# Jackiw-Teitelboim Gravity from the Karch-Randall Braneworld

Hao Genga Andreas Karchb Carlos Perez-Pardavilab Suvrat Rajuc Lisa Randalla
Marcos Riojasb Sanjit Shashib aHarvard University, 17 Oxford St., Cambridge,
MA, 02139, USA.  bTheory Group, Department of Physics, University of Texas,
Austin, TX 78712, USA.  cInternational Centre for Theoretical Sciences, Tata
Institute of Fundamental Research, Shivakote, Bengaluru 560089, India.

###### Abstract

In this letter, we show that Jackiw-Teitelboim (JT) gravity can be naturally
realized in the Karch-Randall braneworld. Notably the role of the dilaton in
JT gravity is played by the radion in a suitably orbifolded version of the
setup. In the classical entanglement entropy calculation, there is an apparent
degeneracy of Ryu-Takayanagi surfaces. We demonstrate how quantum fluctuations
of the radion/dilaton resolves this would-be classical puzzle regarding
entanglement wedge reconstruction.

††preprint: today

##  I Introduction

Recent utilization of the doubly holographic Karch-Randall braneworld [1, 2]
has yielded substantial insights into quantum gravity. These include
analytical models that have been used to produce Page curves describing black
hole radiation in higher-dimensional (i.e in $d>2$ spacetime dimensions)
setups [3, 4]111This is motivated by the earlier work [5, 6, 7, 8].,
demonstrating a connection between the existence of entanglement islands and
massive gravity theories [4, 9, 10], providing simple holographic models to
study quantum field theory in black hole backgrounds [11, 12], and motivating
a novel holographic setup—wedge holography [13, 14, 9, 10].

Wedge holography is a simple canonical deformation of the original Karch-
Randall braneworld which manifests a codimension-two holographic duality [13,
14, 9, 10]. This setup starts with two AdSd Karch-Randall branes in an ambient
AdSd+1 with the two branes intersecting each other at the conformal boundary
of the ambient spacetime. These branes are treated as “end-of-the-world” (EOW)
branes, meaning that the part of the ambient spacetime behind them is excised.
Thus, the leftover bulk spacetime is a wedge (see Fig. 1a). In analogy with
the original Karch-Randall braneworld, the wedge system is doubly holographic
in that there are three equivalent descriptions related to each other by the
AdS/CFT correspondence [15, 16, 17] (see Fig. 1):

  * •

The bulk description: Einstein-Hilbert gravity in the wedge.

  * •

The intermediate description: Two CFTd’s coupled with gravity living on
separate AdSd’s and glued to each other along their asymptotic boundaries by
imposing transparent boundary conditions.

  * •

The boundary description: A CFTd-1 living on the common asymptotic boundary of
the two branes (which is the corner of the wedge in Fig. 1a).

(a)

(b)

(c)

Figure 1: A schematic representation of the doubly holographic nature of the
wedge. (a) is the bulk description, which is just gravity on the
$(d+1)$-dimensional wedge with a codimension-two corner in red. (b) is the
intermediate description in which the two $d$-dimensional theories share a
transparent interface (red). (c) is the boundary description on the interface.

In this letter, we study wedge holography when the bulk is AdS3 with two AdS2
branes. As opposed to the original wedge holography setup in higher dimensions
where the brane fluctuations can be neglected, we will consider fluctuating
branes with orbifold projection so that there is only one fluctuating scalar
degree of freedom. This scalar is the radion in the language of the Randall-
Sundrum I setup [18]. We will derive an action for the radion which is coupled
with the 2d localized braneworld gravity. This can be thought of as the low-
energy effective theory of the intermediate description. We will see that this
theory is precisely the Jackiw-Teitelboim (JT) gravity [19, 20], in which the
dilaton field is the radion.

The importance of our study is that it ensures the consistency of wedge
holography in AdS3. This is because when we ignore the brane fluctuations and
study the entanglement entropy in AdS3 wedge holography using the Ryu-
Takayanagi (RT) formula, we find an infinite number of degenerate RT surfaces.
This renders the entanglement wedge, which is an essential element in
holography [21], as undefined. Taking into account the brane fluctuations,
this degeneracy is lifted, as we show explicitly using our effective JT
description.

