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This Concept Map, created with IHMC CmapTools, has information related to: TGD as ATQFT.cmap, TGD AS ALMOST TOPOLOGICAL QUANTUM FIELD THEORY (ATQFT) 4. Does Kähler action define a topological invariant? a) Kähler action has real part - Kähler function - co- ming from Euclidian regions representing lines of gene- ralized Feynman diagrams and inaginary part - Morse function- coming from Minkowskian regions repre- senting macroscopic space- -time as a deformation of M^4. b) If preferred extremal depends on the 3-surfa- ces at the ends of CD or only on 2-D partonic 2- surfaces plus their 4-D tangent space data, then space-time surface with given topology and ends could be seen as a unique representative for space- time surfaces in this class - a topological invariant. Kähler action reducing to Chern-Simons terms would represent also topological invariants. c) This would not be new. In manifold topology one can assign to given manifold topology highly unique topology (say hyperbolic manifold) such that the volume in hyperbolic metric serves as topological inva- riant. d) In TGD the situation is much richer since one has sub-manifold topology. Strings can get knotted in 3-D sense and string world sheets can get knotted in 4-D sense. This might have direct applications to topolo- gical quantum computation in 4-D sense., TGD AS ALMOST TOPOLOGICAL QUANTUM FIELD THEORY (ATQFT) 2. Motivations for almost TQFT in TGD: a) Weak form of electric- magnetic duality (WFED) implies together with j.A=0 for Kähler action density implies the re- reduction of Kähler action to Chern-Simons (C-S) terms at the 3-D ends of space- time surface at boundaries of causal diamond (CD) and at the light-like 3-D orbits of worm-hole throats (par- ton orbits). C-S action alone would define TQFT. b) By super-conformal symmetry one expects similar reduction to occur for Kähler-Dirac action (K-D) and give Chern- Simons-Dirac action (C-S-D) at the boundaries or its reduction to a curve defining the boundary of string world sheet. c) At boundaries C-S and C-S-D contributions from Euclidian and Minkowskian regions should differ by a multiplication with imagi- nary unit coming from the square root of the determi- nant of induced 4-metric. Does TGD reduce to TQFT? d) No! There are constra- ints bringing in length sca- le and coded by weak elec- tric magnetic duality and one obtains only ATQFT., TGD AS ALMOST TOPOLOGICAL QUANTUM FIELD THEORY (ATQFT) 3. Constraints bringing in induced metric making TGD ATQFT. a) These constraints analogous to those appearing in thermo- dynamics and fixing average energy, par- ticle number etc. and bringing in temperatu- re, chemical potential, etc... TGD is indeed "square root of thermo- dynamics" in ZEO. b) The Lagrange mul- tiplies guaranteeing WFED depend on the induced 4-metric. c) The 3-D constraint guaranteeing that the eigenvalues of quantal four-momen- tum assignable to K-D action is equal to Kähler 4-momentum depends on metric. c) The constraint which reduces the boundary terms resulting from K-D action to the ana- log of M^4 Dirac action with derivatives repla- ced with 4-momenta of fermion lines defined by string ends., TGD AS ALMOST TOPOLOGICAL QUANTUM FIELD THEORY (ATQFT) 1. What TQFTs are? a) TQFTs well-defi- ned quantum field theories trivial from the point of view of conventional physics. b) Only the topology of field configurations of the space in which it is defined matters, magnetic charge, instanton number, winding number ho- lonomies of connect- ion. c) No length scales involved and thus the notion of mass and four-momentum crucial in conventio- nal physics not involved. d) A topological clas- sifation of knots and 3-manifolds using invariants provided by 3-D TQFT charac- terized by Chern-Si- mons action is an example about TQFT. Witten was the pio- neer. e) Important applica- tion: topological quantum computa- tion (TQC).
