Anomalies and index theory

There is at present considerable interest among physicists in various anomalies that occur in gauge theories and their relation to the index theory of the Dirac operator. Numerous papers have appeared on the subject and it is the purpose of this paper to provide a general introduction and guide to the subject. The author concentrates on the main ideas since the detailed calculations are fully available in the literature. He begins by a brief review of the index theorem in its simplest form. This also enables him to establish the notation and terminology he uses which (inevitably) tends to differ slightly from that used in physics. He then goes on to describe a more general version of the index theorem, dealing with parameters. It is this version which is related to the anomalies that turn up in relation to the difficulties of defining gauge invariant determinants. After dealing with determinants he returns to the index theory explaining further refinements all of which help to throw further light on anomalies