What's new inTopological Geometrodynamics: an OverviewNote: Newest contributions are at the top! 
Year 2008 
The relationship between supercanonical and Super KacMoody algebras, Equivalence Principle, and justification of padic thermodynamicsThe relationship between supercanonical algebra (SC) acting at lightcone boundary and Super KacMoody algebra (SKM) acting on lightlike 3surfaces has remained somewhat enigmatic due to the lack of physical insights. This is not the only problem. The question to precisely what extent Equivalence Principle (EP) remains true in TGD framework and what might be the precise mathematical realization of EP is waiting for an answer. Also the justification of padic thermodynamics for the scaling generator L_{0} of Virasoro algebra in obvious conflict with the basic wisdom that this generator should annihilate physical states is lacking. It seems that these three problems could have a common solution. Before going to describe the proposed solution, some background is necessary. The latest proposal for SCSKM relationship relies on nonstandard and therefore somewhat questionable assumptions.
1. New vision about the relationship between SCV and SKMVConsider now the new vision about the relationship between SCV and SKMV.
2. Can SKM be lifted to a subalgebra of SC?A picture introducing only a generalization of coset construction as a new element, realizing mathematically Equivalence Principle, and justifying padic thermodynamics is highly attractive but there is a problem. SKM is defined at lightlike 3surfaces X^{3} whereas SC acts at lightcone boundary dH_{�}=dM^{4}_{�}×CP_{2}. One should be able to lift SKM to imbedding space level somehow. Also SC should be lifted to entire H. This problem was the reason why I gave up the idea about coset construction and SCSKM duality as it appeared for the first time. A possible solution of the lifting problem comes from the observation making possible a more rigorous formulation of HOH duality stating that one can regard spacetime surfaces either as surfaces in hyperoctonionic space HO=M^{8} or in H=M^{4}×CP_{2} [see this]. Consider first the formulation of HOH duality.
3. QuestionsThere are still several open questions.

What goes wrong with string theories?Something certainly goes wrong with superstring models and Mtheory. But what this something is? One could of course make the usual list involving spontaneous compactification, landscape, nonpredictivity, and all that. The point I however want to make relates to the relationship of string models to quantum field theories. The basic wisdom has been that when Feynman graphs of quantum field theories are replaced by their stringy variants everything is nice and finite. The problem is that stringy diagrams do not describe what elementary particles are doing and quantum field theory limit is required at low energies. This nonrenormalizable QFT limit is obtained by the ad hoc procedure called spontaneous compactification, and leads to all this misery that spoils the quality of our life nowadays. My intention is not to ridicule or accuse string theorists. I feel also myself very very stupid since I realized only now what the relationship between superconformal and supersymplectic QFTs and generalized Feynman diagrams is in TGD framework: I described this already in the previous posting but did not want to make noise of my stupidity. This discovery (discovery only at level of my own subjective experience) was just becoming aware about something which should have been absolutely obvious for anyone with IQ above 20;). A very brief summary goes as follows.

Could a symplectic analog of conformal field theory be relevant for quantum TGD?Symplectic (or canonical as I have called them) symmetries of dM^{4}_{+}×CP_{2} (lightcone boundary briefly) act as isometries of the "world of classical worlds". One can see these symmetries as analogs of KacMoody type symmetries with symplectic transformations of S^{2}×CP_{2}, where S^{2} is r_{M}=constant sphere of lightcone boundary, made local with respect to the lightlike radial coordinate r_{M} taking the role of complex coordinate. Thus finitedimensional Lie group G is replaced with infinitedimensional group of symplectic transformations. This inspires the question whether a symplectic analog of conformal field theory at dM^{4}_{+}×CP_{2} could be relevant for the construction of npoint functions in quantum TGD and what general properties these npoint functions would have.1 Symplectic QFT at sphereActually the notion of symplectic QFT emerged as I tried to understand the properties of cosmic microwave background which comes from the sphere of last scattering which corresponds roughly to the age of 5×10^{5} years. In this situation vacuum extremals of Kähler action around almost unique critical RobertsonWalker cosmology imbeddable in M^{4}×S^{2}, where there is homologically trivial geodesic sphere of CP_{2}. Vacuum extremal property is satisfied for any spacetime surface which is surface in M^{4}×Y^{2}, Y^{2} a Lagrangian submanifold of CP_{2} with vanishing induced Kähler form. Symplectic transformations of CP_{2} and general coordinate transformations of M^{4} are dynamical symmetries of the vacuum extremals so that the idea of symplectic QFT emerges natural. Therefore I shall consider first symplectic QFT at the sphere S^{2} of last scattering with temperature fluctuation DT/T proportional to the fluctuation of the metric component g_{aa} in RobertsonWalker coordinates.
2 Symplectic QFT with spontaneous breaking of rotational and reflection symmetriesCMB data suggest breaking of rotational and reflection symmetries of S^{2}. A possible mechanism of spontaneous symmetry breaking is based on the observation that in TGD framework the hierarchy of Planck constants assigns to each sector of the generalized imbedding space a preferred quantization axes. The selection of the quantization axis is coded also to the geometry of "world of classical worlds", and to the quantum fluctuations of the metric in particular. Clearly, symplectic QFT with spontaneous symmetry breaking would provide the soughtfor really deep reason for the quantization of Planck constant in the proposed manner.
3 Generalization to quantum TGDSince number theoretic braids are the basic objects of quantum TGD, one can hope that the npoint functions assignable to them could code the properties of ground states and that one could separate from npoint functions the parts which correspond to the symplectic degrees of freedom acting as symmetries of vacuum extremals and isometries of the 'world of classical worlds'.

