Analytic Computation of Dilaton Black Hole Quasinormal Modes via Seiberg-Witten Theory

TL;DR

This work addresses the problem of determining quasinormal modes of dilaton black holes in Einstein-Maxwell-dilaton gravity by establishing a gauge-theoretic correspondence with the quantum Seiberg-Witten curve for SU(2) with $N_f=3$. By casting both the black hole perturbation equation and the SW curve into the normal form of the confluent Heun equation, the QNM spectrum is extracted from a Seiberg-Witten quantization condition involving the Nekrasov-Shatashvili free energy. The authors demonstrate excellent agreement with traditional semi-analytic methods ($<10^{-3}$ relative error) and reveal clear parametric trends in the spectrum, highlighting a deep link between supersymmetric gauge theory and gravitational dynamics. This framework offers a new analytic lens on black hole perturbations and suggests rich future directions, including extensions to other flavors, perturbations, and brane contexts, via nonperturbative gauge-theoretic data.

Abstract

We study the quasinormal modes (QNMs) of dilaton black holes in Einstein-Maxwell-dilaton gravity through a correspondence with the quantum Seiberg-Witten (SW) curve of $\mathcal{N}=2$ SU(2) gauge theory with $N_f=3$ hypermultiplets. By mapping both the black hole perturbation equation and the quantum SW curve to the confluent Heun form, the QNM problem is reformulated in a gauge-theoretic framework, and the spectrum is obtained via the SW quantization condition. The resulting frequencies show excellent agreement with those computed using the WKB and continued fraction methods, with typical deviations below $10^{-3}$. The QNM spectrum exhibits consistent trends: increasing the black hole charge or scalar field mass raises the oscillation frequency, while higher angular momentum reduces the damping rate. These results demonstrate the precision of the quantum SW framework in describing black hole perturbations and reveal new links between supersymmetric gauge theories and gravitational dynamics.