New Developments in N=2 Supersymmetric Gauge Theories: from Integrability to Black Holes

We explain how new physical results in 4D N = 2 supersymmetric gauge theories can be found by connecting them to 2D quantum integrable models. In particular, we set up an identification between the basic mathematical and physical objects of the two kinds of theories (the Q or Y and T functions of integrability and the two periods of the gauge theories) and then, we derive a stream of concepts and mathematical identities between them. Moreover, we use this new correspondence to prove, understand, and possibly generalise a recent application of gauge theories to black holes perturbation theory. From this, several new insights follow into black holes physics, especially a new powerful way of computing quasinormal mode frequencies (the Thermodynamic Bethe Ansatz nonlinear integral equation), characterising the gravitational wave signal (in the ringdown phase of black hole merging). For simplicity and limits of space, we restrict the discussion to the simplest case of the Liouville integrable model/pure SU(2) gauge theory/D3 brane gravitation background triad.