Poisson-Lie T-Duality beyond the classical level and the renormalization group

TL;DR

This paper explores quantum aspects of Poisson-Lie T-duality in sigma models, demonstrating classical equivalence, analyzing their behavior under renormalization, and revealing non-trivial fixed points indicating complex quantum properties.

Contribution

It constructs specific 2D and 3D models related by Poisson-Lie T-duality, and analyzes their quantum behavior and renormalization group flow, extending classical duality to the quantum level.

Findings

Models have the same 1-loop beta functions

Existence of non-trivial ultraviolet fixed points

Quantum Poisson-Lie T-duality limit does not exist

Abstract

In order to study quantum aspects of $\s$-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.