Fermion Chiral Anomaly and Atiyah-Patodi Index for Spherical Spacetime Boundaries.

We study the structure of the fermion chiral anomaly in four-dimensional Euclidean spacetime bounded by a spherical surface via the Atiyah-Patodi-Singer Index Theorem. The significance of the index theorem in physics is discussed through examples taken from high energy and solid state physics. The proof of the index theorem is then detailed in a relatively pedestrian manner. Two exactly soluble examples involving self-dual gauge fields are presented and the anomaly explicitly evaluated. These two cases consist of a fermion interacting with a U(1) electromagnetic field and an SU(2) instanton, respectively.