Structure of deformations in Jackiw-Teitelboim black holes with matter

Dongsu Bak,${}^{\negthinspace a}$ Chanju Kim,${}^{\negthinspace b}$ Sang-Heon
Yi,${}^{\negthinspace c}$

a) Physics Department & Natural Science Research Institute

University of Seoul, Seoul 02504 KOREA

b) Department of Physics, Ewha Womans University, Seoul 03760 KOREA

c) Center for Quantum Spacetime & Physics Department

Sogang University, Seoul 04107 KOREA

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
ABSTRACT

We consider Jackiw-Teitelboim gravity with a massless matter field and turn on
bulk excitations leading to a nontrivial vev of the corresponding dual
boundary operator. To leading order, we realize the corresponding deformation
of thermofield double state by explicitly identifying their Hilbert space. The
deformed state can be prepared with an operator insertion at the mid-point of
the Euclidean time evolution in the context of Hartle-Hawking construction. We
show that the inserted operators form an SL(2,R) representation. We construct
a specific orthonormal basis that is directly related to the operator basis of
the vev deformations. If we include the higher order corrections, the bulk
geometry is no longer left-right symmetric. We argue that, classically, the
mode coefficients in the bulk deformation cannot be fully recovered from the
data collected along the boundary cutoff trajectories. Then the bulk seems to
contain more information than the cutoff boundary, and this might be
responsible for nontrivial behind-horizon degrees of freedom.

##  1 Introduction

In recent years, there have been remarkable developments in the context of the
Nearly AdS2/Nearly CFT1 (NAdS2/NCFT1) correspondence. NAdS2 arises from an
appropriate dimensional reduction of a near extremal black hole geometry while
NCFT1 may appear as a low energy approximation of a one-dimensional quantum
system like the SYK model (See [1] for a review and also references therein.).
The Jackiw-Teitelboim (JT) model (coupled with a matter field) [2, 3, 4] is a
specific 2d dilaton gravity for the NAdS2 geometry, which may be reduced to
the Schwarzian dynamics along the boundary cutoff trajectories [5]. Combined
with various information-theoretic techniques, this model provides a
computable test bed for various ideas about the resolution of the black hole
information loss problem [6].

A two-sided AdS black hole geometry is well known to be dual to the so-called
thermofield double state in the boundary side [7]. The (undeformed)
thermofield initial state may be prepared by the Euclidean time evolution in
the context of Hartle-Hawking construction [7]. In this note, we are mainly
interested in rather general deformations of the two-sided black hole geometry
by turning on a bulk matter field that is dual to the corresponding boundary
operator. In the so-called standard quantization111In a certain range of mass
parameter of a bulk field, there could be an alternative quantization which is
not our concern in this note [8, 9]., the normalizable modes of the bulk
matter field correspond to the vev’s of the boundary operator while the non-
normalizable modes are dual to the source deformations of the boundary theory
by the same boundary operator. Thus any excitation of a black hole geometry by
the normalizable modes will lead to the corresponding deformations of the
thermofield double state. In this note, specialized to the case of the
massless scalar field, we would like to clarify the structure of deformations
in the 2d bulk as well as in the dual boundary theory side. Especially, we
shall show that the initial state, to the leading order of vev deformations,
may be prepared with an operator insertion at the mid-point of the Euclidean
time evolution [10, 11].

In general, deformations would affect the dynamics of the cutoff trajectories.
One may then try to obtain the information of the deformations by probing the
cutoff trajectories at the boundary. This would, in principle, be possible if
the trajectories contain all the information of the bulk deformations. There
are, however, nontrivial behind-horizon degrees of freedom in the bulk such as
Python’s degrees of freedom [12, 13, 14]. It is not clear at all whether this
hidden information can be fully recovered by collecting boundary data. In
fact, by explicitly solving the equations of motion, we will see that the
cutoff trajectories do not have enough information for the full recovery.

Since AdS2 is rigid even under the bulk matter deformation, the SL$(2,{\bf
R})$ symmetries of the background geometry provide useful information for
understanding the relevant dynamics [16]. In this paper, we explore the
SL$(2,{\bf R})$ symmetry realization of the inserted operators corresponding
to the vev deformations of the thermofield double state. To simplify the
discussion, we consider these vev deformations only up to their leading order.
We show that these inserted operators form a specific SL$(2,{\bf R})$
representation. See [15, 16, 17] for some related discussions.

This paper is organized as follows. In the next section, we review the Jackiw-
Teitelboim model focusing on the 2d two-sided black hole geometries and the
induced Schwarzian dynamics along their cutoff trajectories. In Section 3, we
investigate the structure of generic bulk deformations by turning on a
massless scalar field. We argue that the bulk information may not be fully
recovered from the boundary data collected along the cutoff trajectories.
Sections 4 and 5 are devoted to the realization of SL$(2,{\bf R})$ symmetries
of the inserted operators. To the leading order, we explicitly identify the
operators inserted at the mid-point which reproduce the most general vev
deformations and show that they form a unitary SL(2,R) representation. In the
final section, we summarize our results and comment on some future directions.

##  2 Two-dimensional dilaton gravity

The JT model of our interest is a 2d dilaton gravity with a matter field
described by action

$\displaystyle I=I_{top}+\frac{1}{16\pi
G}\int_{M}d^{2}x\sqrt{-g}\,\phi\left(R+\frac{2}{\ell^{2}}\right)+I_{surf}+I_{M}(g,\chi)\,,$
(2.1)

where $\phi$ is a dilaton field, $\chi$ a matter field and

$\displaystyle I_{top}$ $\displaystyle=\frac{\phi_{0}}{16\pi
G}\int_{M}d^{2}x\sqrt{-g}R\,,$ $\displaystyle I_{surf}$
$\displaystyle=\frac{1}{8\pi G}\int_{\partial
M}\sqrt{\gamma}\,(\phi_{0}+\phi)\,K\,,$ $\displaystyle I_{M}$
$\displaystyle=-\frac{1}{2}\int_{M}d^{2}x\sqrt{-g}\left(\nabla\chi\cdot\nabla\chi+m^{2}\chi^{2}\right)\,.$
(2.2)

In this action, $\ell$ is the AdS radius, and $\gamma_{ij}$ and $K$ denote the
induced metric and the extrinsic curvature on the boundary $\partial M$,
respectively.

The variation of the dilaton field $\phi$ leads to

$R+\frac{2}{\ell^{2}}=0\,,$ (2.3)

which sets the metric to be AdS2. The other equations of motion are obtained
from the variation of the metric $g$ and the scalar field $\chi$,

$\displaystyle\nabla_{a}\nabla_{b}\phi-
g_{ab}\nabla^{2}\phi+\frac{1}{\ell^{2}}g_{ab}\phi$ $\displaystyle=-8\pi
GT_{ab}\,,$ (2.4) $\displaystyle\nabla^{2}\chi-m^{2}\chi$ $\displaystyle=0\,,$
(2.5)

where $T_{ab}$ is the stress tensor of the matter field,

$T_{ab}=\nabla_{a}\chi\nabla_{b}\chi-\frac{1}{2}g_{ab}\left(\nabla\chi\cdot\nabla\chi+m^{2}\chi^{2}\right)\,.$
(2.6)

In the global coordinates, the metric of the AdS2 space is written as

$ds^{2}=\frac{\ell^{2}}{\cos^{2}\mu}\left(-d\tau^{2}+d\mu^{2}\right)\,,$ (2.7)

where $\mu\in[-\frac{\pi}{2},\frac{\pi}{2}]$. The most general vacuum solution
to the dilaton equation of motion (2.4) with $T_{ab}=0$ is given by

$\phi=\phi_{BH}(L,b,\tau_{B})\equiv\bar{\phi}\,L\,\,\frac{(b+b^{-1})\cos(\tau-\tau_{B})-(b-b^{-1})\sin\mu}{2\cos\mu}\,,$
(2.8)

