Analytic thermal bootstrap meets holography

TL;DR

The paper develops an analytic thermal bootstrap for holographic two-point functions of identical scalar operators with integer scaling dimension in AdS black-brane and spherical black-hole backgrounds. By marrying the OPE, KMS condition, and a GFF-based dispersion, it decomposes thermal holographic correlators into a principal, regularized, and arc structure, with the principal part captured exactly by a finite set of multi-stress-tensor data and the tail and non-perturbative horizon effects encoded in the regressed and arc terms. In four dimensions, explicit results for Δφ = 1–5 are provided, including exact principal contributions and asymptotic regularized pieces, with numerical checks showing ~1%–2% agreement for Δφ = 3 against bulk-wave solutions. A bulk interpretation via a graviton-expansion around thermal AdS is established, and Witten-diagram computations validate the principal and regularized pieces at low orders, while higher-curvature corrections are shown to modify the OPE data without altering the overall framework. The approach extends to finite-volume setups and suggests a clear link between horizon physics and arc contributions, offering a new analytic handle on finite-temperature holographic CFTs and guiding future investigations into non-perturbative effects and $1/N$ corrections.

Abstract

We compute thermal holographic correlators by combining their analytic structure with the Kubo-Martin-Schwinger (KMS) condition and multi-stress tensor OPE coefficients determined from the dual AdS description. We focus on two-point functions of identical scalar operators with integer conformal dimensions at zero spatial separation. In the black brane background, we show explicitly that holographic two-point functions split into three contributions: a principal one, computed exactly, plus regularized and arcs contributions, both approximated through the use of OPE coefficients asymptotics. For $Δ_φ=3$, we show that the principal contribution agree with good approximation with the numerical solution of the bulk wave equation. Moreover, we demonstrate that the expansion in generalized free field correlators proposed in [Barrat,6/2025] admits a natural interpretation in terms of Witten diagrams. Finally, we initiate the study of thermal correlators in the spherically symmetric black hole background, computing their principal contributions.

Figures (10)

• Figure 1: Depiction of the setup considered in this paper. Left: In the AdS/CFT framework, we study correlation functions of two scalar operators in a finite temperature CFT dual to a black hole background. The connecting line represents the propagator, which is non-trivial due to the presence of the horizon. Right: We focus on the case where both operators lie on the same thermal circle, corresponding to zero spatial separation.
• Figure 2: Analytic structure of two-point functions at finite temperature and infinite volume in the complex $\tau$-plane for integer spectrum, for the case in which the OPE coefficients do not contain poles. In this case the correlator contains poles at the locations $\beta k$, $k \in \mathbb{Z}$, along the real axis which corresponds to euclidean time. The corresponding correlator is then given by the sum over the residues. In the holographic case this corresponds to the dominant terms in the OPE.
• Figure 3: The regularized part of the correlator, $g_{\text{reg}}(\tau)$, has additional poles in the complex $\tau$-plane called bouncing singularities. The analytic structure of the regularized part therefore consists of KMS poles (colored red) and bouncing singularities (colored orange). The latter are removed from the full correlator by adding the arc contributions.
• Figure 4: Analysis of the boundedness conditions \ref{['eq:boundess']}--\ref{['eq:boundess2']} for the principal contributions and $1 \leq \Delta_\phi \leq 5$. Left: Absolute value of $g_\text{pr}(\tau)$ evaluated along the line $\tau = 1/2 + i \eta$. They are consistently bounded by their values on the real axis. Right: Absolute value of $g_\text{pr}(\tau)$ for the same range of $\Delta_\phi$, evaluated along the line $\tau = \eta +i/2$. They are consistently bounded by their values on the imaginary axis. Both panels use log--log scaling with $\beta=1$ without loss of generality. Note that these terms obey the KMS condition by construction as we vary the real part of $\tau$ and this generates bumps in the log-log scale.
• Figure 5: Analysis of the boundedness conditions \ref{['eq:boundess']}--\ref{['eq:boundess2']} for the approximated correlator with $\Delta_\phi=3$. Left: Absolute value of $g(\tau)$ evaluated along the line $\tau = 1/2 + i \eta$. The correlator is consistently bounded by its values on the real axis. Right: Absolute value of $g(\tau)$ evaluated along the line $\tau = \eta +i/2$. The correlator is not bounded by its values on the imaginary axis. We associate this inconsistency with the fact that we use the asymptotic value of the OPE coefficients instead of their exact one. Both panels use log--log scaling with $\beta=1$ without loss of generality.