Negative R$\acute{\text{e}}$nyi entropy and Brane intersection

TL;DR

This paper investigates the calculation of Rényi entropy in AdS3/(B)CFT2 and discovers that certain gravity solutions involving brane intersections can result in negative Rényi entropy values.

Contribution

It introduces the concept that brane intersection solutions in gravity can produce negative Rényi entropy, challenging previous assumptions.

Findings

Brane intersection solutions lead to negative Rényi entropy.

Revisiting Rényi entropy calculations in AdS3/(B)CFT2.

Implications for holographic entropy interpretations.

Abstract

In this work, we revisit the calculation of R$\acute{\text{e}}$nyi entropy in AdS$_3$/(B)CFT$_2$. We find that gravity solutions brane intersection will lead to negative R$\acute{\text{e}}$nyi entropy.

Equations

$$S_A^n=\frac{1}{1-n}\log \text{Tr}{\rho }_A^n,$$

$$\text{Tr}{\rho }_A^n=\frac{Z_n(A)}{Z_1^n},$$

$$\frac{1}{4}\log \left(\frac{8}{m(m+1)}{\sin }^2(\frac{\pi }{m}){\sin }^2(\frac{\pi }{m+1})\right),$$

$$\text{Tr}{\rho }_A^n=c_n⟨{\sigma }_h^{ϵ}(-L){\sigma }_{\bar{h}}^{ϵ}(L)⟩=c_n{\left(\frac{2L}{ϵ}\right)}^{-\frac{c}{6}(n-\frac{1}{n})},$$

$$S_A^n=\frac{c}{6}\frac{1+n}{n}\log \frac{2L}{ϵ}+g_A+g_B.$$

$$w=f(x)=\log \left(\frac{x+L}{L-x}\right),$$

$$Z_1={\tilde{q}}^{-c/24}\sum _k⟨A|k⟩⟨k|B⟩{\tilde{q}}^{{\Delta }_k},Z_n={\tilde{q}}^{-c/24n}\sum _k⟨A|k⟩⟨k|B⟩{\tilde{q}}^{{\Delta }_k/n},$$

$$\mathrm{tr}{\rho }^n\equiv \frac{Z_n}{Z_1^n}={\left(⟨A|0⟩⟨0|B⟩\right)}^{1-n}{e}^{(\frac{1}{n}-n)\frac{\pi W}{\beta }\frac{c}{6}},$$

$$S_A^{(n)}=\frac{n+1}{n}\frac{\pi c}{6\beta }W+g_A+g_B,$$

$$f(x)=\frac{x+L}{L-x}.$$

$$d{s}_{{\mathcal{M}}_1}^2=d{r}^2+r^{2}d{\theta }^2.$$

$$I({\mathcal{M}}_1)\sim -\frac{c}{6}\log \frac{r_{\text{max}}}{r_{\text{min}}}$$

$$Z_1\sim e^{-I({\mathcal{M}}_1)}=e^{\frac{c}{3}\log \frac{2L}{ϵ}}{Q}_{ϵ},$$

$$d{s}_{{\mathcal{M}}_n}^2=d{r}^2+n^2{r}^{2}d{\theta }^2=d\zeta d\bar{\zeta },$$

$$d{s}_{{\mathcal{M}}_n}^2=n^2{\rho }^2(d{\rho }^2+{\rho }^{2}d{\varphi }^2)=e^{-2\tau }d{s}_{{\hat{\mathcal{M}}}_n}^2,e^{-\tau }=n{\rho }^{n-1},\xi =\rho e^{\text{i}\varphi }.$$

$$\frac{Z[e^{-2\tau }g]}{Z[g]}=\exp \left[-\frac{c}{24\pi }{\int }_{\mathcal{M}}\sqrt{g}[R\tau -(\partial \tau {)}^2]d^{2}x-\frac{c}{12\pi }{\int }_{\partial \mathcal{M}}K\tau \sqrt{\gamma }dy\right],$$

$$Z[{\mathcal{M}}_n]$$

$$=$$

$$\text{Tr}{\rho }_A^n=\exp \left(\frac{c}{6}(\frac{1}{n}-n)\log \frac{2L}{ϵ}\right)={\left(\frac{2L}{ϵ}\right)}^{-\frac{c}{6}(n-\frac{1}{n})}{Q}_{ϵ}^{1-n},$$

