International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 33-36, Pages 1777-1836
doi:10.1155/S0161171204304400

Geometrical and topological foundations of theoretical physics: from gauge theories to string program

Abstract

We study the role of geometrical and topological concepts in the recent developments of theoretical physics, notably in non-Abelian gauge theories and superstring theory, and further we show the great significance of these concepts for a deeper understanding of the dynamical laws of physics. This work aims to demonstrate that the global topological properties of the manifold's model of spacetime play a major role in quantum field theory and that, therefore, several physical quantum effects arise from the nonlocal metrical and topological structure of this manifold. We mathematically argue the need for building new structures of space with different topology. This means, in particular, that the "hidden" symmetries of fundamental physics can be related to the phenomenon of topological change of certain classes of (presumably) nonsmooth manifolds.