How to include fermions into General relativity by exotic smoothness

Abstract

This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, denoted as hyperbolic knot complements in the following. We will discuss the fermionic properties of this class of 3-manifolds, i.e. we are able to identify a fermion with a hyperbolic knot complement. Secondly we will construct a large class of space-times, the exotic $\mathbb{R}^{4}$, containing this class of 3-manifolds naturally. We begin with a topological trivial space, the $\mathbb{R}^{4}$, and change only the differential structure to obtain many nontrivial 3-manifolds. It is known for a long time that exotic $\mathbb{R}^{4}$'s generate extra sources of gravity (Brans conjecture) but here we will analyze the structure of these source terms more carefully. Finally we will state that adding a hyperbolic knot complement will result in the appearance of a fermion as source term in the Einstein-Hilbert action.