Poisson – Lie T - duality beyond the classical level and the renormalization group

In order to study quantum aspects of σ-models related by Poisson–Lie T-duality, we construct three-and two-dimensional models that correspond, in one of the dual faces, to deformations of S 3 and S 2. Their classical canonical equivalence is demonstrated by means of a generating functional, which we explicitly compute. We examine how they behave under the renormalization group and show that dually related models have the same 1-loop beta functions for the coupling and deformation parameters. We find non-trivial fixed points in the ultraviolet, where the theories do not become asymptotically free. This suggests that the limit of Poisson–Lie T-duality to the usual Abelian and non-Abelian T-dualities does not exist quantum mechanically, although it does so classically.