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.

What skyrmions are?

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.

2. 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.

Skyrmion number as winding number

In TGD framework one can regard space-time as 4-surface in either octonionic M⁸_c - "c" refers here to complexification by an imaginary unit i commuting with octonions- 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,σ) 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. I failed to find any elegant formulation. One could however generalize and ask whether skyrmion like entities characterize by winding number are predicted by basic TGD.

1. 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=h_eff/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?
2. For leptons regarded as local 3 anti-quark composites in TGD based view about SUSY \citesusyTGD the same interpretation would make sense. It seems however that the winding number must have both signs. Degree is n is however non-negative.

   Here complexification of M⁸ to M⁸_c is essential. One an allow both holomorphic and anti-holomorphic continuations of real polynomials P (with rational coefficients) using complexification defined by commutative imaginary unit i in M⁸_c so that one has polynomials P(z) resp. P(z*) in turn algebraically continued to complexified octonionic polynomials P(z,o) resp. P(z*,o).

   Particles resp. antiparticles would correspond to the roots of octonionic polynomial P(z,o) resp. P(z*,o) meaning space-time geometrization of the particle-antiparticle dichotomy and would be conjugates of each other. This could give a nice physical interpretation to the somewhat mysterious complex roots of P.

More detailed formulation

To make this formulation more detailed on must ask how 4-D space-time surfaces correspond to 8-D "roots" for the "imaginary" ("real") part of complexified octonionic polynomial as surfaces in M⁸_c.

1. Equations state the simultaneous vanishing of the 4 components of complexified quaternion valued polynomial having degree n and with coefficients depending on the components of O_c, which are regarded as complex numbers x+iy, where i commutes with octonionic units. The coefficients of polynomials depend on complex coordinates associated with non-vanishing "real" ("imaginary") part of the O_c valued polynomial.
2. To get perspective, one can compare the situation with that in catastrophe theory in which one considers roots for the gradient of potential function of behavior variables xⁱ. Potential function is polynomial having control variables as parameters. Now behavior variable correspond "imaginary" ("real") part and control variables to "real" ("imaginary") of octonionic polynomial.

   For a polynomial with real coefficients the solution divides to regions in which some roots are real and some roots are complex. In the case of cusp catastrophe one has cusp region with 3-D region of the parameter defined by behavior variable x and 2 control parameters with 3 real roots, the region in which one has one real root. The boundaries for the projection of 3-sheeted cusp to the plane defined by control variables correspond to degeneration of two complex roots to one real root.

   In the recent case it is not clear whether one cannot require the M⁸_c coordinates for space-time surface to be real but to be in M⁸=M¹+iE⁷ .

3. Allowing complex roots gives 8-D space-time surfaces. How to obtain real 4-D space-time surfaces?
   1. One could project space-time surfaces to real M⁸=M¹+iE⁷ to obtain 4-D real space-time surfaces. In time direction the real part of root is accepted and is same for the root and its conjugate. For E⁷ this would mean that imaginary part is picked up.
   2. If one allows only real roots, the complex conjugation proposed to relate fermions and anti-fermions would be lost.
4. One can select for 4 complex M⁸_c coordinates X^k of the surface and the remaining 4 coordinates Y^k can be formally solved as roots of n:th degree polynomial with dynamical coefficients depending on X^k and the remaining Y^k. This is expected to give rise to preferred extremals with varying dimension of M⁴ and CP₂ projections.
5. It seems that all roots must be complex.
   1. The holomorphy of the polynomials with respect to the complex M⁸_c coordinates implies that the coefficients are complex in the generic point M⁸_c. If so, all 4 roots are in general complex but do not appear as conjugate pairs. The naive guess is that the maximal number of solutions would be n⁴ for a given choice of M⁸ coordinates solved as roots. An open question is whether one can select subset of roots and what happens at t=r_n surfaces: could different solutions be glued together at them.
   2. Just for completeness one can consider also the case that the dynamical coefficients are real - this is true in the E⁸ sector and whether it has physical meaning is not clear. In this case the roots come as real roots and pairs formed by complex root and its conjugate. The solution surface can be divided into regions depending on the character of 4 roots. The n roots consist of complex root pairs and real roots. The members or complex root pairs are mapped to same point in E⁸.

Could skyrmions in TGD sense replicate?

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 would be fascinating if all skyrmions would correspond to space-time surfaces at fundamental M⁸ level. If so, skyrmion property also in magnetic sense would be induced by from a deeper geometric skyrmion property of the MB of the system. 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.

See the article About M⁸-H-duality, p-adic length scale hypothesis and dark matter hierarchy or the chapter Zero energy ontology and matrices.

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

Articles and other material related to TGD.