Abstract

In this paper geometric phases (Berry and Aharonov-Bohm) are generalized to nonlinear topological phase fields on pseudospheres, where the coordinate vector field is parallel transported along the signal/soliton vector field with Levi--Civita connection. Projective PSL(2,R} symmetry describes the relativistic self-interacting bosonic sine-Gordon field. A Coulomb potential can be induced as the stereographic projection of a harmonic oscillator potential mapping angles or phases to distances and vice versa resulting in mutual coupling with a generalized coupling constant given by a nonlinear iteration. With single-valuedness requirement in 137-gonal symmetry it fits within a few ppb uncertainty to the Sommerfeld fine structure constant.