We note that JT gravity [22, 23, 24, 25] has also played an important role in
the recent development of low-dimensional ($d=2$) quantum gravity [5, 6, 26].
This includes providing an analytically controllable model to compute the Page
curve of evaporating black holes [5, 6] and a precise duality between
Euclidean JT gravity to a random matrix ensemble [26]. Our realization of JT
gravity in the Karch-Randall braneworld further argues for its significance in
our study of quantum gravity [27].

##  II Karch-Randall Braneworld and Wedge Holography

In this section, we briefly review the Karch-Randall braneworld and wedge
holography, focusing on the geometric pictures for later holographic study.
The original theory of Karch and Randall is described by the following action

$\begin{split}S=&-\frac{1}{16\pi G_{d+1}}\int
d^{d+1}x\sqrt{-g_{\text{bulk}}}(R[g_{\text{bulk}}]-2\Lambda)\\\
&\qquad\qquad-\frac{1}{8\pi G_{d+1}}\int d^{d}x\sqrt{-h}(K-T)\,,\end{split}$
(1)

where the first term is the standard Einstein-Hilbert action describing the
gravitational in the ambient AdSd+1 with a cosmological constant
$\Lambda=-\frac{d(d-1)}{2}$,222We have set the AdS curvature length to one.
the second term describes the embedding of the brane in the ambient AdSd+1,
$h_{ab}$ is the induced metric on the brane, $K$ is the trace of the extrinsic
curvature of the brane and $T$ is the tension of the brane and satisfies
$T\leq(d-1)$ as a Karch-Randall brane.333The critical value $T=d-1$
corresponds to the Randall-Sundrum brane [18, 28]. The bulk metric
fluctuations satisfy the Neumann boundary condition near the brane

$\nabla_{n}\delta g_{\text{bulk {\hbox{\mu\nu}}}}|_{\text{near brane}}=0\,,$
(2)

where $n$ denotes the normal direction of the brane. With this boundary
condition taken into account, setting the variation of the action Equ. (1) to
zero we obtain two equations of motion—the usual Einstein’s field equation
which determines the bulk geometry and the brane-localized equation of motion
describing the embedding of the brane in the bulk. The latter is given by [29]

$K_{ab}=(K-T)h_{ab}\,.$ (3)

This system of equations can be solved by

$ds^{2}_{\text{bulk}}=dr^{2}+\cosh^{2}(r)ds^{2}_{\text{AdS{\hbox{_d}}}}\,,$
(4)

where $r\in(-\infty,\infty)$, with the brane given by $r=r_{1}=const.$ and
$T=(d-1)\absolutevalue{\tanh r_{1}}$ (we only consider positive tension
branes).444If the brane is given by $r=r_{1}<0$ understood as an EOW brane
with positive tension the domain of the bulk is $r\in(r_{1},\infty)$ and if
the brane is given by $r=r_{2}>0$ understood as an EOW brane with positive
tension the bulk domain is $r\in(-\infty,r_{2})$ [30]. The bulk metric is just
empty AdSd+1 written as a foliation into AdSd slices. This is particularly
useful in the study of the Karch-Randall braneworld as here the brane geometry
$r=const.$ is clearly AdSd. In this letter, we focus on the case $d=2$, in
which case the geometry can be easily visualized as that on the left of Fig.
2a.

In the wedge holography setup [13, 14, 9, 10], we add another (positive
tension) Karch-Randall brane. This system is described by the following action

$\begin{split}S=&-\frac{1}{16\pi G_{3}}\int
d^{3}x\sqrt{-g_{\text{bulk}}}(R[g_{\text{bulk}}]-2\Lambda)\\\
&\qquad\qquad-\frac{1}{8\pi
G_{3}}\int_{i=1,2}d^{d}x\sqrt{-h_{i}}(K_{i}-T_{i})\,,\end{split}$ (5)

where we impose the Neumann boundary conditions Equ. (2) for the bulk metric
fluctuation near both branes. The equations of motion still consist of two
parts—the bulk Einstein’s equation and the two brane-localized equations, with
each one the same as Equ. (3) but with their own tensions $T_{1}$ and $T_{2}$.
Hence, a solution is given by the same bulk metric Equ. (4) with $d=2$ where
$r\in(r_{1},r_{2})$, $T_{1}=\tanh(-r_{1})$, $T_{2}=\tanh r_{2}$, $r_{1}<0$ and
$r_{2}>0$. This geometry is visualized in Fig. 2b.

