Loop groups and quantum fields

This article surveys the application of the representation theory of loop groups to simple models in quantum field theory and to certain integrable systems. The common thread in the discussion is the construction of quantum fields using vertex operators. These examples include the construction and solution of the Luttinger model and other 1+1 dimensional interacting quantum field theories, the construction of anyon field operators on the circle, the `2nd quantization' of the Calogero-Sutherland model using anyons and the geometric construction of quantum fields on Riemann surfaces. We describe some new results on the elliptic Calogero-Sutherland model. (This paper was written in 2001, and we want to make a version of it more accessible because it is a reference for us for subsequent work.)