What's new inTGD: Physics as InfiniteDimensional GeometryNote: Newest contributions are at the top! 
Year 2012 
Could N=2 orN=4 SUSY have something to do with TGD?
N=4 SYM has been the theoretical laboratory of Nima and others. The article Scattering Amplitudes and the Positive Grassmannian by ArkaniHamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form meant to be should be accessible to ordinary physicist. N=4 SYM is definitely a completely exceptional theory and one cannot avoid the question whether it could in some sense be part of fundamental physics. In TGD framework right handed neutrinos have remained a mystery: whether one should assign spacetime SUSY to them or not. Could they give rise to N=2 or N=4 SUSY with fermion number conservation? My latest view is that fully covariantly constant righthanded neutrinos decouple from the dynamics completely. I will repeat first the earlier arguments which consider only fully covariantly constant righthanded neutrinos.
Could massless righthanded neutrinos covariantly constant in CP_{2} degrees of freedom define N=2 or N=4 SUSY? Consider next righthanded neutrinos, which are covariantly constant in CP_{2} degrees of freedom but have a lightlike fourmomentum. In this case fermion number is conserved but this is consistent with N=2 SUSY at both MEs with fermion number conservation. N=2 SUSYs could emerge from N=4 SUSY when one half of SUSY generators annihilate the states, which is a basic phenomenon in supersymmetric theories.
Is the dynamics of N=4 SYM possible in righthanded neutrino sector? Could N=4 SYM be a part of quantum TGD? Could TGD be seen a fusion of a degenerate N=4 SYM describing the righthanded neutrino sector and string theory like theory describing the contribution of string world sheets carrying other leptonic and quark spinors? Or could one imagine even something simpler? What is interesting that the net momenta assigned to the right handed neutrinos associated with a pair of MEs would correspond to the momenta assignable to the particles and obtained by padic mass calculations. It would seem that righthanded neutrinos provide a representation of the momenta of the elementary particles represented by wormhole contact structures. Does this mimircry generalize to a full duality so that all quantum numbers and even microscopic dynamics of defined by generalized Feynman diagrams (Euclidian spacetime regions) would be represented by righthanded neutrinos and MEs? Could a generalization of N=4 SYM with nontrivial gauge group with proper choices of the ground states helicities allow to represent the entire microscopic dynamics?
3vertices for sparticles are replaced with 4vertices for MEs In N=4 SYM the basic vertex is on massshell 3vertex which requires that for real lightlike momenta all 3 states are parallel. One must allow complex momenta in order to satisfy energy conservation and lightlikeness conditions. This is strange from the point of view of physics although number theoretically oriented person might argue that the extensions of rationals involving also imaginary unit are rather natural. The complex momenta can be expressed in terms of two lightlike momenta in 3vertex with one real momentum. For instance, the three lightlike momenta can be taken to be p, k, pka, k= ap_{R}. Here p (incoming momentum) and p_{R} are real lightlike momenta satisfying p⋅ p_{R}=0 with opposite sign of energy, and a is complex number. What is remarkable that also the negative sign of energy is necessary also now. Should one allow complex lightlike momenta in TGD framework? One can imagine two options.
Is SUSY breaking possible or needed? It is difficult to imagine the breaking of the proposed kind of SUSY in TGD framework, and the first guess is that all these 4 superpartners of particle have identical masses. pAdic thermodynamics does not distinguish between these states and the only possibility is that the padic primes differ for the spartners. But is the breaking of SUSY really necessary? Can one really distinguish between the 8 different states of a given elementary particle using the recent day experimental methods?
For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation. 
Scattering amplitudes and the positive Grassmannian
Perhaps I exaggerated a little bit in the previous posting, when I talked about declining theoretical physics. The work of Nima ArkaniHamed and others represents something which makes me very optimistic and I would be happy if I could understand the horrible technicalities of their work. The article Scattering Amplitudes and the Positive Grassmannian by ArkaniHamed, Bourjaily, Cachazo, Goncharov, Postnikov, and Trnka summarizes the recent situation in a form, which should be accessible to ordinary physicist. Lubos has already discussed the article. All scattering amplitudes have on shell amplitudes for massless particles as building bricks The key idea is that all planar amplitudes can be constructed from on shell amplitudes: all virtual particles are actually real. In zero energy ontology I ended up with the representation of TGD analogs of Feynman diagrams using only mass shell massless states with both positive and negative energies. The enormous number of kinematic constraints eliminates UV and IR divergences and also the description of massive particles as bound states of massless ones becomes possible. In TGD framework quantum classical correspondence requires a spacetime correlate for the on mass shell property and it indeed exists. The mathematically illdefined path integral over all 4surfaces is replaced with a superposition of preferred extremals of Kähler action analogous to Bohr orbits, and one has only a functional integral over the 3D ends at the lightlike boundaries of causal diamond (Euclidian/Minkowskian spacetime regions give real/imaginary ChernSimons exponent to the vacuum functional). This would be obviously the deeper principle behind on mass shell representation of scattering amplitudes that Nima and others are certainly trying to identify. This principle in turn reduces to general coordinate invariance at the level of the world of classical worlds. Quantum classical correspondence and quantum ergodicity would imply even stronger condition: the quantal correlation functions should be identical with classical correlation functions for any preferred extremal in the superposition: all preferred extremals in the superposition would be statistically equivalent (see the earlier posting). 4D spin glass degeneracy of Kähler action however suggests that this is is probably too strong a condition applying only to building bricks of the superposition. Minimal surface property is the geometric counterpart for masslessness and the preferred extremals are also minimal surfaces: this property reduces to the generalization of complex structure at spacetime surfaces, which I call HamiltonJacobi structure for the Minkowskian signature of the induced metric. Einstein Maxwell equations with cosmological term are also satisfied. Massless extremals and twistor approach The decomposition M^{4}=M^{2}× E^{2} is fundamental in the formulation of quantum TGD, in the number theoretical vision about TGD, in the construction of preferred extremals, and for the vision about generalized Feynman diagrams. It is also fundamental in the decomposition of the degrees of string to longitudinal and transversal ones. An additional item to the list is that also the states appearing in thermodynamical ensemble in padic thermodynamics correspond to fourmomenta in M^{2} fixed by the direction of the Lorentz boost. In twistor approach to TGD the possibility to decompose also internal lines to massless states at parallel spacetime sheets is crucial. Can one find a concrete identification for M^{2}× E^{2} decomposition at the level of preferred extremals? Could these preferred extremals be interpreted as the internal lines of generalized Feynman diagrams carrying massless momenta? Could one identify the mass of particle predicted by padic thermodynamics with the sum of massless classical momenta assignable to two preferred extremals of this kind connected by wormhole contacts defining the elementary particle? Candidates for this kind of preferred extremals indeed exist. Local M^{2}× E^{2} decomposition and lightlike longitudinal massless momentum assignable to M^{2} characterizes "massless extremals" (MEs, "topological light rays"). The simplest MEs correspond to single spacetime sheet carrying a conserved lightlike M^{2} momentum. For several MEs connected by wormhole contacts the longitudinal massless momenta are not conserved anymore but their sum defines a timelike conserved fourmomentum: one has a bound states of massless MEs. The stable wormhole contacts binding MEs together possess Kähler magnetic charge and serve as building bricks of elementary particles. Particles are necessary closed magnetic flux tubes having two wormhole contacts at their ends and connecting the two MEs. The sum of the classical massless momenta assignable to the pair of MEs is conserved even when they exchange momentum. Quantum classical correspondence requires that the conserved classical rest energy of the particle equals to the prediction of padic mass calculations. The massless momenta assignable to MEs would naturally correspond to the massless momenta propagating along the internal lines of generalized Feynman diagrams assumed in zero energy ontology. Masslessness of virtual particles makes also possible twistor approach. This supports the view that MEs are fundamental for the twistor approach in TGD framework. Scattering amplitudes as representations for braids whose threads can fuse at 3vertices Just a little comment about the content of the article. The main message of the article is that nonequivalent contributions to a given scattering amplitude in N=4 SYM represent elements of the group of permutations of external lines  or to be more precise  decorated permutations which replace permutation group S_{n} with n! elements with its decorated version containing 2^{n}n! elements. Besides 3vertex the basic dynamical process is permutation having the exchange of neighboring lines as a generating permutation completely analogous to fundamental braiding. BFCW bridge has interpretation as a representations for the basic braiding operation. This supports the TGD inspired proposal (TGD as almost topological QFT) that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3vertices. The first 3vertex generalizes the standard stringy 3vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3vertex generalizes the 3vertex of ordinary Feynman diagram (three 4D lines of generalized Feynman diagram identified as Euclidian spacetime regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.