$\bullet$$\circ$$\bullet$$\circ$ (a)

$\bullet$$\circ$$\bullet$$\circ$ (b)

Figure 2: (a) A constant time slice of the empty AdS3 bulk with one Karch-
Randall brane (in black). The brane intersects the asymptotic boundary (in
blue) at the two defects (red dots). The holographic dual is a BCFT2 living on
the blue interval. (b) The empty AdS3 wedge consisting of two KR branes. The
two branes intersect at the two defects (red and blue dots). The dual boundary
description is hence a conformal quantum mechanics living on the two defects.

##  III Realizing JT Gravity

In the original wedge holography papers [13, 14, 9, 10] the branes are treated
as rigid. This assumption is rooted in the original Karch-Randall setup where
we just have one brane and we can take the brane to be sitting at an orbifold
fixed point to model an EOW brane in analog to O-planes in string theory [31,
32]. However, reminding ourselves of the Randall-Sundrum I (RS1) setup [18]
with two RS branes, the relative distance between the branes is a dynamical
variable called the radion, which has important phenomenological implications
[33]. Hence in the case of wedge holography, we should not neglect the
possibility of brane fluctuations.

In this section, we consider small brane fluctuations (relative to the bulk
AdS3 curvature scale) and derive the effective action for the coupling between
these small fluctuations and the brane-localized graviton mode by doing
dimensional reduction. While brane fluctuations can be neglected in higher
dimensions, we will see that this small contribution plays an important role
when the bulk is AdS3. The detailed derivation and a more careful analysis
will be presented in [27].

To be precise, we will consider two branes with tensions $T_{1}$ and $T_{2}$
sitting at $r=r_{1}+\delta\phi_{1}(x)$ and $r=r_{2}+\delta\phi_{2}(x)$ slices
of the following bulk geometry

$ds^{2}=dr^{2}+\cosh^{2}(r)g_{ab}(x)dx^{a}dx^{b}\,,$ (6)

with the relationships $T_{1}=-\tanh r_{1}>0$ and $T_{2}=\tanh r_{2}>0$
satisfied and $\delta\phi_{1},\delta\phi_{2}\ll 1$ (the AdS length scale has
been set to one). The wedge runs from $r=r_{1}+\delta\phi_{1}(x)$ to
$r=r_{2}+\delta\phi_{2}(x)$, and the metric ansatz Equ. (6) describes the
lowest 2d graviton mode (the higher modes would depend on the coordinate $r$
as $g_{ab}(r,x)$).

To obtain the low-energy effective action, we have to plug the above ansatz of
the metric and brane locations into Equ. (5) and keep terms only up to order
$\mathcal{O}(\delta\phi^{2})$. This yields the following action555For details
of the derivation, we refer the readers to [27].

$S_{\text{eff}}=S_{0}-\frac{1}{16\pi G_{3}}\int
d^{2}x\sqrt{-g}\phi(x)\Big{[}R[g]+2\Big{]}+S_{\text{dilaton}}\,,$ (7)

where $\phi(x)=\delta\phi_{2}(x)-\delta\phi_{1}(x)$ is the radion mode,
$S_{0}$ is a purely topological term

$\begin{split}S_{0}&=-\frac{r_{2}-r_{1}}{16\pi G_{3}}\int
d^{2}x\sqrt{-g}R[g]\,,\end{split}$ (8)

and $S_{\text{dilaton}}$ is the kinetic term for the brane fluctuations

$\begin{split}S_{\text{dilaton}}=&-\frac{1}{8\pi G_{3}}\int
d^{2}x\sqrt{-g}\Big{[}\frac{T_{2}}{2}\nabla_{\mu}\delta\phi_{2}\nabla^{\mu}\delta\phi_{2}+T_{2}(\delta\phi_{2})^{2}\\\
&\qquad\qquad\quad\quad\quad+\frac{T_{1}}{2}\nabla_{\mu}\delta\phi_{1}\nabla^{\mu}\delta\phi_{1}+T_{1}(\delta\phi_{1})^{2}\Big{]}\,.\end{split}$
(9)

In analogy to the Randall-Sundrum scenario, to isolate the dynamics of the
radion we have to orbifold our setup. We can project out either
$\delta\phi_{1}(x)$ or $\delta\phi_{2}(x)$ by a canonical orbifold
prescription: start with an array of an infinite number of paired wedges, mod
out the translation symmetry on the array to a single pair of wedges with
$\mathbb{Z}_{2}$ symmetry, and then mod out this $\mathbb{Z}_{2}$ symmetry
(see [27] for details) to get a single wedge with only one fluctuating brane.
The possibility of projecting out either $\delta\phi_{1}(x)$ or
$\delta\phi_{2}(x)$ are gauge-equivalent. To see this gauge equivalence,
without loss of generality we can project out $\delta\phi_{1}(x)$ and redefine
$g_{ab}(x)$ such that the final action is independent of $r_{1}$ and $r_{2}$.
Specifically, by setting

