Quantizing the N=2 super sigma-model in two dimensions

A manifestly covariant background field formalism for the N=2 supersymmetric non-linear σ-model is presented. The formalism allows the symmetries of the model to be exploited to the full in the discussion of the ultraviolet divergences in the quantum theory. This proves the cohomological triviality of the metric counterterms at the l⩾2 loop orders. The formalism confirms the finiteness of models with Ricci-flat metrics through the three-loop order. However, it seems unlikely that these cancellations will persist to higher orders. This general analysis is borne out by a study of the supercurrent structure. It is shown that while there is a component axial U(1) current which obeys an Adler-Bardeen theorem, this current is not in the supercurrent multiplet and its existence cannot therefore be used to prove conformal invariance at the quantum level.