# Meta Generative Attack on Person Reidentification

A V Subramanyam A V Subramanyam is with IIITD, India (e-mail:
subramanyam@iiitd.ac.in). We acknowledge the support of the IHUB-ANUBHUTI-
IIITD FOUNDATION setup under the NM-ICPS scheme of the DST, India.

###### Abstract

Adversarial attacks have been recently investigated in person re-
identification. These attacks perform well under cross dataset or cross model
setting. However, the challenges present in cross-dataset cross-model scenario
does not allow these models to achieve similar accuracy. To this end, we
propose our method with the goal of achieving better transferability against
different models and across datasets. We generate a mask to obtain better
performance across models and use meta learning to boost the generalizability
in the challenging cross-dataset cross-model setting. Experiments on
Market-1501, DukeMTMC-reID and MSMT-17 demonstrate favorable results compared
to other attacks.

###### Index Terms:

Adversarial attacks, Meta learning, ReID.

##  I Introduction

The tremendous performance of deep learning models has led to rampant
application of these systems in practice. However, these models can be
manipulated by introducing minor perturbations [1, 2, 3, 4, 5]. This process
is called adversarial attacks. In case of person re-identification, for a
given query input $x$, a target model $f$ and a gallery, the attack is defined
as,

$\displaystyle\lVert
f(\mathbf{x}+\boldsymbol{\delta})-f(\mathbf{x}_{g})\rVert_{2}>\lVert
f(\mathbf{x}+\boldsymbol{\delta})-f(\bar{\mathbf{x}}_{g})\rVert_{2}\;\;\;\textit{s.t.}\;\lVert\boldsymbol{\delta}\rVert_{p}\leq\epsilon,$
$\displaystyle\mathbf{x}_{g}\ni
topk(\mathbf{x}+\boldsymbol{\delta}),ID(\mathbf{x})=ID(\mathbf{x}_{g})\neq
ID(\bar{\mathbf{x}}_{g})$

where $\mathbf{x}_{g}$ and $\bar{\mathbf{x}}_{g}$ are gallery samples
belonging to different identity and $\boldsymbol{\delta}$ is the adversarial
perturbation with an $l_{p}$ norm bound of $\epsilon$. topk($\cdot$) refers to
the top $k$ retrieved images for the given argument.

Adversarial attacks have been extensively investigated under classification
setting [6] and also studied in other domains [7, 8, 9] in the recent times.
However, to the best of our knowledge, there are very few works which study
these attacks in person re-identification domain. In the following we briefly
discuss some classical attacks under classification setting. Szegedy _et al._
[1] proposed the first work on generation of adversarial sample for deep
neural networks using L-BFGS. Goodfellow _et al._ [2] proposed an efficient
adversarial sample generation method using fast gradient sign method (FGSM).
Kurakin _et al._ [10] proposed an iterative FGSM method. Other prominent works
include [11, 12, 13, 14, 15, 16].

In person re-id [17, 18, 19, 20], both white-box and black box attacks have
been proposed in [21, 22, 23, 24]. These attacks use a labeled source dataset
and show that the attacks are transferable under cross-dataset or cross-model,
or both settings. However, transferabilty of attacks in the challenging cross-
dataset and cross-model setting is still an issue. In this work, we propose to
use a mask and meta-learning for better transferability of attacks. We also
investigate adversarial attacks in a completely new setting where the source
dataset does not have any labels and the target model structure or parameters
are unknown.

##  II Related Works

In [25], authors propose white box and black box attacks. The black box attack
only assumes that the victim model is unknown but the dataset is available.
[26] introduces physically realizable attacks in white box setting by
generating adversarial clothing pattern. [24] proposes a query based attack
wherein the images obtained by querying the victim model are used to form
triplets for triplet loss. [27] proposes white box attack using self metric
attack; wherein the positive sample is obtained by adding noise to the given
input and obtain negative sample from other images. In [21], authors propose a
meta-learning framework using a labeled source and extra association dataset.
This method generalizes well in cross-dataset scenario. In [22], Ding  _et
al._ proposed to use a list-wise attack objective function along with model
agnostic regularization for better transferability. A GAN based framework is
proposed in [23]. Here the authors generate adversarial noise and mask by
training the network using triplet loss.

In this work we use a GAN network to generate adversarial sample. In order to
achieve better transferability of attack across models, we suppress the pixels
that generate large gradients. Suppressing these gradients allows the network
to focus on other pixels. In this way, the network can focus on pixels that
are not explicitly salient with respect to the model used for attack. We
further use meta learning [28] which also allows incorporation of an
additional dataset to boost the transferability. We refer this attack as Meta
Generative Attack (MeGA). Our work is closest in spirit to [21, 23], however,
the mask generation and application of meta learning under GAN framework are
quite distinct from these works.

##  III Methodology

In this work we address both white-box and black-box attacks. We need that the
attack is transferable across models and datasets. Thus if we obtain the
attack sample using a given model $f$, the attack is inherently tied to $f$
[16]. In order that attack does not over-learn, we apply a mask that can focus
on regions that are not highly salient for discrimination. This way the
network can focus on less salient but discriminative regions thereby
increasing the generalizability of attack to other models. On the other hand,
meta learning has been efficiently used in adversarial attacks [29, 21, 30] to
obtain better transferability across datasets. However meta learning has not
been explored with generative learning for attacks in case of PRID. We adapt
the MAML meta learning framework [28] in our proposed method. While the black
box attack works assume the presence of a labeled source dataset, we
additionally present a more challenging setting wherein no labels are
available during attack.

Figure 1: Model architecture. Mask $\mathbf{M}$ is generated using model $f$
and is used to mask the input $\mathbf{x}$. GAN is trained using a meta
learning framework with an adversarial triplet loss and GAN loss.

Our proposed model is illustrated in Figure 1. In case of white-box setting,
the generator $\mathcal{G}$ is trained using the generator loss, adversarial
triplet loss and meta learning loss while the discriminator $\mathcal{D}$ is
trained with the classical binary cross-entropy discriminator loss. The mask
is obtained via self-supervised triplet loss. The network learns to generate
adversarial image. While the GAN loss itself focuses on generating real
samples, the adversarial triplet loss guides the network to generate samples
that will be closer to negative samples and farther away from positive
samples.