$$I_M=-\frac{1}{16\pi G_N}{\int }_M\sqrt{g}(R-2\Lambda )-\frac{1}{8\pi G_N}{\int }_{{\Sigma }_1,{\Sigma }_2}\sqrt{h}(K-T)+\frac{1}{8\pi G_N}{\int }_{\Gamma }(\theta -{\theta }_0)\sqrt{\gamma },$$

$$\delta I_M=-\frac{1}{16\pi G_N}{\int }_M\left(R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\Lambda g_{\mu \nu }\right)\delta g^{\mu \nu }$$

$$-\frac{1}{8\pi G_N}{\int }_{{\Sigma }_{1,2}}\sqrt{h}(K_{ab}-h_{ab}(K-T))\delta h^{ab}+\frac{1}{8\pi G_N}{\int }_{\Gamma }(\theta -{\theta }_0)\delta \sqrt{\gamma }.$$

$$K_{ab}-h_{ab}(K-T)=0,h^{ab}\text{is free},$$

$$d{s}^2=\frac{1}{z^2}\left(f(z)d{\tau }^2+\frac{d{z}^2}{f(z)}+d{x}^2\right),$$

$$Q_{\pm }(x_0,\lambda ):z=\pm \frac{z_H}{\lambda }\sinh \frac{x-x_0}{z_H},\lambda =|\frac{T}{\sqrt{1-T^2}}|,$$

$$Q_1:z=\frac{z_H}{{\lambda }_1}\sinh \frac{x}{z_H},Q_2:z=\frac{z_H}{{\lambda }_2}\sinh \frac{x-x_0}{z_H},x_0>0.$$

$$I_{\text{bulk}}$$

$$=$$

$$I_{\text{bdy}}=-\frac{1}{8\pi G}\int \sqrt{h}(K-T)=-\frac{\beta }{8\pi G}\left(-{\int }_0^{x_1}\frac{1}{z_1^2}dx+{\int }_{x_0}^{x_2}\frac{1}{z_2^2}dx\right).$$

$$I_{ct}$$

$$=$$

$$I_{\text{on-shell}}(z_H)$$

Full text

1*Kavli Institute for Theoretical Sciences (KITS),

University of Chinese Academy of Science, 100190 Beijing, P. R. China

2School of Physics and Technology,

Wuhan University, Wuhan, 430072, P. R. China *

1. 1 Introduction

2. 2 Field theory calculation

3. 2.1 From cylinder partition function

4. 2.2 From Weyl anomaly and Weyl transformation in annulus spacetime

5. 3 Holographic calculation

6. 3.1 set-ups

7. 3.2 Cylinder boundary

8. 3.2.1 Brane intersection

9. 3.2.2 From Dong’s formula

10. 3.3 Annulus boundary

11. 3.3.1 The original partition function $Z_{1}$

12. 3.3.2 Brane intersection

13. 3.3.3 The replica partition function $Z_{n}$

14. 3.3.4 From Dong’s formula

15. 4 Discussion

1 Introduction

R$\acute{\text{e}}$nyi entropy [1] which is a one-parameter generalization of von Neumann entanglement entropy is an important measure of entanglement in quantum field theories. It contains richer physical information. In principle knowing all the R$\acute{\text{e}}$nyi entropies is equivalent to knowing all the eigenvalues of the reduced density matrix of a subsystem. In practice, R$\acute{\text{e}}$nyi entropies are easier to numerically compute and experimentally measure.

R$\acute{\text{e}}$nyi entropy has been extensively studied in many theories [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In this paper we focus on the R$\acute{\text{e}}$nyi entropy in AdS3/(B)CFT2 [54, 55]. Two-dimensional BCFT has an interesting quantity: the boundary entropy (or the logarithm of the g-function) [48] which can also be computed holographically [49]. Even for two-dimensional CFTs without boundaries, the boundary entropy an important role. R$\acute{\text{e}}$nyi entropy can be defined as

[TABLE]

where $A$ is a subsystem and $Z_{n}(A)$ is the path integral on a $n$–sheeted Riemann space $\mathcal{M}_{n}$ obtained by gluing $n$ copies of the original space $\mathcal{M}_{1}$ alone $A$. The reason why boundary entropy arises is that R$\acute{\text{e}}$nyi entropy has a UV singularity and to regularize it we can remove a small disk of radius $\epsilon$ around each branch point with conformal boundary conditions imposed on the circular boundary of the disk. Therefore the definition of R$\acute{\text{e}}$nyi entropy depends on the regularization which is parameterized by conformal boundary condition or boundary entropy [50]. Not like the entanglement entropy is positive definite the boundary entropy can be both positive and negative. For example, the lower bound of boundary entropy of the unitary minimal models with central charge $c_{m}=1-\frac{6}{(m+1)m}$ is [51]

