International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 34, Pages 1777-1836

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

Luciano Boi

Ecole des Hautes Etudes en Sciences Sociales, Centre de Mathématiques, 54 boulevard Raspail, Paris 75006, France

Received 20 April 2003; Revised 27 October 2003

Copyright © 2004 Luciano Boi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.