Wednesday, June 15, 2011

Could Gromov-Witten invariants and braided Galois homology together allow to construct WCW spinor fields?

The challenge of TGD is to understand the structure of WCW spinor fields both in the zero modes which correspond to symplectically invariant degrees of freedom not contributing to the WCW Kähler metric and in quantum fluctuating degrees of freedom parametrized by the symplectic group of δ M4+/-× CP2. Basically the challenge is is to understand the symplectic (or more precisely, contact geometry of δ M4+/-× CP2. It seems that mathematicians and mathematical physicists (Gromov, Witten, Floer, and so on) have developed refined concepts for dealing with this problem.

One can develop good arguments suggesting that an appropriate generalization of Gromov-Witten invariants to covariants combined with braid Galois homology discussed in previous posting could allow do construct WCW spinor fields and at the same time M-matrices defining the rows of the unitary U-matrix between zero energy states. Finite measurement resolution would be the magic notion making everything calculable.

In the proposed framework the view about construction of WCW spinor fields would be roughly following.

  1. One can distinguish between WCW "orbital" degrees of freedom and fermionic degrees of freedom and in the case of WCW degrees of freedom also between zero modes and quantum fluctuating degrees of freedom. Zero modes correspond essentially to the non-local symplectic invariants assignable to the projections of the δ M4+/- and CP2 Kähler forms to the space-time surface. Quantum fluctuating degrees of freedom correspond to the symplectic algebra in the basis defined by Hamiltonians belonging to the irreps of rotation group and color group.

  2. At the level of partonic 2-surfaces finite measurement resolution leads to discretization in terms of braid ends and symplectic triangulation. At the level of WCW discretization replaces symplectic group with its discrete subgroup. This discrete subgroup must result as a coset space defined by the subgroup of symplectic group acting as Galois group in the set of braid points and its normal subgroup leaving them invariant. The group algebra of this discrete subgroup of symplectic group would have interpretation in terms of braided Galois cohomology. This picture provides an elegant realization for finite measurement resolution and there is also a connection with the realization of finite measurement resolution using categorification.

  3. The generating function for Gromow Witten invariants would define an excellent candidate for the part of WCW spinor field defining on zero modes only. The generalization of Gromov-Witten invariants to n-point functions defined by Hamiltonians of δ M4+/-× CP2 are symplectic invariants if net δ M4+/-× CP2 quantum numbers vanish. The most general definition assumes that the vanishing of quantum numbers occurs only for zero energy states having disjoint unions of partonic 2-surfaces at the boundaries of CDs as geometric correlate. A close analogy to the topological string theory of type A emerges. This seems puzzling since in the twistorial approach to N=4 SUSY however topological stringy theory of type B emerges. The celebrated mirror symmetry relating Calabi-Yau-manifolds means that topological string theory of type A is mapped to that of type B in the mirror transformation. The proposal is that the two formulations of TGD in terms of M4× CP2 on one hand and CP3 × CP3 on one hand are related in the similar manner so that the analog of topological string theory of type B would apply in the latter representation of quantum TGD.

  4. The proposed generalized homology theory involving braided Galois group and symplectic group of δ M4+/-× CP2 would realize the "almost" in TGD as almost topological QFT in finite measurement resolution replacing symplectic group with its discretized version. This algebra would relate to the quantum fluctuating degrees of freedom. The braids would carry only fermion number and there would be no Hamiltonians attached with them. The braided Galois homology could define in the more general situation invariants of symplectic isotopies.

  5. One should also add four-momenta and twistors to this picture. The separation of dynamical fermionic and sup-symplectic degrees of freedom suggesets that the Fourier transforms for amplitudes containing the fermionic braid end points as arguments define twistorial amplitudes. The representations of light-like momenta using twistors would lead to a generalization of the twistor formalism. At zero momentum limit one would obtain symplectic QFT with states characterized by collections of Hamiltonians and their super-counterparts.

For details see the new chapter Motives and Infinite Primes of "TGD as a Generalized Number Theory" or the article with same title.


L. Edgar Otto said...

I present today, in conclusion, a slightly wider picture that seems to me to answer some of your questions and confirm the directions you say may be the key.

The PeSla said...