Topological Methods for Gauge Theories

Abstract

In these lectures I will explain and illustrate some topological ideas which give useful structural insight into gauge theories: winding numbers for gauge fields and related zero eigenvalue states of the euclidean Dirac equation. A suitably formulated axial anomaly equation which is similar to the Adler-Bell-Jackiw anomaly relation in quantum field theory provides the key to a pedestrian derivation of the Atiyah-Singer index theorem. The method can be easily generalized to Dirac spinors in euclidean gravitational fields and to other field equations whose index is governed by different winding numbers. The modification of the anomaly relation due to boundary effects and its relation to the Atiyah-Patodi-Singer index theorem will also be briefly explained. Such extensions of the conventional situation allow fractional winding numbers and would be relevant if the “meron” idea of Callan, Gross, and Dashen could be converted into a topological mechanism for quark confinement. In euclidean functional integration the Atiyah-Singer modes lead to a change of the Mathews-Salam rules for the computation of vacuum expectation values of quark fields.