By the coordinate transformation

$\displaystyle\frac{r}{L}$
$\displaystyle=\frac{(b+b^{-1})\cos(\tau-\tau_{B})-(b-b^{-1})\sin\mu}{2\cos\mu}\,,$
$\displaystyle\tanh\frac{tL}{\ell^{2}}$
$\displaystyle=\frac{2\sin(\tau-\tau_{B})}{(b+b^{-1})\sin\mu-(b-b^{-1})\cos(\tau-\tau_{B})}\,,$
(2.9)

we obtain the AdS black hole metric

$ds^{2}=-\frac{r^{2}-L^{2}}{\ell^{2}}dt^{2}+\frac{\ell^{2}}{r^{2}-L^{2}}dr^{2}\,,$
(2.10)

with $\phi=\bar{\phi}\,r$. Utilizing the SL(2, R) isometry of AdS2, we can set
$b=1$ and $\tau_{B}=0$ [18]. This metric describes the Rindler wedge of two-
sided AdS black holes with the radius of black hole horizon $L$. The location
of singularity is defined by the curve $\Phi^{2}\equiv\phi_{0}+\phi=0$ in the
above dilaton field, and $\Phi^{2}$ might be viewed as characterizing the size
of “transverse space” [4]. In this left/right symmetric two-sided black hole
case, one can see that the Gibbons-Hawking temperature, the entropy and energy
are given by

$T=\frac{1}{2\pi}\frac{L}{\ell^{2}}\,,\qquad S=S_{0}+{\cal C}T\,,\qquad
E=\frac{1}{2}{\cal C}T^{2}\,,$ (2.11)

where $S_{0}$ is the ground state entropy given by $S_{0}=\frac{\phi_{0}}{4G}$
and ${\cal C}=\frac{\pi\bar{\phi}\ell^{2}}{2G}$. In general, these physical
quantities could be different for left/right Rindler wedges in the two-sided
black hole case. In the next sections we consider some deformation of black
hole configuration through the dilaton field and show that these quantities
are indeed different for the left/right Rindler wedges.

The boundary time $u$ may be introduced in $\epsilon\rightarrow 0$ limit
through the prescription

$ds^{2}|_{\text{cutoff}}=-\frac{1}{\epsilon^{2}}du^{2},\qquad\phi|_{\text{cutoff}}=\bar{\phi}\frac{\ell}{\epsilon}.$
(2.12)

We will adopt the convention that the right boundary time $t_{r}$ runs upwards
whereas the left boundary time $t_{l}$ runs downward. In other words, we
identify

$u=t_{r}=-t_{l}.$ (2.13)

See Section 4 for the details.

The boundary dynamics is equivalently described by a Schwarzian theory [19,
20, 21],

$S=\int
du\left[-\phi_{l}\left\\{\tan\frac{\tau_{l}(u)}{2},u\right\\}-\phi_{r}\left\\{\tan\frac{\tau_{r}(u)}{2},u\right\\}\right],$
(2.14)

where $\phi_{l}=\phi_{r}$ can be identified with $\bar{\phi}$ in the bulk and
$\tau_{l/r}(u)$ corresponds to the left/right global time coordinate,
respectively, at each cutoff trajectory. If the matter is turned on, the
Schwarzian action would get corrections which, in general, can be asymmetric
at the left and the right boundaries. To the leading order in the deformation,
however, the correction vanishes, as seen in the next section.

We depict the Penrose diagram of a deformed space in Figure 1. In the figure,
the curves near the boundaries represent typical cutoff trajectories of the
boundary dynamics. Given dilaton configurations that are deformed away from
(2.8), one can obtain the cutoff trajectories by using the prescription (2.12)
or from the boundary action with deformed terms.

Figure 1: The left and the right cutoff trajectories are illustrated as curves
near the boundaries in the Penrose diagram of a typical deformed space.

Now we briefly review the general deformation by the matter field. As a simple
left-right asymmetric case, let us recall the eternal Janus deformation [18]
which makes the Hamiltonians of left-right boundaries differ from each other
by turning on exact marginal operators. It is given by

$\chi=\gamma(\mu-\kappa_{0})\,,\qquad\phi=\bar{\phi}L\frac{\cos\tau}{\cos\mu}-4\pi
G\gamma^{2}(1+\mu\tan\mu)\,.$ (2.15)

In this case, though the matter field $\chi$ is asymmetric, the dilaton field
$\phi$ and the black hole temperature are left-right symmetric under the
exchange of $\mu\leftrightarrow-\mu$.

Since the metric is fixed to be AdS2, the matter field equation (2.5) can be
solved. In the global coordinates, the general solution is given by [22]

$\chi=\sum^{\infty}_{n=0}c^{\cal D}_{n}\,{\cal N}_{n}\cos^{\cal
D}\mu\,C_{n}^{\cal D}(\sin\mu)\,e^{-i(n+{\cal D})\tau}+{\rm c.c.}\ ,$ (2.16)

where

${\cal N}_{n}=2^{{\cal D}-1}\Gamma({\cal
D}){\textstyle\sqrt{\frac{\Gamma(n+1)}{\pi\Gamma(n+2{\cal D})}}}\,,$ (2.17)

and $C^{\cal D}_{n}(x)$ denotes the Gegenbauer polynomial. Here, the parameter
${\cal D}$ is defined in terms of the mass of the scalar field $\chi$ as

${\cal D}=\Delta_{\pm}=\frac{1}{2}\left(1\pm\sqrt{1+4m^{2}}\right)\,.$ (2.18)

According to the AdS/CFT correspondence, the bulk matter $\chi$ is dual to a
scalar primary operator $O_{\Delta}(t)$ of a certain dimension $\Delta$. If
$m^{2}\geq 0$, it is identified as $\Delta_{+}$, i.e.,

${\Delta=\Delta_{+}}=\frac{1}{2}\left(1+\sqrt{1+4m^{2}}\right)\,.$ (2.19)

In this case, the deformation by the matter $\chi$ with ${\cal D}=\Delta_{+}$
in (2.16) corresponds to a vev deformation of the dual field theory, while the
other deformation with identifying ${\cal D}=\Delta_{-}$ describes a source
deformation. In AdS2, there is another possibility that $m^{2}$ is negative as
long as the Breitenlohner-Freedman bound [8] holds, namely $-1/4\leq m^{2}<0$
in our case. Then, we have two possible values as the operator dimension
$\Delta$,

$\Delta=\Delta_{\pm}=\frac{1}{2}\left(1\pm\sqrt{1+4m^{2}}\right)\,.$ (2.20)

##  3 Structure of bulk deformations

From this section, we shall consider turning on a massless scalar field dual
to a scalar operator of dimension $\Delta=1$. In this case, the bulk scalar
equation (2.4) can be solved explicitly by [14]

$\displaystyle\chi=\chi_{s}+\chi_{v}\,,$ (3.1)

where

$\chi_{v}=\sum^{\infty}_{n=1}a_{n}\sin n\big{(}\mu+{\frac{\pi}{2}}\big{)}\cos
n(\tau-\tau^{v}_{n})$ (3.2)

and

$\chi_{s}=b_{0}+\sum^{\infty}_{n=1}b_{n}\cos
n\big{(}\mu+{\frac{\pi}{2}}\big{)}\cos n(\tau-\tau^{s}_{n})\,.$ (3.3)

This describes a fully general set of classical solutions to the scalar
equation in the ambient AdS2 space where $\chi_{v/s}$ satisfies the
Dirichlet/Neumann boundary condition at $\mu=\pm\pi/2$. In our JT model, the
metric remains always to be AdS2 and won’t be corrected by any matter
perturbations. For the dilaton field, we shall start from the vacuum solution

$\phi_{bg}=\,\bar{\phi}L\,\frac{\cos\tau}{\cos\mu}$ (3.4)

without loss of any generality. As was mentioned previously, this background
describes a two-sided black hole with temperature
$T=\frac{1}{2\pi}\negthinspace\frac{L}{\,\,\ell^{2}}$. With the scalar
deformation (3.1), the dilaton solution becomes

$\phi=\phi_{bg}+8\pi
G\sum^{\infty}_{m=1}\sum^{\infty}_{n=1}\left(\,a_{m}a_{n}\phi^{v}_{m,n}+b_{m}b_{n}\phi^{s}_{m,n}+2a_{m}b_{n}\phi^{c}_{m,n}\,\right)\,,$
(3.5)

where the full detailed functional forms of $\phi^{v,s,c}_{\,m,n}$ are given
explicitly in [14].