###  III-A GAN training

Given a clean sample $\mathbf{x}$, we use the generator $\mathcal{G}$ to
create the adversarial sample $\mathbf{x}_{adv}$. The overall GAN loss is
given by,
$\mathcal{L}_{GAN}=E_{\mathbf{x}}\log\mathcal{D}(\mathbf{x})+E_{\mathbf{x}}\log(1-\mathcal{D}(\Pi(\mathcal{G}(\mathbf{x}))))$.
Here $\Pi(.)$ denotes the projection into $l_{\infty}$ ball of
$\epsilon$-radius within $\mathbf{x}$ and
$\mathbf{x}_{adv}=\Pi(\mathcal{G}(\mathbf{x}))$. In order to generate
adversarial samples, a deep mis-ranking loss is used [23],

$\displaystyle\mathcal{L}_{adv-
trip}(\mathbf{x}_{adv}^{a},\mathbf{x}_{adv}^{n},\mathbf{x}_{adv}^{p})$
$\displaystyle=\max(\lVert\mathbf{x}_{adv}^{a}-\mathbf{x}_{adv}^{n}\rVert_{2}$
(1)
$\displaystyle-\lVert\mathbf{x}_{adv}^{a}-\mathbf{x}_{adv}^{p}\rVert_{2}+m,0)$

where $m$ is the margin. $\mathbf{x}_{adv}^{a}$ is the adversarial sample
obtained from anchor sample $\mathbf{x}^{a}$. Similarly,
$\mathbf{x}_{adv}^{p}$ and $\mathbf{x}_{adv}^{n}$ are the adversarial samples
obtained from respective positive and negative samples $\mathbf{x}^{p}$ and
$\mathbf{x}^{n}$. This loss tries to push the negatives closer to each other
and pulls the positives farther away. Thus the network learns to generate
convincing adversarial samples.

###  III-B Mask Generation

Attack obtained using the given model $f$ leads to poor generelization to
other networks. In order to have a better tranferability, we first compute the
gradients with respect to self-supervised triplet loss $\mathcal{L}_{adv-
trip}(\mathbf{x},\mathbf{x}^{n},\mathbf{x}^{p})$, where $\mathbf{x}^{p}$ is
obtained by augmentation of $\mathbf{x}$ and $\mathbf{x}^{n}$ is the sample in
the batch which lies at a maximum Euclidean distance from $\mathbf{x}$. Here,
the large gradients are primarily responsible for loss convergence. Since this
way of achieving convergence is clearly coupled with $f$, we mask the large
gradients. Thus, the convergence is not entirely dependent on the large
gradients and focuses on other smaller ones which can also potentially posses
discriminative nature. Thus the overfitting can be reduced by using the mask.
To obtain the mask, we compute,

$\mathbf{grad}_{adv-triplet}=\nabla_{\mathbf{x}}\mathcal{L}_{adv-
trip}(\mathbf{x},\mathbf{x}^{n},\mathbf{x}^{p})$ (2)

Note that, we use the real samples in Eq. 2. The mask is given by
$\mathbf{M}=sigmoid(\lvert\mathbf{grad}_{adv-triplet}\rvert)$, where
$\lvert\cdot\rvert$ denotes absolute value. We mask $\mathbf{x}$ before
feeding as an input to the generator $\mathcal{G}$. The masked input is given
as $\mathbf{x}=\mathbf{x}\odot(1-\mathbf{M})$, where $\odot$ denotes Hadamard
product.

Masking techniques have also been explored in [31, 32] where the idea is to
learn the model such that it does not overfit to the training distribution.
Our masking technique is motivated from the idea that an adversarial example
should be transferbale across different reid models. Our technique is distinct
and can be applied to an individual sample. Whereas, masking technique in [31,
32] seeks agreement among the gradients obtained from all the samples of a
batch. This technique in [31, 32] also suffers from the drawback of tuning
hyperparameter. Further, the masking technique of [31] is boolean while ours
is continuous.

###  III-C Meta Learning

Meta optimization technique allows to learn from multiple datasets for
different tasks while generalizing well on a given task. One of the popular
meta learning approaches, MAML [28], applies two update steps. The first
update happens in an inner loop with a meta-train set while the second update
happens in outer loop with a meta-test set. In our case, we perform the inner
loop update on the discriminator and generator parameters using the meta-train
set and the outer loop update is performed on the parameters of generator
using a meta-test set.

input : Datasets $\mathcal{T}$ and $\mathcal{A}$, model $f$

output : Generator network $\mathcal{G}$ parameters $\boldsymbol{\theta}_{g}$

while  _not converge_ do

for  _samples in $\mathcal{T}$_ do

/* Obtain the mask */

$\mathbf{M}$ $\leftarrow$
$\sigma$($\lvert\nabla_{\mathbf{x}}{\mathcal{L}_{adv-
trip}(\mathbf{x},\mathbf{x}^{n},\mathbf{x}^{p})}\rvert$)

/* Meta train update using $\mathcal{T}$ */

$\boldsymbol{\theta}_{d}\leftarrow\operatorname*{arg\,max}_{\boldsymbol{\theta}_{d}}E_{\mathbf{x}}\log\mathcal{D}(\mathbf{x})+E_{\mathbf{x}}\log(1-\mathcal{D}(\Pi(\mathcal{G}(\mathbf{x}))))$

$\boldsymbol{\theta}_{g}\leftarrow\operatorname*{arg\,min}_{\boldsymbol{\theta}_{g}}\mathcal{L}_{\mathcal{G}}^{\mathcal{T}}+\lambda\mathcal{L}_{adv-
trip}^{\mathcal{T}}(\mathbf{x}_{adv}^{a},\mathbf{x}_{adv}^{n},\mathbf{x}_{adv}^{p})$

$\boldsymbol{\delta}=\mathbf{x}-\Pi(G(\mathbf{x}))$

/* Meta test loss using $\mathcal{A}$ */

Sample triplets from meta-test set $\mathcal{A}$ and compute
$\mathcal{L}=\mathcal{L}_{adv-
trip}^{\mathcal{A}}(\mathbf{x}^{a}-\boldsymbol{\delta},\mathbf{x}^{n},\mathbf{x}^{p})$

/* Meta test update */

$\boldsymbol{\theta}_{g}\leftarrow\operatorname*{arg\,min}_{\boldsymbol{\theta}_{g}}\lambda\mathcal{L}$

Algorithm 1 Training for MeGA

More formally, given a network $\mathcal{D}$ parametrized by
$\boldsymbol{\theta}_{d}$ and $\mathcal{G}$ parametrized by
$\boldsymbol{\theta}_{g}$, we perform the meta-training phase to obtain the
parameters $\boldsymbol{\theta}_{d}$ and $\boldsymbol{\theta}_{g}$. The update
steps are given in Algorithm 1. We also obtain the adversarial perturbation
as, $\boldsymbol{\delta}=\mathbf{x}-\Pi(G(\mathbf{x}))$.

We then apply the meta-testing update using the additional meta-test dataset
${\mathcal{A}}$. In Algorithm 1,
$\mathcal{L}_{\mathcal{G}}^{\mathcal{T}}=E_{\mathbf{x}}\log(1-\mathcal{D}(\Pi(\mathcal{G}(\mathbf{x}))))$.
We discriminate the datasets using superscripts $\mathcal{T}$ for meta-train
set and $\mathcal{A}$ for meta-test set. $\mathcal{L}_{adv-
trip}^{\mathcal{A}}$ draws its samples $\mathbf{x}$ from $\mathcal{A}$. At the
inference stage, we only use $\mathcal{G}$ to generate the adversarial sample.

