  In this paper we continue previous work on counting open string states
between D-branes by considering open strings between D-branes with nonzero
Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example,
when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted
as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so
their interpretation is more interesting -- for example, they often correspond
to nonreduced schemes, which furnishes an important link in understanding old
results relating classical D-brane moduli spaces in orbifolds to Hilbert
schemes, resolutions of quotient spaces, and the McKay correspondence. We give
a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent
Higgs vevs, and check that description by noting that Ext groups between the
sheaves modelling the D-branes, do in fact correctly count open string states.
In particular, our analysis expands the types of sheaves which admit on-shell
physical interpretations, which is an important step for making derived
categories useful for physics.
