AdS/CFT correspondence, quasinormal modes, and thermal correlators in N=4 SYM

TL;DR

This paper uses the Minkowski (Lorentzian) AdS/CFT prescription to compute the poles of retarded thermal Green's functions for operators dual to scalar, vector, and gravitational perturbations in the AdS–Schwarzschild background, thereby defining quasinormal modes in asymptotically AdS spaces. It derives a pragmatic boundary-value framework, solves the scalar sector via Heun equations and continued fractions (identifying true poles and 'false frequencies'), and obtains exact or highly accurate quasinormal spectra for massive scalars, R-currents, and stress-energy tensors. A key result is the emergence of hydrodynamic poles in vector and tensor channels at small momenta, together with an infinite tower of nonhydrodynamic modes, whose dispersion and structure echo the strongly coupled thermal dynamics of ${ m N}=4$ SYM at large $N$ and large 't Hooft coupling. The work highlights both the diagnostic power of holography for real-time thermal physics and the subtleties arising from AdS boundary conditions and the Heun equation, with implications for comparing strong- and weak-coupling regimes.

Abstract

We use the Lorentzian AdS/CFT prescription to find the poles of the retarded thermal Green's functions of ${\cal N=4}$ SU(N) SYM theory in the limit of large N and large 't Hooft coupling. In the process, we propose a natural definition for quasinormal modes in an asymptotically AdS spacetime, with boundary conditions dictated by the AdS/CFT correspondence. The corresponding frequencies determine the dispersion laws for the quasiparticle excitations in the dual finite-temperature gauge theory. Correlation functions of operators dual to massive scalar, vector and gravitational perturbations in a five-dimensional AdS-Schwarzschild background are considered. We find asymptotic formulas for quasinormal frequencies in the massive scalar and tensor cases, and an exact expression for vector perturbations. In the long-distance, low-frequency limit we recover results of the hydrodynamic approximation to thermal Yang-Mills theory.

Figures (14)

• Figure 1: Real part of the eigenfrequencies (solutions of the continued fraction equation (\ref{['seigen']})) at $\textswab{q}=0$ versus the conformal dimension $\Delta$. Black dots correspond to quasinormal frequencies at integer values of $\Delta$, while blank ellipses are the "false frequencies". The dashed line indicates that the sequence presumably continues to infinity.
• Figure 2: (Minus) imaginary part of the eigenfrequencies (solutions of the continued fraction equation (\ref{['seigen']})) at $\textswab{q}=0$ versus the conformal dimension $\Delta$. Black dots correspond to quasinormal frequencies at integer values of $\Delta$, while blank ellipses are the "false frequencies". The dashed line indicates that the sequence presumably continues to infinity.
• Figure 3: $\mathrm{Re}\, \textswab{w}$ of the lowest eight scalar quasinormal frequencies versus the conformal dimension $\Delta$. Dots correspond to integer conformal dimensions.
• Figure 4: -$\mathrm{Im}\, \textswab{w}$ of the lowest eight scalar quasinormal frequencies versus the conformal dimension $\Delta$. Dots correspond to integer conformal dimensions.
• Figure 5: $\mathrm{Re}\, \textswab{w}$ of the scalar fundamental quasinormal frequency vs $\textswab{q}$ for the (integer) conformal dimensions $\Delta \in [2,10]$. The lowest curve corresponds to $\Delta = 2$.