Geometric Models of Matter

The last Friday there was a very interesting paper in arxiv. I am really busy those days (and it will be so until around the 15 of September)so I couldn't still read ot completely. Still I think that I must leave notice of it here.

The paper in question is title like the post entry, Geometric Models of Matter. It has three authors: Michael Atiyah, Nicholas S. Manton, Bernd J. Schroers. Among them the best known one is, of course, sir Michaell Atiyhay, a very well known field medallist in mathemathics.

The abstract of the paper reads:

Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza-Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub-NUT manifold as a model for the electron, the Atiyah-Hitchin manifold as a model for the proton, CP^2 with the Fubini-Study metric as a model for the neutron, and S^4 with its standard metric as a model for the neutrino.

Ok, as I said I still didn't read the full article so I can't say many detaills. But the idea seems simple. They are inspired in the Skyrme modell. There there is a group-valued field from :$$mathbb{R}^3$$

$$U:mathbb{R}^3 \rightarrow G$$.

where the lie group is usually SU(2). In that construction specific characteristics of the proton and neutron (baryon number and so on)are associated to topological constructions, that aare, automatically, conserved quantities.

In the paper they generalize the idea in order to construct another particles, for example the electron. They must choose different kinds of manifolds, and maps. Also they use different topological invariants and so on.

But the idea is that they try to describe matter, and it's associated charges, in basic to purely geometric/topologyc constructions. Of course we are talking about different constructions that the one's involved in gauge theories. The proposal of Atiyah and all somewhat replace the need of an ordinary QFT to begin with. IF I have understood right they only have by now an static construction, that is, they don't have a way to give a dynamics to their theory. That means that it remains a lot of work to be done before they get something remotely similar to the actual world.

But, still, it is a beautiful (at least mathematically) idea. For sure Einstein would have loved it. Let's remember that in GR the space-time has a geometric nature while matter has a non-geometric one. In that sense it is an inelegant theory. If this construction works we would have a fully geometric description of the universe. If that works the immediate answer would be: the resulting theory would be equivalent to ordinary QFT in curved (well, maybe we would first ask for flat space-time9 space time for usual situations? would it give some advantage, other than aesthetic? could it be promoted to a quantum gravity?

Whatever the answer to these questions could be I think that it looks like a theory that deserves some further development. Even if it fails like a viable physical theory it could be a source of new ideas for existing ones.