Exploring low-dimensional QFT using Perturbation Theory and beyond

In quantum field theory (QFT) perturbation theory, despite giving rise to asymptotic series, is a powerful tool that provides a lot of useful information. However, the asymptotic nature of perturbative series demands that they be treated with due care to exploit them to the fullest. To do so, we have to resort to Borel resummation, trans-series, and the theory of resurgence. In this thesis, we focus on QFTs in $d<4$ dimensions and study them starting from their perturbative series. When these series are Borel summable, there is no need to invoke trans-series since Borel resumming the perturbative series alone is enough to reproduce full results. This is the case of Euclidean $O(N)$ symmetric $\phi^4$ vector models in $d=3$ and $N=1$ $\phi^4$ theory in $d=2$. We investigate the critical regime and phase diagrams of these theories paying attention to the renormalization scheme dependence. In particular, we find non-perturbative, finite changes of scheme for a one-parameter family of renormalization schemes. This allows us to determine the exact analytic renormalization dependence of the critical couplings and to investigate in three dimensions a strong-weak duality relation closely linked to the one found by Chang and Magruder long ago. Interestingly, for some schemes, the weak fixed point and the strong one move into the complex plane in complex conjugate pairs, making the phase transition no longer visible from the classically unbroken phase. We verify all our considerations by Borel resumming several perturbative series in the classically unbroken phase. In $d=3$ we computed them up to order eight. We also investigate integrable field theories with renormalons where the perturbative series are not Borel summable. Here, focusing on the free energy in the presence of a chemical potential coupled to a conserved charge, we study in detail the interplay between the resurgent structure and the $1/N$ expansion. Our findings turn out to be different in the three models we analyzed. For some models, we find terms in $1/N$ expansion that can be fully decoded in terms of a resurgent trans-series with one or more IR renormalon corrections, the non-linear sigma model and principal chiral field, respectively. In the Gross-Neveu model, instead, each term in the $1/N$ expansion includes non-perturbative corrections, which can not be predicted by a resurgent analysis of the corresponding perturbative series.