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Abstract

The main focus of this thesis is Quantum Field Theory and our investigation goes in two main directions. In a the first part (chapters 3 and 4) we mainly use the large-N expansion as a tool to investigate some non-perturbative properties of O(N) vector models and O(N)^3 tensor models. Also (chapter 5), we study the critical properties of cubic and quartic fractional Lifshitz field theory using standard perturbation theory. In the second part (chapter 6) we study again the O(N) vector model, but this time as a zero dimensional toy model of a QFT, on which we develop constructive techniques and describe non-perturbative contributions coming from instantons. Chapters 1 and 2 are an introduction to the original research results, where we explain the context in which our work needs to be framed and give an overview of the most important technical tools that we used. In chapter 3 we study a long-range tensor model with O(N)^3 symmetry, which was proven to have four lines of fixed points in the large-N limit. In particular, we test the famous F-theorem on such model, stating that in d = 3 the free energy on the sphere is monotonically decreasing going from the ultraviolet to the infrared along the renormalization group flow. We compute explicitly the free energy at two fixed points connected by a renormalization group trajectory and find that the F-theorem holds, which can be thought of as a further hint for the unitarity of the theory. The computation was technically challenging and it required to sum and infinite family of diagrams by means of the Conformal Partial Waves expansion. Also, we review the O(N) vector model at large-N, that we use as a warm-up exercise to test our technology. In chapter 4 we investigate the effects of compatifying one direction on Conformal Field Theories. The compact direction necessarily introduces an energy scale, i.e. the inverse of its size, that breaks conformal invariance. The main consequence is that one-point functions are no more constrained to be zero, enlarging the set of CFT data needed to fully characterize the theory. We review the long-range O(N) vector model with one compact direction, show the existence of a non-trivial solution of the mass gap equation and of a stable IR fixed point at large-N. We compute the one-point functions of a specific family of bilinear operators at the large-N fixed point. Furthermore, we stress and discuss the difference between a classical statistical field theory and its corresponding quantum version, which becomes manifest when dealing with long-range theories. Indeed, long-range interactions are described by an action with a fractional Laplacian in the space directions, while the usual term with the double derivative with respect to time, coming from the quantum-to-classical mapping, is unchanged. Such kind of theories, which we call quantum fractional Lifshitz field theories, are manifestly anisotropic and the different role of time and space is evident, as opposed to the more common short-range theories in which their difference might be overlooked when working in Euclidean signature. In chapter 5 we investigate the critical properties of quantum fractional Lifshitz field theories with cubic and quartic interaction. One interesting feature of Lifshitz field theories is that, while they break explicitly Lorentz or rotation invariance, they exhibit anisotropic scale invariance at fixed points of the renormalization group. Such anisotropic scale invariance is fully described by the anisotropy exponent z, which we compute to the first non-trivial order in perturbation theory. In chapter 6 we study the quartic O(N) vector model in zero dimensions, i.e. as a purely combi- natorial toy model of a QFT. Working in zero dimensions fully trivialize the problem of computing Feynman amplitudes, making us able to study more easily the properties of the asymptotic perturba- tive expansion. We study both the partition function and the free energy as functions of the coupling constant by using constructive techniques such as the BKAR formula and Loop Vertex Expansion. The Loop Vertex Expansion requires to introduce an Hubbard–Stratonovich auxiliary field and it has the remarkable feature of having a finite radius of convergence. Furthermore, in the case of the O(N) model, it formally corresponds to a small-N expansion. We prove Borel summability of both the partition function, the free energy and the coefficients of their small-N expansion in all the direc- tions of the complex plain. In order to do this, we need to analytically continue such quantities on a multi-sheeted Riemann surface, i.e. past a Stokes line. Finally, we compute their full transseries expansion which, past the Stokes lines, includes instanton-like non-analytic contribution encoding all the non-perturbative information. Despite the fact that some of the results on the topic were already understood by other means, we believe that our work is relevant as it set the ground for repeating the same analysis in QFT.