We now turn to the standard AdS/CFT interpretation of the above bulk
deformation. To this end, note that the scalar solution in any asymptotic
region of Lorentzian AdS2 spacetime may be expanded as222 In this expansion,
we have ignored any logarithmic terms which are not relevant in our discussion
below. (See [23] for instance)

$\chi_{\Delta}=s(\tau)\left(\frac{\ell^{2}}{r}\right)^{1-\Delta}(1+\cdots)+u(\tau)\left(\frac{\ell^{2}}{r}\right)^{\Delta}(1+\cdots)~{},$
(3.6)

where the radial coordinate $r$ is defined by $\phi/\bar{\phi}$ and $\cdots$
denotes higher order contributions of each power series expansion in
$\left(\ell^{2}/r\right)^{2}$. In this note, we are dealing with deformations
of the two-sided black hole which involves two (left and right) asymptotic
regions in general. In the left/right asymptotic region, the presence of
nonvanishing $s_{l/r}(\tau_{l/r}(u))$ represents a source deformation of the
left/right boundary theory by

${\cal L}_{l/r}(u)={\cal L}_{0}(u)+s_{l/r}(\tau_{l/r}(u))O_{\Delta}(u)\,,$
(3.7)

where $\tau_{l/r}(u)$ describes the reparameterization along the left/right
cutoff trajectory respectively. Below for the simplicity of our presentation,
we shall write $\tau_{l/r}(u)$ simply as $\tau$ once it is not confusing. With
our normalization in (2.1), the vev of operator with $s=0$ may be identified
as [23]

$\langle O^{l/r}_{\Delta}(u)\rangle_{s=0}=(2\Delta-1)u_{l/r}(\tau)\,.$ (3.8)

In our case of $\Delta=1$, we shall also introduce normalized vev functions
$w_{l,r}(u)$ by

$\langle
O_{l/r}(u)\rangle_{s=0}=u_{l/r}(\tau)\equiv\frac{2\pi}{\beta}w_{l/r}(u)\,.$
(3.9)

From the above solution in (3.1), the source term for our $\Delta=1$ case may
be identified as

$s_{l/r}=b_{0}+\sum^{\infty}_{n=1}(\pm 1)^{n}b_{n}\cos
n(\tau-\tau^{s}_{n})\,.$ (3.10)

Here and below, the upper/lower sign is for the left/right system
respectively. Similarly the vev function can also be identified as

$w_{l/r}=Q_{l/r}(\tau)\sum^{\infty}_{n=1}(\pm 1)^{n+1}n\,a_{n}\cos
n(\tau-\tau^{v}_{n})\,,$ (3.11)

where $Q_{{{\ell}}/r}$ is defined by

$Q_{l/r}(\tau)=\lim_{\mu\rightarrow\mp\pi/2}\frac{\phi\cos\mu}{\bar{\phi}\,L}\,.$
(3.12)

Without the source deformation, the $Q_{l,r}$ functions may be computed as
[14]

$Q_{l/r}(\tau)=\sqrt{A^{2}_{l/r}+B^{2}_{l/r}}\,\,\Bigl{(}\,\cos(\tau-\tau_{l/r}^{B})-q_{l/r}\Bigr{)}\,,$
(3.13)

where

$\displaystyle A_{l/r}=1+{\cal O}(a^{2}),\ \ B_{l/r}={\cal O}(a^{2}),\ \
q_{l/r}={\cal O}(a^{2}),\ \ \tan\tau_{l/r}^{B}=\frac{B_{l/r}}{A_{l/r}}={\cal
O}(a^{2})\,,$ (3.14)

whose precise functional forms are also given in [14]. Once this function
$Q_{l/r}$ is given, the left/right reparameterization dynamics may be solved
explicitly by [14]

$\cos(\tau-\tau_{l/r}^{B})-q_{l/r}=\frac{1-q^{2}_{l/r}}{\cosh\frac{2\pi
u}{\phantom{ii}\beta_{l/r}}+q_{l/r}}$ (3.15)

with the temperature

$T_{l/r}=T\sqrt{A^{2}_{l/r}+B^{2}_{l/r}}\sqrt{1-q^{2}_{l/r}}=T\,\big{(}1+{\cal
O}(a^{2})\big{)}\,.$ (3.16)

One is then led to

$Q_{l/r}=\frac{\sqrt{1-q^{2}_{l/r}}}{\cosh\frac{2\pi
u}{\phantom{ii}\beta_{l/r}}+q_{l/r}}\frac{{\beta}}{{\phantom{1}\beta_{l/r}}}\,.$
(3.17)

Thus from these overall factors in the vevs, one may see that the vevs decay
exponentially in the boundary time $u$, whose decay rate is precisely the
expected one with our dimension one operator. With these vev deformations, the
boundary Schwarzian dynamics are basically those of asymptotically black hole
spacetimes.

By turning on source deformations only, the boundary reparameterization
dynamics may be solved in a similar manner. The resulting 2d spacetime
describes again left-right asymmetric black holes in general. If one turns on
both the source and the vev deformations at the same time, the left and right
black holes are further excited, which will be reflected in the boundary
Schwarzian dynamics by the excitation of the corresponding reparameterization
modes. In any of the deformations mentioned in the above, one may show that
$\tau_{l/r}(u=\infty)-\tau_{r/l}(u=-\infty)\leq\pi$, which implies that one
cannot send a signal from one side to the other [14]. Namely the two
boundaries are causally disconnected from each other. From the view point of
boundary systems, the left and right systems are completely decoupled from
each other without any direct interactions to permit any information transfer
between them. Note further that in general the left and right black holes
become different from each other as a result of deformation. Especially their
temperatures become different from each other in general.

From now on, let us focus on the above deformations only to the leading order,
ignoring any higher order corrections. In this limit, the left-right black
holes remain unperturbed with $T_{l}=T_{r}=T$. On the left/right cutoff
trajectory, $\tau_{l/r}(u)$ is ranged over $(-\pi/2,\pi/2)$ and the
reparameterization is solved by $\sin\tau_{l/r}=\tanh\frac{2\pi}{\beta}u$,
respectively. The left and right source terms $s_{l,r}$ take the same forms as
(3.10) and the vev functions $w_{l,r}$ become

$w_{l/r}=\cos\tau\,\sum^{\infty}_{n=1}(\pm 1)^{n+1}n\,a_{n}\cos
n(\tau-\tau^{v}_{n})\,.$ (3.18)

Let us first show the independence of the left and right perturbations. By
defining

$s(\tau)=\left[\begin{array}[]{lll}s_{r}(\tau)&\ \ \ {\rm
for}&-\frac{\pi}{2}<\tau<\frac{\pi}{2}\\\ s_{l}(\tau-\pi)&\ \ \ {\rm
for}&\phantom{ii}\frac{\pi}{2}<\tau<\frac{3\pi}{2}\end{array}\right.$ (3.19)

one then finds

$s(\tau)=b_{0}+\sum^{\infty}_{n=1}(-1)^{n}b_{n}\cos n(\tau-\tau^{s}_{n})$
(3.20)

which describes a general $2\pi$-periodic real function defined over the range
$(-\pi/2,3\pi/2)$. This implies that the source functions $s_{l,r}$ defined
over the range $(-\pi/2,\pi/2)$ become totally independent from each other.
This of course agrees with the fact that the source terms in the left and
right boundary actions can be turned on independently from each other. By a
similar argument, one may show that the vev functions $w_{l,r}$ may also be
turned on independently from each other.

