Black Hole Entropy from String Entanglement

Abstract

We discuss the notion of string entanglement in string theory, which aims to study entanglement between worldsheet Hilbert spaces rather than entanglement between spacetime Hilbert spaces defined on a time slice in spacetime. Applying this framework to the FZZ duality and its extension to a three-dimensional black hole, we argue that the thermal entropy of 2d and 3d black holes is accounted for by the string entanglement entropy between folded strings arising in the dual sine-Liouville CFT. We compute this via a worldsheet replica method and show that it decomposes into two parts, which we call the vertex operator contribution and the replica contribution. The former can be evaluated analytically and is shown to coincide with the black hole thermal entropies in the low temperature limit in large D dimensions. Although a computation of the latter is left as an open problem, we present evidence that it captures the remaining portion of the black hole entropy.

Figures (9)

• Figure 1: FZZ duality and stringy ER=EPR. The FZZ duality relates (a) the cigar CFT background and (b) the sine-Liouville background, where the gradient depicts the dilaton profile. Upon continuation from Euclidean signature to Lorentzian signature, in (a), the Euclidean cigar half-disk is continued to a Lorentzian black hole in which two asymptotic regions are connected by an Einstein-Rosen (ER) bridge, while in (b) the Euclidean section, which is topologically is a half-annulus, is continued to two disconnected flat spacetimes. The ER bridge in (a) corresponds to the EPR pairs, which are realized as pairs of open folded strings as shown in red in (b).
• Figure 2: The closed string channel for $s'=1$. (a) The worldsheet picture. The magenta and orange lines represent the worldsheet on two different time slices (constant-$\rho$ curves). (b) The spacetime picture. The winding closed string comes in from $\hat{r}=-\infty$, turns around at some value of $\hat{r}$, and goes back to $\hat{r}=-\infty$. The worldsheet at the two time slices are also shown.
• Figure 3: The open string channel for $s'=1$. (a) The worldsheet picture. The magenta and orange lines represent the worldsheet at two different time slices (constant-$\phi$ curves). (b) The spacetime picture. A folded open string extending along $\hat{r}$ propagates around the $\hat{\theta}$ circle. The worldsheets corresponding to those in the worldsheet picture are also shown.
• Figure 4: The pair of folded open strings to propagate in Lorentzian continuation, for $s'=1$. (a) The pair of open strings in the worldsheet picture. The Euclidean, hemispherical worldsheet with two vertices inserted prepares a pair of open strings, which propagate into the Lorentzian part of the worldsheet shown as vertical wedges. (b) The target space picture. The Euclidean half-cylinder spacetime is connected to two separate copies of flat Lorentzian space (the vertical parts). The red and blue lines represent the folded open strings on $L$ ($\hat{\theta}=0$) and $R$ ($\hat{\theta}=\pi\sqrt{k}$), respectively.
• Figure 5: The location of the $L$ and $R$ curves on a spherical worldsheet for $s'=2$. (a): The OPE fixes the curves only near $W_\pm$ insertions. (b) and (c): two possible ways to connect $L$ and $R$ curves.