On quantum Poisson-Lie T-duality of WZNW models

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title> <jats:p>We study Poisson-Lie T-duality of the Wess-Zumino-Novikov-Witten (WZNW) models which are obtained from a class of Drinfel’d doubles and its generalization. In this case, the resultant WZNW models are known to be classically self-dual under Poisson-Lie T-duality. We describe an explicit construction of the associated currents, and discuss the conformal invariance under this duality. In a concrete example of the SU(2) WZNW model, we find that the self-duality is represented as a chiral automorphism of the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \hat{\mathfrak{su}} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>su</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> </jats:alternatives> </jats:inline-formula>(2) affine Lie algebra, though the transformation of the currents is non-local and non-linear. This classical automorphism can be promoted to the quantum one through the parafermionic formulation of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$ \hat{\mathfrak{su}} $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>su</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> </jats:alternatives> </jats:inline-formula>(2), which in turn induces an isomorphism of the WZNW model. We thus find a full quantum equivalence of the dual pair under Poisson-Lie T-duality. The isomorphism is represented by a sign-change of a chiral boson or the order-disorder duality of the parafermionic conformal field theory as in Abelian T-duality on tori or in the mirror symmetry of the Gepner model.</jats:p>