Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group $H_4$

TL;DR

This paper explores Poisson-Lie symmetry in the $H_4$ WZW model, constructs its T-dual models, confirms their conformality, and analyzes D-branes and boundary conditions within this duality framework.

Contribution

It demonstrates Poisson-Lie symmetry for the $H_4$ WZW model, constructs its T-dual models, and studies D-branes and boundary conditions using duality maps.

Findings

Poisson-Lie symmetry exists only with dual group ${A}_2 igoplus 2{A}_1$

Dual model remains conformal up to two loops

Identifies two types of D-brane boundary conditions via gluing matrices

Abstract

We show that the WZW model on the Heisenberg Lie group $H_4$ has Poisson-Lie symmetry only when the dual Lie group is ${ A}_2 \oplus 2{ A}_1$. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group $H_4$ and its dual pair, ${ A}_2 \oplus 2{ A}_1$, as the target space in such a way that the original model is the same as the $H_4$ WZW model. Furthermore, we show that the dual model is conformal up to two loops order. Finally, we discuss $D$-branes and the worldsheet boundary conditions defined by a gluing matrix on the $H_4$ WZW model. Using the duality map obtained from the canonical transformation description of the Poisson-Lie T-duality transformations for the gluing matrix which locally defines the properties of the $D$-brane, we find two different cases of the gluing matrices for the WZW model based on the Heisenberg Lie group $H_4$ and its dual model.