Bootstrap and non-perturbative methods across dimensions

This thesis collects several explorations in formal aspects of quantum and conformal field theories using non-perturbative methods. The common thread is the "bootstrap" philosophy, in which fundamental principles and mathematical consistency, rather than a microscopic model, are used to constrain the space of physical theories. Alongside it, we use "resurgence" as a complementary method that reconstructs non-perturbative information from the perturbative expansion. We apply these ideas to a range of theories across two, three and four dimensions. In two dimensions, we study the large central charge expansion of conformal field theories. We show that the forbidden singularities of the semi-classical expansion are artefacts of the perturbative series, and that judicious resummation reconstructs the exact correlators. In three dimensions, we apply the numerical conformal bootstrap to the simplest candidate for a deconfined quantum critical point, finding evidence that it is critical and predicting its low-lying spectrum. Bridging three and four dimensions, we consider four-dimensional Yang--Mills theory in a rigid Anti-de Sitter background and bootstrap the conserved currents of its boundary theory to probe proposed mechanisms for the transition to a confining theory in flat space. Finally, in four dimensions, we extend the S-matrix bootstrap to the scattering of two particles of unequal mass, featuring non-analyticities that do not arise for identical particles, and derive bounds on several couplings and on the scalar scattering length.