Question

I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my question is, what are some examples of these applications?

Answer 1

Eric and others have given good answers as to why one expects the index theorem to arise in various physical systems. One of the earliest and most important applications is 't Hooft's resolution of the U(1) problem. This refers to the lack of a ninth pseudo-Goldstone boson (like the pions and Kaons) in QCD that one would naively expect from chiral symmetry breaking. There are two parts to the resolution. The first is the fact that the chiral U(1) is anomalous. The second is the realization that there are configurations of finite action (instantons) which contribute to correlation functions involving the divergence of the U(1) axial current. The analysis relies heavily on the index theorem for the Dirac operator coupled to the SU(3) gauge field of QCD. For a more complete explanation see S. Coleman's Erice lectures "The uses of instantons." There are also important applications to S-duality of N=4 SYM which involve the index theorem for the Dirac operator on monopole moduli spaces