Holographic thermal correlators and quasinormal modes from semiclassical Virasoro blocks

TL;DR

The work develops an exact, semiclassical Virasoro-block framework to solve black-hole perturbation problems in AdS5, recasting QNM spectra and holographic 4d thermal two-point functions as connection problems for Heun-type opers. By exploiting degenerate Virasoro blocks and the locality of fusion transformations, the authors obtain universal s- and t-channel expressions for QNMs and correlators, with the Zamolodchikov relation linking accessory parameters to semiclassical blocks. They further connect these results to Seiberg-Witten prepotentials in a WKB regime, offering a bridge between holographic thermal physics and gauge-theory data, including near-extremal and large-momentum limits. The approach extends to charged backgrounds, five-punctured geometries, and energy-momentum/tensor correlators, and previews extensions to higher-point functions and logarithmic CFTs, highlighting a unifying structure underlying holographic real-time dynamics.

Abstract

Motivated by its relevance for thermal correlators in strongly coupled holographic CFTs, we refine and further develop a recent exact analytic approach to black hole perturbation problem, based on the semiclassical Virasoro blocks, or equivalently via AGT relation, the Nekrasov partition functions in the Nekrasov-Shatashvili limit. Focusing on asymptotically $\text{AdS}_5$ black hole backgrounds, we derive new universal exact expressions for holographic thermal two-point functions, both for scalar operators and conserved currents. Relatedly, we also obtain exact quantization conditions of the associated quasinormal modes (QNMs). Our expressions for the holographic $\text{CFT}_4$ closely resemble the well-known results for 2d thermal CFTs on $\mathbb{R}^{1,1}$. This structural similarity stems from the locality of fusion transformation for Virasoro blocks. We provide numerical checks of our quantization conditions for QNMs. Additionally, we discuss the application of our results to understand specific physical properties of QNMs, including their near-extremal and asymptotic limits. The latter is related to a certain large-momentum regime of semiclassical Virasoro blocks dual to Seiberg-Witten prepotentials.

Figures (3)

• Figure 1: The complex $r$ plane, with the locations of the two boundaries and the horizon marked. The grSK contour is a codimension-1 surface in this plane (drawn at fixed $v$). The direction of the contour is as indicated counter-clockwise, encircling the branch point at the horizon.
• Figure 2: Comparison between QNM expansion and numerics for the first two QNMs ($\mathfrak{w}_0, \mathfrak{w}_1$) at $\mathfrak{q}=3$ for Klein-Gordon scalar with $\Delta=4$ (massless scalar perturbation), and designer scalars with $\mathscr{M}=-1$ (scalar polarization of gauge field perturbation) and $\mathscr{M} = -3$ (vector polarization of metric perturbation). Here $k_{max}$ is the order of truncation in the cross-ratio expansion \ref{['eq-QNM-expansion']}. The $\mathfrak{R}(\mathfrak{w})\geq0$ part of the QNM spectrum is shown. This data again supports \ref{['claim:QNM-log-quant']} that the quantization condition \ref{['eq-QNM-quantization']} continues to holds for $\theta_\mathrm{bdy} \in \mathbb{Z} /2$.
• Figure 3: Numerically computed QNM spectrum for massless uncharged scalar at zero momentum in near-extremal planar black hole background with $Q/Q_{\mathrm{ext}} = .9$. The purely decaying modes are well-described by \ref{['eq:QNMNearExt']} obtained via OPE analysis, equally spaced with gap $\sim Q_{\mathrm{ext}} - Q \sim T$. The oscillation/Christmas-tree-type modes at finite $\mathfrak{w}$ are not visible in the OPE analysis; see main text for more explanations. We also find good agreement between \ref{['eq:QNMNearExt']} and numerics for purely decaying modes in near-extremal limit at more generic masses and momenta. The numerical spectra were computed using the numerical package of Jansen:2017oag.

Theorems & Definitions (7)

• Remark 3.1
• Remark 3.2
• Claim 4.1
• Claim 4.2
• Remark 4.1
• Remark 4.2
• Remark 4.3