Analytic structure of holographic thermal correlators from Fourier series

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 $0<\mathfrak{Re}(τ)<β$. Expanding in $τ$ 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.

Figures (4)

• Figure 1: Absence of non-perturbative contributions in the $1/n$ asymptotic expansion for $\widetilde{G}_E(\zeta_n, k)$\ref{['asymptoticansatz']} when $n = 100$. The black points show the residual of the perturbative sector truncated to order $N$, while the black dashed line shows the expected scale of non-perturbative terms, which are clearly absent. This shows all transseries parameters $\sigma_j$ vanish and hence the exact result is given by the Borel resummation of the perturbative sector only.
• Figure 2: Analytic structure of $f_0(z)$ explored using Padé approximants. Black points show poles of the diagonal Padé approximant for 800 terms in the Fourier series of the holographic calculation at $\Delta = 11/4$, in the $z$-plane (upper panel) and the $\tau$-plane (lower panel). The results are consistent with analyticity of $f_0(z)$ for $z\in \mathbb{C}\setminus [1,\infty[$, equivalent to analyticity of $G_E(\tau, k=0)$ in the strip $0 < \mathfrak{Re}(\tau) < \beta$. The red crosses correspond to 'bouncing singularity' locations $z_\pm = -e^{\pm \pi}$, i.e. \ref{['bouncingtau']}, where the function appears regular.
• Figure 3: The near-singularity behaviour of the Euclidean correlator, using OPE coefficients computed from asymptotic formulae (red dash) and the Padé approximant of 2000 terms in the Fourier series (black solid). We work at $\Delta = 11/4$. Upper panel: Here we show the leading singular behaviour at $\tau = 0$, and the behaviour \ref{['CoincindentPointSingularity']}. Lower panel: After subtracting the leading singular behaviour the next OPE coefficient is $b_0(0)$ corresponding to the leading double-trace sector term, which is constant in $\tau$. For small enough $\tau$ there is a residual corresponding to the breakdown of the fine cancellation of the leading singular behaviour, but this improves with more terms used in the Fourier series.
• Figure 4: The leading double-trace sector coefficient in the OPE expansion \ref{['OPEintro']} at $k=0$, $b_0(0)$, for a range of $\Delta$. The black curve is the coefficient computed from the Borel resummation of the asymptotic expansion of the Fourier series coefficients. The red dots are the same quantities but computed from the Padé approximant of the Fourier series, subtracting the leading singularity, and fitting the remaining constant behaviour (a procedure shown in figure \ref{['fig:subtraction']} for the single case of $\Delta = 11/4$).