  We study fractional 2p-branes and their intersection numbers in non-compact
orbifolds as well the continuation of these objects in Kahler moduli space to
coherent sheaves in the corresponding smooth non-compact Calabi-Yau manifolds.
We show that the restriction of these objects to compact Calabi-Yau
hypersurfaces gives the new fractional branes in LG orbifolds constructed by
Ashok et. al. in hep-th/0401135. We thus demonstrate the equivalence of the
B-type branes corresponding to linear boundary conditions in LG orbifolds,
originally constructed in hep-th/9907131, to a subset of those constructed in
LG orbifolds using boundary fermions and matrix factorization of the
world-sheet superpotential. The relationship between the coherent sheaves
corresponding to the fractional two-branes leads to a generalization of the
McKay correspondence that we call the quantum McKay correspondence due to a
close parallel with the construction of branes on non-supersymmetric orbifolds.
We also provide evidence that the boundary states associated to these branes in
a conformal field theory description corresponds to a sub-class of the boundary
states associated to the permutation branes in the Gepner model associated with
the LG orbifold.