[TABLE]

which approaches $-\infty$ when $m\rightarrow\infty$. Even though from the analysis of RG flow, one can argue that boundary entropy should have a lower bound [52, 53]. One may still wonder can boundary entropy is so negative that the R$\acute{\text{e}}$nyi entropy becomes negative333Usually we treat the boundary entropy as a small correction so this never happens. But from the point of view of AdS/BCFT the boundary entropy can be arbitrarily large. We would like to explore this possibility in AdS3/BCFT2. The bulk theory is AdS Einstein gravity theory with End-Of-the-World (EOW) branes. We will also generalize it to allow the configurations of brane intersections or corners. Such generalization has been explored recently in [57, 58, 56]. However previous studies are restricted in the thermal AdS geometry and global AdS geometry (with a conical defect in the center) in which the static EOW brane configurations play no roles in the R$\acute{\text{e}}$nyi entropy calculation444In [58], the authors also studied the “disk branes” which can reproduce the boundary entropy terms. However, in the Lorentz signature, these branes are not static.. To be complementary, we consider the BTZ black geometry and excited geometry 555which is obtained from the Ba$\tilde{\text{n}}$ados map as defined below.. We find a connection between the negativity of the R$\acute{\text{e}}$nyi entropy and brane intersection. Using these new bulk solutions we can reproduce the field theory results but these geometries also contain some exotic features such as orientation inversion. An inverted orientation results in a negative Euclidean time evolution which is argued to be ill-defined [59, 60]. However similar geometries are also encountered in a recent work [61] for estimating global charge violation.

We consider two following examples: a single interval on the infinite line and the interval at the end of a semi-infinite line in the ground state. These two systems are basically the same in the sense: after we remove small disks around branch points both of these two-dimensional Euclidean manifolds can be transformed into an annulus or a cylinder [50]. The only difference is that for the BCFT system, only one of the boundaries is a physical boundary while for the CFT system, both of the two boundaries are cut-off boundaries. However, when we compute the partition function the cut-off boundary plays the same role as the physical boundary does. Holographically they both correspond to an EOW brane in the gravity theory.

Here we distinguish the annulus geometry and cylinder geometry because their bulk dual geometries are different. In particular, when we consider the BCFT system the regularization we take in the field theory naturally provides a regularization scheme in the holographic calculation and it turns out this regularization is exactly the one recently proposed in [56] so as a by-product our results explain the origin of their cut-off proposal.

This paper is organized as follows. In section 2 we revisit the calculation of R$\acute{\text{e}}$nyi entropy in field theory with an emphasis on boundary entropy contribution which is usually ignored. In section 3, after introducing the gravity theory set-ups, we reproduce the field theory results in two different geometries. We show the subtleties of the gravity solutions with brane intersection and show they lead to negative R$\acute{\text{e}}$nyi entropy.

2 Field theory calculation

Let us start by briefly reviewing the field theory calculation of the R$\acute{\text{e}}$nyi entropy. Let the subsystem $A$ of a CFT be the interval $x\in[-L,L]$. The simplest way to compute (2) is converting it to a two-point function of primary twisted operators $\sigma_{h}$ with conformal dimension $h_{n}=\bar{h}_{n}=(c/24)(n-(1/n))$ [63]. Thus from the general formula of the two-point function of primary fields in two-dimensional CFT, we get [63]

[TABLE]

and the resulting R$\acute{\text{e}}$nyi entropy is

[TABLE]

where $g_{A,B}$ are the boundary entropies that are related to the constant $c_{n}$ but can not be determined by this method directly. To derive the boundary term one we can compute the partition functions.

2.1 From cylinder partition function

The Euclidean surface $\mathcal{M}_{1}$ can be mapped to a cylinder via the conformal transformation

[TABLE]

such that the two cut-off disks at $x=\pm L$ are mapped to two ends located at $w=\pm\log(2L/\epsilon)$ of the cylinder. Applying the open-closed duality the conformal boundary conditions are translated to two boundary states $|A(B)\rangle$ and partition functions on the cylinder can be written as

[TABLE]

where the modular parameter is defined as $\tilde{q}=e^{-4\pi W/\beta}$ and $\beta$ is the circumference of the cylinder. In the limit of $W/\beta>>1$, only the ground state contributes

[TABLE]

so the R$\acute{\text{e}}$nyi entropy is

[TABLE]

where $g_{A,B}=\log\langle A,B|0\rangle$ are the boundary entropies.

