Noncommutativity parameter as a field on the string worldsheet

We consider the noncommutativity parameter of the space–time as a bosonic worldsheet field. By finding a fermionic superpartner for it, we can find star products between the boson–boson, boson–fermion and fermion–fermion fields of superstring worldsheet and also between superfields of the worldsheet superspace. We find a two-dimensional action for the noncommutativity parameter and its fermionic partner. We discuss the symmetries of this action.