The doublet--triplet mass splitting problem is one of the most serious
problems in supersymmetric grand unified theories (GUTs). A class of models
based on a product gauge group, such as the SU(5)_{GUT} times U(3)_H or the
SU(5)_{GUT} times U(2)_H, realize naturally the desired mass splitting that is
protected by an unbroken R symmetry. It has been pointed out that various
features in the models suggest that these product-group unification models are
embedded in a supersymmetric brane world. We show an explicit construction of
those models in the supersymmetric brane world based on the Type IIB
supergravity in ten dimensions. We consider T^6/(Z_{12} times Z_2) orientifold
for the compactified six extra dimensions. We find that all of the particles
needed for the GUT-symmetry-breaking sector are obtained from the D-brane
fluctuations. The three families of quarks and leptons are introduced at an
orbifold singularity, although their origin remains unexplained. This paper
includes extensive discussion on anomaly cancellation in a given orbifold
geometry. Relation to the Type IIB string theory, realization of R symmetry as
a rotation of extra-dimensional space, and effective superpotential at low
energies are also discussed.Comment: 61 page