In our case, $\beta=2\pi$ and $W=2\log(2L/\epsilon)$ so the R$\acute{\text{e}}$nyi entropy matches (5). The first R$\acute{\text{e}}$nyi entropy $S_{A}^{(1)}$ 666Strictly speaking, R$\acute{\text{e}}$nyi entropy is only defined for $n\neq 1$ and $S_{A}^{1}$ should be defined by a proper analytic continuation. is the entanglement entropy where the boundary entropy term can be absorbed into the UV cut-off $\epsilon$ then the result matches the standard result. However, it is pointed out in [50], the boundary entropies are in principle measurable by considering different values of $n$. Therefore one should expect the dual holographic calculation should be able to reproduce them. The advantage of using cylinder geometry is the replicated geometry $\mathcal{M}_{n}$ is also a cylinder so that the bulk dual is easy to find.

2.2 From Weyl anomaly and Weyl transformation in annulus spacetime

The regulated surface can also be mapped to an annulus via the conformal transformation

[TABLE]

Let the metric of the annulus be

[TABLE]

The annulus partition function has a universal contribution from the Weyl anomaly due to the existence of boundaries [65]:

[TABLE]

where $r_{\text{max}}=2L/\epsilon$ ($r_{\text{min}}=\epsilon/(2L)$) is the radius of the outer (inner) boundary. Thus the corresponding partition function is

[TABLE]

where $Q$ is a constant that is regularization dependent [64]. The $n$-sheeted cover of the annulus has the metric

[TABLE]

where $\zeta=re^{\text{i}n\phi}$. Introducing $\xi=\zeta^{1/n}$ we can find that the it relates (12) via a Weyl transformation

[TABLE]

It is well known that the 2d Weyl anomaly is described by the Liouville action [66]

[TABLE]

which implies

[TABLE]

Therefore using (1) and $\rho_{\text{max(min)}}=r^{1/n}_{\text{max(min)}}$ we get the same result as (9)

[TABLE]

if we identify $Q_{\epsilon}=g_{A}g_{B}$. The identification is due to the fact the constant $Q$ is given by the path integral in the holes with state $|A,B\rangle$ thus it is equal to the inner product $\langle A,B|0\rangle$.

3 Holographic calculation

3.1 set-ups

For the holographic calculation, we consider the Euclidean gravity action on a manifold $M$ potentially with a corner. We assume that the corner is bounded by two codimension-1 surfaces $\Sigma_{1}$ and $\Sigma_{2}$ and the corner is a codimension-2 surface $\Gamma$. The full action is given by

[TABLE]

where the last term is the analog of the Hayward term [67] and $\theta_{0}$ is a fixed value that characterizes the corner. The two codimension-1 surfaces can be AdS asymptotic boundaries, hard cut-off surfaces, or EOW boundaries with tension $T$. On $\Sigma_{1,2}$ we need to impose proper boundary conditions such that the variation of the action is well-defined. The variation of the action $I_{M}$ is given by

[TABLE]

If we choose the Dirichlet boundary condition on $\Sigma_{1,2}$ i.e. $\delta\Sigma_{1,2}=0$, then $\delta\sqrt{\gamma}$ is also fixed thus the variation is well defined. However, if we choose the Neumann boundary condition:

[TABLE]

there will be two possible choices $I:\theta-\theta_{0}=0$ or $II:\delta\sqrt{\gamma}=0$ such that the last term (22) vanishes. To reproduce the field theory result we take the first choice and treat $\theta_{0}$ as the parameter of the corner determined by the equation $\theta-\theta_{0}=0$. For this choice, the corner term will not contribute to the on-shell action directly and the gravity theory can be viewed as a generalization of standard AdS/BCFT.

