Resolution of simple singularities yielding particle symmetries in a space–time

Abstract

A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of these cycles is the Dynkin graph of an ADE Lie group, i.e., a Lie group from the cartan series A, D, or E. The deformation of the simple singularity corresponds to ADE symmetry breaking. A (3+1)-dimensional topological model of observation is constructed, transforming consistently under SL(2,C), as an evolving three-dimensional system of world tubes, which connect ‘‘possible points of observation.’’ The existence of an initial singularity for the four-dimensional space–time is related to its global topological structure. Associating the geometry of ADE singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space–time structure.