On quivers, spectral networks and black holes

TL;DR

This work develops a systematic framework to compute connection coefficients for higher-singularity Fuchsian ODEs by combining Liouville crossing symmetry, AGT, and the Nekrasov–Shatashvili ($F^{\rm NS}$) prepotential. It specializes to a five-punctured sphere, derives explicit connection formulas between Frobenius bases using NS blocks and crossing matrices $\mathcal{M}$, and verifies them numerically; the method is anchored in a linear quiver gauge theory realization and Matone relations for moduli. The formalism is then applied to black hole perturbations in AdS backgrounds (Schwarzschild–AdS$_7$ and RN–AdS$_5$), yielding quasinormal mode spectra and AdS/CFT correlators, with large-spin expansions connected to light-cone bootstrap data. Finally, spectral networks are used to diagnose strong-coupling regions in the Seiberg–Witten moduli space relevant to these gravitational problems, indicating potential GMN–TBA techniques for gravity in regimes where instanton parameters are not parametrically small.

Abstract

It was recently found that connection coefficients of the Heun equation can be derived in closed form using crossing symmetry in two-dimensional Liouville theory via the Nekrasov-Shatashvili functions. In this work, we systematize this approach to second-order linear ODEs of Fuchsian type, which arise in the description of N = 2, four-dimensional quiver gauge theories. After presenting the general procedure, we focus on the specific case of Fuchsian equations with five regular singularities and present some applications to black hole perturbation theory. First, we consider a massive scalar perturbation of the Schwarzschild black hole in AdS7. Next, we analyze vector type perturbations of the Reissner-Nordström-AdS5 black hole. We also discuss the implications of our results in the context of the AdS/CFT correspondence and present explicit results in the large spin limit, where we make connection with the light-cone bootstrap. Furthermore, using the spectral network technology, we identify the region of the moduli space in Seiberg-Witten theory that is relevant for the study of black hole quasinormal modes. Our results suggest that, in some cases, this region corresponds to the strong-coupling regime, highlighting the potential applicability of the conformal GMN TBA framework to address scenarios where the gravitational dictionary implies that the instanton counting parameters are not parametrically small.

Figures (4)

• Figure 1: Sphere decomposition.
• Figure 2: Spectral networks for extremal Kerr black hole with parameters in table 5 of Aminov:2020yma. The BH quantum numbers are $l=s=m=0$. On the left we set $\vartheta=0$ while on the right we set $\vartheta=2.46091$. In this case we see two degenerate walls, one connecting two top branch points and the other connecting two bottom branch points.
• Figure 3: Spectral networks for Schwarzschild black hole corresponding to $N_f=3$ theory with parameters in Aminov:2020yma at different quasinormal mode frequencies in \ref{['Schfre0']}-\ref{['Schfre100']}. We observe that as the quantum number increases, the SN topology asymptotically approaches that of a hypermultiplet.
• Figure 4: The first 4 quasinormal modes corresponding to $\Delta=2, \mathfrak{q}=0$ in the table 1 of Nunez:2003eq.