Effective action and superconformal anomalies of $(p, p)$ $\sigma$ models in $(1, 1)$ superspace

A description of $(p,p)$-extended locally supersymmetric $\sigma$ models is given in terms of $(1,1)$ superfields. It is shown that to models with extended supersymmetry there correspond two types of scalar superfields (ordinary and twisted). The effective action of the $(p,p)$ $\sigma$ models is constructed, and conditions under which the theory is free of superconformal anomalies are found.