$\delta\phi_{1}(x)=0\,,\quad g_{ab}(x)\rightarrow
e^{-T_{2}\delta\phi_{2}(x)}g_{ab}(x)\,,$ (10)

the total action becomes

$S_{\text{eff}}=S_{0}-\frac{1}{16\pi G_{3}}\int
d^{2}x\sqrt{-g}\phi(x)\Bigg{[}R[g]+2\Bigg{]}\,,$ (11)

where now the dilaton field is $\phi(x)=\delta\phi_{2}(x)$. This is precisely
Jackiw-Teitelboim gravity [19, 20] for AdS2.

However, there is no nontrivial profile for the dilaton on the AdS2 background
without a cutoff [34]. This cutoff breaks the asymptotic conformal symmetry
and allows the theory to have finite energy excitations [24]. Hence we fix a
small cutoff $\epsilon$ and fix the boundary metric for the nearly-AdS2 and
the dilaton field to be

$g_{ab}(x)|_{\text{bdy}}dx^{a}dx^{b}=-\frac{du^{2}}{\epsilon^{2}}\,,\ \
\phi(x)|_{\text{bdy}}=\phi_{b}=\frac{\phi_{r}}{\epsilon}\,.$ (12)

A subtlety of our derivation is that we have to take $\phi_{b}\sim\epsilon$
(or $\phi_{r}\sim\epsilon^{2}$) to ensure the brane fluctuation is always
small, i.e. $\delta\phi_{i}(x)\ll 1$.

Translating to the AdS3 language Equ. (6), we have to take a cutoff of the
AdS3 wedge and fix the metric there to be (see Fig. 3 for an indication)

$ds^{2}_{\text{bdy}}=dr^{2}-\cosh^{2}(r)\frac{du^{2}}{\epsilon^{2}}\,.$ (13)

$\bullet$$\circ$$\bullet$$\circ$ Figure 3: The shapes of the two branes are
fluctuating (though more rigorously we have also to impose the orbifold
projection). The AdS2 cutoff is induced by the AdS3 cutoff (the red curve).
The shaded regions are excised. Now the regulated asymptotic boundary of the
wedge are two intervals.

Now that we have identified this cutoff and boundary conditions for the metric
in the AdS3 bulk, we have to add the corresponding boundary term to the total
action Equ. (5). The boundary term is the usual Gibbons-Hawking term

$S_{\text{bdy}}=S_{\text{GH}}=-\frac{1}{8\pi G_{3}}\int
d^{2}x_{\text{bdy}}\sqrt{-h^{\text{bdy}}}K^{(3)}\,,$ (14)

which can be shown to be

$\begin{split}S_{\text{bdy}}&=-\frac{2(r_{2}-r_{1})}{16\pi
G_{3}}\int_{\partial\mathcal{M}}dx\sqrt{-h}K-\frac{2\phi_{b}}{16\pi
G_{3}}\int_{\partial\mathcal{M}}dx\sqrt{-h}K\,,\end{split}$ (15)

where $K$ is the trace of the extrinsic curvature of the cutoff boundary of
the AdS2. With this term added to Equ. (11), we get the full-fledged JT
gravity action as that in [35]. For more details we refer the readers to [27].

We emphasize that the scaling $\phi_{b}\sim\epsilon$ is special in our setup.
For example, it does not appear in dimensional reduction for the near-horizon
region of four-dimensional near-extremal black holes [36, 37]. This indicates
that our theory has a different UV completion than previously studied. In our
case, the UV completion is a conformal quantum mechanics system [27].