One straightforward consequence of the above consideration is that one may in
principle recover the full set of source mode-coefficients
$\\{b_{n}e^{-in\tau^{s}_{n}}\\}$ from the left and right boundary data
specified by $s_{l,r}(\tau)$ with $\tau\in(-\pi/2,\pi/2)$. (Of course a
similar statement can also be made for the vev deformations.) Therefore, in
the small deformation limit (i.e. working in the leading order of the above
deformations), probing the left and right boundary perturbations all together,
one may recover the corresponding mode-coefficients completely. However,
including the higher order correction in bulk geometries, the situation
changes completely. Due to the shifts in $\tau_{l/r}(\pm\infty)$, the total
interval size $\Delta\tau_{l}+\Delta\tau_{r}$ of the left right cutoff
trajectories in $\tau$ space becomes in general less than $2\pi$ where
$\Delta\tau_{l/r}$ denotes $\tau_{l/r}(\infty)-\tau_{l/r}(-\infty)$,
respectively. This implies that the mode-coefficients cannot be fully
recovered from the left and right boundary data collected along the full
trajectories. On the other hand, one may try to investigate the corresponding
bulk profiles of the scalar field. For instance consider the two-independent
bulk functions $\chi_{s/v}(\mu,\tau_{1})$ and $\chi_{s/v}(\mu,\tau_{2})$ with
an appropriate choice of $\tau_{1}$ and $\tau_{2}$ ($\tau_{1}\neq\tau_{2}$),
for which the Neumann/Dirichlet boundary condition is imposed at
$\mu=\pm\pi/2$ for the source/vev deformation, respectively. From these two
independent functions, one may verify that the mode-coefficients can be
recovered completely333This does not imply an existence of a bulk observer who
may collect all the required information for the recovery while traveling in
the bulk.. Thus it seems that the bulk in this case contains more information
than the ones that may be probed from the boundaries. One may speculate that
this hidden information in the bulk is responsible for those nontrivial
behind-horizon degrees of freedom such as Python’s lunch degrees of freedom
discussed in [14].

In the next section, we shall identify the deformed boundary states from which
the above leading-order bulk gravity results follow precisely.

##  4 Deformation of thermofield double state

In this section, we shall introduce the thermofield double state [24] in the
boundary theory and its deformations that reproduce the above mentioned bulk
results to the leading order. Especially, we would like to focus on the state
deformations which lead to the vev functions in (3.9) and (3.18). Below we
shall also discuss their relation to the bulk gravity description. In these
vev deformations, we expect that the SL(2,R) symmetries of the boundary system
will be realized with a certain Hilbert space representation [15, 16]. In the
next section we shall present a specific form of orthonormal basis that is
directly related to the operator basis of the vev deformations.

We shall begin with the thermofield double initial state in the boundary
theory defined by

$|\Psi\rangle=\frac{1}{\sqrt{Z}}\sum_{m,n}\langle
n|\,U\,|m\rangle~{}|\bar{m}\rangle_{l}|n\rangle_{r}\,,\qquad Z={\rm
tr}\,U^{\dagger}U\,,$ (4.1)

where $U$ is a Euclidean evolution operator that will be further specified
below and $|\bar{m}\rangle$ denotes the CTP conjugated state of a basis state
$|m\rangle$. In our convention, any operators labeled by $l/r$ will act on the
left/right Hilbert space respectively with an extra transpose operation in the
case of left side operators. The Lorentz time evolution of this initial state
$|\Psi\rangle$ is given by

$|\Psi(t_{l},\,t_{r})\rangle={\cal
T}_{l}\,e^{i\int^{t_{l}}_{0}dt_{l}\,H^{T}_{l}(-t_{l})}\otimes{\cal
T}_{r\,}e^{-i\int^{t_{r}}_{0}dt_{r}\,H_{r}(t_{r})}~{}|\Psi\rangle\,,$ (4.2)

where the left/right time parameters denoted as $t_{l,r}$ may run
independently in general but our boundary time $u$ is related to them by
$t_{r}=-t_{l}=u$. Here, $H_{l/r}(u)$ is the Hamiltonian obtained from (3.7)
and ${\cal T}_{l}/{\cal T}_{r}$ represents the time ordering in the direction
where $-t_{l}/t_{r}$ increases, respectively.

For the undeformed case, the Euclidean evolution operator is given by

$U_{0}=e^{-\frac{\beta}{2}H_{0}}\,,$ (4.3)

and the thermofield double state becomes [7]

$|\Psi\rangle_{0}=\frac{1}{\sqrt{Z_{0}}}\sum_{n}e^{-\frac{\beta}{2}E_{n}}|\bar{n}\rangle_{l}|n\rangle_{r}\,,\qquad
Z_{0}=\sum_{n}e^{-{\beta}E_{n}}\,,$ (4.4)

with an energy eigen-basis of $H_{0}$, from which usual thermal correlation
functions may be obtained as their expectation values.

Now let us deal with the leading order of perturbation. In this case the
Euclidean evolution $U$ consists of the left, the right Euclidean evolution
and a mid-point insertion of the operator for the vev deformation. Namely, it
has a form of

$U=U_{r}\,e^{\cal{V}}\,U_{l}\,.$ (4.5)

The left and right evolution operators are basically obtained by an
appropriate Euclidean continuation of the Lorentzian counterparts in (4.2).
For the right side, we use the usual analytic continuation rule with
$t_{r}=-it_{E}$; The right Lorentzian time ranged over
$(-\infty,0)/(0,\infty)$ is mapped to the Euclidean time $t_{E}$ ranged over
$(-\beta/4,0)/(0,\beta/4)$, respectively. On the right side of Figure 2, we
depict the Euclidean version of the (undeformed) bulk geometry where the
blue/red colored semicircle is for the left/right boundary, respectively. The
red-colored (right-side) semicircle is covered by the Euclidean time range
$t_{E}\in(-\beta/4,\beta/4)$ where the full circle has a circumference
$\beta$. The continuation of the left side is more subtle; We use an analytic
continuation rule $t_{l}\rightarrow it^{l}_{E}$ but with a further shift by
$\pm\frac{\beta}{2}i$ leading to $t_{l}=it_{E}=i(t^{l}_{E}\pm\beta/2)$. Then
the Lorentzian time $t_{l}$ ranged over $(-\infty,0)/(0,\infty)$ is first
mapped to $t^{l}_{E}$ ranged, respectively, over $(-\beta/4,0)/(0,\beta/4)$
but, including the shift, to the Euclidean time $t_{E}$ ranged over
$(\,\beta/4,\,\beta/2)/(-\beta/2,-\beta/4)$, respectively. This range of the
Euclidean time is depicted by the blue-colored semicircle on the right panel
of Figure 2. Combining the left and right semicircles, the full boundary
circle is covered by the range $(-\beta/2,\beta/2)$. With this preparation, it
is proposed in [25] that the left and right Euclidean evolutions are given,
respectively, by

$U_{l}={\cal
T}_{E}\,e^{-\int_{\negthinspace-\negthinspace\frac{\beta}{2}}^{\negthinspace-\negthinspace\frac{\beta}{4}}dt_{E}\,H_{l}(-it_{E})},\
\ \ U_{r}={\cal
T}_{E}\,e^{-\int_{\negthinspace-\negthinspace\frac{\beta}{4}}^{\,\,0}dt_{E}\,H_{r}(-it_{E})}\,.$
(4.6)

This proposal with ${\cal V}=0$ is tested for the two-sided Janus black holes
[26, 25] and shown to reproduce the expected vev function to the leading order
precisely [25]; Indeed, one can also show that the vev for the 2d two-sided
Janus black hole in (2.15) is reproduced by a similar computation, whose
details will be omitted in this note. We shall not test this part of the
proposal any further and, instead, focus on the vev deformation in the
following.

