ON INFINITE DIMENSIONAL SYMMETRY GROUPS IN PHYSICS

Abstract

Recently infinite dimensional Lie algebras have emerged as powerful dynamical tools in physics. After a few general remarks we shall be motivated to study infinite dimensional Lie groups. We restrict ourselves to the infinitely differentiate category as opposed to algebraic or formal Lie groups. The most important groups are not Banach Lie groups, nevertheless gauge groups manage to have many of their properties, on the other hand diffeomorphism groups do not! A third class (Kac-Moody) will be very briefly reviewed. In a second section we shall describe a program devised to understand the “hidden symmetries”. Examples are E 11-d for d-dimensional maximal supergravity (N = 8 for d = 4), the loop groups of σ-models and other “completely integrable” systems. The idea is that free theories admitting an abelian (usually gauge) symmetry can be deformed systematically into interacting theories with non abelian symmetry. A global non abelian symmetry may be used to restrict the deformations. These free theories determine their possible interactions and their non abelian symmetries.