Gravitational sources induced by exotic smoothness and fermions as knot complements

TL;DR

This paper explores how exotic smooth structures on  can act as gravitational sources, leading to the emergence of fermions modeled as spinor fields on hyperbolic knot complements, confirming Brans' conjecture.

Contribution

It demonstrates that exotic  structures induce spinor fields corresponding to fermions, linking exotic smoothness to physical matter sources and confirming Brans' conjecture.

Findings

Exotic  induce spinor fields interpreted as fermions.

Knot complements serve as hyperbolic 3-manifolds with finite volume.

Brans' conjecture is confirmed for both large and small exotic .

Abstract

In this paper we will discuss Brans conjecture that exotic smoothness serves as an additional gravitational source naturally arising from the handlebody construction of the exotic $\mathbb{R}^{4}$. We will consider the two possible classes, the large and the small exotic $\mathbb{R}^{4}$. Then we calculate the Einstein-Hilbert action for both exotic $\mathbb{R}^{4}$ to show the apearance of spinor fields. Then we discuss the physical properties of these spinor fields to relate them to fermions. Finally we identify the corresponding 3-manifolds as knot complements of hyperbolic knots, i.e. the knot complements are hyperbolic 3-manifolds with finite volume. With the help of this result we confirm the Brans conjecture for both kinds of exotic $\mathbb{R}^{4}$.