3.2 Cylinder boundary

When the boundary BCFT is defined on a cylinder geometry, the bulk solution can be thermal AdS or BTZ black holes and there is also a Hawking-Page-like phase transition. Note that the length of the cylinder is very long $2\log(2L/\epsilon)$ then it should be in the high-temperature phase in the open channel. Thus to reproduce the R$\acute{\text{e}}$nyi entropy we should consider the BTZ black hole solutions. The thermal AdS and global AdS with intersecting EOW branes have been studied in [57]. Indeed for these two cases, the on-shell action will not depend on the details of the static EOW branes so boundary entropy terms in the R$\acute{\text{e}}$nyi entropy can not be reproduced. The metric of the (non-rotating) BTZ black hole is

[TABLE]

where $f(z)=1-(z/z_{H})^{2},\tau\sim\tau+2\pi z_{H}$. In the BTZ geometry, the profile of the static EOW brane is [68]

[TABLE]

which ends on the AdS boundary at $x=x_{0}$. If we assume that the BCFT is defined on a cylinder then we need to insert two EOW branes in the bulk as shown in Fig.1

The on-shell action of configurations $(b)$ has been computed in [68]. For other cases, the calculation is almost the same here as an illustration we present the calculation for configurations $(c)$ where the two EOW branes are parameterized as

[TABLE]

The bulk contribution to the action is

[TABLE]

the boundary contribution is

[TABLE]

and the counter term is

[TABLE]

where $\gamma_{mn}$ is the induced metric on the cut-off surface $z=\epsilon$. Adding these terms together we obtain the final regulated on-shell action

[TABLE]

Noticing that the bulk dual of $\mathcal{M}_{n}$ is also described by (24) with the replacement $z_{H}\rightarrow nz_{H}$ then we can reproduce the R$\acute{\text{e}}$nyi entropy

[TABLE]

where we have used the relation $W=2\log\frac{2L}{\epsilon}$ and identification $g_{A}=-\frac{c}{6}\sinh^{-1}(\lambda_{1}),g_{B}=\frac{c}{6}\sinh^{-1}(\lambda_{2})$. One of the observations we want to make is that the EOW brane with negative tension has a negative contribution to the R$\acute{\text{e}}$nyi entropy! The other observation is that if one of the branes has negative tension (configurations $(a),(b),(c)$) then it is possible that the two branes will intersect outside of the horizon. So it is very natural to imagine that there is a connection between the brane intersection and a negative R$\acute{\text{e}}$nyi entropy. Naively we can derive

[TABLE]

where indeed implies that $S_{A}^{(n)}$ becomes negative when $x_{2}<x_{1}$ *i.e. *two branes intersect.

3.2.1 Brane intersection

To study the brane intersection in BTZ geometries in detail let us consider the extremal case: both of the two EOW branes have negative brane tension and their profiles are

[TABLE]

We denote their intersection point by $(x_{*},z_{*})$ which is determined from

[TABLE]

and their ends on the horizon by $x_{1}$ and $x_{2}$, respectively. When $\gamma<1$ the intersection point is outside of the horizon. The intersection angle $\theta_{0}$ can be computed as

[TABLE]

where $\vec{n}$ is the normalized normal vector of the EOW brane. If we fix the brane tensions and require $\gamma<1$ the range of the intersection angle is given by

[TABLE]

As shown in Fig.2 there are two candidates of the bulk dual of the BCFT on $[0,x_{0}]$. Let us examine them one by one.

Region I

Motivated by previous results [57, 58, 56] involved with brane intersection one may think that the region I in Fig.2

is the bulk dual since it is bounded by the EOW branes and the interval $[0,x_{0}]$. To verify this let us compute the on-shell action in the region I. The bulk contribution is

[TABLE]

and the boundary contribution is

[TABLE]

By adding the same counter term (31) we arrive at the regularized on-shell action

[TABLE]

Clearly, this bulk geometry is not the desired one. From the relation between on-shell action and ADM mass $I=2\pi M_{\text{ADM}}$ [69], it implies the bulk geometry is possible to dual to a state with a very high energy $E=M_{\text{ADM}}=\frac{c}{12}W/\beta$.

Region I $\cup$ II

Another obvious guess is that the bulk dual is Region I $\cup$ II. However, according to the rule (the normal vectors of the boundaries should point outward) of AdS/BCFT, the tension of the two EOW branes bounding region II should be positive. It is a little suspicious because it means that after crossing the sign of the brane tension of these two EOW branes gets flipped! Indeed we can show that this guess is not correct by computing the on-shell action. The bulk contribution in region II is

[TABLE]

and the boundary contribution is

[TABLE]

Then we find

[TABLE]

which is different from the expected result (34). However, we find that if we subtract the on-shell action of Region II we can obtain the correct result