###  III-D Training in absence of labels

Deep mis-ranking loss can be used [23] when the labels are available for
$\mathcal{T}$. In this scenario, we present the case where no labels are
available. In the absence of labels and inspired by unsupervised contrastive
loss [33], we generate a positive sample $\mathbf{x}_{adv}^{p}$ by applying
augmentation to the given sample $\mathbf{x}_{adv}^{a}$. The negative sample
$\mathbf{x}_{adv}^{n}$ is generated using batch hard negative sample strategy,
that is we consider all samples except the augmented version of
$\mathbf{x}_{adv}^{a}$ as negative samples and choose the one which is closest
to $\mathbf{x}_{adv}^{a}$. We then use Eq. 1 to obtain the adversarial triplet
loss.

##  IV Experimental Results

###  IV-A Implementation Details

We implemented the proposed method in Pytorch framework. The GAN architecture
is similar to that of the GAN used in [34, 35]. We use the models from Model
Zoo [36] \- OSNet [17], MLFN [18], HACNN [37], ResNet-50 and ResNet-50-FC512.
We also use AlignedReID [38, 39], LightMBN [40], and PCB [41, 42]. We use an
Adam optimizer with a learning rate = $10^{-5}$, $\beta_{1}$ = $0.5$ and
$\beta_{2}=0.999$ and train the model for 40 epochs. We set $m=1$,
$\lambda=0.01$, and $\epsilon=16$. In order to stabilize GAN training, we
apply label flipping with 5% flipped labels. We first present the ablation for
mask and meta learning.

###  IV-B Effect of mask $\mathbf{M}$

We find that when we use mask for Resnet50 and test for different models like
MLFN [18] and HACNN [37], there is a substantial gain in the performance as
shown in Table I. In terms of R-1 accuracy, introduction of mask gives a boost
of 42.10% and 4.8% for MLFN and HACNN respectively. This indicates that mask
provides better transferability. Further, when we evaluate on Resnet50 itself,
there is a minor change in performance which could be because mask is learnt
using Resnet50 itself.

TABLE I: Trained on Market-1501 [43]. Setting Market-1501 $\rightarrow$ Market-1501. $l$ indicates Market-1501 labels are used for training. $\mathbf{M}$ indicates the incorporation of mask. ’Before’ indicates accuracy on clean samples. Model | Resnet50 | MLFN | HACNN
---|---|---|---
| mAP | R-1 | mAP | R-1 | mAP | R-1
Before | 70.4 | 87.9 | 74.3 | 90.1 | 75.6 | 90.9
$l$ | 0.66 | 0.41 | 3.95 | 3.23 | 32.57 | 42.01
$l+\text{AND}$ | 0.56 | 0.35 | 5.39 | 4.55 | 35.13 | 44.20
$l+\text{SAND}$ | 0.51 | 0.33 | 6.01 | 4.89 | 37.50 | 45.11
$l+\mathbf{M}$ | 0.69 | 0.50 | 2.80 | 1.87 | 31.73 | 39.99

###  IV-C Effect of meta learning

We demonstrate the effect of meta learning in Table II. In the case of cross-
dataset (Resnet50) as well as cross-dataset cross-model (MLFN) setting, we
observe that introduction of meta learning gives a significant performance
boost. In terms of R-1 accuracy, there is a boost of 69.87% and 69.29%
respectively for Resnet50 and MLFN. We further observe that Resnet50 does not
have a good transferability towards HACNN. This could be due to two reasons.
First, Resnet50 is a basic model compared to other superior PRID models.
Second, HACNN is built on Inception units [44].

TABLE II: Trained on Market-1501 using MSMT-17 [45] as meta test set. Setting Market-1501 $\rightarrow$ DukeMTMC-reID [46]. $\mathcal{A}$ indicates incorporation of meta learning. Model | Resnet50 | MLFN | HACNN
---|---|---|---
| mAP | R-1 | mAP | R-1 | mAP | R-1
Before | 58.9 | 78.3 | 63.2 | 81.1 | 63.2 | 80.1
$l$ | 17.96 | 24.86 | 18.25 | 24.10 | 42.75 | 58.48
$l+\mathcal{A}$ | 5.80 | 7.49 | 6.15 | 7.4 | 43.12 | 58.97

###  IV-D Adversarial attack performance

We first present the results for cross-model attack in Table III. We use
AlignedReID model, Market-1501 [43] as training set and MSMT-17 [45] as meta
test set. The results are reported for Market-1501 and DukeMTMC-reID [46]. In
case of Market-1501, it is clearly evident that the proposed method is able to
achieve a strong transferability. We can see that incorporation of meta test
set leads to less than halving the mAP and R-1 results compared to case when
only labels are used. For instance, mAP and R-1 of AlignedReID goes down from
7.00% and 6.38% to 3.51% and 2.82% respectively. This is consistently observed
for all three models. Further, the combined usage of mask and meta learning
($l+\mathbf{M}+\mathcal{A}$), denoted as MeGA, achieves best results in cross-
model case of PCB and HACNN. The respective R-1 improvements are 10.00% and
9.10%. Thus our method is extremely effective in generating adversarial
samples.

TABLE III: AlignedReID trained on Market-1501 with MSMT-17 as meta test set. M
is Market-1501 and D is DukeMTMC-reID. MeGA denotes
$l+\mathbf{M}+\mathcal{A}$.

| Model | AlignedReID | PCB | HACNN
---|---|---|---|---
|  | mAP | R-1 | mAP | R-1 | mAP | R-1
M $\rightarrow$ M | Before | 77.56 | 91.18 | 78.54 | 92.87 | 75.6 | 90.9
| $l$ | 7.00 | 6.38 | 16.46 | 29.69 | 16.39 | 20.16
|  $l$ \+ $\mathbf{M}$ | 6.62 | 5.93 | 15.96 | 28.94 | 16.01 | 19.47
| $l+\mathcal{A}$ | 3.51 | 2.82 | 8.07 | 13.86 | 5.44 | 5.28
| MeGA | 5.50 | 5.07 | 7.39 | 12.47 | 4.85 | 4.80
M $\rightarrow$ D | $l$ | 16.04 | 21.14 | 13.35 | 15.66 | 15.94 | 21.85
| $l+\mathbf{M}$ | 16.23 | 21.72 | 13.70 | 15.97 | 16.43 | 22.17
| $l+\mathcal{A}$ | 4.69 | 5.70 | 11.10 | 12.88 | 5.40 | 6.55
| MeGA | 7.70 | 9.47 | 11.81 | 14.04 | 4.73 | 5.40

In case of Market-1501 to DukeMTMC-reID, we observe that simply applying the
meta learning ($l+\mathcal{A}$) generalizes very well. In case of AlignedReID,
mAP and R-1 of 4.60% and 5.70% respectively, are significantly lower compared
to results obtained via $l$ or $l+\mathbf{M}$ settings. The combined setting
of mask and meta learning yields better results for HACNN compared to
AlignedReID and PCB. This may be because of the fact that learning of mask is
still tied to training set and thus may result in overfitting.

