Non-commutativity in a time-dependent background

We compute a time-dependent non-commutativity parameter in a model with a time-dependent background, a spacetime metric of the plane wave type supported by a Neveu–Schwarz two-form potential. This model is the open string version of the WZW model based on a non-semi-simple group previously studied by Nappi and Witten. Like its closed string counterpart, it is exactly conformally invariant to all orders in α′. We quantize the sigma-model in light-cone gauge, compute the worldsheet propagator, and use it to derive the non-commutativity parameter.