. The breaking of rotational invariance corresponds to the selection of a preferred quaternion unit defining the twistor structure and preferred complex sub-space.
   

   In regions of Euclidian signature r could correspond to the radial Eguchi-Hanson coordinate of CP₂ and r=0 corresponds to a fixed point of U(2) subgroup under which CP₂ complex coordinates transform linearly.
   
   

5. Also in this case one can ask whether solutions depending on two complex local coordinates analogous to those for translationally invariant CRF condition are possible. The remain imaginary units would be associated with the surface of sphere allowing complex structure.
   
   

Generalization of CRF conditions?

Could the proposed forms of CRF conditions be special cases of much more general CRF conditions as CR conditions are?

1. Ordinary complex analysis suggests that there is an infinite number of choices of the quaternionic coordinates related by the above described quaternion-analytic maps with 4-D images. The form of of the CRF conditions would be different in each of these coordinate systems and would be obtained in a straightforward manner by chain rule.
   
   
2. One expects the existence of large number of different quaternion-conformal structures not related by quaternion-analytic transformations analogous to those allowed by higher genus Riemann surfaces and that these conformal equivalence classes of four-manifolds are characterized by a moduli space and the analogs of Teichmueller
   parameters depending on 3-topology. In TGD framework strong form of holography suggests that these conformal equivalence classes for preferred extremals could reduce to ordinary conformal classes for the partonic 2-surfaces. An attractive possibility is that by conformal gauge symmetries the functional integral over WCW reduces to the integral over the conformal equivalence classes.
   
   
3. The quaternion-conformal structures could be characterized by a standard choice of quaternionic coordinates reducing to the choice of a pair of complex coordinates. In these coordinates the general solution to quaternion-analyticity conditions would be of form described for the linear ansatz. The moduli space corresponds to that for complex or hyper-complex structures defined in the space-time region.
   
   

Geometric formulation of the CRF conditions

The previous naive generalization of CRF conditions treats imaginary units without trying to understand their geometric content. This leads to difficulties when when tries to formulate these conditions for maps between quaternionic and hyper-quaternionic spaces using purely algebraic representation of imaginary units since it is not clear how these units relate to each other.

In the first article the CRF conditions are formulated in terms of the antisymmetric (1,1) type tensors representing the imaginary units: they exist for almost quaternionic structure and presumably also for almost hyper-quaternionic structure needed in Minkowskian signature.

The generalization of CRF conditions is proposed in terms of the Jacobian J of the map mapping tangent space TM to TN and antisymmetric tensors J_u and J_u representing the quaternionic imaginary units of N and M. The generalization of CRF conditions reads as

J- ∑_u J_u J j_u=0 .

For N=M it reduces to the translationally invariant algebraic form of the conditions discussed above. These conditions seem to be well-defined also when one maps quaternionic to hyper-quaternionic space or vice versa. These conditions are not unique. One can perform an SO(3) rotation (quaternion automorphism) of the imaginary units mediated by matrix Λ^uv to obtain

J- Λ^uv J_uJ j_v=0 .

The matrix Λ can depend on point so that one has a kind of gauge symmetry. The most general triholomorphic map allows the presence of Λ Note that these conditions make sense on any coordinates and complex analytic maps generate new forms of these conditions.

Covariant forms of structure constant tensors reduce to octonionic structure constants and this allows to write the conditions explicitly. The index raising of the second index of the structure constants is however needed using the metrics of M and N. This complicates the situation and spoils linearity: in particular, for surfaces induced metric is needed. Whether local SO(3) rotation can eliminate the dependence on induced metric is an interesting question. Minkowskian imaginary units differ only by multiplication by i from Euclidian so that Minkowskian structure constants differ only by sign from those for Euclidian ones.

In the octonionic case the geometric generalization of CRF conditions does not seem to make sense. By non-associativity of octonion product it is not possible to have a matrix representation for the matrices so that a faithful representation of octonionic imaginary units as antisymmetric 1-1 forms does not make sense. If this representation exists it it must map octonionic associators to zero. Note however that CRF conditions do not involve products of three octonion units so that they make sense as algebraic conditions at least.

Does residue calculus generalize?

