Generalized K$\ddot{a}$hler Geometry and current algebras in classical N=2 superconformal WZW model

I examine the Generalized K$\ddot{a}$hler geometry of classical $N=(2,2)$ superconformal WZW model on a compact group and relate the right-moving and left-moving Kac-Moody superalgebra currents to the Generalized K$\ddot{a}$hler geometry data using Hamiltonian formalism. It is shown that canonical Poisson homogeneous space structure induced by the Generalized K$\ddot{a}$hler geometry of the group manifold is crucial to provide $N=(2,2)$ superconformal sigma-model with the Kac-Moody superalgebra symmetries. Biholomorphic gerbe geometry is used to prove that Kac-Moody superalgebra currents are globally defined.