Holographic thermal correlators: A tale of Fuchsian ODEs and integration contours

TL;DR

This paper studies real-time thermal correlators in holographic theories using the grSK geometry, analyzing Fuchsian ODEs and singularities to understand physical features and limitations of the approach.

Contribution

It provides a detailed analysis of linearized wave equations in AdS black holes, clarifies the role of apparent singularities, and demonstrates the importance of the grSK geometry in evaluating thermal correlators.

Findings

Apparent singularities do not affect physical higher-point functions.

Schwinger-Keldysh and KMS conditions hold despite singularities.

Energy density operator lacks exponentially growing modes above critical charge.

Abstract

We analyze real-time thermal correlation functions of conserved currents in holographic field theories using the grSK geometry, which provides a contour prescription for their evaluation. We demonstrate its efficacy, arguing that there are situations involving components of conserved currents, or derivative interactions, where such a prescription is, in fact, essential. To this end, we first undertake a careful analysis of the linearized wave equations in AdS black hole backgrounds and identify the ramification points of the solutions as a function of (complexified) frequency and momentum. All the equations we study are Fuchsian with only regular singular points that for the most part are associated with the geometric features of the background. Special features, e.g., the appearance of apparent singular points at the horizon, whence outgoing solutions end up being analytic, arise at higher codimension loci in parameter space. Using the grSK geometry, we demonstrate that these apparent singularities do not correspond to any interesting physical features in higher-point functions. We also argue that the Schwinger-Keldysh collapse and KMS conditions, implemented by the grSK geometry, continue to hold even in the presence of such singularities. For charged black holes above a critical charge, the energy density operator does not possess an exponentially growing mode, associated with `pole-skipping' (from one such apparent singularity). Our analysis suggests that the connection between the scrambling physics of black holes and energy transport has, at best, a limited domain of validity.