CRF conditions allow to generalize Cauchy formula allowing to express value of analytic function in terms of its boundary values. This would give a concrete realization of the holography in the sense that the physical variables in the interior could be expressed in terms of the data at the light-like partonic orbits and and the ends of the space-time surface. Triholomorphic function satisfies d'Alembert/Laplace equations - in induced metric in TGD framework- so that the maximum modulus principle holds true. The general ansatz for a preferred extremals involving Hamilton-Jacobi structure leads to d'Alembert type equations for preferred extremals.

Could the analog of residue calculus exist? Line integral would become 3-D integral reducing to a sum over poles and possible cuts inside the 3-D contour. The space-like 3-surfaces at the ends of CDs could define natural integration contours, and the freedom to choose contour rather freely would reflect General Coordinate Invariance. A possible choice for the integration contour would be the closed 3-surface defined by the union of space-like surfaces at the ends of CD and by the light-like partonic orbits.

Poles and cuts would be in the interior of the space-time surface. Poles have co-dimension 2 and cuts co-dimension 1. Strong form of holography suggests that partonic 2-surfaces and perhaps also string world sheets serve as candidates for poles. Light-like 3-surfaces (partonic orbits) defining the boundaries between Euclidian and Minkowskian regions are singular objects and could serve as cuts. The discontinuity would be due to the change of the signature of the induced metric. There are CDs inside CDs and one can also consider the possibility that sub-CDs define cuts, which in turn reduce to cuts associated with sub-CDs.

Could one understand the preferred extremals in terms of quaternion-analyticity?

Could one understand the preferred extremals in terms of quaternion-analyticity or its possible generalization to an analytic representation for co-quaternionicity expected in space-time regions with Euclidian signature? What is the generalization of the CRF conditions for the counterparts of quaternion-analytic maps from hyper-quaternionic X⁴ to quaternionic CP₂ and from quaternionic X⁴ to hyper-quaternionic M⁴? It has already become clear that this problem can be probably solved by using the the geometric representation for quaternionic imaginary units.

The best thing to do is to look whether this is possible for the known extremals: CP₂ type vacuum extremals, vacuum extremals expressible as graph of map from M⁴ to a Lagrangian sub-manifold of CP₂, cosmic strings of form X²× Y²⊂ M⁴ × CP₂ such that X² is string world sheet (minimal surface) and Y² complex sub-manifold of CP₂. One can also check whether Hamilton-Jacobi structure of M⁴ and of Minkowskian space-time regions and complex structure of CP₂ have natural counterparts in the quaternion-analytic framework.

1. Consider first cosmic strings. In this case the quaternionic-analytic map from X⁴ = X²× Y² to M⁴× CP₂ with octonion structure would be map X⁴ to 2-D string world sheet in M² and Y² to 2-D complex manifold of CP₂. This could be achieved by using the linear variant of CRF condition. The map from X⁴ to M⁴ would reduce to ordinary hyper-analytic map from X² with hyper-complex coordinate to M⁴ with hyper-complex coordinates just as in string models. The map from X⁴ to CP₂ would reduce to an ordinary analytic map from X² with complex coordinates. One would not leave the realm of string models.
   
   
2. For the simplest massless extremals (MEs) CP₂ coordinates are arbitrary functions of light-like coordinate u=k•m, k constant light-like vector, and of v=ε • m, ε a polarization vector orthogonal to k. The interpretation as classical counterpart of photon or Bose-Einstein condense of photons is obvious. There are good reasons to expect that this ansatz generalizes by replacing the variables u and v with coordinate along the light-like and space-like coordinate lines of Hamilton-Jacobi structure. The non-geodesic motion of photons with light-velocity and variation of the polarization direction would be due to interactions with the space-time sheet to which it is topologically condensed. Note that light-likeness condition for the coordinate curve gives rise to Virasoro conditions. This observation led long time ago to the idea that 2-D conformal invariance must have a non-trivial generalization to 4-D case.
   

   Now space-time surface would have naturally M⁴ coordinates and the map M⁴→ M⁴ would be just identity map satisfying the radial CRF condition. Can one understand CP₂ coordinates in terms of quaternion- analyticity? The dependence of CP₂ coordinates on u=t-x only can be formulated as CFR condition ∂_u* s^k=0 and this could is expected to generalize in the formulation using the geometric representation of quaternionic imaginary units at both sides. The dependence on light-light coordinate u follows from the translationally invariant CRF condition.
   

