  In this paper we explicitly work out the precise relationship between Ext
groups and massless modes of D-branes wrapped on complex submanifolds of
Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex
operators for massless Ramond sector states, in open string B models describing
Calabi-Yau manifolds at large radius, directly in BCFT using standard methods.
Naively these vertex operators are in one-to-one correspondence with certain
sheaf cohomology groups (as is typical for such vertex operator calculations),
which are related to the desired Ext groups via spectral sequences. However, a
subtlety in the physics of the open string B model has the effect of physically
realizing those spectral sequences in BRST cohomology, so that the vertex
operators are actually in one-to-one correspondence with Ext group elements.
This gives an extremely concrete physical test of recent proposals regarding
the relationship between derived categories and D-branes. We check these
results extensively in numerous examples, and comment on several related
issues.
