Bouncing singularities and thermal correlators on line defects

Abstract

Thermal correlators in holographic conformal field theories are known to exhibit singularities in complex time, sometimes referred to as ``bouncing singularities", which are believed to be related to bulk geodesics probing the black hole interior. These singularities correspond to exponentially suppressed contributions in the high-frequency limit of the thermal correlators. We revisit in detail the calculation of retarded two-point functions of local operators dual to bulk scalar fields in the planar AdS black hole background. We confirm that these correlators develop bouncing singularities, and highlight the agreement of two independent methods: a large frequency WKB analysis with infalling boundary conditions at the horizon; and an asymptotic OPE analysis that relies only on the near-boundary expansion, without any direct input from the black hole interior. We then extend these calculations to the case of the retarded two-point function of displacement operators on a Wilson line in the finite temperature gauge theory. This is computed holographically by solving the wave equation for the transverse fluctuations of the dual string worldsheet in the planar AdS black hole background. We find that these defect correlators also exhibit bouncing singularities, and again observe exact agreement between the WKB analysis sensitive to the black hole interior and the asymptotic OPE analysis. This agreement suggests that the bouncing singularities and the corresponding OPE data encode a universal high-frequency structure of the retarded correlators, and we propose a factorization formula that encodes the deviations from this universality.

Figures (7)

• Figure 1: A schematic figure to illustrate the regime of validity of near-horizon solution with infalling boundary condition and near-boundary solution. The overlap region is enclosed by dashed blue line, where the matching procedure is performed.
• Figure 2: We choose the steepest descent contour that begins at $r=-ir_h$ and analytically continues to the real axis, reaching $r\to\infty$ after crossing the horizon below $r=r_h-i0$.
• Figure 3: The real time contour (blue) defining the Fourier transform of the retarded correlator $G_R(t)$ can be deformed into the purple contour: the segment along the imaginary axis implements the canonical Borel resummation, while the portions encircling the branch cuts generate the nonperturbative corrections.
• Figure 4: A schematic figure illustrating the procedure for constructing the steepest descent contour solution that is regular at $r=-i r_h$. A sequence of matching steps is performed to impose the regularity condition, connect to the WKB phase, and ultimately match to the near-boundary solution.
• Figure 5: The convergence of $\widehat{a}_{4n}^{Rt}/\widehat{a}_{4n}^{Rt0}$ as $n$ increases, where $\widehat{a}_{4n}^{Rt}$ is obtained numerically. Comparisons are shown for several values of $\Delta$.