ABSTRACTS
OF 
PART I: THE RECENT VIEW ABOUT FIELD EQUATIONS 
PART II: GENERAL THEORY 
Category Theory and Quantum TGD Possible applications of category theory to quantum TGD are discussed. The so called 2plectic structure generalizing the ordinary symplectic structure by replacing symplectic 2form with 3form and Hamiltonians with Hamiltonian 1forms has a natural place in TGD since the dynamics of the lightlike 3surfaces is characterized by ChernSimons type action. The notion of planar operad was developed for the classification of hyperfinite factors of type II_{1}and its mild generalization allows to understand the combinatorics of the generalized Feynman diagrams obtained by gluing 3D lightlike surfaces representing the lines of Feynman diagrams along their 2D ends representing the vertices. The fusion rules for the symplectic variant of conformal field theory, whose existence is strongly suggested by quantum TGD, allow rather precise description using the basic notions of category theory and one can identify a series of finitedimensional nilpotent algebras as discretized versions of field algebras defined by the fusion rules. These primitive fusion algebras can be used to construct more complex algebras by replacing any algebra element by a primitive fusion algebra. Trees with arbitrary numbers of branches in any node characterize the resulting collection of fusion algebras forming an operad. One can say that an exact solution of symplectic scalar field theory is obtained. Conformal fields and symplectic scalar field can be combined to form symplectoformal fields. The combination of symplectic operad and Feynman graph operad leads to a construction of Feynman diagrams in terms of npoint functions of conformal field theory. Mmatrix elements with a finite measurement resolution are expressed in terms of a hierarchy of symplectoconformal npoint functions such that the improvement of measurement resolution corresponds to an algebra homomorphism mapping conformal fields in given resolution to composite conformal fields in improved resolution. This expresses the idea that composites behave as independent conformal fields. Also other applications are briefly discussed. Back to the table of contents 
PART III: TWISTORS, BOSONIC EMERGENCE, SPACETIME SUPERSYMMETRY 
Twistors, N=4 SuperConformal Symmetry, and Quantum TGD Twistors  a notion discovered by Penrose  have provided a fresh approach to the construction of perturbative scattering amplitudes in YangMills theories and in N=4 supersymmetric YangMills theory. This approach was pioneered by Witten. The latest step in the progress was the proposal by Nima ArkaniHamed and collaborators that super Yang Mills and super gravity amplitudes might be formulated in 8D twistor space possessing real metric signature (4,4). The questions considered in this chapter are following.
The arguments of this chapter suggest that some these questions might have affirmative answers. The idea about lightlike loop momenta however fails but led to a first precise proposal for how Feynman diagrammatics could emerge from TGD where only fermions are elementary particles discussed in a separate chapter. 
Yangian Symmetry, Twistors, and Quantum TGD There has been impressive steps in the understanding of N=4 maximally sypersymmetric YM theory possessing 4D superconformal symmetry. This theory is related by AdS/CFT duality to certain string theory in AdS_{5}× S^{5} background. Second stringy representation was discovered by Witten and is based on 6D CalabiYau manifold defined by twistors. The unifying proposal is that so called Yangian symmetry is behind the mathematical miracles involved. In the following I will discuss briefly the notion of Yangian symmetry and suggest its generalization in TGD framework by replacing conformal algebra with appropriate superconformal algebras. Also a possible realization of twistor approach and the construction of scattering amplitudes in terms of Yangian invariants defined by Grassmannian integrals is considered in TGD framework and based on the idea that in zero energy ontology one can represent massive states as bound states of massless particles. There is also a proposal for a physical interpretation of the Cartan algebra of Yangian algebra allowing to understand at the fundamental level how the mass spectrum of nparticle bound states could be understood in terms of the nlocal charges of the Yangian algebra. Twistors were originally introduced by Penrose to characterize the solutions of Maxwell's equations. Kähler action is Maxwell action for the induced Kähler form of CP_{2}. The preferred extremals allow a very concrete interpretation in terms of modes of massless nonlinear field. Both conformally compactified Minkowski space identifiable as so called causal diamond and CP_{2} allow a description in terms of twistors. These observations inspire the proposal that a generalization of Witten's twistor string theory relying on the identification of twistor string world sheets with certain holomorphic surfaces assigned with Feynman diagrams could allow a formulation of quantum TGD in terms of 3dimensional holomorphic surfaces of CP_{3}× CP_{3} mapped to 6surfaces dual CP_{3}× CP_{3}, which are sphere bundles so that they are projected in a natural manner to 4D spacetime surfaces. Very general physical and mathematical arguments lead to a highly unique proposal for the holomorphic differential equations defining the complex 3surfaces conjectured to correspond to the preferred extremals of Kähler action. 