On the left side of Figure 2, we depict the lower half of Euclidean geometry
combined with the subsequent Lorentzian evolution where the former is used to
generate the deformed thermofield initial state. To prepare a thermofield
initial state by the Hartle-Hawking construction, we need to patch the
Euclidean part to the Lorentzian one along an appropriate hypersurface. To the
leading order of deformation, the bulk geometry will not be deformed and one
may patch the geometries along the time-reversal symmetric slice at $\tau=0$.

$t_{l}$$t_{r}$ Figure 2: On the left, we depict the lower half of the
Euclidean geometry combined with the subsequent Lorentzian evolution. The
former is used to generate the deformed thermofield initial state. The right
figure illustrates the full Euclidean evolution which may be used to compute
the normalization factor $Z$ or the thermal expectation value (vev) of
operators with an appropriate insertion of operator $O(t)$.

Let us now include the vev deformation and construct the corresponding mid-
point inserted operator given by

${\cal{V}}=\frac{\beta}{2}\sum^{\infty}_{n=1}a_{n}\,e^{in\tau^{v}_{n}}\,O_{n}\big{(}\text{\footnotesize$\frac{\beta}{4}$}i\big{)}$
(4.7)

with

$O_{n}(-it_{E})={\cal P}_{n-1}\,O(-it_{E})\,,$ (4.8)

where a differential operator ${\cal P}_{n-1}\,(n=1,2,\cdots)$ will be further
specified below. The operator $e^{\cal V}$ may also be realized as a Euclidean
evolution operator

$e^{\cal{V}}={\cal
T}_{E}\,e^{+\int^{\,\,\epsilon-\frac{\beta}{4}}_{-\epsilon-\frac{\beta}{4}}dt_{E}g(t_{E})O(-it_{E})}$
(4.9)

with

$g(t_{E})=\text{\footnotesize$\frac{\beta}{2}\,$}\delta\big{(}t_{E}+\text{\footnotesize$\frac{\beta}{4}$}\big{)}\sum^{\infty}_{n=1}a_{n}e^{in\tau^{v}_{n}}\,{\cal{P}}_{n-1}\,,$
(4.10)

where we take the $\epsilon\rightarrow 0$ limit in the end. In Figure 2, these
insertions are represented by the black dots in the lower or the upper half of
the Euclidean evolution.

One may also introduce a Hamiltonian along the lower half of the Euclidean
evolution by

$H(-it_{E})=\left[\begin{array}[]{lll}\ H_{l}(-it_{E})\,,&&\
\,\,-\frac{\beta}{2}\,\,<\,t_{E}<\negthinspace\negthinspace-\epsilon\negthinspace-\negthinspace\frac{\beta}{4}\\\
-g(t_{E})O(-it_{E})\,,&&-\epsilon\negthinspace-\negthinspace\frac{\beta}{4}\negthinspace<\,t_{E}<\,\,\epsilon\negthinspace-\negthinspace\frac{\beta}{4}\\\
\ H_{r}(-it_{E})\,,&&\ \
\epsilon\negthinspace-\negthinspace\frac{\beta}{4}<\,t_{E}\,<\,\,\,0\end{array}\right.\,.$
(4.11)

Then $U$ will be given by

$U={\cal
T}_{E}\,e^{-\int^{\,0}_{{\negthinspace}-{\negthinspace}\frac{\beta}{2}}dt_{E}H(-it_{E})}$
(4.12)

including the contribution from the mid-point insertion. In the Euclidean
space, the counterpart of the Lorentzian unitarity requires the reflection
positivity

$H^{\dagger}(-it_{E})=H(it_{E})\,,$ (4.13)

by which one may also introduce the Euclidean Hamiltonian $H(-it_{E})$ for the
upper half of the full thermal circle ranged over $(0,\beta/2)$. Based on
this, it is straightforward to show that

$U^{\dagger}={\cal T}_{E}\,e^{-\int_{\,0}^{\frac{\beta}{2}}dt_{E}H(-it_{E})}$
(4.14)

and the Euclidean evolution along the full-circle is then given by
$U^{\dagger}U$.

We shall now come to the purely vev deformation without any source terms
introduced. The Euclidean evolution operator $U$ in this case reads

$U=e^{-\frac{\beta}{4}H_{0}}\,e^{\cal V}e^{-\frac{\beta}{4}H_{0}}\,,$ (4.15)

from which we would like to reproduce (3.9) with (3.18). Below we shall focus
on $\langle O_{r}(t)\rangle$; Of course $\langle O_{l}(t)\rangle$ may be
treated in the same way, but we shall not repeat the latter computation in
this note. Let us first note that444Similarly $\langle{O_{l}}(t)\rangle$ is
given by $\langle\Psi|\,{O}(t)^{T}\otimes{\bf 1}|\Psi\rangle$.

$\langle{O_{r}}(t)\rangle=\langle\Psi|{\bf 1}\otimes{O}(t)|\Psi\rangle$ (4.16)

which may be evaluated perturbatively. With the fact that the undeformed vev
of ${O}$ vanishes, it is straightforward to show that

$\langle{O_{r}}(t)\rangle={\cal A}_{-}(t)+{\cal A}_{+}(t)+{\cal
O}(a_{n}^{2})\,,$ (4.17)

where

${\cal A}_{\pm}(t)=\lim_{\epsilon\rightarrow
0}\int^{\,\,\epsilon\pm\frac{\beta}{4}}_{-\epsilon\pm\frac{\beta}{4}}dt_{E}\,g(t_{E})\,\frac{1}{Z_{0}}\text{tr}\Big{[}e^{-\beta
H_{0}}\,{O}(t)\,{O}(-it_{E})\Big{]}\,.$ (4.18)

For the boundary CFT, the thermal two-point correlation is well known as

$\frac{1}{Z_{0}}\text{tr}\Big{[}e^{-\beta
H_{0}}\,{O}(t)\,{O}(-it_{E})\Big{]}=\frac{\text{\footnotesize$2\pi/\beta^{2}$}}{1-\cosh\frac{2\pi}{\beta}({t}+i{t_{E}})}\,,$
(4.19)

where its normalization is worked out in [27] (See also [11]). In order to
reproduce (3.18) from the above perturbative computation with only $a_{1}$
turned on, we need to take $g(t_{E})$ in (4.18) as

$g_{1}(t_{E})=\frac{\beta}{2}\Big{[}e^{i\tau^{v}_{1}}\,\delta(t_{E}+\beta/4)+e^{-i\tau^{v}_{1}}\,\delta(t_{E}-\beta/4)\Big{]}\,,$
(4.20)

where we have introduced a notation
$g(t_{E})=\sum^{\infty}_{n=1}a_{n}g_{n}(t_{E})$ and used the identity (Recall
that $\cosh\frac{2\pi}{\beta}t=1/\cos\tau$)

$\frac{1}{1-\cosh\frac{2\pi}{\beta}\big{(}t-\frac{\beta}{4}i\big{)}}=e^{-i\tau}\cos\tau\,.$
(4.21)

From this, we conclude that ${\cal{P}}_{0}=1$. One may also check that no
choice other than the above insertion point works in reproducing (3.18).

