The study of non-abelian gauge theories in compact or non-flat spaces can be useful to gather insights and new perspectives on the confinement problem. We consider Yang-Mills theory on four dimensional Anti-de Sitter space and wonder how signals of confinement in the bulk can be detected from boundary observables. The Dirichlet boundary condition cannot exist at arbitrarily large radius because it would give rise to colored asymptotic states in flat space and hence a deconfinement-confinement transition has to occur as the radius is increased. By perturbative computations we provide evidence for the scenario of merger and annihilation. Namely, the theory with Dirichlet boundary condition stops existing because it merges and annihilates with another theory. We also derive a general result for the leading-order anomalous dimension of the so called displacement operator for a generic perturbation in Anti-de Sitter, showing that it is related to the beta function of bulk couplings.

Four-dimensional gauge theories can flow in the IR to non-trivial CFTs. By employing Borel resummation techniques both to the ordinary perturbative series and to the Banks-Zaks conformal expansion, we first analyze the conformal window of QCD and find substantial evidence that QCD with n_f=12 flavours flows in the IR to a CFT. We then study UV fixed points for SU(n_c) gauge theories with fundamental fermion matter in 4+2epsilon dimensions. Using resummation techniques similar to those used in the 4d QCD case, we provide evidence for the existence of non-supersymmetric CFTs in d=5 space-time dimensions in a certain range of colors and flavours.