  Building on earlier work, we construct linear sigma models for strings on
curved spaces in the presence of branes. Our models include an extremely
general class of brane-worldvolume gauge field configurations. We explain in an
accessible manner the mathematical ideas which suggest appropriate worldsheet
interactions for generating a given open string background. This construction
provides an explanation for the appearance of the derived category in D-brane
physics complementary to that of recent work of Douglas.
