Scalar quasinormal modes of Kerr--AdS$_{\bf 7}$ via accessory parameter expansions

Abstract

We study scalar perturbations of a seven-dimensional Kerr--AdS black hole with three independent rotation parameters. In this geometry, the Klein--Gordon equation separates into one radial and two angular second-order linear ordinary differential equations, each with five regular singular points. We apply the Hill determinant method to compute analytic expansions of the accessory parameters in two different parameter regimes. The resulting accessory parameter expansions coincide with those obtained from the instanton part of the Nekrasov--Shatashvili function of four-dimensional $\mathcal{N}=2$ quiver gauge theories. These expansions are then used to derive perturbative expressions for the angular eigenvalues in the slowly rotating limit and for the quasinormal mode frequencies in the small black hole limit.