Skyrmions in TGD

I received a link to an article telling about research carried out for condensed matter skyrmions, which are very general condensed matter quasiparticles. They were found to replicate like DNA and cells. I realized that I have not clarified myself the possibility of skyrmions on TGD world and decided to clarify my thoughts.

Consider first what skyrmions are.

1. Skyrmions are topological entities. One has some order parameter having values in some compact space S. This parameter is defined in say 3-ball such that the parameter is constant at the boundary meaning that one has effectively 3-sphere. If the 3rd homotopy group of S characterizing topology equivalence classes of maps from 3-sphere to S is non-trivial, you get soliton-llike entities, stable field configurations not deformable to trivial ones (constant value). Skyrmions can be assigned to space S which is coset space SU(2)_L× SU(2)_R/SU(2)_V, essentially S³ and are labelled by conserved integer-valued topological quantum number.

One can imagine variants of this. For instance, one can replace 3-ball with disk. SO(3)=S³ with 2-sphere S². The example considered in the article corresponds to discretized situation in which one has magnetic dipoles/spins at points of say discretized disk such that spins have same direction about boundary circle. The distribution of directions of spin can give rise to skyrmion-like entity. Second option is distribution of molecules which do not have symmetry axis so that as rigid bodies the space of their orientations is discretized version of SO(3). The field would be the orientation of a molecule of lattice and one has also now discrete analogs of skyrmions.

The map S³→ S³ defining skyrmion could be taken as a phenomenological consequence of M⁸-H duality implying the old-fashioned description of hadrons involving broken SO(4) symmetry (PCAC) and unbroken symmetry for diagonal group SO(3)_V (CCV). The analog of (π,sigma) field could correspond to a B-E condensate of pions (π,σ).

The obvious question is whether the map S³→ S³ defining skyrmion could have a deeper interpretation in TGD framework. M⁸-H (H=M⁴× CP₂) duality suggests perhaps the most plausible and most general identification of skyrmions.

1. Consider first the situation in M⁸. M⁸ picture predicts besides 4-D space-time surfaces inside CDs also brane like entities as 6-spheres having 3-ball as projection to CD with t=r_n, where r_n is a root of the real polynomial defining space-time surface. r_n equals also to the radius r(E⁴) of sphere S³ in E⁴ appearing as analog of fiber space over 3-ball.

Outside the 6-spheres the points x of 3-balls intersecting space-time surface having t≠ r_n correspond to finite number of points in E⁴. At t=r_n this discrete set of points associated with x explodes to S³. This is somewhat like turning of the graph of map x→ f(x) to vertical direction.

The third homotopy group is not necessarily trivial. For instances for Cartesian products X₃= X₂× Y₁ π₃ is direct sum of homotopy groups of X₂ and Y₁. For X₃= S²× S¹ one has π₃(X)= π₃(S²)=π₃(S³)=Z so that for string like objects one can have skyrmions. For tori X₃= S¹× S¹× S² π³(X is trivial.

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