Geometry of dual two-dimensional nonlinear $\sigma$ models

Geometrical aspects of duality in two-dimensional nonlinear $\sigma$ models are considered. The metric and torsion potential are found explicitly for the dual versions of two theories: a) the dimensional reduction to $d=2$ of the self-interaction of an $N=2$, $d=4$ tensor supermultiplet represented by a sum of an “unimproved” (linear) and “improved” (nonlinear) free action, b) the two-dimensional Freedman–Townsend model. The single- and two-loop $\beta$ functions have been calculated (on a computer).