I will couple two-dimensional N=(2,2) gauge theories to boundary matter in order to make them flow to worldsheet CFTs for open strings. In particular, I will focus on models for strings ending on BPS D-branes on even-dimensional cycles of Calabi-Yau threefolds. This framework provides an effective way of thinking about the role of the derived category of coherent sheaves in D-brane physics. Applications include a study of degenerations and singularities of the CFT, monodromy in closed string moduli space, local models for marginal stability behavior, and a possible matrix definition of these backgrounds.