In Table IV we discuss the results for cross-dataset and cross-model case
against more models. Here also we can see that both AlignedReID and PCB lead
to strong attacks against other models in a different dataset.

In Table V, we present the results for MSMT-17. Here, the model is trained
using AlignedReID and PCB using Market-1501 and DukeMTMC-reID as meta test
set. When trained and tested using AlignedReID, the R-1 accuracy drops from
67.6% on clean samples to 17.69%. On the other hand when trained using PCB and
tested on AlignedReID, the performance drops to 16.70%. This shows that our
attack is very effective in large scale datasets such as MSMT-17.

TABLE IV: AlignedReID and PCB trained on Market with MSMT-17 as meta test set. Setting Market-1501 $\rightarrow$ DukeMTMC-reID. Model | OSNet | LightMBN | ResNet50 | MLFN | ResNet50FC512 | HACNN
---|---|---|---|---|---|---
| mAP | R-1 | R-10 | mAP | R-1 | R-10 | mAP | R-1 | R-10 | mAP | R-1 | R-10 | mAP | R-1 | R-10 | mAP | R-1 | R-10
Before | 70.2 | 87.0 | - | 73.4 | 87.9 | - | 58.9 | 78.3 | - | 63.2 | 81.1 | - | 64.0 | 81.0 | - | 63.2 | 80.1 | -
AlignedReID | 15.31 | 22.30 | 35.00 | 16.24 | 24.13 | 39.65 | 5.17 | 6.64 | 13.77 | 12.28 | 16.38 | 29.39 | 6.97 | 9.69 | 19.38 | 4.77 | 5.61 | 11.98
PCB | 12.27 | 14.45 | 27.49 | 12.88 | 15.70 | 28.54 | 7.14 | 8.55 | 20.01 | 11.95 | 16.54 | 30.92 | 9.45 | 11.46 | 23.90 | 3.97 | 4.66 | 10.00
TABLE V: Trained on Market-1501 using DukeMTMC-reID as meta test set. Setting Market-1501 $\rightarrow$ MSMT-17. Model | AlignedReID
---|---
| mAP | R-1 | R-10
MeGA (AlignedReID) | 9.37 | 17.69 | 33.42
MeGA (PCB) | 8.82 | 16.70 | 31.98

Figure 2: Left column: Red and blue box show the given image from Market-1501
and its mask ($1-M$) respectively. Right column: Attacked (top) and clean
(bottom) images from MSMT-17

###  IV-E Comparison with SOTA models

In Table VI we present the comparison with TCIAA [23], UAP [47] and Meta-
attack [21]. We observe that our method outperforms TCIAA by a huge margin. We
can also see that when mis-ranking loss is naively applied in case of TCIAA†
[21], the model’ performance degrades. Our attack has better performance
compared to both TCIAA and Meta-attack.

TABLE VI: AlignedReID trained on Market with MSMT-17 as meta test set. Setting Market-1501 $\rightarrow$ DukeMTMC-reID. † uses PersonX [48] as extra dataset.∗ uses PersonX for meta learning.  Model | Aligned reid
---|---
| mAP | R-1 | R-10
Before | 67.81 | 80.50 | 93.18
TCIAA [23] | 14.2 | 17.7 | 32.6
MeGA∗ (Ours) | 11.34 | 12.81 | 24.11
MeGA (Ours) | 7.70 | 9.47 | 19.16
| PCB
Before | 69.94 | 84.47 | -
TCIAA [23] | 31.2 | 45.4 | -
TCIAA† [23] | 38.0 | 51.4 | -
UAP [47] | 29.0 | 41.9 | -
Meta-attack∗ ($\epsilon=8$) [21] | 26.9 | 39.9 |
MeGA∗ ($\epsilon=8$) (Ours) | 22.91 | 31.70 | -
MeGA ($\epsilon=8$) (Ours) | 18.01 | 21.85 | 44.29

###  IV-F Subjective Evaluation

We show the example images obtained by our algorithm in Figure 2 and top-5
retrieved results in Figure 3 for the OSNet model. We can see that in the case
of clean samples the top-3 retrieved images match the query ID, however, none
of the retrieved images match query ID in the presence of our attack.

Figure 3: Query image marked in blue border. Top 5 retrieved mages from OSNet
for Market-1501 (top). Green colored boxes are correct match and red ones are
incorrect. Retrieved images after attacking query sample (bottom).

###  IV-G Attack using unlabelled source

In this section we discuss the attack when source dataset $\mathcal{T}$ is
unlabeled and neither the victim model nor the dataset used for training
victim model are available. This is a very challenging scenario as supervised
models cannot be used for attack. Towards this, we use unsupervised trained
models on Market-1501 and MSMT-17 from [49]. In Table VII, we present results
for training using MSMT-17 and testing on Market. We observe that IBNR50
obtains a mAP and R-1 accuracy of 40.7% and 52.34% when both labels and mask
are not used. When mask is incorporated there is a substantial boost of 3.82%
in mAP and 4.81% in R-1 accuracy in case of OSNet. These gains are even higher
for MLFN and HACNN.

In case of Market-1501 to MSMT-17 in Table VIII, we see that the attack using
only mask performs reasonably well compared to that of attacks using label or
both label and mask. Due to the comparatively small size of Market-1501, even
the attacks using labels are not very efficient.

TABLE VII: MSMT-17 $\rightarrow$ Market-1501. R50 denotes Resnet50. Model | OSNet | MLFN | HACNN
---|---|---|---
| mAP | R-1 | mAP | R-1 | mAP | R-1
Before | 82.6 | 94.2 | 74.3 | 90.1 | 75.6 | 90\. 9
$l$ (R50) | 30.50 | 39.45 | 26.37 | 38.03 | 31.15 | 39.34
$l+\mathbf{M}$ (R50) | 24.50 | 33.07 | 21.76 | 32.18 | 18.81 | 23.66
$\mathbf{M}$ (R50) | 36.5 | 47.56 | 34.92 | 52.61 | 31.15 | 39.34
IBN R50 | 40.7 | 52.34 | 40.62 | 61.46 | 35.44 | 44.84
$\mathbf{M}$ (IBN R50) | 36.88 | 47.53 | 35.01 | 52.79 | 30.98 | 38.98
TABLE VIII:  Market-1501 $\rightarrow$ MSMT-17. Model | OSNet | MLFN | HACNN
---|---|---|---
| mAP | R-1 | mAP | R-1 | mAP | R-1
Before | 43.8 | 74.9 | 37.2 | 66.4 | 37.2 | 64.7
$l$ (R50) | 31.78 | 60.43 | 25.17 | 49.33 | 28.9 | 54.91
$l+\mathbf{M}$ (R50) | 29.04 | 56.11 | 22.02 | 43.57 | 28.26 | 53.53
$\mathbf{M}$ (R50) | 35.16 | 66.28 | 29.16 | 56.65 | 29.69 | 57.81

##  V Conclusion

We present a generative adversarial attack method using mask and meta-learning
techniques. The mask allows better transferability across different networks,
whereas, meta learning allows better generalizability. We present elaborate
results under various settings. Our ablation also shows the importance of mask
and meta-learning. Elaborate experiments on Market-1501, MSMT-17 and DukeMTMC-
reID shows the efficacy of the proposed method.