Infinite primes and algebraic Brahman Atman identityThe hierarchy of infinite primes (and of integers and rationals) was the first mathematical notion stimulated by TGD inspired theory of consciousness. The construction recipe is equivalent with a repeated second quantization of supersymmetric arithmetic quantum field theory with bosons and fermions labeled by primes such that the many particle states of previous level become the elementary particles of new level. The hierarchy of spacetime sheets with many particle states of spacetime sheet becoming elementary particles at the next level of hierarchy and also the hierarchy of n:th order logics are also possible correlates for this hierarchy. For instance, the description of proton as an elementary fermion would be in a well defined sense exact in TGD Universe. This construction leads also to a number theoretic generalization of spacetime point since given real number has infinitely rich number theoretical structure not visible at the level of the real norm of the number a due to the existence of real units expressible in terms of ratios of infinite integers. This number theoretical anatomy suggest kind of number theoretical Brahman=Atman principle stating that the set consisting of number theoretic variants of single point of the imbedding space (equivalent in real sense) is able to represent the points of the world of classical worlds or even quantum states of the Universe . Also a formulation in terms of number theoretic holography is possible. Just for fun and to test these ideas one can consider a model for the representation of the configuration space spinor fields in terms of algebraic holography. I have considered guesses for this kind of map earlier and it is interesting to find whether additional constraints coming from zero energy ontology and finite measurement resolution might give. The identification of quantum corrections as insertion of zero energy states in time scale below measurement resolution to positive or negative energy part of zero energy state and the identification of number theoretic braid as a spacetime correlate for the finite measurement resolution give considerable additional constraints.
For background see the chapter Was von Neumann right after all?. See also the article "Topological Geometrodynamics: an Overall View".

Configuration space gamma matrices as hyperoctonionic conformal fields having values in HFF?The fantastic properties of HFFs of type II_{1} inspire the idea that a localized version of Clifford algebra of configuration space might allow to see spacetime, embedding space, and configuration space as emergent structures. Configuration space gamma matrices act only in vibrational degrees of freedom of 3surface. One must also include center of mass degrees of freedom which appear as zero modes. The natural idea is that the resulting local gamma matrices define a local version of HFF of type II_{1} as a generalization of conformal field of gamma matrices appearing super string models obtained by replacing complex numbers with hyperoctonions identified as a subspace of complexified octonions. As a matter fact, one can generalize octonions to quantum octonions for which quantum commutativity means restriction to a hyperoctonionic subspace of quantum octonions . Nonassociativity is essential for obtaining something nontrivial: otherwise this algebra reduces to HFF of type II_{1} since matrix algebra as a tensor factor would give an algebra isomorphic with the original one. The octonionic variant of conformal invariance fixes the dependence of local gamma matrix field on the coordinate of HO. The coefficients of Laurent expansion of this field must commute with octonions. The world of classical worlds has been identified as a union of configuration spaces associated with M^{4}_{�} labeled by points of H or equivalently HO. The choice of quantization axes certainly fixes a point of H (HO) as a point remaining fixed under SO(1,3)×U(2) (SO(1,3)×SO(4)). The condition that hyperquaternionic inverses of M^{4} � HO points exist suggest a restriction of arguments of the npoint function to the interior of M^{4}_{�}. Associativity condition for the npoint functions forces to restrict the arguments to a hyperquaternionic plane HQ=M^{4} of HO. One can also consider the commutativity condition by requiring that arguments belong to a preferred commutative subspace HC of HO. Fixing preferred real and imaginary units means a choice of M^{2}=HC interpreted as a partial choice of quantization axes. This has quite strong implications.

