The first problem that is analysed in this thesis is the possibility of the existence of a parity-odd trace anomaly in 4d in the presence of a curved background. We show some evidence by using Feynman diagram techniques that this anomaly is present in the theory of a free chiral fermion and that it has the curious feature of having a purely imaginary coefficient in Lorentzian signature.
We also analyse the theory of a free massive fermion in 3d. We compute two- and three-point functions of a gauge current and the energy-momentum tensor and, for instance, obtain the well-known result that in the IR limit we reconstruct the relevant CS action in the effective action. We then couple the model to higher spin currents and explicitly work out the spin 3 case. In the UV limit we obtain an effective action which was proposed many years ago as a possible generalization of spin 3 CS action. In the IR limit we derive a different higher spin action. This analysis can evidently be generalized to higher spins. We also discuss the conservation and properties of the correlators we obtain in the intermediate steps of our derivation