[TABLE]

and it leads to the desired R$\acute{\text{e}}$nyi entropy using (35). Therefore the result suggests that we should not flip the sign of the brane tension and propose that when the normal vectors of the boundaries point inward the orientation of this bulk region should be inverted. In other words, the correct bulk dual is Region I$\cup{\text{II}^{-1}}$. One attitude to this bizarre result is that the bulk solution with brane intersection is non-physical just like the bulk solution with brane self-intersection [56, 70, 71, 72, 73]. It implies that the simple gravity model (21) is not adequate when the brane tension is large. From the field theory side, the brane self-intersection is excluded by a bootstrap analysis [56]. Naively our results suggest that the brane intersection can be excluded by the positivity of R$\acute{\text{e}}$nyi entropy. Note that R$\acute{\text{e}}$nyi entropy of the interval at the end of a semi-infinite line is related to a one-point function of the BCFT which is a crucial input of the bootstrap analysis. Then it is possible to understand the brane non-intersection also from the bootstrap analysis but it is beyond the scope of this paper.

3.2.2 From Dong’s formula

In the end, let us show how this result can also be obtained from Dong’s formula. It is proposed by Dong Xi, the holographic R$\acute{\text{e}}$nyi entropy of a subsystem $A$ can be computed from a refined version of R$\acute{\text{e}}$nyi entropy defined as

[TABLE]

which is dual to the area of a bulk codimension-2 cosmic brane homologous to the subregion $A$:

[TABLE]

The cosmic brane has a tension $(n-1)/(4nG)$ so it back reacts on the bulk geometry by creating a conical defect with an opening angle $2\pi/n$. So the goal is to find such a conical geometry with the boundary to be our cylinder. Starting from the BTZ back hole metric we can make the replacement $h(z)\rightarrow h_{n}(z)=1-z^{2}/(n^{2}z_{H}^{2})$ and fix the period $\tau\sim\tau+2\pi z_{H}$ then the resulting geometry will have a required conical singularity at $z=nz_{H}$. This geometry is exactly the conical geometry that we are looking for. Thus the refined R$\acute{\text{e}}$nyi entropy is

[TABLE]

which exactly leads to (36). It is interesting to see that Dong’s formula not only gives the leading log terms but also the boundary entropy terms 777as far as we notice this is the first time to reproduce the boundary entropy using Dong’s formula. When $x_{2}<x_{1}$, *i.e. *when the branes intersect we find that the refined R$\acute{\text{e}}$nyi entropy becomes negative but it does not mean the area of the cosmic brane is negative. Our interpretation is that the area is still positive but the orientation is inverted! Next, we consider another possible bulk solution for reproducing the R$\acute{\text{e}}$nyi entropy.

3.3 Annulus boundary

In the field theory analysis described in section 2.2, we have shown when the original spacetime $\mathcal{M}_{1}$ has a geometry of annulus the replicated spacetime $\mathcal{M}_{n}$ will have a conical singularity which should be regularized. We will show that bulk dual of $\mathcal{M}_{n}$ can be constructed from the Ba$\tilde{\text{n}}$ados map [74, 75].

3.3.1 The original partition function $Z_{1}$

When the boundary BCFT is defined on an annulus (or a disk), the bulk geometry can be Poincar$\acute{\text{e}}$ AdS with metric

[TABLE]

and the EOW brane profile888When the normal vector of the brane points towards the center the brane tension is negative for the choice $(+)$ and positive for the choice $(-)$ . [68]

[TABLE]

where $\lambda$ is still defined as (25) and $r_{D}$ is a free parameter. It is helpful to consider the case of a disk with radius $r_{D}$ first. The bulk dual geometry is a part of the round sphere described by (59). Then the on-shell action in this region can be easily computed as the following. The bulk contribution is

[TABLE]

the boundary contribution

[TABLE]

and the counter term is

[TABLE]

thus the final result is [55, 68]

[TABLE]

The logarithm term is still divergent but expected because it is responsible for the Weyl anomaly. To obtain a finite result we can consider an annulus instead of a disk on the boundary. Let the radius of the annulus be $r_{+}$ and $r_{-}$ then there are still two situations: the two branes intersect or not in the bulk. When they do not intersect the on-shell action of the corresponding bulk region is simply given by

[TABLE]

which matches the field theory result (13).

3.3.2 Brane intersection

As shown in Fig.3, when we tune the brane tension two EOW branes are possible to intersect.

To be concrete let us consider two EOW branes

[TABLE]

where we assume that $0<r_{-}<r_{+}$. The two branes will intersect when

[TABLE]

at the position $(r_{*},z_{*})$

[TABLE]

Similar to (40) the intersection angle is given by

[TABLE]

Naively we guess that the bulk dual is region I in Fig.3 however it is straightforward to show that the on-shell action in region I vanishes thus we have to also consider region II. Note that in region I the normal vector of $Q_{1}(Q_{2})$ should point inward (outward) so if we preserve the sign of brane tensions we should change the orientation of region II. Thus the on-shell action of region II*-1* is computed as

[TABLE]

which exactly gives the desired on-shell action $Z_{1}$ (64).

