RESOLUTION OF SIMPLE SINGULARITIES YIELDING PARTICLE SYMMETRIES IN A SPACE-TIME

A finite subgroup of the conformal group SL(2,C) can be related to inv ariant polynomials on a hypersurface in C3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycl es of deprojections. The homological intersection graph of these cycle s 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 co rresponds to ADE symmetry breaking. A (3+1)-dimensional topological mo del of observation is constructed, transforming consistently under SL( 2,C), as an evolving three-dimensional system of world tubes, which co nnect ''possible points of observation.'' The existence of an initial singularity for the four-dimensional space-time is related to its glob al topological structure. Associating the geometry of ADE singularitie s to the vertex structure of the topological model puts forward the co njecture on a likewise relation of inner symmetries of elementary part icles to local space-time structure.