For $g_{n}$ with $n>1$, one could adopt the following recursive strategy. The
integration in (4.18) together with the expression in (4.19) can be used to
find the differential operator ${\cal{P}}_{n}$ acting on a function of $t_{E}$
as

${\cal{P}}_{n}(x)\frac{1}{1-\cosh\frac{2\pi}{\beta}(t+it_{E})}\bigg{|}_{t_{E}=-\beta/4}=(-1)^{n}(n+1)~{}e^{-i(n+1)\tau}\cos\tau\,,$
(4.22)

where $x$ denotes the differential operator
$-\frac{\beta}{2\pi}\frac{d}{dt_{E}}$. By a change of variables
$\cosh\frac{2\pi}{\beta}t=1/\cos\tau$, the above expression becomes

${\cal{P}}_{n}(x)e^{-i\tau}\cos\tau=(-1)^{n}(n+1)~{}e^{-i(n+1)\tau}\cos\tau\,.$
(4.23)

with $x=-i\cos\tau\frac{d}{d\tau}$. Acting with the differential operator
$-i\cos\tau\frac{d}{d\tau}$ on this defining equation once more, one may
obtain the following recursion relation

$2x\,{\cal{P}}_{n}(x)=(n+1){\cal{P}}_{n+1}(x)+(n-1){\cal{P}}_{n-1}(x)$ (4.24)

with initial conditions

${\cal{P}}_{0}(x)=1\,,\qquad{\cal{P}}_{1}(x)=2x\,.$ (4.25)

One may notice that the solutions to this recursion relation are given by a
special type of the Meixner-Pollaczek polynomials [28] ,

${\cal{P}}_{n}(x)={\cal{P}}^{\lambda=1}_{n}(x\,;\,{\textstyle\phi=\frac{\pi}{2}})\,,$
(4.26)

which can be defined through a hypergeometric function as

${\cal{P}}^{\lambda}_{n}(x\,;\,\phi)=\frac{(2\lambda)_{n}}{n!}e^{in\phi}F(-n,\lambda+ix\,;\,2\lambda\,|\,1-e^{-2i\phi})\,.$
(4.27)

This completes the identification of the inserted operator ${\cal{V}}$ that
reproduces the vev in (3.18). In the above construction of the thermofield
double state, we have not included higher order corrections as we mentioned
repeatedly. For instance, consider the left-right asymmetric black hole
spacetime, which arises in quadratic order of the above deformations
generically. In this case, the above mid-point insertion will not be working
anymore and, then, one needs another prescription which, unfortunately, we do
not know how to arrange. We leave this issue to the future study.

##  5 SL(2,R) representation in the operator space

In the previous section, we have constructed the mid-point inserted operator
$\cal V$ which is realized as a linear combination of the operators
$\\{O_{1},O_{2},\cdots\\}$. By each of this insertion, one may obtain the
corresponding initial state $|\psi\rangle_{\text{\tiny$\cal V$}}$. This
amounts to a variation of operator-state maps in general CFTs. This
realization of states is highly nonlinear in terms of the coefficients
$\\{a_{1}e^{i\tau_{1}^{v}},a_{2}e^{2i\tau_{2}^{v}},\cdots\\}$ especially
including the gravity correction via the deformation of the dilaton field
which is quadratic in $a_{n}$ as shown in (3.5).

Since our AdS2 dynamics involves SL(2,R) symmetries in general, it is expected
that the inserted operators also transform under the symmetries. In this
section, we would like to clarify how these operators $O_{n}$ form a
representation of the SL(2,R) algebra

$[B,E]=iP\,,\ \ \ [E,P]=iB\,,\ \ \ [B,P]=iE\,,$ (5.1)

where $B$, $P$ and $E$ denote the three SL(2,R) generators. For later purpose,
let us also introduce a Casimir operator ${\cal
C}=B^{2}+P^{2}-E^{2}=K_{+}K_{-}-E(E-1)$ where the raising and lowering
operators $K_{\pm}$ are defined by $K_{\pm}=B\mp iP$. We note that this
realization of the symmetries is already explored in Section  4.2 of Ref.
[16], where the generators of SL(2,R) are identified as555These generators are
related to the ones in Ref. [16] by an automorphism $\tau\rightarrow\tau+\pi$.

$B=-i\cos\tau\partial_{\tau}\,,\qquad
P=-i\Big{(}\sin\tau\partial_{\tau}+\frac{1}{\cos\tau}\Big{)}\,,\qquad
E=i\Big{(}\partial_{\tau}+\tan\tau\Big{)}\,.$ (5.2)

Below, we shall show how these generators are acting upon the space of
${\cal{P}}_{n}$ explicitly. First, using the orthogonality property of
Meixner-Pollaczek polynomials given by [28]

$\displaystyle\int^{\infty}_{-\infty}dx\,P^{\lambda}_{n}(x)P^{\lambda}_{m}(x)w(x\,;\lambda,\phi)$
$\displaystyle=\frac{2\pi\,\text{\footnotesize$\Gamma(n+2\lambda)$}}{\,\,n!\,(2\sin\phi)^{2\lambda}}\,\delta_{nm}\,,\quad\lambda>0\,,\
0<\phi<\pi\,,$

with $w(x\,;\lambda,\phi)=|\Gamma(\lambda\negthinspace+\negthinspace
ix)|^{2}e^{(2\phi-\pi)x}$, one may introduce a new set of orthonormal
polynomials $e_{m}(x)$ defined by

$e_{m}(x)\equiv\sqrt{\text{\footnotesize$\frac{2}{m\pi}$}}\,{\cal{P}}_{m-1}(x)\,,\qquad
m=1,2,\cdots\,.$ (5.3)

These normalized polynomials $e_{m}(x)$ satisfy then the following recursion
relation

$x\,e_{m}(x)={\textstyle\frac{\sqrt{m(m+1)}}{2}\,e_{m+1}(x)+\frac{\sqrt{m(m-1)}}{2}\,e_{m-1}(x)}\,.$
(5.4)

This also implies that

$\displaystyle B\cdot e_{m}\cdot e^{-i\tau}\cos\tau$
$\displaystyle=\left[{\textstyle\frac{\sqrt{m(m+1)}}{2}\,e_{m+1}+\frac{\sqrt{m(m-1)}}{2}\,e_{m-1}}\right]\cdot
e^{-i\tau}\cos\tau\,,$ $\displaystyle P\cdot e_{m}\cdot e^{-i\tau}\cos\tau$
$\displaystyle=i\Big{[}{\textstyle\frac{\sqrt{m(m+1)}}{2}\,e_{m+1}-\frac{\sqrt{m(m-1)}}{2}\,e_{m-1}}\Big{]}\cdot
e^{-i\tau}\cos\tau\,,$ $\displaystyle E\cdot e_{m}\cdot e^{-i\tau}\cos\tau$
$\displaystyle=m\,e_{m}\cdot e^{-i\tau}\cos\tau\,,$ (5.5)

where the actions of $P$ and $E$ are computed using (4.23) followed by the
$\tau$-space operations given by (5.2). Thus these latter two computations
require an auxiliary function $e^{-i\tau}\cos\tau$ on which $e_{m}$ is acting
upon.

In fact, one may construct the SL(2,R) generators that are acting on
$e_{m}(x)$ directly. This may be achieved by additionally including a finite
translation operation given by (See [29, 30] for mathematical precedents)

$e^{\pm i\frac{d}{dx}}f(x)=f(x\pm i)\,.$ (5.6)

Then the three generators may be realized as

$B=x\,,\ \ \ P=i\cos({\textstyle\frac{d}{dx}})\,x\,,\ \ \
E=\sin({\textstyle\frac{d}{dx}})\,x\,,$ (5.7)

which act on an arbitrary square-integrable complex function $f(x)$. To verify
these expressions, we first note that $\\{e_{1}(x),e_{2}(x),\cdots\\}$ forms
an orthonormal basis for any square-integrable function $f(x)$. The
identification of $B=x$ is already introduced in the above construction. The
expression for $E$ can be found as follows; One starts from the generating
function of ${\cal{P}}_{n}(x)$ given by

$G(t,x)\equiv\sum_{n=0}^{\infty}t^{n}{\cal{P}}_{n}(x)=\frac{1}{1+t^{2}}\,e^{2x\arctan
t}\,.$ (5.8)

By integration of the both sides with respect to $t$ followed by a
multiplication of $x$, one obtains

$\sum_{m=1}^{\infty}\frac{1}{m}t^{m}\,x\,{\cal{P}}_{m-1}(x)=\frac{1}{2}\,\left(e^{2x\arctan
t}-1\right)$ (5.9)

and, by a further action of $\sin({\textstyle\frac{d}{dx}})$ on both sides, is
led to