3.3.3 The replica partition function $Z_{n}$

Now we want to find the bulk geometry for computing $Z_{n}$. Note that the 2d CFT is defined on conical manifold $\mathcal{M}_{n}$ with the metric

[TABLE]

Introducing $\xi=\zeta^{1/n}$ then the metric can be written as

[TABLE]

Thus the 2d CFT defined on $\mathcal{M}_{n}$ is conformally related to the 2d CFT defined on $\hat{\mathcal{M}}_{n}$. Considering the vacuum state of the latter theory, the 3d bulk dual is the Poincar$\acute{\text{e}}$ AdS3 with metric

[TABLE]

The conformal transformation $\omega=\xi^{n}\equiv p(\xi),\quad\bar{w}=\bar{\xi}^{n}\equiv q(\bar{\xi})$ can be lifted to a 3d diffeomorphism known as the Ba$\tilde{\text{n}}$ados map999It is a little different from the ones [75] because we work in the Euclidean signature.

[TABLE]

The corresponding bulk geometry has the metric

[TABLE]

where

[TABLE]

In our case, the map and $T_{\pm}$ are

[TABLE]

The metric can be written in the form

[TABLE]

with

[TABLE]

At the position $\eta_{c}=\frac{2\rho}{\sqrt{n^{2}-1}}$ the metric degenerates so we should treat it as the center (or horizon) of the bulk space [76] (see also [23]). First, it is useful to compute the on-shell action in the whole spacetime with an IR cut-off $\rho=\Lambda_{\rho}$:

[TABLE]

Next, we replace the IR cut-off $\rho=\Lambda_{\rho}$ with an EOW brane as shown in Fig.4. Using the map (77) the profile of EOW branes becomes

[TABLE]

where $\rho_{D}=r_{D}^{\frac{1}{n}}$. For simplicity, let us first choose $\lambda=0$. Then the profile is

[TABLE]

which implies that $\rho$ has to be in the range

[TABLE]

For convenience, we divide the dual bulk spacetime into 2 regions and the on-shell action in region II of Fig.4 can be easily computed as

[TABLE]

Since $\rho$ will not approach [math] so we can safely drop the first term in (99). The on-shell action in region I is given by (92) with the replacement $\Lambda_{\rho}=\rho_{*}$:

[TABLE]

thus the final result is

[TABLE]

The leading term matches the field theory calculations of the disk partition function. For the annulus, we get precisely the same result as (19)

[TABLE]

where non-universal subleading terms are canceled. Because we have set $\lambda=0$ thus the corresponding boundary entropy also vanishes. The analytic result of the on-shell action is hard to obtain for non-vanishing $\lambda$ because of the complicity of the expression of the EOW profile. So here we will make a perturbative analysis with respect to $\lambda$ and the goal is to reproduce the boundary entropy terms.

$\lambda\neq 0$: perturbative results

For non-vanishing $\lambda$, we can still separate the spacetime into two parts and compute the on-shell action separately according to the line

[TABLE]

The on-shell action in the region I is still

[TABLE]

The bulk contribution to the on-shell action in the region II is

[TABLE]

where $\eta_{E}$ is the profile of the EOW

[TABLE]

In order to evaluate the boundary contribution we first work out the determinant of the induced metric

[TABLE]

Then we find that leading order correction is101010Note that we should regularize $\rho_{D}\rightarrow\rho_{D}-\frac{\rho_{D}\lambda^{2}}{2}-\frac{\epsilon^{2}}{2\rho_{D}}+\epsilon\lambda$.

[TABLE]

and

[TABLE]

thus

[TABLE]

Adding the leading term of (108):

[TABLE]

we get

[TABLE]

as expected considering that

[TABLE]

We also work out the subleading correction

[TABLE]

which also matches the expansion of the field theory result (63). Thus we conclude that the bulk metric (84) with the EOW brane profile (93) gives the correct bulk dual for computing the replica partition function $Z_{n}$.

Brane intersection

For the annulus boundary, the bulk geometry may also have an intersection phase as shown in Fig.5. It is similar to the situation in the BTZ case discussed in 3.2.1. Even though the shaded region in Fig.5 is not the proper bulk dual for computing $Z_{n}$ but it is still interesting to see what is its possible dual state.

