This chapter is devoted to introducing infinite-dimensional Lie groups and algebras and cohomology (in contrast with the finite-dimensional case) as a preparation for chapter 10. Special attention is given to gauge groups and current algebras (an example of the same kind of generalization was already given in chapter 8 in the context of supersymmetric extended objects).

A second set of examples, in which the extension properties of the Lie algebras involved are studied, is provided by the Virasoro and Kac- Moody algebras, as well as the two-dimensional conformal group. It is also shown that Polyakov's induced two-dimensional gravity (which itself is not discussed) provides yet another example of a Wess-Zumino term, obtained from the group Diff S1 of the diffeomorphisms of the circle.

Introduction

Most of the infinite-dimensional groups that appear in physics are defined by endowing the space of mappings of a (finite-dimensional) manifold M into another finite-dimensional manifold Q with a group structure. For instance, this is the case for the groups Map(M,G) ≡ G(M) that correspond to the smooth mappings M → G and which (in physics) are sometimes referred to as ‘local’ groups; this is also the case of the gauge groups, as well as of the loop groups LG or Map(S1,G) given by the smooth mappings S1 → G. The loop groups can be generalized to groups of mappings Sn → G; these generalized loop groups or sphere groups may be denoted by LnG (with L1G ≡ LG and L0G = G), and were already encountered in section 8.8 in a particular case in which G was the (graded) supertranslation group.