"Almost topological" has also a meaning usually not assigned with it. Thurston's geometrization conjecture stating that geometric invariants of canonical representation of manifold as Riemann geometry, defined topological invariants, could generalize somehow. For instance, the geometric invariants of preferred extremals could be seen as topological or more refined invariants (symplectic, conformal in the sense of 4D generalization of conformal structure). If quantum ergodicity holds true, the statistical geometric invariants defined by the classical correlation functions of various induced classical gauge fields for preferred extremals could be regarded as this kind of invariants for submanifolds. What would distinguish TGD from standard topological QFT would be that the invariants in question would involve length scale and thus have a physical content in the usual sense of the word! For background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation. 
Is there a connection between preferred extremals and AdS_{4}/CFT correspondence?
The preferred extremals satisfy Einstein Maxwell equations with a cosmological constant and have negative curvature for negative value of Λ. 4D spacetimes with hyperbolic metric provide canonical representation for a large class of fourmanifolds and an interesting question is whether these spaces are obtained as preferred extremals and/or vacuum extremals. 4D hyperbolic space with Minkowski signature is locally isometric with AdS_{4}. This suggests a connection with AdS_{4}/CFT correspondence of Mtheory. The boundary of AdS would be now replaced with 3D lightlike orbit of partonic 2surface at which the signature of the induced metric changes. The metric 2dimensionality of the lightlike surface makes possible generalization of 2D conformal invariance with the lightlike coordinate taking the role of complex coordinate at lightlike boundary. AdS would presumably represent a special case of a more general family of spacetime surfaces with constant Ricci scalar satisfying EinsteinMaxwell equations and generalizing the AdS_{4}/CFT correspondence. For the ordinary AdS_{5} correspondence empty M^{4} is identified as boundary. In the recent case the boundary of AdS_{4} is replaced with a 3D lightlike orbit of partonic 2surface at which the signature of the induced metric changes. String world sheets have boundaries along lightlike 3surfaces and spacelike 3surfaces at the lightlike boundaries of CD. The metric 2dimensionality of the lightlike surface makes possible generalization of 2D conformal invariance with the lightlike coordinate taking the role of hyper complex coordinate at lightlike 3surface. AdS_{5}× S^{5} of Mtheory context is replaced by a 4surface of constant Ricci scalar in 8D imbedding space M^{4}× CP_{2} satisfying EinsteinMaxwell equations. A generalization of AdS_{4}/CFT correspondence would be in question. There is however a strong objection from cosmology: the accelerated expansion of the Universe requires positive value of Λ and favors De Sitter Space dS_{4} instead of AdS_{4}. These observations give motivations for finding whether AdS_{4} and/or AdS_{4} allows an imbedding as vacuum extremal to M^{4}× S^{2}⊂ M^{4}× CP_{2}, where S^{2} is a homologically trivial geodesic sphere of CP_{2}. It is easy to guess the general form of the imbedding by writing the line elements of, M^{4}, S^{2}, and AdS_{4}.
The conclusion is that AdS_{4} allows a local imbedding as a vacuum extremal. Whether also an imbedding as a nonvacuum preferred extremal to homologically nontrivial geodesic sphere is possible, is an interesting question. The only modification in the case of De Sitter space dS_{4} is the replacement of the function A= 1+y^{2} appearing in the metric of AdS_{4} with A=1y^{2}. Also now the imbedded portion of the metric is a spherical shell. This brings in mind TGD inspired model for the final state of the star which is also a spherical shell. pAdic length scale hypothesis motivates the conjecture that stars indeed have onionlike layered structure consisting of shells, whose radii are consistent with padic length scale hypothesis. This brings in mind also TitiusBode law. For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Do geometric invariants of preferred extremals define topological invariants of spacetime surface and code for quantum physics?". 
Could correlation functions, Smatrix, and coupling constant evolution be coded the statistical properties of preferred extremals?
Quantum classical correspondence states that all aspects of quantum states should have correlates in the geometry of preferred extremals. In particular, various elementary particle propagators should have a representation as properties of preferred extremals. This would allow to realize the old dream about being able to say something interesting about coupling constant evolution although it is not yet possible to calculate the Mmatrices and Umatrix. Hitherto everything that has been said about coupling constant evolution has been rather speculative arguments except for the general vision that it reduces to a discrete evolution defined by padic length scales. General first principle definitions are much more valuable than ad hoc guesses even if the latter give rise to explicit formulas. In quantum TGD and also at its QFT limit various correlation functions in given quantum state code for its properties. These correlation functions should have counterparts in the geometry of preferred extremals. Even more: these classical counterparts for a given preferred extremal ought to be identical with the quantum correlation functions for the superposition of preferred extremals.

Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants
The recent progress in the understanding of the preferred extremals led to a reduction of the field equations to conditions stating for Euclidian signature the existence of Kähler metric. The resulting conditions are a direct generalization of corresponding conditions emerging for the string world sheet and stating that the 2metric has only nondiagonal components in complex/hypercomplex coordinates. Also energy momentum of Kähler action and has this characteristic (1,1) tensor structure. In Minkowskian signature one obtains the analog of 4D complex structure combining hypercomplex structure and 2D complex structure. The construction lead also to the understanding of how Einstein's equations with cosmological term follow as a consistency condition guaranteeing that the covariant divergence of the Maxwell's energy momentum tensor assignable to Kähler action vanishes. This gives T= kG+Λ g. By taking trace a further condition follows from the vanishing trace of T: R = 4Λ/k . That any preferred extremal should have a constant Ricci scalar proportional to cosmological constant is very strong prediction. Note however that both Λ and k∝ 1/G are both parameters characterizing one particular preferred extremal. One could of course argue that the dynamics allowing only constant curvature spacetimes is too simple. The point is however that particle can topologically condense on several spacetime sheets meaning effective superposition of various classical fields defined by induced metric and spinor connection. The following considerations demonstrate that preferred extremals can be seen as canonical representatives for the constant curvature manifolds playing central role inThurston's geometrization theorem known also as hyperbolization theorem implying that geometric invariants of spacetime surfaces transform to topological invariants. The generalization of the notion of Ricci flow to Maxwell flow in the space of metrics and further to Kähler flow for preferred extremals in turn gives a rather detailed vision about how preferred extremals organize to oneparameter orbits. It is quite possible that Kähler flow is actually discrete. The natural interpretation is in terms of dissipation and self organization. A. The geometrical invariants of spacetime surfaces as topological invariants An old conjecture inspired by the preferred extremal property is that the geometric invariants of the spacetime surface serve as topological invariants. The reduction ofKähler action to 3D ChernSimons terms gives support for this conjecture as a classical counterpart for the view about TGD as almost topological QFT. The following arguments give a more precise content to this conjecture in terms of existing mathematics.
