New Phases of D ≥ 2 Current and Diffeomorphism Algebras in Particle Physics

Abstract

As one who thought deeply about all aspects of symmetries Hermann Weyl [1] had traced their origin in nature to the mathematical character of physical laws. In the last thirty years, the developments in particle physics have been dominated by one single theme, the exploitation of symmetries. They can be either exact or approximate, ultimately fundamental or effective[2]. With its unqualified successes the use of symmetries has become synonymous with that of Lie algebras and groups. In the early seventies the mathematician, Jean Dieudonné [3] wrote: “ Les groupes de Lie sont devenus le centre des mathematiques; on ne peut rien faire de sérieux sans eux”. In this era of gauge and string theories, we may, without much exaggeration, assert the preeminent role at the frontiers of physics of infinite dimensional Lie group theory by replacing the words “des mathematiques” above by “ de la physique théorique”.