  A generic half-BPS surface defect of ${\mathcal N}=4$ supersymmetric U$(N)$
Yang-Mills theory is described by a partition of $N = n_1 + \ldots + n_M$ and a
set of $4M$ continuous parameters. We show that such a defect can be realized
by $n_I$ stacks of fractional D3-branes in Type II B string theory on a
$\mathbb{Z}_M$ orbifold background in which the brane world-volume is partially
extended along the orbifold directions. In this set up we show that the $4M$
continuous parameters correspond to constant background values of certain
twisted closed string scalars of the orbifold. These results extend and
generalize what we have presented for the simple defects in a previous paper.