B. Generalizing Ricci flow to Maxwell flow for 4geometries and K\"ahler flow for spacetime surfaces The notion of Ricci flow has played a key part in the geometrization of topological invariants of Riemann manifolds. I certainly did not have this in mind when I choose to call my unification attempt "Topological Geometrodynamics" but this title strongly suggests that a suitable generalization of Ricci flow could play a key role in the understanding of also TGD. B.1. Ricci flow and Maxwell flow for 4geometries The observation about constancy of 4D curvature scalar for preferred extremals inspires a generalization of the wellknown volume preserving Ricci flow introduced by Richard Hamilton and defined in the space of Riemann metrics as dg_{αβ}/dt= 2R_{αβ}+ (2/D)R_{avg}g_{αβ} . Here R_{avg} denotes the average of the scalar curvature, and D is the dimension of the Riemann manifold. The flow is volume preserving in average sense as one easily checks (<g^{αβ}dg_{αβ}/dt> =0). The volume preserving property of this flow allows to intuitively understand that the volume of a 3manifold in the asymptotic metric defined by the Ricci flow is topological invariant. The fixed points of the flow serve as canonical representatives for the topological equivalence classes of 3manifolds. These 3manifolds (for instance hyperbolic 3manifolds with constant sectional curvatures) are highly symmetric. This is easy to understand since the flow is dissipative and destroys all details from the metric. What happens in the recent case? The first thing to do is to consider what might be called Maxwell flow in the space of all 4D Riemann manifolds allowing Maxwell field.
B.2. Maxwell flow for spacetime surfaces One can consider Maxwell flow for spacetime surfaces too. In this case Kähler flow would be the appropriate term and provides families of preferred extremals. Since spacetime surfaces inside CD are the basic physical objects are in TGD framework, a possible interpretation of these families would be as flows describing physical dissipation as a fourdimensional phenomenon polishing details from the spacetime surface interpreted as an analog of Bohr orbit.
B.3. Dissipation, self organization, transition to chaos, and coupling constant evolution A beautiful connection with concepts like dissipation, selforganization, transition to chaos, and coupling constant evolution suggests itself.
B.4 Does a 4D counterpart of thermodynamics make sense? The interpretation of the Kähler flow in terms of dissipation, the constancy of R, and almost constancy of L_{K} suggest an interpretation in terms of 4D variant of thermodynamics natural in zero energy ontology (ZEO), where physical states are analogs for pairs of initial and final states of quantum event are quantum superpositions of classical time evolutions. Quantum theory becomes a "square root" of thermodynamics so that 4D analog of thermodynamics might even replace ordinary thermodynamics as a fundamental description. If so this 4D thermodynamics should be qualitatively consistent with the ordinary 3D thermodynamics.
For details and background see the chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the title "Preferred extremals of Kähler action as manifolds with constant Ricci scalar whose geometric invariants are topological invariants". 
Electron as a trefoil or something more general?
The progress in the understanding of the modified Dirac equation led to the conclusion that the mere conservation of em charge in spinorial manner rather than from theorem of Gauss leads the conclusion that the solutions of the modified Dirac equation must be localized at string world sheets and partonic 2surfaces: righthanded neutrino is expection and delocalized into entire spacetime surface. Looking more closely the implications of this led to the conclusion that every ordinary elementary fermion is accompanied by closed string. Ordinary elementary bosons can be accompanied by two such strings. If lightlike wormhole throat orbots carry several fermions one can have several closed strings. These closed strings can get knotted and braided. I do not bother to type more but attach the abstract of a little article Electron as a trefoil or something more general?. There have been suggestions that elementary particle could be braided structure and that standard model quantum numbers could be reduced to topology. In TGD framework this option does not look plausible. The braiding at the level of wormhole throat orbits is however in principle possible but need not be significant for the known elementary particles. In TGD framework elementary particle is identified as a closed Kähler magnetic flux tube carrying monopole magnetic field. This flux tube is accompanied by a closed string representing the end of string world sheet carrying induced spinor field. This string can be homologically nontrivial curve and can also get knotted. Bosons even braiding becomes possible and in the general case knotting, braiding, and nontrivial homology are possible. Therefore an extremely rich topological structure is predicted, which might corresponds to relatively low energy scale. Topological sum for knots and reconnection are the basic topological reactions for these strings and can change the knotting of the string. These reactions represent basic vertices for closed strings so that closed string model could give at least idea about the dynamics of knotting and unknotting. For details and background see the chapter Knots and TGD, or the already mentioned article . 
Realization of large N SUSY in TGD
The generators large N SUSY algebras are obtained by taking fermionic currents for second quantized fermions and replacing either fermion field or its conjugate with its particular mode. The resulting super currents are conserved and define super charges. By replacing both fermion and its conjugate with modes one obtains c number valued currents. Therefore N=∞ SUSY  presumably equivalent with superconformal invariance  or its finite N cutoff is realized in TGD framework and the challenge is to understand the realization in more detail. Superspace viz. Grassmann algebra valued fields Standard SUSY induces superspace extending spacetime by adding anticommuting coordinates as a formal tool. Many mathematicians are not enthusiastic about this approach because of the purely formal nature of anticommuting coordinates. Also I regard them as a nonsense geometrically and there is actually no need to introduce them as the following little argument shows. Grassmann parameters (anticommuting theta parameters) are generators of Grassmann algebra and the natural object replacing superspace is this Grassmann algebra with coefficients of Grassmann algebra basis appearing as ordinary real or complex coordinates. This is just an ordinary space with additional algebraic structure: the mysterious anticommuting coordinates are not needed. To me this notion is one of the conceptual monsters created by the overpragmatic thinking of theoreticians. This allows allows to replace field space with super field space, which is completely welldefined object mathematically, and leave spacetime untouched. Linear field space is simply replaced with its Grassmann algebra. For nonlinear field space this replacement does not work. This allows to formulate the notion of linear superfield just in the same manner as it is done usually. The generators of supersymmetries in superspace formulation reduce to super translations , which anticommute to translations. The super generators Q_{α} and Qbar_{dotβ} of super Poincare algebra are Weyl spinors commuting with momenta and anticommuting to momenta: {Q_{α},Qbar_{dotβ}}=2σ^{μ}_{αdotβ}P_{μ} . One particular representation of super generators acting on super fields is given by D_{α}=i∂/∂θ_{α} , D_{dotα}=i∂/∂_{θbardotα}+ θ^{β}σ^{μ}_{βdotα} ∂_{μ} Here the index raising for 2spinors is carried out using antisymmetric 2tensor ε^{αβ}. Superspace interpretation is not necessary since one can interpret this action as an action on Grassmann algebra valued field mixing components with different fermion numbers. Chiral superfields are defined as fields annihilated by D_{dotα}. Chiral fields are of form Ψ(x^{μ}+iθbarσ^{μ}θ, θ). The dependence on θbar_{dotα} comes only from its presence in the translated Minkowski coordinate annihilated by D_{dotα}. Superspace enthusiast would say that by a translation of M^{4} coordinates chiral fields reduce to fields, which depend on θ only. The space of fermionic Fock states at partonic 2surface as TGD counterpart of chiral super field As already noticed, another manner to realize SUSY in terms of representations the super algebra of conserved supercharges. In TGD framework these super charges are naturally associated with the modified Dirac equation, and anticommuting coordinates and superfields do not appear anywhere. One can however ask whether one could identify a mathematical structure replacing the notion of chiral super field. I have proposed that generalized chiral superfields could effectively replace induced spinor fields and that second quantized fermionic oscillator operators define the analog of SUSY algebra. One would have N=∞ if all the conformal excitations of the induced spinor field restricted on 2surface are present. For righthanded neutrino the modes are labeled by two integers and delocalized to the interior of Euclidian or Minkowskian regions of spacetime sheet. The obvious guess is that chiral superfield generalizes to the field having as its components manyfermions states at partonic 2surfaces with theta parameters and their conjugates in oneone correspondence with fermionic creation operators and their hermitian conjugates.
