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This Concept Map, created with IHMC CmapTools, has information related to: Vacuum functional.cmap, VACUUM FUNCTIONAL 3. Kähler action involves also 3-D "bo- undary" terms. a) These terms are analogous to the Lagrangian multiplier terms appearing in the definition of thermodynamical parti- tion function (for instance, fixing the va- lues of energy and particle number). These terms are proportional to Lagran- gian multipliers (for instance, tempera- ture and chemical potential). b) Quantum classical corresponden- ce suggests that the charges in the Car- tan algebra of isometries are identical to the eigenvalues of corresponding fermio- nic charges. This contraint selects speci- fic spinor modes and implies that space- time geometry codes information about quantum numbers. c) Weak form of electric magnetic duality is realized as boundary terms assignable to the light-like orbits of par- tonic 2-surfaces and reducing the Käh- ler acaction for preferred extremals to Chern-Simons terms if j.A term in the action vanishes identically as it does for the know extremals. This reduction me- ans enormous calculational simplificati- on since it is not necessary to know all the details of the preferred extremal. It is correlate for holography and gives hopes about the calculability of the the- ory. It also supports the idea that TGD is almost topological QFT., VACUUM FUNCTIONAL 1. In Zero Energy Ontology (ZEO) va- cuum functional is analogous to a com- plex square root of exponent of free energy. a) Vacuum functional decomposes to a product of real exponent of Kähler function exponent of imaginary term analogous to action in quantum field theories. In topological picture the exponent of imaginary term is analogous to exponent of Morse function and its extrema have topological interpretation in terms of topology change at the level of WCW. b) Simplest example about Morse func- tion is height function for torus having maximum, minimum and two saddle points where the signature of the quad- ratic form associated with height func- tion changes. They all correspond to topology change for h=constant section. b) Kähler- resp. Morse function is ex- pressible as Kähler action for Euclidian resp. Minkowkian regions of space-time surface. Square root for the determinant of the induced metric is indeed imagina- ry in Minkowskian regions. c) Kähler function should not have saddle points since this is signature for metric, which is not positive definite. Kähler function log(1+r^2) for CP_2 is good example in this respect. Same should hold true also in Euclidian regi- ons. The properties of Kähler action in Euclidian signature guarantee that it is non-negative. This alone does not yet guarantee Euclidian signature of WCW metric, VACUUM FUNCTIONAL 4. The maxima of Kähler function have special physical importance. a) In ZEO they these maxima correspond to especially favoured 4-D field patterns since they connect 3-D surfaces at the bo- undaries of CD. b) This interpretation is especially relevant in quantum biology where temporal pat- terns for many-sheeted space-time surfa- ces (in particular, for topologically quantized classical fields) serve as templates for the time evolution of living system. The idea that temporal evolution of magnetic body carrying dark matter realized as large h_eff phases serves as template for the develop- ment of organism, is natural in this frame- work., VACUUM FUNCTIONAL 2. Some properties of Kähler action and Kähler function. a) Kähler action is inversely proportio- nal to Kähler coupling strength a_K. Thermodynamical analogy motivates the postulate is that α_K is analogous to critical temperature so that TGD Uni- verse would be quantum critical and vacuum function unique (if several cri- tical points exist, then the minimum of a_K is a natural choice, assuming that it exists). b) Kähler function depends on zero mo- des which do not contribute to the line element and appear as parameters in the Kähler metric defined in terms of Kähler function using standard formula. The quantum fluctuating degrees of freedom contributing to the line element can be expressed in terms of complex coordina- tes whereas zero modes are real para- meters. Various fluxes associated with induced Kähler form define symplectic invariants and also zero modes if sym- plectic transformations of δCDxCP_2 are isometries of WCW metric. The space of quantum fluctuating degrees of freedom corresponds to symplectic group or its coset space. c) The notion of finite measurement re- solution suggets strongly a hierarchy of coset spaces related to the hierarchy of inclusions of hyperfinite factors of type II_1. Also the hierarchy of Planck con- stants h_eff=n×h possibly realized in terms of of inclusion hierarchy of sub- algebras of conformal algebra isomorphic with the algebra itself suggests this.