How quantum classical correspondence is realized at parton level?Quantum classical correspondence must assign to a given quantum state the most probable spacetime sheet depending on its quantum numbers. The spacetime sheet X^{4}(X^{3}) defined by the Kähler function depends however only on the partonic 3surface X^{3}, and one must be able to assign to a given quantum state the most probable X^{3}  call it X^{3}_{max}  depending on its quantum numbers. X^{4}(X^{3}_{max}) should carry the gauge fields created by classical gauge charges associated with the Cartan algebra of the gauge group (color isospin and hypercharge and electromagnetic and Z^{0} charge) as well as classical gravitational fields created by the partons. This picture is very similar to that of quantum field theories relying on path integral except that the path integral is restricted to 3surfaces X^{3} with exponent of Kähler function bringing in genuine convergence and that 4D dynamics is deterministic apart from the delicacies due to the 4D spin glass type vacuum degeneracy of Kähler action. Stationary phase approximation selects X^{3}_{max} if the quantum state contains a phase factor depending not only on X^{3} but also on the quantum numbers of the state. A good guess is that the needed phase factor corresponds to either ChernSimons type action or a boundary term of YM action associated with a particle carrying gauge charges of the quantum state. This action would be defined for the induced gauge fields. YM action seems to be excluded since it is singular for lightlike 3surfaces associated with the lightlike wormhole throats (not only (det(g_{3})^{1/2} but also det(g_{4})^{1/2} vanishes). The challenge is to show that this is enough to guarantee that X^{4}(X^{3}_{max}) carries correct gauge charges. Kind of electricmagnetic duality should relate the normal components F_{ni} of the gauge fields in X^{4}(X^{3}_{max}) to the gauge fields F_{ij} induced at X^{3}. An alternative interpretation is in terms of quantum gravitational holography. The difference between ChernSimons action characterizing quantum state and the fundamental ChernSimons type factor associated with the Kähler form would be that the latter emerges as the phase of the Dirac determinant. One is forced to introduce gauge couplings and also electroweak symmetry breaking via the phase factor. This is in apparent conflict with the idea that all couplings are predictable. The essential uniqueness of Mmatrix in the case of HFFs of type II_{1} (at least) however means that their values as a function of measurement resolution time scale are fixed by internal consistency. Also quantum criticality leads to the same conclusion. Obviously a kind of bootstrap approach suggests itself. For background see the chapter Overall View about Quantum TGD. 
How padic coupling constant evolution and padic length scale hypothesis emerge from quantum TGD proper?What padic coupling constant evolution really means has remained for a long time more or less open. The progress made in the understanding of the Smatrix of theory has however changed the situation dramatically. 1. Mmatrix and coupling constant evolution The final breakthrough in the understanding of padic coupling constant evolution came through the understanding of Smatrix, or actually Mmatrix defining entanglement coefficients between positive and negative energy parts of zero energy states in zero energy ontology (see this). Mmatrix has interpretation as a "complex square root" of density matrix and thus provides a unification of thermodynamics and quantum theory. Smatrix is analogous to the phase of Schrödinger amplitude multiplying positive and real square root of density matrix analogous to modulus of Schrödinger amplitude. The notion of finite measurement resolution realized in terms of inclusions of von Neumann algebras allows to demonstrate that the irreducible components of Mmatrix are unique and possesses huge symmetries in the sense that the hermitian elements of included factor N subset M defining the measurement resolution act as symmetries of Mmatrix, which suggests a connection with integrable quantum field theories. It is also possible to understand coupling constant evolution as a discretized evolution associated with time scales T_{n}, which come as octaves of a fundamental time scale: T_{n}=2^{n}T_{0}. Number theoretic universality requires that renormalized coupling constants are rational or at most algebraic numbers and this is achieved by this discretization since the logarithms of discretized mass scale appearing in the expressions of renormalized coupling constants reduce to the form log(2^{n})=nlog(2) and with a proper choice of the coefficient of logarithm log(2) dependence disappears so that rational number results. 2. pAdic coupling constant evolution One can wonder how this picture relates to the earlier hypothesis that padic length coupling constant evolution is coded to the hypothesized log(p) normalization of the eigenvalues of the modified Dirac operator D. There are objections against this normalization. log(p) factors are not number theoretically favored and one could consider also other dependencies on p. Since the eigenvalue spectrum of D corresponds to the values of Higgs expectation at points of partonic 2surface defining number theoretic braids, Higgs expectation would have log(p) multiplicative dependence on padic length scale, which does not look attractive. Is there really any need to assume this kind of normalization? Could the coupling constant evolution in powers of 2 implying time scale hierarchy T_{n}= 2^{n}T_{0} induce padic coupling constant evolution and explain why padic length scales correspond to L_{p} propto p^{1/2}R, p≈ 2^{k}, R CP_{2} length scale? This looks attractive but there is a problem. pAdic length scales come as powers of 2^{1/2} rather than 2 and the strongly favored values of k are primes and thus odd so that n=k/2 would be half odd integer. This problem can be solved.
