What skyrmions are?
Consider first what skyrmions are.
- 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.
Skyrmion number as winding number
In TGD framework one can regard space-time as 4-surface in either octonionic M8c - "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.
- 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 \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 M8c 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 M8c 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.
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 M8c.
- 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 Oc, 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 Oc valued polynomial.
- 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 M8c coordinates for space-time surface to be real but to be in M8=M1+iE7 .
- Allowing complex roots gives 8-D space-time surfaces. How to obtain real 4-D space-time surfaces?
- 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.
- If one allows only real roots, the complex conjugation proposed to relate fermions and anti-fermions would be lost.
- One can select for 4 complex M8c coordinates Xk of the surface and the remaining 4 coordinates Yk can be formally solved as roots of n:th degree polynomial with dynamical coefficients depending on Xk and the remaining Yk. This is expected to give rise to preferred extremals with varying dimension of M4 and CP2 projections.
- It seems that all roots must be complex.
- The holomorphy of the polynomials with respect to the complex M8c coordinates implies that the coefficients are complex in the generic point M8c. 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=rn surfaces: could different solutions be glued together at them.
- 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.
What about the observation that condensed matter skyrmions replicate? Could this have analog at fundamental level?
- 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.
- 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.
For a summary of earlier postings see Latest progress in TGD.