As Lubos tells, the monopoles are not Dirac monopoles with quantized magnetic charge, which are not allowed by the gauge invariance of Maxwell equations in topologically trivial space-time. They do not seem to be GUT monopoles either. Rather, they seem to correspond to magnetic flux tubes having opposite effective magnetic charges at their ends. This kind of objects are imaginable even in the standard Maxwellian theory. It is however important to notice that the flux must continue beyond the ends by a conservation of magnetic flux if Maxwell's equation in flat space are true so that the ends are fictive to some extent. What one can think in Minkowski space is the analog of a bar magnet with return flux dispersing at the ends of magnet.
These Dirac strings are very interesting from TGD point of view since CP2 has a non-trivial topology. The fact that the second homology group is non-trivial (integers) means that there are non-contractible 2-surfaces in CP2. These surfaces are characterized homologically by integer valued Kähler magnetic flux. These monopoles are not Dirac monopoles but homological monopoles for which the topological half of Maxwell's equations remains true (Faraday's induction law and vanishing of divergence of magnetic field).
Since the projection of Kähler form to space-time defines Maxwell's field (part of electromagnetic field and under some conditions all of it) these homological monopoles can appear at space-time level.
- If space-time surfaces are orientable, these monopoles appear always as pairs with opposite Kähler magnetic charges.
- For CP2 type vacuum extremals representing elementary fermions the flux is confined inside CP2 rather than flowing radially out. Same is true for a piece of CP2 type vacuum extremal representing gauge boson as a wormhole contact.
- If one allows non-orientable space-time surfaces so that homology becomes Z2 valued also genuine monopoles carrying even value of magnetic charge are possible. I would guess that CP2 projection of space-time surface must be non-orientable in this case. Klein bottle and Möbius strip imbedded in CP2 could server as basic examples. By gluing handles one obtains more complex non-orientable 2-surfaces.
Let us restrict the consideration mostly to the orientable case.
- Magnetic flux tubes at the ends of string like objects (cosmic strings with enormous string tension) would be the basic realization. Simplest extremal is just Cartesian product X2× Y2, where X2 is string orbit in M4 and Y2 a complex sub-manifold of CP2. Cosmic strings are basic objects in TGD inspired very early cosmic string dominated cosmology when space-time sheets had not yet emerged. During cosmic evolution these flux tubes would widen to magnetic flux tubes carrying arbitrary weak magnetic field. These flux tubes could (dare I say "would"?) be still present. I would guess that non-orientable 2-surfaces of CP2 do not allow a representation as complex manifolds so that I would not expect them to give rise to cosmic string like objects.
- Fermions correspond in TGD Universe to light-like 3-surfaces separating so called CP2 type vacuum extremal with Euclidian signature of metric from space-time sheet with Minkowskian signature of metric. This light-like 3-surface can be interpreted as an orbit of a partonic 2-surface and besides fermionic quantum numbers could carry magnetic charge so that magnetically charged variants leptons and quarks could exist. A good guess is that these states are very massive. One would have a long ranged radial magnetic field and the second member of monopole pair could correspond to the outer boundary of the space-time sheet. Whether boundary conditions allow this is not clear.
- Also bosonic variants of monopoles are possible. Gauge bosons are identified as wormhole contacts (piece of CP2 type vacuum extremal) connecting space-time sheets with Minkowskian signature of metric and there would be pair of throats carrying opposite magnetic charges. Throats could carry fermionic and anti-fermionic quantum numbers so that all gauge bosons and possible Higgs particle could have magnetically charged variants. In two-sheeted space-time picture one would have magnetic dipoles with magnetic charges extremely near to each other . From the point of view of either space-time sheet one would have monopole carrying classical magnetic field giving rise to a genuine force.
- I have proposed that magnetic flux in super-conductors could be transferred between space-time sheets through these kind of wormhole contacts carrying at its throats opposite magnetic charges possibly carrying also other elementary particle numbers and behaving as bosons. The magnetic flux tubes observed in superconductors could enter to the super-conductor from a larger space-time sheet and return to it through this kind of wormhole throats.
There is also a much more mundane possibility but still representing something new. Flux tubes could be just flux tubes with outer boundary (of form X2× S2, D2 disk, rather than of form X1× Y2. Y2 closed manifold in CP2) and beginning from the boundary of space-time sheet or from wormhole throat and carrying magnetic flux which need not satisfy Dirac condition. Condensed matter monopoles could be this kind of objects.
Notice that spin glasses in which the Dirac fluxes are observed are systems in which the direction of magnetization is frozen in a given patch but varies from patch to patch. In TGD framework the notion of 4-D spin glass landscape emerges naturally from the vacuum degeneracy of Kähler action meaning same for 4-D space-time region so that kind of dynamical spin glass would be in question.
For background see the chapters General View About Physics in Many-Sheeted Space-Time: Part I and General View About Physics in Many-Sheeted Space-Time: Part II of "Physics in Many-Sheeted Space-time".