In the TGD framework the solution of the problem is simple. At the fundamental level space-time is replaced with a collection of space-time surfaces of finite size in H= M4×CP2. The corresponding 3-surfaces or at least their M^4 projections have finite size
The homology of CP2 makes possible 3-surfaces, which are flux tubes with cross section which is a closed 2-surface carrying quantized magnetic flux. This is not possible in Minkowski space. The associated magnetic fields require no current. The flux tubes are stable against splitting and can be arbitrarily long. U-shaped flux tubes give rise to tentacles which can reconnect and play a key role in biocatalysis.
The monopole part of the magnetic field, having also the Maxwellian part, explains the magnetic fields in cosmic scales. Of course, also the stability of the magnetic field of Earth is a mystery and finds a similar explanation. The strength of the monopole part is about 2/5 of that for the entire magnetic field of Earth.
For a summary of earlier postings see Latest progress in TGD.
For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.