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# Causal connectability between quantum systems and the black hole interior in
holographic duality

Samuel Leutheusser and Hong Liu  Center for Theoretical Physics, Massachusetts
Institute of Technology,
77 Massachusetts Ave., Cambridge, MA 02139

###### Abstract

In holographic duality an eternal AdS black hole is described by two copies of
the boundary CFT in the thermal field double state. This identification has
many puzzles, including the boundary descriptions of the event horizons, the
interiors of the black hole, and the singularities. Compounding these
mysteries is the fact that, while there is no interaction between the CFTs,
observers from them can fall into the black hole and interact. We address
these issues in this paper. In particular, we (i) present a boundary
formulation of a class of in-falling bulk observers; (ii) present an argument
that a sharp bulk event horizon can only emerge in the infinite $N$ limit of
the boundary theory; (iii) give an explicit construction in the boundary
theory of an evolution operator for a bulk in-falling observer, making
manifest the boundary emergence of the black hole horizons, the interiors, and
the associated causal structure. A by-product is a concept called causal
connectability, which is a criterion for any two quantum systems (which do not
need to have a known gravity dual) to have an emergent sharp horizon
structure.

††preprint: MIT-CTP/5335

##  I Introduction

Figure 1: The Penrose diagram of an eternal black hole. The dashed lines are
event horizons and the wavy lines are the singularities. Two observers from
$R$ and $L$ can meet and interact behind the horizon despite the fact that
there is no interaction between the left and right CFTs.

Understanding the emergence of causal structure in the bulk gravity theory
from the boundary system in holographic duality has been an outstanding
challenge. This issue can be formulated at many different levels. One of the
most conspicuous features of bulk causal structure is the event horizon of a
black hole. Understanding how these horizons and the regions beyond them
emerge from the boundary theory should be a first step to address finer
questions regarding bulk causal structure. Consider an eternal black hole,
which is dual to two copies of the boundary CFT in the thermal field double
state ${\lvert{\rm TFD}\rangle}$ Maldacena:2001kr . See Fig. 1. The boundary
time, $t$, for each copy of the CFT coincides with the bulk Schwarzschild time
which ends at the horizon. How is it then possible to construct an explicit
evolution operator in the CFT describing a bulk observer originally in region
$R$ falling through the horizon into the $F$ region?111 In Jackiw-Teitelboim
gravity it is possible to take an operator behind the horizon using
symmetries, as discussed in Maldacena:2018lmt ; Lin:2019qwu . There are also
many different ways that boundary observables can probe regions behind the
horizon see e.g. Kraus:2002iv ; Fidkowski:2003nf ; Festuccia:2005pi ;
Hartman:2013qma ; Liu:2013iza ; Liu:2013qca ; Susskind:2014rva ;
Grinberg:2020fdj ; Zhao:2020gxq ; Haehl:2021emt ; Haehl:2021prg ;
Haehl:2021tft , but in these discussions neither an emergent Kruskal-type time
nor the casual structure of the horizon was visible from the boundary.
Similarly, ER$=$EPR type arguments VanRaamsdonk:2010pw ; Maldacena:2013xja
are largely concerned with a single time slice, not casual structure. See
Jafferis:2020ora  for an interesting recent discussion of in-falling observers
using modular flows. Earlier discussions of bulk reconstruction in the $F$
region include Hamilton:2005ju ; Hamilton:2006fh ; Papadodimas:2012aq . See
also Nomura:2018kia ; Nomura:2019qps ; Nomura:2019dlz ; Langhoff:2020jqa ;
Nomura:2020ska  for a description of the black hole interior from the
perspective of coarse-graining.  How should we interpret the black hole
singularities in the bulk $F$ and $P$ regions from the boundary theory?
Compounding these mysteries is the observation, first emphasized in
Marolf:2012xe , that while there is no interaction between the two CFTs,
observers from the $R$ and $L$ regions can fall into the $F$ region and
interact with each other.

In this paper we address these questions. We first introduce a formulation of
in-falling observers, which naturally leads to the concept of casual
connectability: a boundary criterion for an emergent sharp horizon in the dual
gravity system. We then provide an explicit boundary construction of a one-
parameter family of unitary operators, $U(s)$, that play the role of evolution
operators along an in-falling trajectory. More explicitly, $U(s)$ has the
following properties:

  1. 1.

It is generated by a Hermitian operator with a spectrum bounded from below.

  2. 2.

Consider a scalar field $\phi(X)$ in the $R$ region, i.e. with $X\in R$, and
its evolution under $U(s)$: $\Phi(X;s)\equiv U^{\dagger}(s)\phi(X)U(s)$. In
this set-up, there exists an $s_{0}>0$, such that for $s>s_{0}$, $\Phi(X;s)$
can start having nonzero commutators with operators in CFTL. Furthermore,
$s_{0}$ coincides with the null Kruskal coordinate distance from $X$ to the
horizon.

  3. 3.

In the geometric optics limit (i.e. if the mass of $\phi$ is large), and with
zero momentum along boundary spatial directions, $\Phi(X;s)=\phi(X_{s})$ where
$X_{s}$ is a bulk point. For $s<s_{0}$, $X_{s}\in R$, while for $s>s_{0}$,
$X_{s}\in F$.

The key to our discussion is the emergence, in the large $N$ limit of the
boundary theory, of a type III1 von Neumann algebraic structure from the type
I boundary operator algebra and the half-sided modular translation structure
associated to this type III1 algebra. The black hole horizons, interiors, and
singularities can all be understood as consequences of it. The type III1
structure also sheds new light on the origin and nature of ultraviolet
divergences in gravity. In this paper we outline the general ideas and the
main results, leaving detailed expositions to LL .

##  II causal connectability: a boundary formulation of bulk horizon structure

In this section we introduce a boundary formulation of a class of bulk in-
falling observers and the associated signature of a sharp horizon.

###  II.1 A boundary formulation of in-falling observers

In Marolf:2012xe  a puzzle regarding the duality between the TFD state and the
bulk eternal black hole geometry was raised. Consider an initial state of the
form

$\lvert{\Psi_{0}}\rangle=e^{iA_{L}}{\lvert{\rm TFD}\rangle}$ (1)

where $A_{L}$ is a Hermitian operator in CFTL and we assume that its insertion
only changes the energy of the system by an $O(1)$ amount such that its
backreaction on the geometry can be neglected. Since operators from the $R$
and $L$ sides commute, any measurement operator $M$ of the $R$ observer should
commute with $e^{iA_{L}}$, i.e.

