In a three dimensional U(1) gauge theory with massive fermions, one can integrate out the fermion fields with the result that the effective action for the gauge field now has an additional Chern Simons term whose level depends on the sign of the fermion masses and the number of fermion flavours. I have been following the calculation is Gerald Dunne's lecture notes ( "Aspects of Chern-Simons Theory" (http://arxiv.org/abs/hep-th/9902115) ) that demonstrated this by expanding the functional determinant of the Dirac operator into a series of Feynman diagrams. I have the following questions about it:
1) It seems to me, however that such an expansion would be justifiable only in the regime where the fermion masses are much larger than the gauge coupling $e^2$. That's when you could say that a perturbative expansion of the functional determinant gives you a reliable answer. Is that right?
2) If that is right, then how would you show that Chern Simons terms are induced as they are upon integrating out fermionic fields? Would it even be true in that case?
I'd highly appreciate if anybody can provide a satisfactory answer here.