On the geometrization of matter by exotic smoothness

TL;DR

This paper explores how exotic smoothness structures on 4-manifolds can give rise to matter-like fields in spacetime, linking geometric modifications to Dirac and gauge field actions.

Contribution

It demonstrates that knot surgery on elliptic surfaces induces matter and gauge fields from geometric structures, connecting smoothness changes to physical field equations.

Findings

Knot surgery alters the Einstein-Hilbert action.

Knotted tori correspond to solutions of the Dirac equation.

Geometric structures produce gauge field actions.

Abstract

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 mass-less 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)xSU(2)xSU(3).