Heun equation and its applications in ADS black hole physics

Abstract

The study of black hole perturbations in anti-de Sitter (AdS) spacetime has gained a renewed interest due to the application of analytical techniques inspired by four - dimensional N = 2 supersymmetric gauge theory. In particular, such techniques have illuminated deep connections between black hole observables (quasinormal modes, holographic thermal correlators, greybody factors, etc.) and the mathematical structures like Heun di!erential equation and classical conformal blocks in two dimensional Liouville conformal field theory (CFT). These developments have facilitated the exact analytical computation of physical observables within black hole framework. This thesis aims to derive exact analytical expressions for the holographic retarded Green’s functions and the greybody factors considering scalar and vector perturbations in various AdS5 black hole geometries. One of the crucial steps involves reformulating the radial part of perturbation equations as Heun equation, whose connection formulae among it’s solutions (Heun functions) can be deduced from the crossing symmetry of conformal blocks in Liouville CFT, facilitated by Alday-Gaiotto-Tachikawa (AGT) correspondence. This correspondence maps classical Liouville conformal blocks to the Nekrasov-Shatashvili (NS) limit of instanton partition function, which can be computed in terms of the combinatorial expressions, in four dimensional N = 2 supersymmetric gauge theory. We apply this methodology to compute exact analytical expressions of the holographic thermal correlators for a scalar operator with conformal dimension ” in thermal CFTs and its counterpart with chemical potential in R1→H3, by solving scalar perturbations in the dual AdS hyperbolic black hole geometries and its charged version. Furthermore, we extend our analysis to Kerr-AdS5 black hole and Reissner-Nordstrom-AdS5 black holes by considering scalar perturbations and we compute holographic retarded Green’s functions and the greybody factors. Moreover, to cure the pole appearing in the nontrivial NS partition function, we propose a resummation techniques based on Catalan numbers, leading to closed form expression to the correction terms in the holographic retarded Green’s functions and in the greybody factors under suitable assumptions. Finally, we investigate the greybody factors for scalar and Maxwell field perturbations in AdS5 black brane background. The exact results obtained in this thesis provide new insights into the analytical structure of holographic thermal correlators and the greybody factor in AdS black hole spacetimes. These findings deepen our understanding of strongly coupled quantum field theories via holography and highlights the rich interplay between black hole physics and the supersymmetric gauge theory by virtue of AGT correspondence.