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.
Categories: hep-th gr-qc math-ph math.GT math.MP