Anti-de Sitter space acts as an infrared cutoff for asymptotically free theories, allowing interpolation between a weakly coupled small-sized regime and a strongly coupled flat-space regime. We scrutinize the interpolation for theories in two dimensions from the perspective of boundary conformal theories. We show that the appearance of a singlet marginal operator destabilizes a gapless phase existing at a small size, triggering a boundary renormalization group flow to a gapped phase that smoothly connects to flat space. We conjecture a similar mechanism for confinement in gauge theories.
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