Analytic structure of holographic thermal correlators from Fourier series

TL;DR

This paper develops a Fourier series approach to compute holographic thermal correlators, revealing their distributional convergence and extracting OPE coefficients, including double-trace sectors, with insights into non-perturbative structures.

Contribution

It introduces a Fourier series method for holographic thermal correlators that captures distributional convergence and directly yields OPE coefficients, including double-trace data.

Findings

Fourier series converges as a distribution, not a function.

All OPE coefficients, including double-trace sectors, are obtained.

Bouncing singularities are identified as non-perturbative sectors with zero transseries parameters.

Abstract

We compute the holographic Euclidean two-point function of scalar operators in a thermal state. We work directly using the Fourier series on the thermal circle. The Fourier series does not converge as a function, but instead converges as a distribution, consistent with QFT expectations. The result is manifestly periodic and consistent with analyticity in the strip $0<\mathfrak{Re}(\tau)<\beta$. Expanding in $\tau$ we obtain all OPE coefficients, including the double-trace sector. Thus our approach has an advantage compared to recent work where double-traces were bootstrapped from stress-tensor data. Bouncing singularities appear as non-perturbative sectors in the transseries for Fourier coefficients, but their transseries parameters are all zero in the case of the Euclidean correlator.