How the fermionic anticommutation relations are determined? Understanding the fermionic anticommutation relations is not trivial since all fermion fields except righthanded neutrino are assumed to be localized at 2surfaces. Since fermionic conserved currents must give rise to welldefined charges as 3D integrals the spinor modes must be proportional to a square root of delta function in normal directions. Furthermore, the modified Dirac operator must act only in the directions tangential to the 2surface in order that the modified Dirac equation can be satisfied. The square root of delta function can be formally defined by starting from the expansion of delta function in discrete basis for a particle in 1D box. The product of two functions in xspace is convolution of Fourier transforms and the coefficients of Fourier transform of delta function are apart from a constant multiplier equal to 1: δ (x)= K∑_{n} exp(inx/2π L). Therefore the Fourier transform of square root of delta function is obtained by normalizing the Fourier transform of delta function by N^{1/2}, where N→ ∞ is the number of plane waves. In other words: (δ (x))^{1/2}= (K/N)^{1/2}∑_{ n}exp(inx/2π L). Canonical quantization defines the standard approach to the second quantization of the Dirac equation.
Can these conditions be applied both at string world sheets and partonic 2surfaces.
One can of course worry what happens at the limit of vacuum extremals. The problem is that Γ^{t} vanishes for spacetime surfaces reducing to vacuum extremals at the 2surfaces carrying fermions so that the anticommutations are inconsistent. Should one require  as done earlier that the anticommutation relations make sense at this limit and cannot therefore have the standard form but involve the scalar magnetic flux formed from the induced Kähler form by permuting it with the 2D permutations symbl? The restriction to preferred extremals, which are always nonvacuum extremals, might allow to avoid this kind of problems automatically. In the case of righthanded neutrino the situation is genuinely 3dimensional and in this case nonvacuum extremal property must hold true in the regions where the modes of ν_{R} are nonvanishing. The same mechanism would save from problems also at the partonic 2surfaces. The dynamics of induced spinor fields must avoid classical vacuum. Could this relate to color confinement? Could hadrons be surrounded by an insulating layer of Kähler vacuum? For details and background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the same title. 
M^{8}H duality, preferred extremals, criticality, and Mandelbrot fractals
M^{8}H duality (see this) represents an intriguing connection between number theory and TGD but the mathematics involved is extremely abstract and difficult so that I can only represent conjectures. In the following the basic duality is used to formulate a general conjecture for the construction of preferred extremals by iterative procedure. What is remarkable and extremely surprising is that the iteration gives rise to the analogs of Mandelbrot fractals and spacetime surfaces can be seen as fractals defined as fixed sets of iteration. The analogy with Mandelbrot set can be also seen as a geometric correlate for quantum criticality. M^{8}H duality M^{8}H duality states the following. Consider a distribution of two planes M^{2}(x) integrating to a 2surface N^{2} with the property that a fixed 1plane M^{1} defining time axis globally is contained in each M^{2}(x) and therefore in N^{2}. M^{1} defines real axis of octonionic plane M^{8} and M^{2}(x) a local hypercomplex plane. Quaternionic subspaces with this property can be parameterized by points of CP_{2}. Define quaternionic surfaces in M^{8} as 4surfaces, whose tangent plane is quaternionic at each point x and contains the local hypercomplex plane M^{2}(x) and is therefore labelled by a point s(x)∈ CP_{2}. One can write these surfaces as union over 2D surfaces associated with points of N^{2}: X^{4}= ∪_{x∈ N2} X^{2}(x)⊂ E^{6} . These surfaces can be mapped to surfaces of M^{4}× CP_{2} via the correspondence (m(x),e(x))→ (m,s(T(X^{4}(x)). Also the image surface contains at given point x the preferred plane M^{2}(x) ⊃ M^{1}. One can also write these surfaces as union over 2D surfaces associated with points of N^{2}: X^{4}= ∪_{x∈ N2} X^{2}(x)⊂ E^{2}× CP_{2} . One can also ask what are the conditions under which one can map surfaces X^{4}= ∪_{x∈ N2} X^{2}⊂ E^{2}× CP_{2} to 4surfaces in M^{8}. The map would be given by (m,s)→ (m,T^{4}(s) and the surface would be of the form as already described. The surface X^{4} must be such that the distribution of 4D tangent planes defined in M^{8} is integrable and this gives complicated integrability conditions. One might hope that the conditions might hold true for preferred extremals satisfying some additional conditions. One must make clear that these conditions do not allow most general possible surface. The point is that for preferred extremals with Euclidian signature of metric the M^{4} projection is 3dimensional and involves light like projection. Here the fact that lightlike line L⊂ M^{2} spans M^{2} in the sense that the complement of its orthogonal complement in M^{8} is M^{2}. Therefore one could consider also more general solution ansatz for which one has X^{4}= ∪_{x∈ Lx⊂ N2} X^{3}(x)⊂ E^{2}× CP_{2} . One can also consider coquaternionic surfaces as surfaces for which tangent space is in the dual of a quaternionic subspace containing the preferred M^{2}(x). The integrability conditions The integrability conditions are associated with the expression of tangent vectors of T(X^{4}) as a linear combination of coordinate gradients ∇ m^{k}, where m^{k} denote the coordinates of M^{8}. Consider the 4 tangent vectors e_{i)} for the quaternionic tangent plane (containing M^{2}(x)) regarded as vectors of M^{8}. e_{i)} have components e_{i)}^{k}, i=1,..,4, k=1,...,8. One must be able to express e_{i)} as linear combinations of coordinate gradients ∇ m^{k}: e_{i)}^{k}= e_{i)}^{α}∂_{α}m^{k} . Here x^{α} and e^{k} denote coordinates for X^{4} and M^{8}. By forming inner products of of e_{i)} one finds that matrix e_{i)}^{α} represents the components of vierbein at X^{4}. One can invert this matrix to get e^{i)}_{α} satisfying e^{i)}_{α}e_{i)}^{β}=δ_{α}^{β} and e^{i)}_{α}e_{j)}^{α}=δ^{i}_{j}. One can solve the coordinate gradients ∇ m^{k} from above equation to get ∂_{α}m^{k} = e^{i)}_{α}e_{i)}^{k}== E_{α}^{k} . The integrability conditions follow from the gradient property and state D_{α}E^{k}_{β}= D_{β}E^{k}_{α} . One obtains 8× 6=48 conditions in the general case. The slicing to a union of twosurfaces labeled by M^{2}(x) reduces the number of conditions since the number of coordinates m^{k} reduces from 8 to 6 and one has 36 integrability conditions but still them is much larger than the number of free variables essentially the six transversal coordinates m^{k}. For coquaternionic surfaces one can formulate integrability conditions now as conditions for the existence of integrable distribution of orthogonal complements for tangent planes and it seems that the conditions are formally similar. How to solve the integrability conditions and field equations for preferred extremals? The basic idea has been that the integrability condition characterize preferred extremals so that they can be said to be quaternionic in a welldefined sense. Could one imagine solving the integrability conditions by some simple ansatz utilizing the core idea of M^{8}H duality? What comes in mind is that M^{8} represents tangent space of M^{4}× CP_{2} so that one can assign to any point (m,s) of 4surface X^{4}⊂ M^{4}× CP_{2} a tangent plane T^{4}(x) in its tangent space M^{8} identifiable as subspace of complexified octonions in the proposed manner. Assume that s∈ CP_{2} corresponds to a fixed tangent plane containing M^{2}_{x}, and that all planes M^{2}_{x} are mapped to the same standard fixed hyperoctonionic plane M^{2}⊂ M^{8}, which does not depend on x. This guarantees that s corresponds to a unique quaternionic tangent plane for given M^{2}(x). Consider the map Tοs. The map takes the tangent plane T^{4} at point (m,e)∈ M^{4}× E^{4} and maps it to (m,s_{1}=s(T^{4}))∈ M^{4}× CP_{2}. The obvious identification of quaternionic tangent plane at (m,s_{1}) would be as T^{4}. One would have Tοs=Id. One could do this for all points of the quaternion surface X^{4}⊂ E^{4} and hope of getting smooth 4surface X^{4}⊂ H as a result. This is the case if the integrability conditions at various points (m,s(T^{4})(x))∈ H are satisfied. One could equally well start from a quaternionic surface of H and end up with integrability conditions in M^{8} discussed above. The geometric meaning would be that the quaternionic surface in H is image of quaternionic surface in M^{8} under this map. Could one somehow generalize this construction so that one could iterate the map Tοs to get Tοs=Id at the limit? If so, quaternionic spacetime surfaces would be obtained as limits of iteration for rather arbitrary spacetime surface in either M^{8} or H. One can also consider limit cycles, even limiting manifolds with finitedimension which would give quaternionic surfaces. This would give a connection with chaos theory.
