Duality in $d=2$ $\sigma$ models of chiral field with anomaly

Using a manifestly geometric language of Cartan forms, we expose continuous dual symmetry of general $d=2$ models of chiral field with anomaly (ACF) (also called the $d=2$ sigma-models with multivalued action) and find the corresponding duality transformations. Both ordinary and supersymmetric ACF are treated. Like in the case of the standard chiral field, duality transformations of ACF prove to be related intimately to its integrability. We introduce also the notions of dual algebra and dual sigma-model and demonstrate their important role for understanding the classical and quantum structure of ACF. It is shown, in particular, that the transition to infrared fixed points of ACF models can be described in a pure algebraic way as a contraction of the dual algebra, as a result of which the coset space of the dual sigma-model becomes flat. Finally, we analyze, in an analogous context, the classical equations of the anomalous version of $CP^1$-model. These are found to yield a trivial dynamics.