Wednesday, November 22, 2023

About the universality of the holomorphic solution ansatz

The explicit solution of field equations in terms of the generalized holomorphy is now known. Also the emergence of supersymplectic symmetry is understood: it emerges as symmetries of Chern-Simons-Kähler action at the 3-D partonic orbits defining part of 3-D holographic data.

The solution ansatz is independent of action as long it is general coordinate invariance depending only on the induced geometric structures. Space-time surfaces would be minimal surfaces apart from lower-dimensional singular surfaces at which the field equations involve the entire action. Only the singularities, classical charges and positions of topological interaction vertices depend on the choice of the action (see this). Kähler action plus volume term is the choice of action forced by twistor lift making the choice of H unique.

The universality has a very intriguing implication. One can assign to any action of this kind conserved Noether currents and their fermionic counterparts (also super counterparts). One would have a huge algebra of conserved currents characterizing the space-time geometry. The corresponding charges need not be conserved since the conservation conditions at the partonic orbits and other singularities depend on the action. The discussion of the symplectic symmetries leads to the conclusion that they give rise to conserved charges at the partonic 3-surfaces obeying Chern-Simons-Kähler dynamics, which is non-deterministic.

Partonic 3-surfaces could be in the same role as space-like 3-surfaces as initial data: the time coordinate for this time evolution would be dual to the light-like coordinate of the partonic orbit. Could one say that the measurement localizing the partonic orbit leads to a phase characterized by a particular action? The classical conserved quantities are determined by the action. The WCW K\"ahler function should correspond to this action and different actions would correspond to different regions of WCW. Could phase transition between these regions take place when the 4-surface determined by the partonic orbit belongs to regions corresponding to two different effective actions. The twistor lift suggests that action must be the sum of the Kähler action and volume term so that only Kähler couplings strength, the coefficient of instanton term and the dynamically determined cosmological constant would vary.

See the article Symmetries and Geometry of the "World of Classical Worlds" or the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds".

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

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