Some integrability results for wronskians in gravity and supersymmetric gauge theories

Abstract In this thesis we study the Confluent Heun equation (CHE) with quantum integrability methods, with particular focus on the relevant wronskians solving it’s monodromy problem. It is known that CHE is in correspondence with the quantization of Seiberg-Witten (SW) differential for N = 2 Super-Yang-Mills (SYM) with number of flavours Nf = 3 in the Nekrasov-Shatashvili (NS) background. Besides, the same equation is crucial in black hole perturbation of Schwarzschild and Kerr geometry theory. In this context, the wronskians between the Floquet functions are computed in a new way, for both Nf = 0 and Nf = 3 theories and compared with results in the literature. By using them, we compute the wronskians between regular solutions, as well, which are their connection coefficients and play the role of the so called Q-functions in quantum integrability. To reach this goal, we built a new algorithmic way to compute the quantum momentum of the Floquet and regular solutions by means of some special polynomials that can be computed in a recursive way and enjoy interesting properties.