Holographic Correlators at Finite Temperature

TL;DR

<3-5 sentence high-level summary>This work develops a controlled construction of holographic thermal two-point functions for a scalar operator in AdS/CFT at finite temperature, using a finite-temperature extension of the HPPS program. The authors fix leading bulk quartic corrections with arbitrary derivatives by enforcing analyticity, OPE compatibility, and the KMS condition, and they show these corrections can be organized into a meromorphic thermal Mellin-like amplitude M_{\beta}(s) acting on mean-field theory data; for derivatives they introduce a differential operator framework to generate higher-spin contributions. They validate the framework by matching the simplest (L=0) case to an explicit thermal AdS Witten diagram and derive dispersion relations that reconstruct the correlator from its discontinuity, connecting to the thermal Lorentzian inversion formula. The results provide a new, infinite family of solvable thermal correlators and suggest a thermal Mellin amplitude program applicable to finite-temperature holography, with promising avenues toward bulk exchange, black hole backgrounds, and a deeper understanding of uniqueness and analyticity at finite temperature.

Abstract

We consider weakly-coupled QFT in AdS at finite temperature. We compute the holographic thermal two-point function of scalar operators in the boundary theory. We present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without any derivatives we show agreement with an explicit diagrammatic computation. The structure of the answer is suggestive of a thermal Mellin amplitude. Additionally, we derive a simple dispersion relation for thermal two-point functions which reconstructs the function from its discontinuity.

Figures (2)

• Figure 1: The thermal AdS Witten diagram for the leading order correction to the $\langle\phi(x_1)\phi(x_2)\rangle_\beta$ thermal correlator from a bulk $\lambda_0 \Phi^4$ interaction. The interaction point (red) is integrated over thermal AdS. The gray circle in the center is meant to remind us that there is a periodic direction in the bulk, and propagators (blue) can wind around that direction. In computing the diagram, one sums over these thermal images of the endpoints of propagators.
• Figure 2: Analytic structure of the integrand of (\ref{['eq:dispersion starting relation']}) in the complex $w$-plane. The thermal two-point function $g(r/w,r w)$ has OPE cuts starting at $\pm r$ and $\pm 1/r$, where $r = \sqrt{z \overline{z}}$. The kernel $K(z,\overline{z}; w)$ has poles at $w=0, \pm \sqrt{z/\overline{z}}$. The initial contour $\mathcal{C}_0$ wraps the poles at $w = \pm \sqrt{z/\overline{z}}$ as pictured above. Deforming the contour, one obtains the contour along the cuts as depicted below the arrow, yielding an integral over the discontinuities of $g$.