In this paper we consider type I string theory compactified on a ${\mathbf{Z}}_7$ orbifold. The model has $N=1\mathrm{}$ supersymmetry, a U(4)$\otimes \mathrm{U}\mathrm{}\left(4\right)\mathrm{}\otimes \mathrm{U}\mathrm{}\left(4\right)\mathrm{}\otimes \mathrm{SO}\mathrm{}\left(8\right)\mathrm{}$ gauge group, and chiral matter. There are only $D9\mathrm{}$-branes (for which we discuss tadpole cancellation conditions) in this model corresponding to a perturbative heterotic description in a certain region of the moduli space. We construct the heterotic dual, match the perturbative type I and heterotic tree-level massless spectra via giving certain scalars appropriate vacuum expectation values, VEVs, and point out the crucial role of the perturbative superpotential (on the heterotic side) for this matching. The relevant couplings in this superpotential turn out to be nonrenormalizable (unlike the $Z$-orbifold case discussed by Kakushadze, where Yukawa couplings sufficed for duality matching). We also discuss the role of the anomalous U(1) gauge symmetry present in both type I and heterotic models. In the perturbative regime we match the (tree-level) moduli spaces of these models. We point out possible generalizations of the ${\mathbf{Z}}_3$ and ${\mathbf{Z}}_7$ cases to include $D5\mathrm{}$-branes which would help in understanding nonperturbative five-brane dynamics on the heterotic side.

• Received 21 May 1997

DOI:https://doi.org/10.1103/PhysRevD.56.3686