One might hope that the selfreferentiality condition sοT=Id for the CP_{2} projection of (m,s) or its fractal generalization could solve the complicated integrability conditions for the map T. The image of the spacetime surface in tangent space M^{8} in turn could be interpreted as a description of spacetime surface using coordinates defined by the local tangent space M^{8}. Also the analogy for the duality between position and momentum suggests itself. Is there any hope that this kind of construction could make sense? Or could one demonstrate that it fails? If s would fix completely the tangent plane it would be probably easy to kill the conjecture but this is not the case. Same s corresponds for different planes M^{2}_{x} to different point tangent plane. Presumably they are related by a local G_{2} or SO(7) rotation. Note that the construction can be formulated without any reference to the representation of the imbedding space gamma matrices in terms of octonions. Complexified octonions are enough in the tangent space of M^{8}. Connection with Mandelbrot fractal and fractals as fixed sets for iteration The occurrence of iteration in the construction of preferred extremals suggests a deep connection with the standard construction of 2D fractals by iteration  about which Mandelbrot fractal is the canonical example. X^{2}(x) (or X^{3}(x)) could be identified as a union of orbits for the iteration of sοT. The appearance of the iteration map in the construction of solutions of field equation would answer positively to a long standing question whether the extremely beautiful mathematics of 2D fractals could have some application at the level of fundamental physics according to TGD. X^{2} (or X^{3}) would be completely analogous to Mandelbrot set in the sense that it would be boundary separating points in two different basis of attraction. In the case of Mandelbrot set iteration would take points at the other side of boundary to origin on the other side and to infinity. The points of Mandelbrot set are permuted by the iteration. In the recent case sοT maps X^{2} (or X^{3}) to itself. This map need not be diffeomorphism or even continuous map. The criticality of X^{2} (or X^{3}) could be seen as a geometric correlate for quantum criticality. In fact, iteration plays a very general role in the construction of fractals. Very general fractals can be defined as fixed sets of iteration and simple rules for iteration produce impressive representations for fractals appearing in Nature. Therefore it would be highly satisfactory if spacetime surfaces would be in welldefined sense fixed sets of iteration. This would be also numerically beautiful aspect since fixed sets of iteration can be obtained as infinite limit of iteration for almost arbitrary initial set. What is intriguing and challenging is that there are several very attractive approaches to the construction of preferred extremals and the challenge of unifying them still remains. For details and background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation, or the article with the same title. 
The analog of AdS^{5} duality in TGD frameworkThe generalization of AdS^{5} duality of N=4 SYMs to TGD framework is highly suggestive and states that string world sheets and partonic 2surfaces play a dual role in the construction of Mmatrices. In the following I give an argument providing a "proof" of this duality and also demonstrating that for singular string world sheets and partonic 2surfaces perturbative description of generalized Feynman diagrams is especially simple since string effectively reduces to point like particles. Some terminology first.
For background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as InfiniteDimensional Geometry" or the article with the same title. 
Twistor revolution and TGDLubos wrote a nice summary about the talk of Nima Arkani Hamed about twistor revolution in Strings 2012 and gave also a link to the talk. It seems that Nima and collaborators are ending to a picture about scattering amplitudes which strongly resembles that provided bt generalized Feynman diagrammatics in TGD framework TGD framework is much more general than N=4 SYM and is to it same as general relativity for special relativity whereas the latter is completely explicit. Of course, I cannot hope that TGD view could be taken seriously  at least publicly. One might hope that these approaches could be combined some day: both have a lot to give for each other. Below I compare these approaches. The origin of twistor diagrammatics In TGD framework zero energy ontology forces to replace the idea about continuous unitary evolution in Minkowski space with something more general assignable to causal diamonds (CDs), and Smatrix is replaced with a square root of density matrix equal to a hermitian l square root of density matrix multiplied by unitary Smatrix. Also in twistor approach unitarity has ceased to be a star actor. In pAdic context continuous unitary time evolution fails to make sense also mathematically. Twistor diagrammatics involves only massless on mass shell particles on both external and internal lines. Zero energy ontology (ZEO) requires same in TGD: wormhole lines carry parallely moving massless fermions and antifermions. The mass shell conditions at vertices are enormously powerful and imply UV finiteness. Also IR finiteness follows if external particles are massive. What one means with mass is however a delicate matter. What does one mean with mass? I have pondered 35 years this question and the recent view is inspired by padic mass calculations and ZEO, and states that observed mass is in a welldefined sense expectation value of longitudinal mass squared for all possible choices of M^{2} ⊂ M^{4} characterizing the choices of quantization axis for energy and spin at the level of "world of classical worlds" (WCW) assignable with given causal diamond CD. The choice of quantization axis thus becomes part of the geometry of WCW. All wormhole throats are massless but develop nonvanishing longitudinal mass squared. Gauge bosons correspond to wormhole contacts and thus consist of pairs of massless wormhole throats. Gauge bosons could develop 4D mass squared but also remain massless in 4D sense if the throats have parallel massless momenta. Longitudinal mass squared is however nonvanishing andpadic thermodynamics predicts it. The emergence of 2D subdynamics at spacetime level Nima et al introduce ordering of the vertices in 4D case. Ordering and related braiding are however essentially 2D notions. Somehow 2D theory must be a part of the 4D theory also at spacetime level, and I understood that understanding this is the challenge of the twistor approach at this moment. The twistor amplitude can be represented as sum over the permutations of n external gluons and all diagrams corresponding to the same permutation are equivalent. Permutations are more like braidings since they carry information about how the permutation proceeded as a homotopy. YangBaxter equation emerge. The allowed braidings are minimal braidings in the sense that the repetitions of permutations of two adjacent vertices are not considered to be separate. Minimal braidings reduce to ordinary permutations. Nima also talks about affine braidings which I interpret as analogs of KacMoody algebras meaning that one uses projective representations which for KacMoody algebra mean nontrivial central extension. Perhaps the condition is that the square of a permutation permuting only two vertices which each other gives only a nontrivial phase factor. Lubos suggests an alternative interpretation for "affine" which would select only special permutations and cannot be therefore correct. There are rules of identifying the permutation associated with a given diagram involving only basic 3gluon vertex with white circle and its conjugate. Lubos explains this "Mickey Mouse in maze" rule in his posting in detail: to determine the image p(n) of vertex n in the permutation put a mouse in the maze defined by the diagram and let it run around obeying single rule: if the vertex is black turn right and if the vertex is white turn left. Eventually the mouse ends up to external vertex. The mouse cannot end up with loop: if it would do it, the rule would force it to run back to n after the full loop and one would have fixed point: p(n)=n. The reduction in the number of diagrams is enormous: the infinity of different diagrams reduces to n! diagrams!
The emergence of Yangian symmetry Yangian symmetry associated with the conformal transformations of M^{4} is a key symmetry of Grassmannian approach. Is it possible to derive it in TGD framework?
Twistor approach has also its problems and here TGD suggests how to proceed. Signature problem is the first problem.
For background see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as InfiniteDimensional Geometry" or the article with the same title. 
