Bulkcone Singularities and Complex Geodesics

Abstract

Thermal correlators in holographic CFTs on a sphere exhibit bulk-cone singularities at points connected by null geodesics in the bulk. The operator product expansion analysis of the stress-tensor sector of the correlator shows that there are analogous singularities at spacelike separation for thermal CFTs on a plane. We show that these are associated with complex null geodesics. There is a phase transition between the real and complex spacelike geodesics underpinning this picture. We also provide a phase-shift calculation of the position of these generalised bulk-cone singularities.

Figures (10)

• Figure 1: Log-log plots of the ratio of successive coefficients $a^n_{0, 2n}$ for various values of $\Delta$, alongside the linear fits of the data for large $n$ following (\ref{['eq:t_0_asymp_coeff']}). Notice that the $y$-intercept, given by $\log{B}$, appears independent of $\Delta$. We see that the fit to (\ref{['eq:t_0_asymp_coeff']}) is precise, with the fit parameters for various $\Delta$ displaying standard deviations of $\mathcal{O}(10^{-5})$. For smaller $n$, corresponding to more negative values of $\log{(n/n+1)}$, the data ceases to fit (\ref{['eq:t_0_asymp_coeff']}) as expected.
• Figure 2: Plot of the position of singularity $|\vec{x}|$ as a function of $t$, as predicted by (\ref{['eq:corr_sing_t_val']}) for $\Delta = 7/3$, alongside the lightcone $|\vec{x}| = t$. Gridlines are shown for $t = \pm \frac{\pi}{2}, \pm \pi$ and $|\vec{x}| = \frac{\sqrt{\pi}}{2}\frac{\Gamma(1/4)}{\Gamma(3/4)}, 3.37$. The upper $|\vec{x}|$ gridline is the singularity position in $|\vec{x}|$ when $t=0$. The gridlines show that the singularity curve is symmetric about $t = \frac{\pi}{2}$. The singularity curve asymptotes from above to the lightcone at large $t$. Values of $|t| < 0.2$ are excluded due to a loss in accuracy for small $t$, see appendix \ref{['app_small_t_coeffs']} for further details.
• Figure 3: Log-Log plot of the ratio of successive coefficients $\Lambda_n$ for $t=\pi/2$, for various values of $\Delta$. Shown also are the linear fits for large $n$.
• Figure 4: Effective potential for null geodesics, shown for $b=1/2$ (blue) and $b=1$ (orange). We have set $|P| = 1/\sqrt{2}$.
• Figure 5: Spatial displacement for null geodesic $|\vec{X}|(b)$ as a function of the impact parameter $b$.