Poisson-Lie T-duality and Loop Groups of Drinfeld Doubles

TL;DR

This paper constructs a duality-invariant first order action on the loop group of a Drinfeld double, describing Poisson-Lie T-duality between sigma models and revealing a symplectic form as a generating function for their canonical transformation.

Contribution

It introduces a new duality-invariant action on loop groups that unifies Poisson-Lie T-duality and incorporates a WZW-term on the Drinfeld double, applicable beyond conformally invariant models.

Findings

The action describes both sigma models related by Poisson-Lie T-duality.

The difference between dual model actions is a total derivative linked to the Semenov-Tian-Shansky symplectic form.

The symplectic form acts as a generating function for the canonical transformation.

Abstract

A duality invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both of the pair of $\sigma$-models related by Poisson-Lie T-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant $\si$-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shansky symplectic form on the Drinfeld double and it gives directly a generating function of the canonical transformation relating the $\si$-models from the dual pair.