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Skyrmions in TGD

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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.

  • More generally, skyrmions emerge naturally in old-fashioned hadron physics, where SU(2)L× SU(2)R/SU(2)V involves left-handed, right-handed and vectorial (diagonal) subgroups of SO(4)=SU(2)L× SU(2)R. The realization would be in terms of 4-component field (π,σ), where π is charged pion with 3 components – axial vector – and σ which is scalar. The additional constraint π•π +σ2= constant defines 3-sphere so that one has field with values in S3. There are models assigning this kind of skyrmion with nucleon, atomic nuclei, and also in the bag model of hadrons bag can be thought of as a hole inside skyrmion. These models seem to have something to do with reality so that a natural question is whether skyrmions might appear in TGD. In TGD framework one can regard space-time as 4-surface in either octonionic M8c, c refers here to complexification by commuting imaginary unit, or in M4× CP2. For the solution surfaces M8 has natural decomposition M8=M2× E6 and E6 has SO(6) as isometry group containing subgroup SU(3) having automorphisms of octonions as subgroup leaving M2 invariant. SO(6)=SU(4) contains SU(3) as subgroup, which has interpretation as isometries of CP2 and counterpart of color gauge group. This supports M8-H duality, whose most recent form is discussed here.
  • 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=rn, where rn is a root of the real polynomial defining space-time surface. rn 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≠ rn correspond to finite number of points in E4. At t=rn 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.

  • It is natural to assume that the points at the boundary of B3 at boundary of light-cone correspond ti single point so that B3 effectively corresponds to S3. Could this map S3→ E4 define approximately a map S3→ S3× D allowing skyrmion interpretation? Could the value of t=r correspond to values of r(E4) in the range [rn-1,rn]? A more general condition allowing skyrmion interpretation would be that the E4 projection of 3-surfaces with t∈ [rn-1,rn] is compact and has non-trivial 3rd homotopy group.
  • 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.

  • What about the situation in H? The space-time surfaces in M8 are mapped to those in H and t=rn sections have at each point 3-sphere of CP2 as an analog of fiber space. For 4-D space-time surfaces inside CD the t=rn 3 balls have for given point of B3 finite number of points as CP2 projection and the surface in question is compact and can have non-trivial homotopy group π3. Therefore the situation is same as for M8.
  • In the models of nucleon and nuclei the interpretation of conserved topological skyrmion number is as baryon number. This number should correspond to the homotopy class of the map in question, essentially winding number. For polynomials of complex number degree corresponds to winding number. Could the degree n=heff/h0 of polynomial P having interpretation as effective Planck constant and measure of complexity – kind of number theoretic IQ – be identifiable as skyrmion number? Could it be interpreted as baryon number too?
  • For leptons regarded as local 3 anti-quark composites in TGD based view about SUSY(see this) the same interpretation would make sense. It seems however that the winding number must have both signs. One must in allow also anti-holomorphic continuations of real polynomials using complexification with respect to commutative imaginary unit i in M8c. Particles and antiparticles would correspond to conjugate roots of octonionic P so that fermion meaning geometrization of the particle-antiparticle dichotomy. Particles which are their own antiparticles would correspond to real roots. This would give a nice physical interpretation to the somewhat mysterious complex roots of P. What about the observation that condensed matter skyrmions replicate? Could this have analog at fundamental level?
    1. The assignment of conserved topological quantum number to the skyrmion is not consistent with replication unless the skyrmion numbers of outgoing states sum up to that of the initial state. If the system is open one can circumvent this objection. The replication would be like replication of DNA in which nucleotides of new DNA strands are brought to the system to form new strands.
    2. It is perhaps unrealistic to think that also the skyrmions studied in the article could correspond to space-time surfaces and that skyrmion property in magnetic sense could be induced by from a deeper geometric skyrmion property of the MB of the system. As noticed, magnetic flux tubes have π3=Z. The openness of the system would be essential to guarantee conservation of baryon number. Here the fact that leptons and baryons have opposite baryon numbers helps in TGD framework. Note also ordinary DNA replication could correspond to replication of MB and thus of skyrmion sequences.

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

  • Articles and other material related to TGD.

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