Foliation-dependence of CFTs in Lorentzian-AdS/CFT

In the Lorentzian $\mathrm{AdS}/\mathrm{CFT}$ correspondence, CFTs are identified by asymptotic boundary surfaces and the boundary conditions imposed on those surfaces. However, AdS can be foliated in various ways to give different boundaries. We show that the CFTs obtained using certain distinct foliations are different. This happens because the asymptotic region of a foliation overlaps with the deep interior region of another. In particular, we focus on the CFTs defined on surfaces of large constant radius in global coordinates, Rindler-AdS coordinates, and Poincaré coordinates for ${\mathrm{AdS}}_3$. We refer to these as global-CFT, Rindler-CFT and Poincaré-CFT, respectively. We demonstrate that the correlators for these CFTs are different and argue that the bulk duals to these should agree up to very close to the respective horizons but then start differing. Since the BTZ black hole is obtained as a quotient of ${\mathrm{AdS}}_3$, we discuss the implications of our results for bulk duals of periodically identified Poincaré and Rindler-CFTs. Our results are consistent with some recent proposals suggesting a modification of the semiclassical BTZ geometry close to the horizons.

Figure 1

The boundary of AdS for global foliation is shown in gray. This is a surface of constant radius in global coordinates for large values of the radius. The yellow surfaces are surfaces of constant Rindler-AdS radius. It is easy to see that fixing boundary conditions on the global boundary imposes conditions on small Rindler-AdS radial surfaces also. Conversely, putting boundary conditions on just large Rindler-AdS radial surfaces does not put any boundary conditions on the global boundary outside a finite domain. Thus, the global-CFT and the Rindler-CFT are different and imply different dynamics for the bulk.

Figure 2

(a) The global cutoff surface $\rho ={\rho }_c$ is shown in red, and the Rindler cutoff surface $r_{\mathrm{R}}={r_{\mathrm{R}}}_c$ is shown in blue. (b) When we take ${\rho }_c$ to infinity, all the $r_{\mathrm{R}}$ surfaces bunch up along the edges of the “causal diamond.” Two of them are shown in the figure.

Figure 3

We open up the global boundary cylinder for better visualization. The colored regions are the causal diamonds and are the interior of the curve the bulk Rindler-AdS horizons trace on the boundary cylinder. The global boundary cylinder is taken to be large with $\rho \sim \mathcal{O}({\epsilon }^{-1})$ and $\epsilon \ll 1$. The region where the Rindler-AdS radial coordinate $r_{\mathrm{R}}\sim \mathcal{O}(1)$ is shown in blue. This is the region where the Fefferman-Graham expansions in $\rho$ and $r_{\mathrm{R}}$ are not consistent and the Rindler-CFT cannot be approximated by the global-CFT.

Figure 4

(a) Global cutoff surface $\rho ={\rho }_c$ is shown in red and Poincaré cutoff surface $r_{\mathrm{P}}={r_{\mathrm{P}}}_c$ is shown in blue. (b) When we take ${\rho }_c$ to infinity, all the $r_{\mathrm{P}}$ surfaces bunch up along the edges of the causal diamond. Two of them are shown in the figure.

Figure 5

We open up the boundary cylinder for better visualization. The colored region is the causal diamond and is the interior of the curve the bulk Poincaré horizon traces on the global boundary cylinder. The boundary cylinder is taken to be large with $\rho \sim \mathcal{O}({\epsilon }^{-1})$ and $\epsilon \ll 1$. The region where the Poincaré radial coordinate $r_{\mathrm{P}}\sim \mathcal{O}(1)$ is shown in blue. This is the region where the Fefferman-Graham expansions in $\rho$ and $r_{\mathrm{P}}$ are not consistent and the Poincaré-CFT cannot be approximated by the global-CFT.

Figure 6

We open up the boundary cylinder for better visualization. In (a) one formally divides the global-CFT into two halves and follows their causal development inside the causal diamonds. This is analogous to “Rindlerizing the global-CFT.” This is not the same as the Rindler-CFT (b). While the two theories are approximately equal deep inside the diamonds, they start differing at the edges. The global-CFT is defined everywhere, but the Rindler-CFT is defined only inside the diamonds. Correlation functions of the latter do not give those of the former under analytic continuation.

Figure 7

We open up the boundary cylinder for better visualization. In (a) we show the causal development of the interval $(0,2\pi )$. Correlation functions inside can be analytically continued to the full cylinder [32]. This is not the same as the Poincaré-CFT (b). While the two theories are approximately equal deep inside the diamond, they start differing at the edges. The global-CFT is defined everywhere but the Poincaré-CFT is defined only inside the diamond. Correlation functions of the latter do not give those of the former under analytic continuation.

Figure 8

(a) The geometry of the D1-D5 black hole has an outer flat space connected by a neck to the massless BTZ black hole. (b) The geometry of a generic state has outer flat space connected by a neck to a throat which ends in a smooth cap without horizons and singularities.

Figure 9

The massive BTZ has two asymptotically AdS regions. These regions share the future and past regions behind the event horizons.

Figure 10

According to the proposals in [16, 19, 20] the bulk dual to two decoupled CFTs on $S^1\times R$ that are in the thermofield double state (which is equivalent to our PIRCs) are quantum geometries which resemble the massive BTZ at large distances but get capped outside the would-be horizons. Thus, there are no shared future or past regions. Our results are consistent with these proposals.

Figure 11

(a) The global cutoff surface $\rho ={\rho }_c$ is shown in red, and the spherical Rindler-AdS cutoff surface $r_{\mathrm{SR}}={r_{\mathrm{SR}}}_c$ is shown in blue. (b) When we take ${\rho }_c$ to infinity, all the $r_{\mathrm{SR}}$ surfaces bunch up along the top and bottom edges of the “causal strip” which may be thought of as the union of causal diamonds for all the Rindler-AdS observers [51]. Two such constant $r_{\mathrm{SR}}$ surfaces are shown in the figure.