The role of complex structures in w-symmetry

In a symplectic framework, the infinitesimal action of symplectomorphisms together with suitable reparametrizations of the two dimensional complex base space generate some type of W-algebras. It turns out that complex structures parametrized by Beltrami differentials play an important role in this context. The construction parallels very closely two dimensional Lagrangian conformal models where Beltrami differentials are fundamental.