   The dependence on the real coordinate v is however problematic since the dependence is naturally on complex coordinate w assignable to the polarization plane of form z= f(w). This would give dependence on 2 transversal coordinates and CP₂ projection would be 3-D rather than 2-D. One can of course ask whether this dependence is actually present for preferred extremals? Could the polarization vector be complex local polarization vector orthogonal to the light-like vector? In quantum theory complex polarization vectors are used of routinely and become oscillator operators in second quantization and in TGD Universe MEs indeed serve as space-time correlates for photons or their BE condensates.
   
   

3. Vacuum extremals with Lagrangian manifold as (in the generic case 2-D) CP₂ projection are not expected to be preferred extremals for obvious reasons. One one can however try similar approach. Hyper-quaternionic structure for space-time surface using Hamilton-Jacobi structure is the first guess. CP₂ should allow a quaternionic coordinate decomposing to a pair of complex coordinates such that second complex coordinate is constant for 2-D Lagrangian manifold and second parameterizes it. Any 2-D surface allows complex structure defined by the induced metric so that there are good hopes that these coordinates exist. The quaternion-analytic map would map in the most general case is trivial for both hypercomplex and complex coordinate of M⁴ but the quaternionic Taylor coefficients reduce to real numbers to that the image is 2-D.
   
   
4. For CP₂ type vacuum extremals the M⁴ projection is random light-like curve. Now one expects co-quaternionicity and that quaternion-analyticity is not the correct manner to formulate the situation. "Co-" suggests that instead of expressing surface as graph one should perhaps express it in terms of conditions stating that some quaternionic analytic functions in H are vanish.
   

   One can fix the coordinates of X⁴ to be complex coordinates of CP₂ so that one gets rid of the degeneracy due to the choice of coordinates. M⁴ allows hyper-quaternionic coordinates and Hamilton-Jacobi structures define different choices of hyper-quaternionic coordinates. Now the second light- like coordinate would vary along random light-like curves providing slicing of M⁴ by 3-D surfaces.
   Hamilton-Jacobi structure defines at each point a plane M²(x) fixed by the light-like vector at the point and the 2-D orthogonal plane. In fact 4-D coordinate grid is defined. This local choice must be integrable, which means that one has slicing by 2-D string world sheets and polarization planes orthogonal to them.
   

   The problem is that the mapping of quaternionic CP₂ coordinate to hyper-quaternionic coordinates of M⁴ (say v=0, w=0) in terms of quaternionic analyticity is not easy. "Co-" suggets that , one could formulate light-likeness condition using Hamilton-Jacobi structure as conditions w*-constant=0 and v-constant=0. Note that one has u*=v.
   
   

5. In the naive generalization CRF conditions are linear. Whether this is the case in the formulation using the geometric representation of the imaginary units is not clear since the quaternionic imaginary units depend on the vielbein of the induced 3-metric (note however that the SO(3) gauge rotation appearing in the conditions could allow to compensate for the change of the tensors in small deformations of the spaced-time surface). If linearity is real and not true only for small perturbations, one could have linear superpositions of different types of solutions, which looks strange. Could the superpositions describe perturbations of say cosmic strings and massless extremals?
   
   
6. Both forms of algebraic C-R-F conditions generalize to the octonionic situation and right multiplication of powers of octonion by Taylor coefficients plus linearity imply that there are no problems with associativity. This inspires several questions.
   

   Could octonion analytic maps of imbedding space allow to construct new solutions from the existing ones? Could quaternion analytic maps applied at space-time level act as analogs of holomorphic maps and generalize conformal gauge invariance to 4-D context?
   

Conclusions

To sum up, connections between different conjectures related to the preferred extremals - M⁸-H duality, Hamilton-Jacobi structure, induced twistor space structure, quaternion-Kähler property and its Minkowskian counterpart, and even quaternion analyticity, are clearly emerging. The underlying reason is strong form of GCI forced by the construction of WCW geometry and implying strong from of holography posing extremely powerful quantization conditions on the extremals of Kähler action in ZEO. Without the conformal gauge conditions the mutual inconsistency of these conjectures looks rather infeasible.

See the chapter Classical part of the twistor story or the article Classical part of the twistor story.