${\left\langle{\Psi_{0}|M|\Psi_{0}}\right\rangle}={\left\langle{{\rm
TFD}|M|{\rm TFD}}\right\rangle}$ (2)

so the presence of $e^{iA_{L}}$ cannot have any consequence on the
measurement. But this appears to be in contradiction with the ability of the
insertion of $e^{iA_{L}}$ to influence a right observer who has fallen into
the $F$ region of the eternal black hole geometry, see Fig. 1.

The above argument by itself does not directly pose a contradiction, as it
assumes that the evolution of an in-falling observer from the $R$ region
remains in CFTR. It highlights, however, a seemingly counterintuitive
requirement: for the identification of Fig. 1 to be correct, the description
of an in-falling observer originally from the $R$ region must involve both the
$R$ and $L$ systems. Indeed, from the causal structure of the black hole
geometry, any operator in the $F$ region should involve degrees of freedom
from both CFTR and CFTL. Thus whatever measurement operator, $M$, the observer
uses in the $F$ region must involve degrees of freedom in CFTL, and we cannot
assume that $M$ commutes with $e^{iA_{L}}$.

In this paper we will show that the evolution of a family of in-falling
observers on the gravity side can be described by a boundary “evolution
operator” $U(s)=e^{-iGs},\,s\in{\mathbb{{R}}}$ that satisfies the following
properties:

  1. 1.

$G$ involves degrees of freedom from both CFTR and CFTL.

  2. 2.

The Hermitian generator $G$ has a spectrum that is bounded from below,

$G\geq 0\ .$ (3)

The first property is needed for the in-falling evolution $\Phi(X;s)\equiv
U(-s)\phi(X)U(s)$ of a bulk operator $\phi(X)$ with $X\in R$ to have support
in the $F$ region. The second property is natural from the following
perspectives: (i) if we interpret the eigenvalues of $G$ as energies for a
family of bulk observers, they should be bounded from below to ensure
stability,222Note that $G$ should be understood as the “energy” associated
with a full Cauchy slice in the black hole geometry rather than some local
region. While some of the in-falling observers may only have a finite
“lifetime” due to the presence of the singularity, they should nevertheless
have a well-defined quantum mechanical description before hitting the
singularity. (ii) the spectrum condition distinguishes $G$, as a generator of
“time” flow, from operators generating spacelike displacements (such as
momentum operators).

###  II.2 Sharp horizon structure only at infinite $N$

We will now show that the property (3) has important general implications
regardless of the specific form of $U(s)$: a sharp event horizon can only
emerge in the large $N$ limit of the boundary theory.

For this purpose, consider again the state (1), and the probability $p(s)$ for
an in-falling observer originally from the $R$ region to observe the existence
of $e^{iA_{L}}$ along their “trajectory” parameterized by $s$. To reproduce
the causal structure of the black hole spacetime, $p(s)$ should have the form

$p(s)=\begin{cases}0&s<s_{0}\cr\neq 0&s>s_{0}\end{cases},$ (4)

with $s_{0}>0$, as it is only possible to detect the influence of $e^{iA_{L}}$
after the horizon has been crossed. The existence of such an $s_{0}$ and the
non-smooth behavior of $p(s)$ at $s_{0}$ reflect the sharp causal structure
from a sharp horizon.

There is a simple quantum mechanical argument Hegerfeldt:1993qe  that the
behavior (4) is in fact not possible. Denote the projection operator that can
detect the possible existence of $e^{iA_{L}}$ as $P_{R}$. The subscript $R$
emphasizes that this is an operator in CFTR. The probability $p(s)$ can then
be written as

$p(s)={\left\langle{\Psi_{0}|U^{\dagger}(s)P_{R}U(s)|\Psi_{0}}\right\rangle}={\left\langle{\phi(s)|\phi(s)}\right\rangle},\quad\lvert{\phi(s)}\rangle=P_{R}e^{-iGs}\lvert{\Psi_{0}}\rangle\
.$ (5)

From (3), we can analytically continue $U(s)$ to the lower half complex-$s$
plane. Accordingly, $\lvert{\phi(s)}\rangle$ is a vector-valued analytic
function of $s$ in the lower half complex $s$-plane, and is continuous along
the real $s$-axis. Equation (4) means that $\lvert{\phi(s)}\rangle$ vanishes
for a finite segment, $s\in(0,s_{0})$, of the real $s$-axis. Cauchy’s theorem
then says if $\lvert{\phi(s)}\rangle$ is zero for any finite segment of $s$,
it has to be identically zero for all $s$, incompatible with (4). Thus $p(s)$
can be zero only at isolated values of $s$ or identically zero, but cannot
obey (4).

This argument is very general, independent of details of specific states or
quantum systems. For example, the two CFTs can interact and have a bulk
geometry described by a traversable wormhole Gao:2016bin .

Figure 2: Rindler regions of Minkowski spacetime.

Since the bulk gravity theory does have a sharp light cone, the above no-go
argument must somehow be avoided in the duality relation. To understand a
possible resolution, consider a closely related case: Rindler patches for a
quantum field theory in Minkowski spacetime. See Fig. 2. If we discretize the
theory by putting it on a lattice, the Minkowski vacuum
$\lvert{\Omega}\rangle$ can be expressed as a TFD state for the $R$ and $L$
Rindler patches. In the discrete case there are no sharp light-cones. Any
evolution on a lattice system has a small tail which gives rise to a nonzero
commutator between two space-like separated operators. Indeed, in the discrete
case, the above no-go argument applies: observers from the $R$ and $L$ Rindler
systems are either always connected (for $p(s)$ having only isolated zeros) or
can never be connected (for $p(s)$ identically zero). However, in the
continuum limit, they are separated by sharp light-cones, and can meet in the
$F$ region only after evolution by some nonzero $s_{0}$. This difference in
the sharpness of the light-cone structure between the discrete case and the
continuum limit can be attributed to a fundamental difference in the structure
of their operator algebras. In the discrete case, the full Hilbert space
factorizes into a tensor product of those of the $R$ and $L$ systems, and the
operator algebras associated with the $R$ and $L$ systems are type I von
Neumann algebras. In the continuum limit, there is no local Hilbert space
associated with the $R$ or $L$ patch, and the local operator algebra
associated to a Rindler region is a type III1 von Neumann algebra.333For
reviews on the classification of von Neumann algebras see chapter III.2 of
Haag:1992hx  or section 6 of Witten:2018zxz . In the continuum case, the no-go
argument does not apply, since for a type III1 von Neumann algebra, there does
not exist any projector $P_{R}$ that can be used to detect the influence of an
$L$ observer.444Any projector in a type III von Neumann algebra is infinite,
and it is not possible to use such a projector to measure local excitations
Buchholz:1994eb ; Yngvason:2014oia . Heuristically, due to the lack of a local
Hilbert space associated with a Rindler region, there is no way to form finite
projectors.