$\displaystyle\sum_{m=1}^{\infty}\frac{1}{m}t^{m}\,\sin({\textstyle\frac{d}{dx}})\big{[}\,x\,{\cal{P}}_{m-1}(x)\big{]}=\frac{1}{2}e^{2x\arctan
t}\sin 2\arctan t$ $\displaystyle=\frac{t}{1+t^{2}}e^{2x\arctan
t}=\sum_{m=1}^{\infty}t^{m}{\cal{P}}_{m-1}(x)\,.$ (5.10)

Thus, with $E=\sin({\textstyle\frac{d}{dx}})x$, $E\,e_{m}(x)=m\,e_{m}(x)$
which is the desired result reproducing the third line of (5). This
demonstrates our expression for $E$ in (5.7). Finally, the expression for $P$
in (5.7) can be found from $-i[B,E]$, which is acting on $e_{m}$ by

$P\,e_{m}=i\Big{[}{\textstyle\frac{\sqrt{m(m+1)}}{2}\,e_{m+1}-\frac{\sqrt{m(m-1)}}{2}\,e_{m-1}}\Big{]}\,.$
(5.11)

It is also straightforward to check the generators constructed in this way
satisfy the SL(2,R) algebra in (5.1) with ${\cal C}=0$. Among unitary
representations of SL(2,R), there is the so called discrete representation
${\cal D}^{+}_{j}$ (See for example [31]), which is realized in the Hilbert
space

${\cal D}^{+}_{j}=\\{|jm\rangle;m=j,j+1,j+2,\cdots\\}$ (5.12)

with $j$ real and positive, $K_{-}|jj\rangle=0$, and ${\cal C}=-j(j-1)$. Now
one may straightforwardly confirm that the above representation belongs to
${\cal D}^{+}_{j=1}$ with ${\cal C}=0$.

It is natural to anticipate that the dimension $\Delta$ operator dual to the
massive scalar field is represented by ${\cal D}^{+}_{j=\Delta}$ with ${\cal
C}=-\Delta(\Delta-1)=-m^{2}$. It may be interesting to construct all the
representations of operators dual to the massive scalar fields by our method.
For a more group-theoretic approach to this topic, one may refer to the
reference [15].

##  6 Conclusions

In this work, we have considered the most general normalizable and
nonnormalizable bulk deformations in Jackiw-Teitelboim gravity with a massless
field which corresponds to either vev deformations or source deformations of
the thermofield double state in the dual boundary theory. Such deformations
are, in general, left-right asymmetric, resulting in different black hole
temperatures. Moreover, we have argued that, classically, the bulk profiles
may not be fully recovered from the data collected along the boundary cutoff
trajectories. Then the bulk seems to contain more information than the cutoff
boundaries and this might be responsible for the behind-horizon degrees of
freedom such as those of Python’s lunches.

The deformed state can be prepared by inserting operators on the boundary of
Euclidean AdS2 in the context of Hartle-Hawking construction. In the limit of
small vev deformations, we have explicitly identified the operators for the
bulk deformations which are all inserted at the mid-point during the Euclidean
time evolution along the lower half of the boundary of the thermal disk. We
have found that inserted operators form a discrete SL(2,R) representation
$\mathcal{D}_{j=1}^{+}$ with vanishing Casimir. Since the boundary system has
SL(2,R) symmetries, it is natural to anticipate such a realization in the
operator space. If the matter is massive with mass $m$ instead of massless,
the corresponding SL(2,R) representation would be $\mathcal{D}_{j=\Delta}^{+}$
with $\mathcal{C}=-m^{2}$. See [15] for this conclusion from a slightly
different perspective.

In constructing the operators corresponding to vev functions, we have ignored
higher order corrections. The approximation allows us to work in left-right
symmetric undeformed geometries, since the asymmetry arises starting from the
second order in deformations. Inclusion of higher order terms would involve
asymmetric black hole spacetime, and the mid-point insertion of operators
would no longer be a valid prescription. We leave this issue to the future
study. Finally, it would be interesting to generalize the results of this
paper by turning on both the vev and the source deformations at the same time,
for which black holes are further excited as discussed in [14].

### Acknowledgement

We would like to thank Andreas Gustavsson for careful reading of the
manuscript. DB was supported by the 2023 Research Fund of the University of
Seoul. C.K. was supported by NRF Grant 2022R1F1A1074051. S.-H.Y. was supported
by NRF Grant 2021R1A2C1003644 and supported by Basic Science Research Program
through the NRF funded by the Ministry of Education (NRF-2020R1A6A1A03047877).

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# FastFlows: Flow-Based Models for Molecular Graph Generation

Nathan C. Frey
MIT
<EMAIL_ADDRESS>
Vijay Gadepally
MIT
<EMAIL_ADDRESS>
Bharath Ramsundar
Deep Forest Sciences
<EMAIL_ADDRESS>

###### Abstract

We propose a framework using normalizing-flow based models, SELF-Referencing
Embedded Strings, and multi-objective optimization that efficiently generates
small molecules. With an initial training set of only 100 small molecules,
FastFlows generates thousands of chemically valid molecules in seconds.
Because of the efficient sampling, substructure filters can be applied as
desired to eliminate compounds with unreasonable moieties. Using easily
computable and learned metrics for druglikeness, synthetic accessibility, and
synthetic complexity, we perform a multi-objective optimization to demonstrate
how FastFlows functions in a high-throughput virtual screening context. Our
model is significantly simpler and easier to train than autoregressive
molecular generative models, and enables fast generation and identification of
druglike, synthesizable molecules.

##  1 Introduction

The goal of generative modeling of small molecules is to discover structurally
novel molecules with optimal physicochemical properties. Prior work using
variational autoencoders (VAEs) [1], generative adversarial networks (GANs)
[2], and reinforcement learning [3] has shown the promise of generative
modeling in the chemical sciences. Normalizing flows (NFs) [4] have emerged as
a promising model architecture for chemical space distribution learning and
molecular graph generation [5; 6; 7; 8]. Unlike the generative models
discussed above, NFs do not rely on a compressed latent space representation
for generative modeling. Instead, an NF learns an invertible mapping between a
simple base distribution and a target distribution.

Previous work applying NFs to molecule generation [5] has shown that NFs with
post-hoc corrections to enforce chemical validity achieve high validity,
novelty, and uniqueness scores on benchmark datasets like QM9 [9] and ZINC250K
[10]. However, NFs require deep architectures with many bijective
transformations to model complex target distributions [11; 12; 13], which
results in a prohibitive computational cost to training flows [14]. Some
applications of generative modeling may call for learning immense,
heterogeneous regions of chemical space, which deep DGMs seem well-suited for.
It remains to be seen whether DGMs have any utility in exploring more compact,
well-defined chemical spaces, where simpler methods like genetic algorithms
[15] and particle swarm optimization [16] are easier to use and just as, if
not more, performative.

In this paper, we present FastFlows, a normalizing flow-based approach for
fast and efficient molecular graph sampling with DGMs. Through careful choice
of the underlying flow architecture, FastFlows avoids the common difficulties
and instabilities of training other generative models like GANs and VAEs.
Using the 100% robust SELF-referencing Embedded Strings (SELFIES)
representation [17] ensures that all generated samples are chemically valid,
so we can train FastFlows in low-data limits (10s or 100s of training
datapoints) to "memorize" a target distribution, while still generating novel,
unique, and valid samples. Because FastFlows uses a simple molecular graph
representation (SELFIES) and does not use autoregressivity, tens of thousands
of molecules can be generated each second, avoiding the prohibitive time
complexity of sampling from autoregressive DGMs. FastFlows uses substructure
filters to discard unreasonable molecules, and easily computed metrics for
druglikeness and synthetic accessibility/complexity are used to identify
Pareto optimal molecules from thousands of generated candidates. This work
presents an effective scheme for practical, robust molecular generation with
DGMs, and a path towards discovering novel druglike compounds.