About the definition of HamiltonJacobi structureI have talked in previous postings a lot about HamiltonJacobi structure without bothering to write detailed definitions. In the following I discuss the notion in more detail. Thanks for Hamed who asked for more detailed explanation. Hermitian and hyperHermitian structures The starting point is the observation that besides the complex numbers forming a number field there are hypercomplex numbers. Imaginary unit i is replaced with e satisfying e^{2}=1. One obtains an algebra but not a number field since the norm is Minkowskian norm x^{2}y^{2}, which vanishes at lightcone x=y so that lightlike hypercomplex numbers x+/ e) do not have inverse. One has "almost" number field. Hypercomplex numbers appear naturally in 2D Minkowski space since the solutions of a massless field equation can be written as f=g(u=tex)+h(v=t+ex) whith e^{2}=1 realized by putting e=1. Therefore Wick rotation relates sums of holomorphic and antiholomorphic functions to sums of hyperholomorphic and antihyperholomorphic functions. Note that u and v are hypercomplex conjugates of each other. Complex ndimensional spaces allow Hermitian structure. This means that the metric has in complex coordinates (z_{1},....,z_{n}) the form in which the matrix elements of metric are nonvanishing only between z_{i} and complex conjugate of z_{j}. In 2D case one obtains just ds^{2}=g_{zz*}dzdz*. Note that in this case metric is conformally flat since line element is proportional to the line element ds^{2}=dzdz* of plane. This form is always possible locally. For complex nD case one obtains ds^{2}=g_{ij*}dz^{i}dz^{j*}. g_{ij*}=(g_{ji*})* guaranteing the reality of ds^{2}. In 2D case this condition gives g_{zz*}= (g_{z*z})*. How could one generalize this line element to hypercomplex ndimensional case? In 2D case Minkowski space M^{2} one has ds^{2}= g_{uv}dudv, g_{uv}=1. The obvious generalization would be the replacement ds^{2}=g_{uivj}du^{i}dv^{j}. Also now the analogs of reality conditions must hold with respect to u^{i}↔ v^{i}. HamiltonJacobi structure Consider next the path leading to HamiltonJacobi structure. 4D Minkowski space M^{4}=M^{2}× E^{2} is Cartesian product of hypercomplex M^{2} with complex plane E^{2}, and one has ds^{2}= dudv+ dzdz* in standard Minkowski coordinates. One can also consider more general integrable decompositions of M^{4} for which the tangent space TM^{4}=M^{4} at each point is decomposed to M^{2}(x)× E^{2}(x). The physical analogy would be a position dependent decomposition of the degrees of freedom of massless particle to longitudinal ones (M^{2}(x): lightlike momentum is in this plane) and transversal ones (E^{2}(x): polarization vector is in this plane). Cylindrical and spherical variants of Minkowski coordinates define two examples of this kind of coordinates (it is perhaps a good exercize to think what kind of decomposition of tangent space is in question in these examples). An interesting mathematical problem highly relevant for TGD is to identify all possible decompositions of this kind for empty Minkowski space. The integrability of the decomposition means that the planes M^{2}(x) are tangent planes for 2D surfaces of M^{4} analogous to Euclidian string world sheet. This gives slicing of M^{4} to Minkowskian string world sheets parametrized by euclidian string world sheets. The question is whether the sheets are stringy in a strong sense: that is minimal surfaces. This is not the case: for spherical coordinates the Euclidian string world sheets would be spheres which are not minimal surfaces. For cylindrical and spherical coordinates hower M^{2}(x) integrate to plane M^{2} which is minimal surface. Integrability means in the case of M^{2}(x) the existence of lightlike vector field J whose flow lines define a global coordinate. Its existence implies also the existence of its conjugate and together these vector fields give rise to M^{2}(x) at each point. This means that one has J= Ψ∇ Φ: Φ indeed defines the global coordinate along flow lines. In the case of M^{2} either the coordinate u or v would be the coordinate in question. This kind of flows are called Beltrami flows. Obviously the same holds for the transversal planes E^{2}. One can generalize this metric to the case of general 4D space with Minkowski signature of metric. At least the elements g_{uv} and g_{zz*} are nonvanishing and can depend on both u,v and z,z*. They must satisfy the reality conditions g_{zz*}= (g_{zz*})* and g_{uv}= (g_{vu})* where complex conjugation in the argument involves also u↔ v besides z↔ z*. The question is whether the components g_{uz}, g_{vz}, and their complex conjugates are nonvanishing if they satisfy some conditions. They can. The direct generalization from complex 2D space would be that one treats u and v as complex conjugates and therefore requires a direct generalization of the hermiticity condition g_{uz}= (g_{vz*})*, g_{vz}= (g_{uz*})* . This would give complete symmetry with the complex 2D (4D in real sense) spaces. This would allow the algebraic continuation of hermitian structures to HamiltonJacobi structures by just replacing i with e for some complex coordinates. For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title. 
The importance of being lightlikeThe singular geometric objects associated with the spacetime surface have become increasingly important in TGD framework. In particular, the recent progress has made clear that these objects might be crucial for the understanding of quantum TGD. The singular objects are associated not only with the induced metric but also with the effective metric defined by the anticommutators of the modified gamma matrices appearing in the modified Dirac equation and determined by the Kähler action. The singular objects associated with the induced metric Consider first the singular objects associated with the induced metric.
The singular objects associated with the effective metric There are also singular objects assignable to the effective metric. According to the simple arguments already developed, string world sheets and possibly also partonic 2surfaces are singular objects with respect to the effective metric defined by the anticommutators of the modified gamma matrices rather than induced gamma matrices. Therefore the effective metric seems to be much more than a mere formal structure.
These arguments provide genuine a support for the notion of quaternionicity and suggest a connection with the twistor approach. For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title. 
Quantum criticality and electroweak gauge symmetriesQuantum criticality is one of the basic guiding principles of Quantum TGD. What it means mathematically is however far from clear.
