Integrabilità e Teorie di Gauge

This thesis concerns the use of tools originally developed in the context of low-dimensional exactly solvable models to study the properties of supersymmetric gauge theories in three and four space-time dimensions. The research topic is interdisciplinary: it involves exact scattering matrix theory and quantum groups, perturbed conformal field theory, the thermodynamics of spin chains and many-body systems, supersymmetric gauge theories and the AdS/CFT correspondence that links the latter to string theory. The first part of the thesis contains review material on exactly solvable models, integrability in the AdS/CFT correspondence, the Thermodynamic Bethe Ansatz method (TBA) and its application in gauge theory. The infinite set of TBA equations describing the anomalous dimensions of gauge invariant composite operators in N =4 super Yang-Mills (SYM) has recently been mapped into a finite dimensional matrix nonlinear Riemann-Hilbert problem, named Quantum Spectral Curve or Pμ-system. The Pμ-system has significant advantages over the original TBA formulation and it has led to many exact results both at strong and weak coupling. So far, only two examples in this special class of nonlinear matrix Riemann-Hilbert problems have been discussed in the literature and not much is known on the relationship between these systems and other, more standard, approaches to integrability. Therefore, finding other examples would be highly desirable. The original part of the thesis contains the derivation of the Pμ-system for a model of considerable physical interest on its own: the Hubbard model describing the transport properties of electrons in one-dimensional alloys. There are many similarities between the equations for the spectrum of the Hubbard model and those for anomalous dimensions in AdS/CFT and a detailed study of this simpler system may shed light on many open issues and unexplored features of the P μ-systems related to gauge theory.