Dirac operators on coset spaces

The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kahler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin(c)-structures. When a manifold is spin(c) and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors, as shown by Avis, Isham, Cahen, and Gutt. Likewise, for manifolds like SU(3)/SO(3), which are not even spin(c), we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S-n=SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen (C) 2003 American Institute of Physics.