Some fresh ideas about twistorialization of TGD I found from web an article by Tim Adamo titled "Twistor actions for gauge theory and gravity". The work considers the formulation of N=4 SUSY gauge theory directly in twistor space instead of Minkowski space. The author is able to deduce MHV formalism, tree level amplitudes, and planar loop amplitudes from action in twistor space. Also local operators and null polygonal Wilson loops can be expressed twistorially. This approach is applied also to general relativity: one of the challenges is to deduce MHV amplitudes for Einstein gravity. The reading of the article inspired a fresh look on twistors and a possible answer to several questions (I have written two chapters about twistors and TGD giving a view about development of ideas). Both M^{4} and CP_{2} are highly unique in that they allow twistor structure and in TGD one can overcome the fundamental "googly" problem of the standard twistor program preventing twistorialization in general spacetime metric by lifting twistorialization to the level of the imbedding space containg M^{4} as a Cartesian factor. Also CP_{2} allows twistor space identifiable as flag manifold SU(3)/U(1)× U(1) as the selfduality of Weyl tensor indeed suggests. This provides an additional "must" in favor of submanifold gravity in M^{4}× CP_{2}. Both octonionic interpretation of M^{8} and triality possible in dimension 8 play a crucial role in the proposed twistorialization of H=M^{4}× CP_{2}. It also turns out that M^{4}× CP_{2} allows a natural twistorialization respecting Cartesian product: this is far from obvious since it means that one considers spacelike geodesics of H with lightlike M^{4} projection as basic objects. pAdic mass calculations however require tachyonic ground states and in generalized Feynman diagrams fermions propagate as massless particles in M^{4} sense. Furthermore, lightlike Hgeodesics lead to noncompact candidates for the twistor space of H. Hence the twistor space would be 12dimensional manifold CP_{3}× SU(3)/U(1)× U(1). Generalisation of 2D conformal invariance extending to infiniteD variant of Yangian symmetry; lightlike 3surfaces as basic objects of TGD Universe and as generalised lightlike geodesics; lightlikeness condition for momentum generalized to the infinitedimensional context via superconformal algebras. These are the facts inspiring the question whether also the "world of classical worlds" (WCW) could allow twistorialization. It turns out that center of mass degrees of freedom (imbedding space) allow natural twistorialization: twistor space for M^{4}× CP_{2} serves as moduli space for choice of quantization axes in Super Virasoro conditions. Contrary to the original optimistic expectations it turns out that although the analog of incidence relations holds true for KacMoody algebra, twistorialization in vibrational degrees of freedom does not look like a good idea since incidence relations force an effective reduction of vibrational degrees of freedom to four. The Grassmannian formalism for scattering amplitudes generalizes practically as such for generalized Feynman diagrams. The vision about what BCFW approach to generalized Feynman diagframs could mean has been fluctuating wildly during last months. The Grassmannian formalism for scattering amplitudes is expected to generalize for generalized Feynman diagrams: the basic modification is due to the possible presence of CP_{2} twistorialization and the fact that 4fermion vertex  rather than 3boson vertex  and its super counterparts define now the fundamental vertices. Both QFT type BFCW and stringy BFCW can be considered. The recent vision is as follows.
As both ArkaniHamed and Trnka state "everything is positive". This is highly interesting since padicization involves canonical identification, which is well defined only for nonnegative reals without further assumptions. This raises the conjecture that positivity is necessary in order to achieve number theoretical universality. 
Quantum Field Theory Limit of TGD from Bosonic Emergence This chapter summarizes the basic mathematical realization of the modified Feynman rules hoped to give rise to a unitary Mmatrix (recall that Mmatrix is product of a positive square root of density matrix and unitary Smatrix in TGD framework and need not be unitary in the general case). The basic idea is that bosonic propagators emerge as fermionic loops. The approach is bottom up and leads to a precise general formulation for how the counterpart of YM action emerges from Dirac action coupled to gauge bosons and to modified Feynman rules. An essential element of the approach is a physical formulation for UV cutoff. Actually cutoff in both mass squared and hyperbolic angle is needed since Wick rotation does not make sense in TGD framework. This approach predicts all gauge couplings and assuming a geometrically very natural hyperbolic UV cutoff motivated by zero energy ontology one can understand the evolution of standard model gauge couplings and reproduce correctly the values of fine structure constant at electron and intermediate boson length scales. Also asymptotic freedom follows as a basic prediction. The UV cutoff for the hyperbolic angle as a function of padic length scale is somewhat ad hoc element of the model and a quantitative model for how this function could follow from the requirement of quantum criticality is formulated and discussed. These considerations and numerical calculations lead to a general vision about how real and padic variants of TGD relate to each other and how padic fractalization takes place. As in case of twistorialization Cutkosky rules allowing unitarization of the tree amplitudes in terms of TT^{+} contribution involving only lightlike momenta seems to be the only working option and requires that TT^{+} makes sense padically. The vanishing of the fermionic loops defining bosonic vertices for the incoming massless momenta emerges as a consistency condition suggested also by quantum criticality and by the fact that only BFF vertex is fundamental vertex if bosonic emergence is accepted. The vanishing of on mass shell Nvertices gives an infinite number of conditions on the hyperbolic cutoff as function of the integer k labeling padic length scale at the limit when bosons are massless and IR cutoff for the loop mass scale is taken to zero. It is not yet clear whether dynamical symmetries, in particular superconformal symmetries, are involved with the realization of the vanishing conditions or whether hyperbolic cutoff is all that is needed.

