Geometry anomalies

When a global symmetry G breaks down to some subgroup H, the low-energy phenomenological lagrangian is given by a σ-model defined on the homogeneous space G/H. We show that when chiral fermions are coupled to this σ-model, there are in general anomalies in the Ward identities which express the non-conservation of the G-currents, thus breaking the standard current algebra predictions based on G/H. We discuss the anomaly cancellation conditions in detail. We then consider the extension of these results to the possible presence of anomalies under arbitrary reparametrizations of the Goldstone boson manifold. We relate these to anomalies in the holonomy group of the manifold and again exhibit the conditions under which these anomalies are cancelled.