For simplicity, we assume that one of the EOW branes is tensionless

[TABLE]

The intersection point is at

[TABLE]

where

[TABLE]

The on-shell action can be computed from the difference

[TABLE]

which is totally different from the expected result (19). According to the identification $I=2\pi M_{\text{ADM}}$, it suggests this bulk geometry is dual to a state with energy with energy

[TABLE]

3.3.4 From Dong’s formula

In the end, let us argue how to use Dong’s formula to directly obtain the result 111111The argument is similar in spirit if not in detail to one in [56]. The replica manifold $\mathcal{M}_{n}$ has a conical singularity so we first introduce a uniform cover $\hat{\mathcal{M}}_{n}$ defined in (75) to regularize it. Now we can put a flat metric on the uniform cover thus the bulk dual $\hat{\mathcal{B}}_{n}$ is simply the Poincar$\acute{\text{e}}$ metric. Then the required conical geometry can be obtained by taking the $\mathbb{Z}_{n}$ quotient of $\hat{\mathcal{B}}_{n}$. Thus the cosmic brane which is the stabilizer of $\mathbb{Z}_{n}$ quotient just starts from the center of the small EOW brane and ends at the center of the big EOW brane so that refined R$\acute{\text{e}}$nyi entropy which is given by the length of this cosmic brane which is exactly (19)

[TABLE]

From Dong’s formula it is easier to see when $r^{1/n}_{+}(\sqrt{1+\lambda_{2}^{2}}+\lambda_{2})<r^{1/n}_{-}(\sqrt{1+\lambda_{1}^{2}}+\lambda_{1})$ *i.e. *when branes intersect the refined R$\acute{\text{e}}$nyi entropy is negative.

4 Discussion

In this paper, we revisit the calculation of R$\acute{\text{e}}$nyi entropy in AdS3/(B)CFT2 and we find that there is a connection between the negativity of R$\acute{\text{e}}$nyi entropy and brane intersection. In some cases [58, 57], gravity solutions with brane intersection seem to be physical because they give the correct spectrum of boundary-condition-changing operators in BCFT. In some cases [70, 71, 56], the brane (self-)intersection is not physical because the energy of the corresponding state is beyond the black hole threshold. Our results also suggest that the brane intersection should be regularized or the semi-classical gravity theory is not the full theory. Of course, our result is far away from a proof. Following [56], more rigorous results may be obtained from a bootstrap perspective.

We also explain the origin of the cut-off proposal [56] given by the EOW branes. We check it in both the BTZ geometry and Ba$\tilde{\text{n}}$ados geometry. Applying Dong’s formula we also obtain the finite part of the R$\acute{\text{e}}$nyi entropy, the boundary entropy. When the brane intersection happens, Dong’s formula also gives a negative result since the orientation of the spacetime is changed. We believe that similar results can be found in the higher dimensional cases. Another direction of generalization is to consider multiple intervals.

We show that the replica partition function can be obtained by considering a Ba$\tilde{\text{n}}$ados geometry with EOW branes. The crucial point is that in the Ba$\tilde{\text{n}}$ados geometry, the EOW brane profile should be modified accordingly. In [78], a similar geometry is proposed to dual the Virasoro coherent state. So our results may help to study such states.

The geometries with brane intersection can also be constructed from a cut-and-glue procedure [61] (see also [79]) as shown in Fig.6. The orientation of the central region gets inverted. However, it is argued in [61] that this ill-defined geometry can be cured by including a suitable period of Lorentzian evolution.

It is interesting to explore whether the geometries which are studied in this paper can be regularized in a similar way.

In a recent work [80], AdS/BCFT is generalized to include a brane-localized scalar that can describe non-conformal boundary conditions. Interestingly by adding the scalar degree of freedoms the EOW brane can have a large number of new solutions such as the connected EOW solution in the Poincar$\acute{\text{e}}$ metric which is not possible without the brane-localized scalar field. One important application of this setup is to study the boundary RG flow. Similar to the field theory arguments [52, 53] it is possible to use this setup to show that boundary entropy has a lower bound from a holographic point of view such that the brane intersection never happens.

Acknowledgments

We thank Cheng Peng for the valuable discussion and comments on an early version of the draft. We thank many of the members of KITS for interesting related discussions. JT is supported by the National Youth Fund No.12105289 and funds from the UCAS program of special research associate. XX is supported by NSFC NO. 12175237 and the Fundamental Research Funds for the Central Universities.