The Bulk-to-Boundary Propagator in Black Hole Microstate Backgrounds

Abstract

First-quantized propagation in quantum gravitational AdS$_3$ backgrounds can be exactly reconstructed using CFT$_2$ data and Virasoro symmetry. We develop methods to compute the bulk-to-boundary propagator in a black hole microstate, $\langle φ_L \mathcal{O}_L \mathcal{O}_H \mathcal{O}_H\rangle$, at finite central charge. As a first application, we show that the semiclassical theory on the Euclidean BTZ solution sharply disagrees with the exact description, as expected based on the resolution of forbidden thermal singularities, though this effect may appear exponentially small for physical observers.

Figures (7)

• Figure 1: This figure depicts a Euclidean bulk-boundary correlator in a black hole microstate. Although we have forced the correlator to live on the Euclidean BTZ geometry, due to violations of the KMS condition the correlator will be multivalued on the Euclidean time circle, and so must have a branch cut. Thus semiclassical predictions for bulk correlators must breakdown. In particular, as the Euclidean time circle shrinks to vanishing size at the horizon, it would seem that exact bulk correlators must differ signficantly from their semiclassical limits at the Euclidean horizon.
• Figure 2: These plots compare the exact (blue, $\log(|\mathcal{V}_0^{\mathrm{exact}}|)$) and semiclassical (pink, $\log(|\mathcal{V}_0^{\mathrm{semi}}|)$) correlators for different values of $r$. The parameters for these plots are $c=30.1, h_L=0.505, \frac{h_H}{c}=4$, so that $r_+ \approx 9.7$. The semiclassical approximation is excellent for these values of $t_E$ and $r$. The gray dashed lines are $\pm \beta/2$. We used the exact result from recursion up to order $z^{60}\bar{z}^{60}$, with convergence $\left|\frac{\mathcal{V}_0^{\mathrm{exact}}(\text{60 orders})-\mathcal{V}_0^{\mathrm{exact}}(\text{59 orders})}{\mathcal{V}_0^{\mathrm{exact}}(\text{60 orders})}\right|<10^{-12}$.
• Figure 3: Left: This figure depicts a Euclidean bulk-boundary correlator $|\mathcal{V}_0^\mathrm{semi}|$ on the BTZ 'cigar' geometry, focusing on slices at fixed $r$, where we can easily study Euclidean time periodicity. Right: These plots display the semiclassical bulk-boundary correlator $\mathcal{V}_0^\mathrm{semi}$ on constant-$r$ slices. The semiclassical correlator is periodic in $t_E$, and its range of variation becomes smaller as we approach the horizon $r=r_+$, where it is constant in $t_E$. The red dashed line is $t_E=\beta$ and the parameters are $\frac{h_H}{c}=1, h_L=1$.
• Figure 4: The blue lines are the exact result $|\mathcal{V}_0^{\text{exact}}|$ and the yellow lines are the semiclassical $|\mathcal{V}_0^{\text{semi}}|$. From top to bottom the rows of plots correspond to $c=8.1, 16.1, 32.1, 64.1$, respectively. Other parameters for these plots are $h_L=0.01, \frac{h_H}{c}=100$, and $r_+ \approx 50$. The first two plots in each row are in the region whose distance from the horizon is much smaller than the AdS radius. The red dashed line is $t_E=\beta$ and the gray dashed line is $t_E=\pm\beta/2$. The exact results in the visible plot range have converged to better than $10^{-13}$ precision (the precision of convergence is defined as in figure \ref{['fig:SemiMatchesExact']}).
• Figure 5: This is a plot of $|\mathcal{V}_0^{\text{semi}}|$ zoomed in to the tip of the Euclidean 'cigar', with $r_+<r<1.025r_+$ and $0<t_E<\beta$. The radial coordinate of the disk is $r - r_+$ and the angular direction is $\frac{2 \pi}{\beta} t_E$; the BTZ angular coordinate $\theta = 0$. The center of the plot is the position of the Euclidean horizon and $r_+ \approx 49$.