Light scalars from a compact fifth dimension

We consider a general five-dimensional sigma-model coupled to gravity, with any number of scalars and general sigma-model metric and potential. We discuss in detail the problem of the boundary conditions for the scalar fluctuations, in the case where the fifth dimension is compact, and provide a simple (and very general) algorithmic procedure for computing the spectrum of physical scalar fluctuations of the fully back-reacted system. Focusing in particular on the conditions under which the spectrum of scalar excitations (glueballs) contains parametrically light states, we apply the formalism to some especially simple toy models, which can be thought of as the gauge/gravity duals of strongly-coupled, non-conformal four-dimensional gauge theories. Our examples are chosen both within the context of phenomenological effective field theory constructions (bottom-up approach), and within the context of consistent truncations of ten-dimensional string theories in the supergravity limit (top-down approach). In one of the examples, a light dilaton is present in the spectrum in spite of the presence of a bad naked singularity in the deep IR, near which the RG flow of the dual theory is certainly very far away from any fixed point. If this feature were to persist in a complete model in which the singularity is resolved, this would prove that a light dilaton is to be expected in at least certain walking technicolor theories. We provide here all the technical details for testing this statement, once such a complete model is identified.