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This Concept Map, created with IHMC CmapTools, has information related to: Kaehler-Dirac action.cmap.cmap, KÄHLER-DIRAC ACTION 3. How string world sheets could emerge? a) Additional consistency condition to neutrality of string world sheets is that Kähler-Dirac gamma matri- ces have no components orthogo- nal to the 2-surface in question. Hence various fermionic would flow along string world sheet. b) If the Kähler-Dirac gamma matrices at string world sheet are expressible in terms of two non-vanishing gamma matri- ces parallel to string world sheet and sheet and thus define an integrable distribution of tangent vectors, this is achieved. What is important that modified gamma matrices can indeed span lower than 4-D space and often do so. c) String models suggest that string world sheets are minimal surfaces of space-time surface or of imbedding space but might not be necessary to pose this condition separately., KÄHLER-DIRAC ACTION 1. Kähler-Dirac (KD) action is super-symmetric concomi- tant of Kähler action. a) Modified gamma matrices replace induced gamma mat- rices. Defined as contractions of canonical momentum cur- rents with imbedding space gamma matrices. b) Mixing of M^4 and CP_2 gamma matrices cause mix- ing of M^ chiralities and is a signature for massivation. c) K-D gamma matrices va- nish for vacuum extremals with vanishing induced Käh- ler field. d) Anticommutators of K-D gammas define effective met- ric. Physical interpretation? e) Kähler-Dirac equation in- ternally consistent only if an extremal of Kähler action in question. Means super- symmetry: one obtains infi- nite number of super-charge associated with the modes of the induced spinor field., KÄHLER-DIRAC ACTION 2. Well-definedness of em char- ge highly non-trivial condi- tion. a) Presence of induced W fields means that the modes of K-D equation in general cannot have well-defined em charge. b) In TGD one cannot locally re- define em charge by going to unitary gauge since the geomet- ry of CP_2 fixes uniquely the di- rection of em charge as covari- antly constant generator of electro-weak gauge algebra. c) For spinor modes in CP_2 it is possible to speak about well- defined em charge despite the mixing but it seems that this is not possible for induced spinor fields. d) The problem disappears if the induced W gauge fields vanish. This does not yet guarantee that couplings to classical gauge fields are physical in long scales. Also classical Z^0 field should vanish so that couplings would be purely vectorial. This might be true in long enough scales only. e) For instance, the condition e^1= a×e^0 and e^2=-a×e^3 with a^2= 2 also to satisfy the conditions. This implies that CP_2 projection is 2-dimensio- nal. More general solutions with non-vanishing Z^0 field are al- so possible: now a can depend on position. e) The vanishing of the weak part of induced gauge field impli- es that the CP_2 projection of the region carrying spinor mode is 2-D. Therefore the condition that the modes if induced spinor field are restricted to em neutral string world sheets guarantees well- definedness of em charge., KÄHLER-DIRAC ACTION 4. In the proposed scenario string world sheets emerge rather than being postulated from beginning. a) Em neutrality conditions allow also 4-D spinor modes if they are true for entire spa- time surface. This is true if the space-time surface has 2-D CP_2 projection. b) If CP_2 projection is homo- logically non-trivial geodesic sphere, this is guaranteed. Are all homologically non-tri- vial geodesic spheres obtain- ed by color rotations em ne- utral? c) If CP_2 projection is comp- plex manifold as it is for string like objects, this might be also achieved. d) Does the phase of cosmic strings assumed to dominate primordial cosmology corre- spond to this phase. During radiation dominated phase 4-D string like objects would transform to string world sheets!, KÄHLER-DIRAC ACTION 5. Right handed neutrino is exceptional. a) Electroweak gauge poten- tials do not couple to at all. Therefore em neutrality con- dition is un-necessary if the induced gamma matrices do not mix right handed neutrino with left-handed one. This is guaranteed if the M^4 and CP_2 parts of Kähler-Dirac operator annihilate separately right-handed neutrino spinor. b) For covariantly constant right-handed neutrino mode defining a generator of super- symmetries K-D equation is certainly true. Are there also other solutions?
