D-terms and D-strings in open string models

We study the Fayet-Iliopoulos (FI) D-terms on D-branes in type II Calabi-Yau backgrounds. We provide a simple worldsheet proof of the fact that, at tree level, these terms only couple to scalars in closed string hypermultiplets. At the one-loop level, the D-terms get corrections only if the gauge group has an anomalous spectrum, with the anomaly cancelled by a Green-Schwarz mechanism. We study the local type IIA model of D6-branes at SU(3) angles and show that, as in field theory, the one-loop correction suffers from a quadratic divergence in the open string channel. By studying the closed string channel, we show that this divergence is related to a closed string tadpole, and is cancelled when the tadpole is cancelled. Next, we study the cosmic strings that arise in the supersymmetric phases of these systems in light of recent work of Dvali et. al. In the type IIA intersecting D6-brane examples, we identify the D-term strings as D4-branes ending on the D6-branes. Finally, we use N=1 dualities to relate these results to previous work on the FI D-term of heterotic strings.