##  2 FastFlows: Efficient Flow-Based Generative Models

To circumvent a common problem with molecular generative models - invalid
outputs due to chemical rules not being encoded in the model architecture - we
first encode molecules as SELFIES strings (Figure 1a). The SELFIES grammar and
bond constraints enforce chemical valency rules, guaranteeing that generated
SELFIES are syntactically and semantically valid, without requiring post-hoc
corrections or complex model architectures that are difficult to train. In a
normalizing flow (NF) (Figure 1b), vectors from a simple base distribution
like a multi-dimensional Gaussian are passed through the flow, and a sequence
of invertible transformations maps the vector to a sample from the target
distribution.

We use an NF to model target distributions of molecules. Because NFs are
composed of bijective transformations, they provide an easily interpretable
one-to-one mapping between inputs and outputs, without lossy compression to a
latent space representation. NFs offer both generative sampling and exact
likelihood calculation, unlike VAEs which provide only a lower-bound on log-
likelihood and GANs, which do not provide likelihood estimation. Here, the NF
provides a simple and straightforward means of generating new molecules.

Figure 1: a Representative molecule encoded as SMILES and SELFIES strings. b
Normalizing flow architecture that maps an input distribution to a target
distribution through invertible transformations.  Figure 2: Dequantized one-
hot encodings of SELFIES representations are inputs to the normalizing flow.
The normalizing flow maps a simple base distribution to a complex target
distribution.

A schematic of the data pre-processing steps and mapping between the base and
target distributions is shown in Figure 2. SELFIES strings are one-hot encoded
and dequantized [11] by adding random noise from the interval $[0,0.95)$ to
each element. The original inputs can be recovered by applying a floor
function, and the continuous dequantized inputs are used to train the model.
The NF uses real-valued non-volume preserving (Real NVP) transformations [11]
with 32 layers and 8 residual blocks per layer with 16 hidden feature maps,
and checkerboard masking.

Real NVP has the advantages of efficient and fast training and sampling,
unlike autoregressive models, because sampling is parallelized over input
dimensions [11]. Sampling efficiency is a key factor in enabling FastFlows,
although Real NVP is less expressive and requires a deeper network
architecture than more recently developed autoregressive and residual flows
[12; 18]. The base distribution is a multi-dimensional standard normal. When
training FastFlows, a small batch size of 4 is used, the learning rate is set
to 1e-3, and the dataset is restricted to 100 samples from the target
distribution.

Figure 3: FastFlows workflow diagram. 100K molecules are sampled from a
trained FastFlow in 4.2 seconds. Substructure filters are applied as needed,
and objective targets are calculated for each sample. Pareto optimal samples
are identified according to trade-offs between druglikeness and
synthesizability.

The molecular graph generation workflow is shown schematically in Figure 3.
The generative model is trained on an initial subset of training data. Due to
the parallelized, efficient sampling of Real NVP, we generate samples and
transform them back into SMILES strings at a rate of 23,981 molecules/second
on a single NVIDIA Tesla V100 GPU. Samples are filtered for uniqueness, pan-
assay interfering compounds [19], and unstable or reactive moieties [20].
Again, because of the cheap computational and time costs of molecular
generation, we can generate chemically valid samples from the target
distribution and filter out undesirable compounds as needed. Using a more
sophisticated model architecture or larger training set may increase the
diversity and unique percentage of generated samples, but FastFlows is
designed for the practical purpose of rapid sampling that enables high-
throughput virtual screening of compact target distributions, rather than
generalized distribution learning.

For simplicity, we score generated molecules with quantitative estimate of
druglikeness (QED) [21], synthetic accessibility (SA) [22], and a synthetic
complexity score (SCScore) [23] learned from a corpus of chemical reactions.
The scored molecules are used to find the Pareto frontier of optimal
candidates, using the _PyePAL_ [24] package. The algorithm for finding the
Pareto frontier is given in Algorithm 1. A molecule is Pareto dominated if
there is another molecule that is at least as performative along every metric,
and more performative for at least one metric. Pareto optimal molecules define
the Pareto frontier, where target metrics cannot be improved without degrading
others.

Algorithm 1 Pareto frontier algorithm.

Input: generated molecule score values

for scores _s_ in score list do

if any score > s then $\triangleright$ Find if point is dominated

add s to dominated list

end if

end for

Return: dominated list

For example, we might intuitively expect that synthetically complex molecules
are less synthetically accessible, which introduces a trade-off between the
two properties. The Pareto frontier solves the multi-objective optimization
problem of identifying generated molecules that are maximally druglike,
synthesizable, and complex (more like a reaction product than a reactant). We
also note that QED and synthesizability metrics are correlated with model
quality [25], whereas many other commonly used generative model metrics can be
trivially satisfied by a judicious choice of molecular representation [15], or
are not correlated with model quality [25].

##  3 Experiments

Figure 4: Distributions of a synthetic accessibility, b quantitative estimate
of druglikeness, and c synthetic complexity score for generated samples.
Pareto frontiers (d-f) for generated samples.

### QM9

Our goal is to provide a framework for solving the multi-objective
optimization problem of finding druglike, synthetically accessible, and
synthetically complex molecules. While optimizing for single objectives may
lead to molecules that are too simple (synthetic accessibility), too similar
to already-known drugs (QED), or too complex (SCScore), simultaneously
optimizing for all three metrics leads to a richer exploration of the
generated samples. Figure 4a-c shows the distributions of SA, QED, and SCScore
for over 2,000 generated molecules after applying filters.

Two-dimensional Pareto frontiers are shown in Figure 4d-f, comparing trade-
offs between each combination of metrics. In Figure 4e, we observe that
calculating the Pareto frontier with respect to SCScore and SA reduces the
candidate molecule space from over 2,000 to only three molecules. Each
frontier provides a short list of Pareto optimal candidates for further
investigation, via more computationally demanding techniques like docking and
molecular dynamics simulations.

### ChEMBL

We repeat the above FastFlow experiment trained on 100 compounds from ChEMBL
[26]. ChEMBL molecules are, on average, much larger than those from QM9, which
impacts training and generation. After training on ChEMBL molecules, FastFlows
generates 481 molecules per second (versus nearly 24,000 molecules per second
when trained on QM9). Still, the extremely fast sampling means that sampling
molecules is memory-limited rather than time-limited.

While the FastFlows approach extends to more relevant target distributions for
drug discovery like ChEMBL, the increased dimensionality of interesting
chemical spaces presents a difficulty for flow-based models. For 100 random
samples from ChEMBL, the largest SELFIES string is 552 characters and the
SELFIES vocabulary includes 33 characters, yielding a target distribution with
dimensionality of 18216. Mapping the base 18216-dimensional standard normal
distribution to the target with an NF requires many successive bijective
transformations, and again highlights the need for more expressive flows that
preserve fast sampling and training.

##  4 Discussion

In this work, we presented FastFlows, a framework for generative modeling of
small molecules using normalizing flows. We showed that lightweight
normalizing flows trained in the low-data limit on SELFIES representations can
be used as computationally cheap, efficient deep generative models for
sampling chemical matter from target distributions. Fast training and the
ability to sample more than 20,000 molecules / second enable coupling of the
generative model to downstream filtering and multi-objective optimization to
identify chemically valid, unique, synthetically accessible, and complex
druglike molecules. We demonstrated this workflow on target distributions from
the QM9 and ChEMBL datasets. Importantly, the code used for deep generative
modeling in this work is available in the _DeepChem_ package [27] implemented
with TensorFlowProbability [28], and in the _nflows_ PyTorch library [29] and
the SELFIES library [17] at https://github.com/aspuru-guzik-group/selfies. We
identified challenges with using normalizing flows for distribution learning,
namely, the need for deep networks with many bijective transformations to
adequately learn a mapping to the target distribution. For this reason, it is
prohibitive from a computational cost standpoint for current flow-based models
to be competitive with other architectures on distribution learning tasks
[14].

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