Non-perturbative aspects of gauge theories from gauge-gravity dualities

In this Ph.D. Thesis we consider two specific supergravities which are well-established within the literature on holography, and which are known to provide the low-energy effective description of either superstring theory or M-theory: the six-dimensional half-maximal theory of Romans, and the maximal supergravity in seven dimensions. We implement their dimensional reduction by compactifying on an S1 and T2, respectively, to obtain a five-dimensional sigma-model coupled to gravity. Spectra of bosonic excitations are computed numerically by considering field fluctuations on background geometries which holographically realise confinement. We furthermore propose a diagnostic tool to detect mixing effects between scalar resonances and the pseudo-NambuGoldstone boson associated with spontaneous breaking of conformal invariance: the dilaton. This test consists of neglecting a certain component of the spin-0 fluctuation variables, effectively disregarding their back-reaction on the underlying geometry; where discrepancies arise compared to the complete calculation we infer dilaton mixing. For both theories this analysis evinces a parametrically light dilaton. For each supergravity we uncover a tachyonic instability within their parameter space; motivated by these pathological findings we proceed to conduct an investigation into their respective phase structures, reasoning that there must necessarily exist some mechanism by which these instabilities are rendered physically inaccessible. We compile a comprehensive catalogue of geometrically distinct backgrounds admissible within each theory, and derive general expressions for their holographically renormalised free energy F. Another numerical routine is employed to systematically extract data for some special deformation parameters, and F is plotted in units of an appropriate universal scale. Our analysis proves fruitful: each theory exhibits clear evidence of a first-order phase transition which induces the spontaneous decompactification of the shrinking circular dimension before the instability manifests, favouring instead a class of singular solutions. The aforementioned dilaton resonance appears only along a metastable portion of the branch of confining backgrounds.