Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group H4H4

We show that the WZW model on the Heisenberg Lie group H4H4 has Poisson–Lie symmetry only when the dual Lie group is A2⊕2A1A2⊕2A1. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group H4H4 and its dual pair, A2⊕2A1A2⊕2A1, as the target space in such a way that the original model is the same as the H4H4 WZW model. Furthermore, we show that the dual model is conformal up to two-loop order. Finally, we discuss D  -branes and the worldsheet boundary conditions defined by a gluing matrix on the H4H4 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 H4H4 and its dual model.