Supersymmetric 4d gauge theories and Integrability

This thesis is devoted to some particular aspects of integrability in $4d$ SUSY gauge theories. Taking advantage of the integrable structures emergent in the theory, non-local observables such as null polygonal Wilson loops are studied in $4d$ planar $\mathcal{N}=4$ Super Yang-Mills. Their duality with the $4d$ gluon scattering amplitudes makes the analysis even more interesting. The so-called \emph{Pentagon approach}, an application of the Operator Product Expansion (OPE) method to the null polygonal Wilsol loops, makes possible a non-perturbative evaluation of these objects. They are recast as an OPE series over the $2d$ GKP flux-tube excitations, a description reminescent of the QCD flux-tube stretching between quarks. The integrability of the flux-tube allows us to evaluate the series, in principle, for any value of the coupling constant. From this analysis, several results have been obtained. In the strong coupling regime we reproduced the TBA-like equations expected from the minimal area problem in string theory, finding agreement with the $AdS/CFT$ prediction. In this respect, of fundamental importance is the emergence of effective bound states between elementary fermionic excitations. Along the way, we encountered some intriguing analogies between these null polygonal Wilson loops and the Nekrasov instanton partition function $\mathcal{Z}$ for $\mathcal{N}=2$ theories. Furthermore, a new non-perturbative enhancement of the classical string argument has been confirmed, stemming from the dynamics of the string on the five sphere $S^5$ and described by the non-linear $\sigma$-model $O(6)$. Some properties of a fundamental building block of the OPE series, the $SU(4)$ structure of the form factors for a specific twist operator $\hat{P}$, have been analysed. This $SU(4)$ matrix part is given a representation in terms of rational functions, organized in a Young diagrams pattern.