On the geometrization of matter by exotic smoothness

摘要

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and Stern to change the smoothness structure. The influence of this surgery to the Einstein–Hilbert action is discussed. Using the Weierstrass representation, we are able to show that the knotted torus used in knot surgery is represented by a spinor fulfilling the Dirac equation and leading to a Dirac term in the Einstein–Hilbert action. For sufficient complicated links and knots, there are “connecting tubes” (graph manifolds, torus bundles) which introduce an action term of a gauge field. Both terms are genuinely geometrical and characterized by the mean curvature of the components. We also discuss the gauge group of the theory to be U(1) × SU(2) × SU(3).