The variation of modes of the induced spinor field in a variation of spacetime surface respecting the preferred extremal property Consider first the variation of the induced spinor field in a variation of spacetime surface respecting the preferred extremal property. The deformation must be such that the deformed modified Dirac operator D annihilates the modified mode. By writing explicitly the variation of the modified Dirac action (the action vanishes by modified Dirac equation) one obtains deformations and requiring its vanishing one obtains δ Ψ=D^{1}(δ D)Ψ . D^{1} is the inverse of the modified Dirac operator defining the analog of Dirac propagator and δ D defines vertex completely analogous to γ^{k}δ A_{k} in gauge theory context. The functional integral over preferred extremals can be carried out perturbatively by expressing Δ D in terms of δ h^{k} and one obtains stringy perturbation theory around X^{2} associated with the preferred extremal defining maximum of Kähler function in Euclidian region and extremum of Kähler action in Minkowskian region (stationary phase approximation). What one obtains is stringy perturbation theory for calculating npoints functions for fermions at the ends of braid strands located at partonic 2surfaces and representing intersections of string world sheets and partonic 2surfaces at the lightlike boundaries of CDs. δ D or more precisely, its partial derivatives with respect to functional integration variables  appear atthe vertices located anywhere in the interior of X^{2} with outcoming fermions at braid ends. Bosonic propagators are replaced with correlation functions for δ h^{k}. Fermionic propagator is defined by D^{1}. After 35 years or hard work this provides for the first time a reasonably explicit formula for the Npoint functions of fermions. This is enough since by bosonic emergence(se this) these Npoint functions define the basic building blocks of the scattering amplitudes. Note that bosonic emergence states that bosons corresponds to wormhole contacts with fermion and antifermion at the opposite wormhole throats. What critical modes could mean for the induced spinor fields? What critical modes could mean for the induced spinor fields at string world sheets and partonic 2surfaces. The problematic part seems to be the variation of the modified Dirac operator since it involves gradient. One cannot require that covariant derivative remains invariant since this would require that the components of the induced spinor connection remain invariant and this is quite too restrictive condition. Right handed neutrino solutions delocalized into entire X^{2} are however an exception since they have no electroweak gauge couplings and in this case the condition is obvious: modified gamma matrices suffer a local scaling for critical deformations: δ Γ^{μ} = Λ(x)Γ^{μ} . This guarantees that the modified Dirac operator D is mapped to Λ D and still annihilates the modes of ν_{R} labelled by conformal weight, which thus remain unchanged. What is the situation for the 2D modes located at string world sheets? The condition is obvious. Ψ suffers an electroweak gauge transformation as does also the induced spinor connection so that D_{μ} is not affected at all. Criticality condition states that the deformation of the spacetime surfaces induces a conformal scaling of Γ^{μ} at X^{2}, It might be possible to continue this conformal scaling of the entire spacetime sheet but this might be not necessary and this would mean that all critical deformations induced conformal transformations of the effective metric of the spacetime surface defined by {Γ^{μ}, Γ^{ν}}=2 G^{μν}. Thus it seems that effective metric is indeed central concept (recall that if the conjectured quaternionic structure is associated with the effective metric, it might be possible to avoid problem related to the Minkowskian signature in an elegant manner). Note that one can consider even more general action of critical deformation: the modes of the induced spinor field would be mixed together in the infinitesimal deformation besides infinitesimal electroweak gauge transformation, which is same for all modes. This would extend electroweak gauge symmetry. Modified Dirac equation holds true also for these deformations. One might wonder whether the conjecture dynamically generated gauge symmetries assignable to finite measurement resolution could be generated in this manner. Thus the critical deformations would induce conformal scalings of the effective metric and dynamical electroweak gauge transformations. Electroweak gauge symmetry would be a dynamical symmetry restricted to string world sheets and partonic 2surfaces rather than acting at the entire spacetime surface. For 4D delocalized righthanded neutrino modes the conformal scalings of the effective metric are analogous to the conformal transformations of M^{4} for N=4 SYMs. Also ordinary conformal symmetries of M^{4} could be present for string world sheets and could act as symmetries of generalized Feynman graphs since even virtual wormhole throats are massless. An interesting question is whether the conformal invariance associated with the effective metric is the analog of dual conformal invariance in N=4 theories. Critical deformations of spacetime surface are accompanied by conserved fermionic currents. By using standard Noetherian formulas one can write J^{μ}_{i}= Ψbar Γ^{μ}δ_{i} Ψ + δ_{i} ΨbarΓ^{μ}Ψ . Here δ Ψ_{i} denotes derivative of the variation with respect to a group parameter labeled by i. Since δ Ψ_{i} reduces to an infinitesimal gauge transformation of Ψ induced by deformation, these currents are the analogs of gauge currents. The integrals of these currents along the braid strands at the ends of string world sheets define the analogs of gauge charges. The interpretation as KacMoody charges is also very attractive and I have proposed that the 2D Hodge duals of gauge potentials could be identified as KacMoody currents. If so, the 2D Hodge duals of J would define the quantum analogs of dynamical electroweak gauge fields and KacMoody charge could be also seen as nonintegral phase factor associated with the braid strand in Abelian approximation (the interpretation in terms of finite measurement resolution is discussed earlier). One can also define super currents by replacing Ψbar or Ψ by a particular mode of the induced spinor field as well as cnumber valued currents by performing the replacement for both Ψbar and Ψ. As expected, one obtains a superconformal algebra with all modes of induced spinor fields acting as generators of supersymmetries restricted to 2D surfaces. The number of the charges which do not annihilate physical states as also the effective number of fermionic modes could be finite and this would suggest that the integer N for the supersymmetry in question is finite. This would conform with the earlier proposal inspired by the notion of finite measurement resolution implying the replacement of the partonic 2surfaces with collections of braid ends. Note that KacMoody charges might be associated with "long" braid strands connecting different wormhole throats as well as short braid strands connecting opposite throats of wormhole contacts. Both kinds of charges would appear in the theory. What is the interpretation of the critical deformations? Critical deformations bring in an additional gauge symmetry. Certainly not all possible gauge transformations are induced by the deformations of preferred extremals and a good guess is that they correspond to holomorphic gauge group elements as in theories with KacMoody symmetry. What is the physical character of this dynamical gauge symmetry?
Note that criticality suggests that one must perform functional integral over WCW by decomposing it to an integral over zero modes for which deformations of X^{4} induce only an electroweak gauge transformation of the induced spinor field and to an integral over moduli corresponding to the remaining degrees of freedom. For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title. 
Constraints on superconformal invariance from padic mass calculations and ZEOThe generalization of superconformal symmetry to 4D context is basic element of quantum TGD. Several variants for the realization of supersymmetry has been proposed. Especially problematic has been the question about whether the counterpart of standard SUSY is realized in TGD framework or not. Thanks to the progress made in the understanding of preferred extremals of Kähler action and of solutions of modified Dirac equation, it has become to develop the vision in considerable detail. As a consequence some existing alternative visions have been eliminated from consideration. One of them the proposal that Equivalence Principle might have realization in terms of coset representations. Second one is the idea that righthanded neutrino generating spacetime supersymmetry might be in color partial wave so that sparticles would be colored: this would explain why sparticles are not observed at LHC. The new view providing a possible alternative explanation for the absence of sparticles has been discussed in previous posting. Concerning the general understanding of superconformal invariance in TGD framework, an important physical constraint comes from the success padic thermodynamics: superconformal invariance indeed forms a core element of padic mass calculations (see this, especially this).
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equationarticle with the same title. 
The role of the righthanded neutrino in TGD based view about SUSYThe general ansatz for the preferred extremals of Kähler action and application of the conservation of em charge to the modified Dirac equation have led to a rather detailed view about classical and TGD and allowed to build a bridge between general vision about superconformal symmetries in TGD Universe and field equations.
A highly interesting aspect of SuperKacMoody symmetry is the special role of right handed neutrino.
1. How particle and right handed neutrino are bound together? Ordinary SUSY means that apart from kinematical spin factors sparticles and particles behave identically with respect to standard model interactions. These spin factors would allow to distinguish between particles and sparticles. But is this the case now?
2. Taking a closer look on sparticles It is good to take a closer look at the delocalized right handed neutrino modes.
What can happen in spin degrees of freedom in supersymmetric interaction vertices if one accepts this interpretation? As already noticed, this depends solely on what one assumes about the correlation of the fourmomenta of particle and ν_{R}.
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation of "Quantum TGD as InfiniteDimensional Geometry" or the article with the same title. 
The emergence of Yangian symmetry and gauge potentials as duals of KacMoody currentsYangian symmetry plays a key role in N=4 supersymmetric gauge theories. What is special in Yangian symmetry is that the algebra contains also multilocal generators. In TGD framework multilocality would naturally correspond to that with respect to partonic 2surfaces and string world sheets and the proposal has been that the SuperKacMoody algebras assignable to string worlds sheets could generalize to Yangian. Witten has written a beautiful exposition of Yangian algebras (see this). Yangian is generated by two kinds of generators J^{A} and Q^{A} by a repeated formation of commutators. The number of commutations tells the integer characterizing the multilocality and provides the Yangian algebra with grading by natural numbers. Witten describes a 2dimensional QFT like situation in which one has 2D situation and KacMoody currents assignable to real axis define the KacMoody charges as integrals in the usual manner. It is also assumed that the gauge potentials defined by the 1form associated with the KacMoody current define a flat connection: ∂_{μ}j^{A}_{ν} ∂_{ν}j^{A}_{ν} +[j^{A}_{μ},j^{A}_{ν}]=0 . This condition guarantees that the generators of Yangian are conserved charges. One can however consider alternative manners to obtain the conservation.
How to generalize this to the recent context?
