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

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 $T{S}^3\oplus T^{\star }{S}^3$ 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.

54 pages, 11 Figures, WS RMP style, complete revision with many background material, an error in the argumentation was fixed (many thanks to L. Taylor), the proof about the relation between foliations and exotic R^4 was completed, an construction of a foliation from a Casson handle was added

References in corpus (13)

• Corks, Plugs and exotic structures
• On the geometrization of matter by exotic smoothness
• Quantum Geometry and Wild embeddings as quantum states
• Cosmic censorship of smooth structures
• Exotic smooth R^4, noncommutative algebras and quantization
• Brackets, forms and invariant functionals
• Quantum general relativity and the classification of smooth manifolds
• Quantum D-branes and exotic smooth R^4
• Gerbes on orbifolds and exotic smooth R^4
• Constructing a quantum field theory from spacetime
• Gravitational sources induced by exotic smoothness
• Dark energy and 3-manifold topology
• Gravitational sources induced by exotic smoothness and fermions as knot complements