The anomalous effective action in three-dimensional gauge theories

We analyse the effective action for gauge fields in odd dimensions, obtained by integrating out the fermions in the Feynman path integral. In particular, we discuss the generation of a Chern-Simons term by massless fermions minimally coupled to an Abelian gauge field. We review two methods of revealing the presence of a Chern-Simons term in the effective action: first, as the consequence of a nontrivial holonomy of the fermionic ground state, then as the result of the generation of an anomalous imaginary part of the effective action. We derive the most general form of the anomalous effective action at the lowest nontrivial order of a derivative expansion in time. We discuss the implications of our analysis for the theory of the fractional quantum Hall effect as well as for the quantization of anomalous theories. © 1989 Società Italiana di Fisica.