Constraints on direct gauge mediation models with complex representations

We discuss constraints from cosmology on the models with direct gauge mediation based on the ISS supersymmetry breaking models. It is a generic feature that the ISS models possess a U(1) symmetry which is eventually spontaneously broken, when the models are based on the gauge symmetries with complex representations. We show that the resultant pseudo Nambu-Goldstone boson causes problems in cosmology, which leads to severe constraints on the models. We also show that most parameter space of the models can be probed by observing gravitational waves emitted from the cosmic string network when the U(1) symmetry is a gauge symmetry.