$q\to \infty$ limit of the quasitriangular WZW model

TL;DR

This paper investigates the $q oinite$ limit of the $q$-deformed WZW model, revealing its underlying affine Poisson structure, symplectic groupoid phase space, and duality properties, advancing understanding of quantum group limits in conformal field theory.

Contribution

It introduces the affine Poisson structure governing the $q oinite$ limit of the $q$-WZW model and constructs its symplectic groupoid, providing new insights into the model's geometric and algebraic structure.

Findings

Identified the affine Poisson structure underlying the $q oinite$ current algebra.

Constructed the symplectic groupoid as the phase space of the limit model.

Demonstrated a duality and chiral decomposition compatible with Poisson-Lie symmetries.

Abstract

We study the $q\to\infty$ limit of the $q$-deformation of the WZW model on a compact simple and simply connected target Lie group. We show that the commutation relations of the $q\to\infty$ current algebra are underlied by certain affine Poisson structure on the group of holomorphic maps from the disc into the complexification of the target group. The Lie algebroid corresponding to this affine Poisson structure can be integrated to a global symplectic groupoid which turns out to be nothing but the phase space of the $q\to\infty$ limit of the $q$-WZW model. We also show that this symplectic grupoid admits a chiral decomposition compatible with its (anomalous) Poisson-Lie symmetries. Finally, we dualize the chiral theory in a remarkable way and we evaluate the exchange relations for the $q\to\infty$ chiral WZW fields in both the original and the dual pictures.