Scales and hierarchies in warped compactifications and brane worlds

Warped compactifications with branes provide a new approach to the hierarchy problem and generate a diversity of four-dimensional thresholds. We investigate the relationships between these scales, which fall into two classes. Geometrical scales, such as thresholds for Kaluza-Klein, excited string, and black hole production, are generically determined solely by the spacetime geometry. Dynamical scales, notably the scale of supersymmetry breaking and moduli masses, depend on other details of the model. We illustrate these relationships in a class of solutions of type IIB string theory with imaginary self-dual fluxes. After identifying the geometrical scales and the resulting hierarchy, we determine the gravitino and moduli masses through explicit dimensional reduction, and estimate their value to be near the four-dimensional Planck scale. In the process we obtain expressions for the superpotential and K\"ahler potential, including the effects of warping. We identify matter living on certain branes to be effectively sequestered from the supersymmetry breaking fluxes: specifically, such ``visible sector'' fields receive no tree-level masses from the supersymmetry breaking. However, loop corrections are expected to generate masses, at the phenomenologically viable TeV scale.