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

TL;DR

This paper explores how simple singularities and their resolutions relate to particle symmetries and space-time topology, proposing a topological model connecting ADE singularities to elementary particle symmetries.

Contribution

It introduces a topological 3+1D model linking ADE singularities to particle symmetries and space-time structure, offering a new perspective on symmetry breaking and singularity resolution.

Findings

Resolution of simple singularities via cycles corresponds to ADE Lie groups.

Topological model connects space-time singularities with particle symmetries.

Conjecture relating inner particle symmetries to local space-time geometry.

Abstract

A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C^3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of this cycles is the Dynkin graph of an ADE Lie group. 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 3-dimensional system of world tubes, which connect ``possible points of observation". The existence of an initial singularity for the 4-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.