Integrability structures in SU(2) supersymmetric gauge theory with four flavours and AdS black holes

In this thesis we study through the Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence the deformed Seiberg-Witten curve of four flavour N = 2 SU(2) supersymmetric Yang-Mills theory. This theory, compared with the ones with fewer flavours of quarks, is very different as there is no dinamically generated scale and thus the theory is asymptotically conformal. At the ODE level this is manifested in the absence of irregular singular points in the equation and thus no Stokes phenomenon is present. While in the eyes of the AGT correspondence this is the most natural setting, for the ODE/IM correspondence this leads to the absence of the transfer matrices of integrability. However we introduce the monodromy group of the ODE into the problem and show its equivalence with the system of Q-functions. We stress how the gauge theory prepotential should be seen as the generating function of a canonical transformation on the space of monodromy data which is endowed with a symplectic structure. Using this fact we achieve a definition of the prepotential directly from the differential equation, using the Matone relation, without requiring, in principle, the usage, but only the matching with gauge theory. The same differential equation shows up as the equation describing a scalar perturbation in the five-dimensional AdS-Schwarzschild background. The applications of this is two-fold: first integrability could be used to derive Bethe ansatz equation for the quasi-normal modes of AdS black holes. Second this gravity theory is the holographic dual of four-dimensional N = 4 SYM and the connection coefficients for the ODE can then be connected to an exact expression for the two-point function at finite temperature in this theory