A 4D IIB Flux Vacuum and Supersymmetry Breaking. II. Bosonic Spectrum and Stability

We recently constructed type-IIB compactifications to four dimensions depending on a single additional coordinate, where a five-form flux $\Phi$ on an internal torus leads to a constant string coupling. Supersymmetry is fully broken when the internal manifold includes a finite interval of length $\ell$, which is spanned by a conformal coordinate in a finite range $0 < z < z_m$. Here we examine the low-lying bosonic spectra and their classical stability, paying special attention to self-adjoint boundary conditions. Special boundary conditions result in the emergence of zero modes, which are determined exactly by first-order equations. The different sectors of the spectrum can be related to Schr\"odinger operators on a finite interval, characterized by pairs of real constants $\mu$ and $\tilde{\mu}$, with $\mu$ equal to ${1}/{3}$ or ${2}/{3}$ in all cases and different values of $\tilde{\mu}$. The potentials behave as $\frac{\mu^2-1/4}{z^2}$ and $\frac{\tilde{\mu}^2-1/4}{\left(z_m-z\right)^2}$ near the ends and can be closely approximated by exactly solvable trigonometric ones. With vanishing internal momenta, one can thus identify a wide range of boundary conditions granting perturbative stability, despite the intricacies that emerge in some sectors. For the Kaluza--Klein excitations of non-singlet vectors and scalars the Schr\"odinger systems couple pairs of fields, and the stability regions, which depend on the background, widen as the ratio ${\Phi}/{\ell^4}$ decreases.