Sunday, July 09, 2006

Super Kac-Moody symmetry and TGD as almost topological conformal field theory

I have discussed in previous postings how N=4 super-conformal invariance might emerge in TGD framework (at least in the leptonic sector) and suggested that TGD might reduce to an almost-topological super conformal field theory (SCFT). Superconformal symmetry involves two ingredients: super Kac-Moody and super-canonical invariance and in the following it is found that the super Kac-Moody symmetries acting as deformations of partonic boundary component allow N=4 super-conformal symmetry as a maximal super-conformal symmetry and that TGD indeed reduces to an almost topological SCFT with parton dynamics defined by the Chern-Simons action for the induced Kähler gauge potential.

  1. Consider first the modified Dirac equation for the induced spinor fields. Ordinary Γ matrices are replaced with modified Γ matrices which are linear combinations of the imbedding space Γ matrices with coefficients which are canonical momentum densities defined by the action determining the dynamics of the surface. The divergence of the vector field of X3l defined by the modified Γ matrices vanishes by classical field equations and this guarantees the vanishing of the mass term in the modified Dirac equation necessary for the existence of super currents identified as products of the modified Γ matrices with the solutions of the modified Dirac equation. The conjecture is that the Dirac determinant defined by the product of generalized eigenvalues of the modified Dirac operator equals to the exponent of Kähler function so that the action assignable to partonic 3-surfaces would determine the dynamics of quantum TGD completely.

  2. The classical action is in general ill-defined for light-like 3-surfaces if it contains induced metric becoming degenerate to X3l. The only possibility is that the action is Chern-Simons form for the induced Kähler gauge potential. This would indeed mean that TGD reduces to almost topological field theory in the sense that the induced metric would appear only implicitly via the condition that partonic 3-surface is light-like (I have however no intention to replace TGD with ATGD;-)

  3. The Chern-Simons action involves only CP2 Γ matrices so that SL(2,C) replaces the group SU(2) acting as rotations of spinors generating super-conformal symmetries. This would guarantee Lorentz invariance. Therefore the Kac-Moody algebra would be more general than for N=4 super-conformal invariance but would restrict naturally to SU(2) when one chooses rest frame at lightcone boundary.

  4. Super Kac-Moody symmetries should correspond to solutions of the modified Dirac equation which are in some sense holomorphic. Holomorphy makes sense since light-like partonic 3-surfaces can be provided with coordinates z, z*, r, where r is lightlike coordinate: one can speak about degenerate complex, Kähler, and symplectic, and metric structures at partonic 3-surface. The analyticity of spinors with respect to z is the obvious condition to be satisfied. The modified Γ matrix Γz must annihilate the solutions. Covariant constancy with respect to z* and r are also natural conditions. The corresponding integrability conditions state that induce spinor curvature contracted with the modified Γ matrices annihilates the physical states (just as it annihilates right handed neutrinos). Each solution of the integrability conditions corresponds to a representation of N=4 super-conformal algebra. A breaking of super-conformal symmetry can occur and would mean that less than one half of the components of the induced spinor field satisfy the integrability conditions. In the leptonic sector N=4 supersymmetry associated with right handed neutrino is always present but super-conformal symmetries could be lost.

To sum up, it seems that it that almost-topological SCFT emerges from TGD framework in a natural manner and is the only possibility if one accepts the identification of the interior dynamics of the space-time surface in terms of the classical degrees of freedom postulated in quantum measurement theory.

The last section of chapter Construction of Quantum Theory of "Towards S-Matrix" represents the detailed form of the argument above.

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