Duality in d=2 ? models of chiral field with anomaly

Abstract

It is shown that the equations of the chiral field with anomaly in d = 2 (the equations of nonlinear sigma models with multiply valued action) possess continuous dual symmetry, and a realization of the duality transformations is found in the explicitly geometrical language of Cartan forms. The connection between this symmetry and the integrability for the chiral field with anomaly is established. Both the ordinary and the supersymmetric cases are considered. The concepts of dual algebra and dual sigma model are introduced, and it is shown that they play an important part in clarifying the classical and quantum structure of the d = 2 models of the chiral field with anomaly. In particular, it is shown that the transition to the infrared stability points of the chiral field with anomaly can be described purely algebraically as a contraction of the dual algebra which has the consequence that the factor space of the corresponding dual sigma model becomes flat. The equations of the model of the n field with anomaly are also analyzed from an analogous point of view. The method of Cartan forms makes it possible to show that the classical dynamics of this model is trivial.