The resulting algebra satisfies the basic commutation relations [J^{A},J^{B}]=f^{AB}_{C}J^{C} , [J^{A},Q^{B}]=f^{AB}_{C}Q^{C} . plus the rather complex Serre relations described in Witten's article). The connection between KacMoody symmetries and gauge symmetries is suggestive and in this case it would be realized in terms of 2D Hodge duality. Also finite measurement resolution realized in the sense that the points at the ends of given braid strand are regarded to be effectively infinitesimally close so that the gauge algebra is effectively Abelian is essential. Yangian symmetry is crucial for the success of the twistor approach. Zero energy ontology implies that generalized Feynman diagrams contain only massless partonic 2surfaces with propagators defined by longitudinal momentum components defined in terms of M^{2}⊂ M^{4} characterizing given causal diamond. There there are excellent hopes that twistor approach applies also in TGD framework. Note that also the conformal transformations of M^{4} might allow Yangian variants. For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title. 
The recent vision about preferred extremals and solutions of the modified Dirac equationThe understanding of preferred exrremals of preferred extremals of Kähler action and solutions of the modified Dirac equation has increased dramatically during last months and I have been busily deducing the consequences for the quantum TGD. This process led also to a new chapter of the book about physics as WCW geometry. I attach below the extended abstract of the chapter. During years several approaches to what preferred extremals of Kähler action and solutions of the modified Dirac equation could be have been proposed and the challenge is to see whether at least some of these approaches are consistent with each other. It is good to list various approaches first.
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title. 
How quaternicity of spacetime could be consistent with Hermitian/HamiltonJacobi structure?The recent progress in the understanding of preferred extremals of Kähler action suggests also an interesting connection to the number theoretic vision about field equations. In particular, it might be possible to understand how one can have Hermitian/HarmiltonJacobi structure simultaneously with quaternionic structure and how quaternionic structure is possible for the Minkowskian signature of the induced metric. One can imagine two manners of introducing octonionic and quaternionic structures. The first one is based on the introduction of octonionic representation of gamma matrices and second on the notion of octonion realanalycity.
Could one pose the additional requirement that the signature of the effective metric G defined by the modified gamma marices (and to be distinguished from Einstein tensor) is Euclidian in the sense that all four eigenvalues of this tensor would have same sign.

The recent vision about preferred extremals and solutions of the modified Dirac equationDuring last months a considerable progress in understanding of both the preferred extremals of Kähler action and solutions of modified Dirac equation has taken place and there are good reasons to believe that various approaches are converging. Instead of trying to describe the results in detail here I just give the abstract of the article The recent vision about preferred extremals and solutions of the modified Dirac equation. The text appears also in the chapter Does the modified Dirac action define the fundamental variational principle? of the online book "TGD: Physics as Infinitedimensional Geometry". Here is the short abstract. During years several approaches to what preferred extremals of Kähler action and solutions of the modified Dirac equation could be have been proposed and the challenge is to see whether at least some of these approaches are consistent with each other. It is good to list various approaches first.
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article already mentioned. 
Under what conditions electric charge is conserved for the modified Dirac equation?One might think that talking about the conservation of electric charge at 21st century is a waste of time. In TGD framework this is certainly not the case.
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title. 
What could be the counterpart of Tduality in TGD framework?Stephen Crowell sent me a book of Michel Lapidus about zeros of Riemann zeta and also about his own ideas in this respect. The book has been written in avery lucid manner and looks very interesting. The big idea is that the Tduality of string models could correspond to the functional equation for Riemann zeta relating the values of zeta at different sides of the critical line. Tduality is formulated for strings in space M^{d}× S^{1} or its generalization replacing S^{1} with higherdimensional torus and generalized to fractal strings. Duality states that the transformation R→ 1/R with suitable unit for R defined by string tension is a duality: the physics for these different values of R is same. Intuitively this is due to the fact that the contributions of the string modes representing nfold winding and those representing vibrations labelled by integer n are transformed to each other in the transformation R→ 1/R. Lapidus is a mathematician and mathematicians often do not care too much about the physical meaning of the numbers. For a physicist like me it is extremely painful to type the equation R→ 1/R without explicitly explaining that it should actually read as R→ R_{0}^{2}/R, where R_{0} is length unit, which must represent fundamental length scale remaining invariant under the duality transformation. Only after this physicist could reluctantly put R_{0}=1 but still would feel himself guilty of unforgivable sloppiness. R_{0}=1 simplifies the formulas but one must not forget that there are three scales involved rather than only two. The question inspired by this nitpicking is how the physics in the length scales R_{1} and R relates to the physics in length scale R. Are dualities  or perhaps holography like relations in question so that Tduality would follow from these dualities? Could one replace winding number with magnetic charge and Tduality with canonical identification? How could one generalize Tduality to TGD framework? One should identify the counterpart of the winding number, the three fundamental scales, and say something about the duality transformation itself.
Is the physics of life dual to the physics in CP_{2} scale? The duality of life with elementary particle physics at CP_{2} length scale  the TGD counterpart of Planck scale  looks rather farfetched idea. There is however already earlier support for this idea.
The unavoidable and completely crazy looking question raised by Tduality is whether there is intelligent life in the Euclidian realm below the CP_{2} length scale  inside the lines of generalized Feynman graphs. This kind of possibility cannot be avoided if one takes holography absolutely seriously. In purely mathematical sense TGD suggests even stronger form of holography based on the notion of infinite primes. In this holography the number theoretic anatomy of given spacetime point is infinitely complex and evolves. The notion of quantum mathematics replacing numbers by Hilbert spaces representing ordinary arithmetics in terms of direct sum and tensor product suggest the same. Spacetime point would be in this picture its own infinitely complex Universe  the Platonia. Could one get expression for Kähler coupling strength from restricted form of modular invariance? The contributions to the exponent of the vacuum functional, which is proportional to Kähler action for preferred extremal, are real resp. imaginary in Euclidian resp. Minkowskian regions. Under rather general assumptions (weak form of electricmagnetic duality defining boundary conditions at wormhole throats plus additional intuitively plausible assumption) these contributions are proportional to the same ChernSimons term but with possibly different constant of proportionality. These terms sum up to a ChernSimons term with a coefficient analogous to the complex inverse gauge coupling τ=θ/2π + i4π/g^{2}_{K} . The real part would correspond to Kähler function coming from Euclidian regions defining the lines of generalized Feynman diagrams and imaginary part to Minkowskian regions. There are could arguments suggesting that With the conventions that I have used θ/2π is counterpart for 1/α_{K} and there are good arguments that it corresponds to finte structure constant in electron length scale. Furthermore, Tduality would suggest that τ is proportional to 1+i so that one would have θ/2π = 4π/g^{2}_{K} . This condition would fit nicely with the fact that ChernSimons contributions from Minkowskian and Euclidian regions are identical. If this equation holds true the modular transformations must reduce to those leaving this relationship invariant and can only permute the complex and real parts and thus leave τ invariant. One could also interpret this value of τ as physically especially interesting representation and assign to all values of τ related by modular transformation an isotropy group leaving it fixed. All other physically equivalent values would be obtained as SL(2,Z) orbit of this value. The counterpart of Tduality should somehow relate dynamics in Minkowskian and Euclidian regions and this raises the question whether it corresponds to τ→ iτ and is represented by some duality transformation τ→ (aτ+b)/(cτ+d) , where (a,b;c,d) defines a unimodular matrix (adbc=1) with integer elements, that is in SL(2,Z). The electricmagnetic duality τ→ 1/τ and the shift τ→ τ+1 are the generators of this group. It is not however quite clear whether they can be regarded as gauge symmetries in TGD framework. If they are gauge symmetries, then the critical values of Kähler coupling strength defined as fixed points of coupling constant evolution must form an orbit of SL(2,Z). It could be also that modular symmetry is broken to a subgroup of SL(2,Z) and this subgroup leaves τ invariant in the case of minimal symmetry.
