Regular and Floquet bases for gauge and gravity theories: a non perturbative approach

The main topic of the paper is represented by the change of basis, in Heun-type equations, from the one of decaying (at two singular points) solutions to that of Floquet solutions. Crucial in the connection relations is the phase acquired by the Floquet solutions by going from a (ir)regular singularity to another. The new 'kink method' is exploited to compute the quantum momentum of the Floquet solutions as convergent series, explicitly at all orders. Hence, upon integrating it term by term, the acquired phase can be derived explicitly as a similar series. Since Heun equation and its confluences describe ${\cal N}=2$ SYM theories in the NS background as quantisation of Seiberg-Witten differentials and also appear in perturbations of gravity solutions, tests and predictions in both domains can be made. A very encouraging test is the matching of the acquired phase with the dual gauge period, $A_D$, as given by the Nekrasov instanton function. Actually, the procedure can be considered an alternative to instanton computations. On the gravity side, results for the wave functions are at leading order compatible with analogous expressions found by studying the Teukolsky (or others, like Regge-Wheeler) equation. Besides, the whole construction can represent an useful approach to these equations at all orders, shedding also light on non-perturbative contributions, which should reveal very interesting for the consequences in gravity perturbations.