Topological Origin of the Coupling Constants Hierarchy in Kaluza–klein Approach

Abstract

We consider an Abelian BF-model in the frame of ten-dimensional Kaluza–Klein approach on the space T2×X×M, where X belongs to the class of four-dimension decorated plumbed cobordisms (dp-cobordisms) and M is an An-1-singularity resolution manifold homeomorphic to a compactified ALE space. These four-dimensional manifolds with boundaries possess nontrivial cohomology properties that lead to a specific generalization of the Dirac quantization conditions and enables us to express classical partition functions in terms of 4-form fluxes through the direct product of nontrivial 2-cycles associated with the manifolds X and M. The intersection matrices of these manifolds play the role of coupling constants for the fluxes. We build several examples of dp-cobordisms containing in their intersection matrices the hierarchy of dimensionless low-energy coupling constants of interactions which are available in the real universe. We also consider the phenomenon of "running coupling constants," in particular the cosmological constant evolution induced by the topology changes of internal space X.

Algorithm for Fast Calculation of Hirzebruch-Jung Continued Fraction Expansions to Coding of Graph Manifolds

“…By its numerical value, 134 2.66 10 − × , this constant may be identify with the cosmological constant in the contemporary universe. Also in [13] we argue that the constant 11 9 4.48 10 …”

“…in Kaluza-Klein approach [13]. By its numerical value, 134 2.66 10 − × , this constant may be identify with the cosmological constant in the contemporary universe.…”

“…The fact is that our universe contains at least five fundamental interactions (strong, electro-magnetic, weak, gravitational and cosmological) with the corresponding coupling constants. In our model [13] each coupling constant is connected with the rational Euler number of proper Bh-sphere, so to obtain five coupling constants it is necessary to paste together (by plumbing) five certain Bh-spheres. The graph corresponding to this situation is shown in Figure 3 …”

“…Specific examples are given in [13] [14]. The characteristic feature of all these schemes is the emergence (in the structure of rational Euler invariants) of irreducible fractions with enormous numerators and denominators, which continued fraction expansions are too large to be obtained by classical methods.…”

Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

“…This calculation can be useful in multidimensional models of Kaluza-Klein type, where coupling constants of gauge interactions are simulated by the rational linking matrices of the internal space [1]. We constructed various models [2] [3] where the role of internal spaces is played by a specific family of 3-dimensional graph manifolds, whose rational linking matrices describe the hierarchy of gauge coupling constants of the real universe. The basic structure blocks of these graph manifolds are Seifert fibered Brieskorn homology spheres, defined as the link of Brieskorn singularity ( ) ( ) { } 1 2 3 , , a a a pairwise relatively prime positive numbers [4].…”