Speaker
Description
We study the linear problems in $z,t$ ($t$ the time) associated to the Painlev\'e III$_3$, III$_1$ and V and VI equations when the Painlevé solution $q(t)$ approaches a pole or a zero. In this limit the problem in $z$ for the Painlev\'e III$_3$ reduces to the modified Mathieu equation, that for the Painlevé III$_1$ to the Doubly Confluent Heun Equation and the ones for the Painlevé V and VI to the Confluent Heun Equation and to the Heun equation, respectively. These equations are Nekrasov-Shatashvili quantisations/deformations of Seiberg-Witten differentials for $SU(2)$ ${\cal N}=2$ super Yang-Mills gauge theory with number of multiplets $N_f=0,2,3,4$, respectively. On the gravity side, the Heun equation is the form to which (angular and radial) Teukolsky equations for Kerr
Newman-de Sitter geometries can be reduced, while the confluent Heun equation is the form of the Teukolsky equation for a Schwarzschild geometry.
In this talk I will discuss the problem of computing quantities - related to Heun equations - that can be relevant for both SUSY theories and classical gravity. From the technical point of view, one will benefit from techniques imported from quantum integrable models, in specific the 'kink method' of TBA.
This talk is based on
D. Fioravanti, M. Rossi,
From Painlevé equations to ${\cal N}=2$ susy gauge theories: prolegomena,
arXiv: 2412.21148
