  We study the quantization of chiral fermions coupled to generalized Dirac
operators arising in NCG Yang-Mills theory. The cocycles describing chiral
symmetry breaking are calculated. In particular, we introduce a generalized
locality principle for the cocycles. Local cocycles are by definition
expressions which can be written as generalized traces of operator commutators.
In the case of pseudodifferential operators, these traces lead in fact to
integrals of ordinary local de Rham forms. As an application of the general
ideas we discuss the case of noncommutative tori. We also develop a gerbe
theoretic approach to the chiral anomaly in hamiltonian quantization of NCG
field theory.
