Abelian gerbes, generalized geometries and foliations of small exotic R^4

TL;DR

This paper establishes a link between small exotic ^4, foliations of S^3 characterized by the Godbillon-Vey invariant, and twisted generalized geometries, revealing how exotic smoothness influences charge quantization.

Contribution

It demonstrates a strict relation between small exotic ^4 and codimension-one foliations of S^3 via the Godbillon-Vey invariant, connecting exotic smoothness to generalized geometries and charge quantization.

Findings

Exotic ^4 are distinguished by the Godbillon-Vey number, proportional to the square of the foliation's radius.

Integer Godbillon-Vey invariants relate to flat PSL(2,) bundles.

Exotic smooth structures induce charge quantization effects in spacetime.

Abstract

In the paper we prove the existence of the strict but relative relation between small exotic $\mathbb{R}^{4}$ for a fixed radial family of DeMichelis-Freedman type, and cobordism classes of codimension one foliations of $S^{3}$ distinguished by the Godbillon-Vey invariant, $GV\in H^{3}(S^{3},\mathbb{R})$ (represented by a 3-form). This invariant can be integrated to get the Godbillon-Vey number. For a fixed radial family, we will show that the isotopy classes (invariance w.r.t. small diffeomorphisms or coordinate transformations) of all members in this family are distinguished by the Godbillon-Vey number of the foliation which is equal to the square of the radius of the radial family. The special case of integer Godbillon-Vey invariants $GV\in H^{3}(S^{3},\mathbb{Z})$ is also discussed and is connected to flat $PSL(2,\mathbb{R})-$bundles. Next we relate these distinguished small exotic smooth $\mathbb{R}^{4}$'s to twisted generalized geometries of Hitchin on $TS^{3}\oplus T^{\star}S^{3}$ and abelian gerbes on $S^{3}$. In particular the change of the smoothness on $\mathbb{R}^{4}$ corresponds to the twisting of the generalized geometry by the abelian gerbe. We formulate the localization principle for exotic 4-regions in spacetime and show that the existence of these domains causes the quantization of electric charge, the effect usually ascribed to the existence of magnetic monopoles.