The Universe as a Set of Topological Fluids with Hierarchy and Fine Tuning of Coupling Constants in Terms of Graph Manifolds

Abstract

The hierarchy and fine tuning of the gauge coupling constants are described on the base of topological invariants (Chern classes interpreted as filling factors) characterizing a collection of fractional topological fluids emerging from three dimensional graph manifolds, which play the role of internal spaces in the Kaluza-Klein approach to the topological BF theory. The hierarchy of BF gauge coupling constants is simulated by diagonal elements and eigenvalues of rational linking matrices of tree graph manifolds pasted together from Brieskorn (Seifert fibered) homology spheres. Specific examples of graph manifolds are presented which contain in their linking matrices the hierarchy of coupling constants distinctive for the dimensionless coupling constants in our Universe. The fine tuning effect is simulated owing to the special numerical properties of diagonal elements of the linking matrices. We pay a particular attention to fine tuning problem for the cosmological constant.