The above Rindler story suggests a way to go around the no-go argument
regarding (4). The argument implicitly used that the full operator algebra of
bounded operators of a CFT is type I (with the existence of a finite rank
projector $P_{R}$), which is the case for the theory at finite $N$.555$N$ is
the quantity that characterizes the number of boundary degrees of freedom,
such as the rank of the gauge group or the central charge of the boundary CFT.
But the duality with the classical black hole geometry and the associated
sharp causal structure needs to hold only in the large $N$ limit. We will
argue that in the $N\to\infty$ limit there is a pair of emergent type III1
algebras, ${{\mathcal{M}}}_{R,L}$, in the boundary theory.666Note that
operator algebra associated with a local region in a QFT is type III1. But
here the emergent type III1 algebras refer to those associated with the full
boundary spacetime. The event horizons, black hole interior, and singularities
are all consequences of this emergence.

Given that the conditions (4)–(5) for a sharp horizon structure cannot be
defined for a type III1 algebra, we need a generalization. We consider the
function Buchholz:1994eb ; Yngvason:2014oia

$F(s)=\sup\left\\{\left|{\left\langle{\Psi_{0}|U^{\dagger}(s){{\mathcal{O}}}_{R}U(s)|\Psi_{0}}\right\rangle}-{\left\langle{{\rm
TFD}|U^{\dagger}(s){{\mathcal{O}}}_{R}U(s)|{\rm
TFD}}\right\rangle}\right|,\;{{\mathcal{O}}}_{R}\in{{\mathcal{M}}}_{R},\;||{{\mathcal{O}}}_{R}||\leq
1\right\\}\ .$ (6)

Existence of a sharp bulk horizon structure implies the existence of an
$s_{0}>0$ and the behavior

$F(s)=\begin{cases}0&s<s_{0}\cr\neq 0&s>s_{0}\end{cases}\ .$ (7)

For infinitesimal $A_{L}$, the above equation is the same as the existence of
an $s_{0}>0$ and ${{\mathcal{O}}}_{R}$ such that

$[A_{L},U^{\dagger}(s){{\mathcal{O}}}_{R}U(s)]\neq 0\ ,\quad s>s_{0}\ .$ (8)

The condition (7) can be used to describe an emergent sharp horizon for any
two quantum systems and general states, even those without a known gravity
dual. We will refer to two systems in a state which satisfies (7) as being
causally connectable.

###  II.3 Emergent type III1 von Neumann algebras

There is a natural candidate for the emergent type III1 von Neumann algebras.
Consider the algebra ${{\mathcal{A}}}_{R}$ generated by single-trace operators
of CFTR in the large $N$ limit, and the GNS Hilbert space
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$ of ${{\mathcal{A}}}_{R}$ with
respect to the thermal density matrix $\rho_{\beta}={1\over
Z_{\beta}}e^{-{\beta}H_{R}}$ on CFTR (${\beta}$ is the inverse temperature of
${\lvert{\rm TFD}\rangle}$). We denote the representation of
${{\mathcal{A}}}_{R}$ on ${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$ as
${{\mathcal{M}}}_{R}$. Here are some features of
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$:

  1. 1.

The representation of $\rho_{\beta}$ in
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$ is a pure state which we denote as
$\lvert{\Omega_{0}}\rangle$. $\lvert{\Omega_{0}}\rangle$ is cyclic and
separating for ${{\mathcal{M}}}_{R}$.

  2. 2.

${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$ coincides with the GNS Hilbert
space of the union of ${{\mathcal{A}}}_{L}$ and ${{\mathcal{A}}}_{R}$ with
respect to ${\lvert{\rm TFD}\rangle}$. In particular, ${{\mathcal{M}}}_{L}$,
the commutant of ${{\mathcal{M}}}_{R}$ in the operator algebra on
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$, can be viewed as the
representation of ${{\mathcal{A}}}_{L}$ on
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$.777The emergence of this commutant
algebra in ${{\mathcal{H}}}^{(GNS)}_{\rho_{\beta}}$ is also discussed in
appendix A of Magan:2020iac .

We conjecture that ${{\mathcal{M}}}_{R}$ and its commutant
${{\mathcal{M}}}_{L}$ are type III1 in the large $N$ limit. Indications of
this include the fact that the finite temperature spectral functions of
single-trace operators have a continuous spectrum supported on the full real
frequency axis despite the boundary CFT being defined on a compact space.888In
Festuccia:2005pi  the continuous spectrum has been argued to be responsible
for the emergence of black hole horizons and singularities. See also
Festuccia:2006sa  for general arguments regarding the emergence of such a
continuous spectrum.

On the gravity side we quantize small metric and matter perturbations around
the eternal black hole geometry. The resulting Fock space built on the Hartle-
Hawking vacuum $\lvert{HH}\rangle$ is denoted as ${{\mathcal{H}}}_{\rm
BH}^{\rm(Fock)}$ and the algebras of operators for the bulk theory in the $R$
and $L$ regions of the black hole are denoted as
$\widetilde{{\mathcal{M}}}_{R},\widetilde{{\mathcal{M}}}_{L}$. Under the
duality we identify:

${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}={{\mathcal{H}}}_{\rm
BH}^{\rm(Fock)},\quad\lvert{HH}\rangle=\lvert{\Omega_{0}}\rangle,\quad{{\mathcal{M}}}_{R}=\widetilde{{\mathcal{M}}}_{R},\quad{{\mathcal{M}}}_{L}=\widetilde{{\mathcal{M}}}_{L}\
.$ (9)

The above identifications essentially consist of the statement of bulk
reconstruction for the $R$ and $L$ regions of the black hole. They should hold
perturbatively in the $1/N$ expansion (or bulk $G_{N}$ expansion). That
${{\mathcal{M}}}_{R,L}$ should be type III1 is also required by the duality,
as the bulk field algebras $\widetilde{{\mathcal{M}}}_{R,L}$, being associated
with local bulk regions in a quantum field theory in curved spacetime, must be
type III1.

At the leading order in the $1/N$ expansion, the bulk algebras
$\widetilde{{\mathcal{M}}}_{R,L}$ are generated by a free field theory while
the boundary algebras ${{\mathcal{M}}}_{R,L}$ are generated by a generalized
free field theory (since representations of single-trace operators on
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$ are generalized free fields). More
explicitly, suppose a bulk field $\phi$ is dual to a boundary operator
${{\mathcal{O}}}$. The restrictions $\phi_{R,L}$ of $\phi$ to the $R,L$
regions of the black hole are dual respectively to the representations in
${{\mathcal{H}}}^{\rm(GNS)}_{\rho_{\beta}}$ of ${{\mathcal{O}}}_{R,L}$. They
can be expanded in modes as (the sums below should be viewed as a proxy for
integrals)

