Modular Invariant Anomalous U(1) Breaking
  We describe the effective supergravity theory present below the scale of spontaneous gauge symmetry breaking due to an anomalous U(1), obtained by integrating out tree-level interactions of massive modes. A simple case is examined in some detail. We find that the effective theory can be expressed in the linear multiplet formulation, with some interesting consequences. Among them, the modified linearity conditions lead to new interactions not present in the theory without an anomalous U(1). These additional interactions are compactly expressed through a superfield functional. 
Mary K. Gaillard and Joel Giedt