Anomalies of minimal $N=(0,1)$ and $N=(0,2)$ sigma models on homogeneous spaces

We study chiral anomalies in $N=(0,1)$ and $(0,2)$ two-dimensional minimal sigma models defined on the generic homogeneous spaces G/H. Such minimal theories contain only (left) chiral fermions and in certain cases are inconsistent because of 'incurable' anomalies. We explicitly calculate the anomalous fermionic effective action and show how to remedy it by adding a series of local counterterms. In this procedure, we derive a local anomaly matching condition, which is demonstrated to be equivalent to the well-known global topological constraint on $p_1\left(G/H\right)$, the first Pontryagin class. More importantly, we show that these local counterterms further modify and constrain 'curable' chiral models, some of which, for example, flow to the nontrivial infrared superconformal fixed point. Finally, we also observe an interesting relation between $N=(0,1)$ and $(0,2)$ two-dimensional minimal sigma models and supersymmetric gauge theories. (paper)