Isomonodromy, Painlevé transcendents and scattering off of black holes

Abstract

Abstract:We apply the method of isomonodromy to study the scattering of a generic Kerr-NUT-(A)dS black hole. For generic values of the charges, the problem is related to the connection problem of the Painlevé VI transcendent. We review a few facts about Painlevé VI, Garnier systems and the Hamiltonian structure of flat connections in the Riemann sphere. We then outline a method for computing the scattering amplitudes based on Hamilton-Jacobi structure of Painlevé, and discuss the implications of the generic result to black hole complementarity.

Warped symmetries of the Kerr black hole

“…The monodromy parameters α ± connect in an elegant manner in the scattering coefficients to the analytic properties of the geometry, see e.g. [48,49,52,53]. Their use here will be as energies eigenstates that will motivate our choice of vector fields in Sec.…”

On the Kerr-AdS/CFT correspondence

“…the relation between scattering coefficients of some Kerr-de Sitter [10,11] black holes in four dimensions to exact c = 1 conformal blocks using deformations of the flat holomorphic connection -isomonodromy deformations. These conformal blocks were in itself computed exactly [12] in terms of the Painlevé VI τ -function, capitalizing on recent developments in gauge/gravity correspondence in the form of the Alday-Gaiotto-Tachikawa (AGT) conjecture [13,14].…”

“…At the same time, confinement and asymptotic freedom are two aspects of one single phenomenon of conformal symmetry breaking. The latter may be illustrated by a naive geometric "hard wall" construction, for which the discrete would-be meson mass spectrum is given by 10) where Λ = 1/z 0 is an infrared cutoff and x ν,n is the n-th zero of the regular Bessel function J ν (x), has nevertheless phenomenologically appealing features and could still serve as a description of the deconfined phase of QCD, in the presence of mass (temperature) perturbations. In other words, the successes in describing quark-gluon plasma resonances…”

“…Details can be found in [10,11], and we will present here a summary of the ideas. Take a generic Fuchsian second order ordinary differential equation (ODE),…”

Isomonodromic approach to scattering by black holes

“…Being more explicit, we start with a Kerr-NUT-(A)dS black hole in D = 4 dimensions and study linear conformally coupled scalar perturbations around this class of black holes. Two important facts in this setup are well-known: (1) The equations for linear perturbations of Kerr-NUT-(A)dS black holes with spin 0, 1 and 2, also called Teukolsky master equations, are separable for any dimension 6-8 ; (2) The D = 4 radial equation for a conformally coupled field is reducible to a Heun equation (4 regular singular points) 1,6 . Thus, our scattering problem reduces to the study of the connection coefficients of an ordinary differential equation with 4 regular singular points, the Heun equation,…”