PART IV: HYPERFINITE FACTORS AND HIERARCHY OF PLANCK CONSTANTS 
Does TGD Predict the Spectrum of Planck Constants? The quantization of Planck constant has been the basic them of TGD since 2005. The basic idea was stimulated by the finding of Nottale that planetary orbits could be seen as Bohr orbits with enormous value of Planck constant given by hbar_{gr}= GM_{1}M_{2}/v_{0}, where the velocity parameter v_{0} has the approximate value v_{0}≈ 2^{11} for the inner planets. This inspired the ideas that quantization is due to a condensation of ordinary matter around dark matter concentrated near Bohr orbits and that dark matter is in macroscopic quantum phase in astrophysical scales. The second crucial empirical input were the anomalies associated with living matter. The recent version of the chapter represents the evolution of ideas about quantization of Planck constants from a perspective given by seven years's work with the idea. A very concise summary about the situation is as follows. Basic physical ideas The basic phenomenological rules are simple and there is no need to modify them.
Spacetime correlates for the hierarchy of Planck constants The hierarchy of Planck constants was introduced to TGD originally as an additional postulate and formulated as the existence of a hierarchy of imbedding spaces defined as Cartesian products of singular coverings of M^{4} and CP_{2} with numbers of sheets given by integers n_{a} and n_{b} and hbar=nhbar_{0}. n=n_{a}n_{b}. With the advent of zero energy ontology, it became clear that the notion of singular covering space of the imbedding space could be only a convenient auxiliary notion. Singular means that the sheets fuse together at the boundary of multisheeted region. The effective covering space emerges naturally from the vacuum degeneracy of Kähler action meaning that all deformations of canonically imbedded M^{4} in M^{4}×CP_{2} have vanishing action up to fourth order in small perturbation. This is clear from the fact that the induced Kähler form is quadratic in the gradients of CP_{2} coordinates and Kähler action is essentially Maxwell action for the induced Kähler form. The vacuum degeneracy implies that the correspondence between canonical momentum currents ∂L_{K}/∂(∂_{α}h^{k}) defining the modified gamma matrices and gradients ∂_{α} h^{k} is not onetoone. Same canonical momentum current corresponds to several values of gradients of imbedding space coordinates. At the partonic 2surfaces at the lightlike boundaries of CD carrying the elementary particle quantum numbers this implies that the two normal derivatives of h^{k} are manyvalued functions of canonical momentum currents in normal directions. Multifurcation is in question and multifurcations are indeed generic in highly nonlinear systems and Kähler action is an extreme example about nonlinear system. What multifurcation means in quantum theory? The branches of multifurcation are obviously analogous to single particle states. In quantum theory second quantization means that one constructs not only single particle states but also the many particle states formed from them. At spacetime level single particle states would correspond to N branches b_{i} of multifurcation carrying fermion number. Twoparticle states would correspond to 2fold covering consisting of 2 branches b_{i} and b_{j} of multifurcation. Nparticle state would correspond to Nsheeted covering with all branches present and carrying elementary particle quantum numbers. The branches coincide at the partonic 2surface but since their normal space data are different they correspond to different tensor product factors of state space. Also now the factorization N= n_{a}n_{b} occurs but now n_{a} and n_{b} would relate to branching in the direction of spacelike 3surface and lightlike 3surface rather than M^{4} and CP_{2} as in the original hypothesis. Multifurcations relate closely to the quantum criticality of Kähler action. Feigenbaum bifurcations represent a toy example of a system which via successive bifurcations approaches chaos. Now more general multifurcations in which each branch of given multifurcation can multifurcate further, are possible unless on poses any additional conditions. This allows to identify additional aspect of the geometric arrow of time. Either the positive or negative energy part of the zero energy state is "prepared" meaning that single nsubfurcations of Nfurcation is selected. The most general state of this kind involves superposition of various nsubfurcations. 