##  IV A Puzzle and Its Resolution

A canonical quantity to study in AdS3 wedge holography is the entanglement
entropy between the two asymptotic defects which support the dual quantum
mechanics (see Fig. 2b). This can be calculated using the Ryu-Takayanagi (RT)
formula [38, 39] by studying the RT surfaces that connect the two branes [13,
9]. Choosing global AdS2 for those AdS2 slices, the bulk metric Equ. (4) can
be written as

$ds^{2}=dr^{2}+\cosh^{2}(r)\big{[}-\cosh^{2}(\eta)d\tau^{2}+d\eta^{2}\big{]}\,,$
(16)

if we ignore the brane fluctuations $\delta\phi_{i}(x)$. The RT surface can be
parametrized as $\eta=\eta(r)$, $\tau=const.$ which leads to the area
functional

$A=\int_{r_{1}}^{r_{2}}dr\sqrt{1+\cosh^{2}(r)\eta^{\prime}(r)^{2}}\,.$ (17)

It is easy to show that, with the boundary term carefully taken into account,
the variation of the area functional vanishes if and only if
$\eta^{\prime}(r)=0$. Therefore, any constant $\eta$ surface is an RT surface
and they all have equal area $A=r_{2}-r_{1}$ (see Fig. 4 for an illustration).
According to this result, there would be no definite RT surface for the
entanglement entropy between the two defects and as a corollary there would be
no definite entanglement wedge for either of the defects. This would be
problematic because it would preclude a holographic interpretation from the
point of view of bulk (or more precisely, entanglement wedge [40])
reconstruction.

$\bullet$$\circ$$\bullet$$\circ$ Figure 4: The empty AdS3 wedge consisting of
two KR branes. The bulk geometry is the AdS3 global patch. The green curves
are representatives of the degenerate RT surfaces calculating the entanglement
entropy between the two defects (the blue dot and the red dot).

Fortunately, this puzzling result occurs only when the branes are treated as
rigid. We expect brane fluctuations to lift the degeneracy of the RT surfaces.
Because the brane fluctuation is in general time-dependent, we should use the
HRT formula [41] to compute the entanglement entropy from the wedge. However,
this would be complicated as the result is heavily dependent on properly
treating the boundary condition near the brane.

This challenge of computing entanglement entropy with brane fluctuations is
readily met if we instead do it using our effective braneworld theory Equ.
(11). In that case we are just computing the entanglement entropy between the
two cutoff boundaries of the nearly-AdS2 space. By solving the equation of
motion of the dilaton and 2d metric, we have that

$\displaystyle ds^{2}_{2d}$
$\displaystyle=-\cosh^{2}(\eta)d\tau^{2}+d\eta^{2}\,,$ (18)
$\displaystyle\phi(x)$ $\displaystyle=\phi_{r}\cosh\eta\cos\tau\,,$ (19)

where $\phi_{r}$ is a constant. Thus, using the maxmin prescription [42] of
the HRT formula we have

$S_{\text{EE}}=\max_{\tau}\min_{\eta}\Big{(}\frac{\phi_{0}+\phi(x)}{4G_{3}}\Big{)}=\frac{\phi_{0}+\phi_{r}}{4G_{3}}\,,$
(20)

where the minimization has a solution $\eta=0$, $\tau=0$. As a result, from
the bulk AdS3 perspective Equ. (16), the RT surface is just the $\eta=0$
surface which sits precisely in the middle of the bulk (i.e. the middle green
surface in Fig. 4). Thus, the entanglement wedge of either defect is the half
bulk region next to it, which is consistent with the symmetry between the two
defects.

##  V Conclusions

In this letter, we studied wedge holography when the bulk is three-
dimensional. We found that it is necessary to consider brane fluctuations,
identified as the radion mode in the braneworld language [18, 28], for the
apparent consistency of the holographic setup. We found that the low-energy
dynamics of the brane fluctuations with orbifold projection is precisely
Jackiw-Teitelboim gravity.

## Acknowledgements

We are grateful to Andreas Blommaert, Kristan Jensen, Hong Liu and Douglas
Stanford for useful discussions. HG is very grateful to his parents and
recommenders. The work of HG is supported by the grant (272268) from the Moore
Foundation “Fundamental Physics from Astronomy and Cosmology.” The work of AK
and MR was supported, in part, by a grant from the Simons Foundation (Grant
651440, AK). The work of CP was supported in part by the National Science
Foundation under Grant No. PHY-1914679 and by the Robert N. Little Fellowship.
The work of SR was partially supported by a Swarnajayanti fellowship,
DST/SJF/PSA-02/2016-17 from the Department of Science and Technology (India).
Research at ICTS-TIFR is supported by the government of India through the
Department of Atomic Energy grant RTI4001. The work of LR is supported by NSF
grants PHY-1620806 and PHY-1915071, the Chau Foundation HS Chau postdoc award,
the Kavli Foundation grant “Kavli Dream Team,” and the Moore Foundation Award
8342. SS is supported by NSF Grants No. PHY-1914679 and PHY-2112725.

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