$\displaystyle\phi_{R}(t,{\vec{x}},w)=\sum_{\omega,{\vec{k}}}e^{-i\omega
t+i{\vec{k}}\cdot{\vec{x}}}v_{\omega{\vec{k}}}(w)\,a_{\omega{\vec{k}}}^{(R)},\qquad\phi_{L}(t,{\vec{x}},w)=\sum_{\omega,{\vec{k}}}e^{i\omega
t-i{\vec{k}}\cdot{\vec{x}}}v_{\omega{\vec{k}}}(w)\,a_{\omega,{\vec{k}}}^{(L)},$
(10)
$\displaystyle{{\mathcal{O}}}_{R}(t,{\vec{x}})=\sum_{\omega,{\vec{k}}}e^{-i\omega
t+i{\vec{k}}\cdot{\vec{x}}}N_{\omega{\vec{k}}}\,a_{\omega{\vec{k}}}^{(R)},\qquad{{\mathcal{O}}}_{L}(t,{\vec{x}})=\sum_{\omega,{\vec{k}}}e^{i\omega
t-i{\vec{k}}\cdot{\vec{x}}}N_{\omega{\vec{k}}}\,a_{\omega,{\vec{k}}}^{(L)},$
(11)

where ${\vec{x}}$ collectively denotes the spatial coordinates on the boundary
and $w$ denotes the bulk radial coordinate. $v_{\omega{\vec{k}}}(w)$ are the
bulk mode functions and the $N_{\omega{\vec{k}}}$ are constants. The
identifications (9) are reflected in the fact that $\phi$ and
${{\mathcal{O}}}$ share the same creation/annihilation operators
$a_{\omega{\vec{k}}}^{(R,L)}$, and thus $\phi_{R,L}$ can be viewed as elements
of boundary algebras ${{\mathcal{M}}}_{R,L}$.

The identifications (9) are shown in Fig. 3. In subsequent sections we will
show that the type III1 nature of ${{\mathcal{M}}}_{R,L}$ leads to the
emergence of the $F$ and $P$ regions and the associated causal structure.

Figure 3: The identifications (9) establish duality of CFTR,L with the
exteriors of the black hole spacetime, but they do not directly say anything
about the existence of the $F,P$ regions of a connected bulk.

##  III Emergent new times in the boundary

In this section we discuss how to generate new times in the boundary theory.
Our main tool is half-sided modular translation Borchers:1991xk ;
Wiesbrock:1992mg , and an extension of it. Suppose ${{\mathcal{M}}}$ is a von
Neumann algebra and the vector $\Omega$ is cyclic and separating for
${{\mathcal{M}}}$. We denote the corresponding modular flow and conjugation
operators as ${{\Delta}}_{{\mathcal{M}}}$ and $J_{{\mathcal{M}}}$. Now suppose
there exists a von Neumann subalgebra ${{\mathcal{N}}}$ of ${{\mathcal{M}}}$
with the properties:

  1. 1.

$\Omega$ is cyclic for ${{\mathcal{N}}}$ (it is automatically separating for
${{\mathcal{N}}}$ as ${{\mathcal{N}}}\subset{{\mathcal{M}}}$).

  2. 2.

The half-sided modular flow of ${{\mathcal{N}}}$ under
${{\Delta}}_{{\mathcal{M}}}$ lies within ${{\mathcal{N}}}$, i.e.

${{\Delta}}^{-it}_{{\mathcal{M}}}{{\mathcal{N}}}{{\Delta}}^{it}_{{\mathcal{M}}}\subset{{\mathcal{N}}},\quad
t\leq 0\ .$ (12)

It can then be shown Borchers:1991xk ; Wiesbrock:1992mg ; Borchers:1998ye
that there exists a unitary group $U(s),\,s\in{\mathbb{{R}}}$ with the
following properties:

  1. 1.

$U(s)$ has a positive generator, i.e.

$U(s)=e^{-iGs},\qquad G\geq 0$ (13)

  2. 2.

It leaves $\Omega$ invariant

$U(s)\Omega=\Omega,\quad\forall s\in{\mathbb{{R}}}$ (14)

  3. 3.

Half-sided inclusion

$U^{\dagger}(s){{\mathcal{M}}}U(s)\subseteq{{\mathcal{M}}},\quad\forall s\leq
0\ .$ (15)

  4. 4.

${{\mathcal{N}}}$ can be obtained from ${{\mathcal{M}}}$ with an action of $U$

${{\mathcal{N}}}=U^{\dagger}(-1){{\mathcal{M}}}U(-1)$ (16)

and with
${{\mathcal{N}}}_{t}\equiv\Delta_{{{\mathcal{M}}}}^{-it}{{\mathcal{N}}}\Delta_{{{\mathcal{M}}}}^{it}$

$e^{iGs}{{\mathcal{N}}}_{t}e^{-iGs}={{\mathcal{N}}}_{f_{1}(s,t)},\quad
f_{1}(s,t)=-{1\over 2\pi}\log(e^{-2\pi t}-s)\ .$ (17)

We also note that

$J_{{\mathcal{M}}}U(s)J_{{\mathcal{M}}}=U(-s)$ (18)

and since
${{\mathcal{M}}}^{\prime}=J_{{\mathcal{M}}}{{\mathcal{M}}}J_{{\mathcal{M}}}$

$U^{\dagger}(s){{\mathcal{M}}}^{\prime}U(s)\subseteq{{\mathcal{M}}}^{\prime},\quad
s\geq 0\ .$ (19)

The above structure is called half-sided modular translation and exists only
if ${{\mathcal{M}}}$ is a type III1 von Neumann algebra Borchers:1998 .

In our context, we take ${{\mathcal{M}}}={{\mathcal{M}}}_{R}$,
${{\mathcal{M}}}^{\prime}={{\mathcal{M}}}_{L}$, and $\Omega=\Omega_{0}$
(introduced in Sec. II.3), with the modular operator
${{\Delta}}_{{\mathcal{M}}}={\beta}(H_{R}-H_{L})$ where $H_{R,L}$ are,
respectively, the Hamiltonians of CFTR,L. Thus modular evolutions of
${{\mathcal{M}}}_{R}$ under ${{\Delta}}_{{\mathcal{M}}}$ are simply the
standard boundary time translations. By choosing different ${{\mathcal{N}}}$
we can construct different one-parameter group evolutions. These are
candidates for new emergent “times” as the corresponding generators are
bounded from below, as in (13). At leading order in the $1/N$ expansion,
${{\mathcal{M}}}_{R}$ and ${{\mathcal{M}}}_{L}$ are generated by generalized
free fields. With the mode expansion (11), using (15) the action of $U(s)$ on
${{\mathcal{O}}}_{R}$ can be written in terms of a linear transform on
$a_{\omega{\vec{k}}}^{(R)}$,

$U^{\dagger}(s)a_{k}^{(R)}U(s)=\sum_{k^{\prime}}C_{kk^{\prime}}(s)a_{k^{\prime}}^{(R)},\quad
k=(\omega,{\vec{k}}),\quad s\leq 0\ .$ (20)

When $s>0$, $U(s)$ takes ${{\mathcal{O}}}_{R}$ outside ${{\mathcal{M}}}_{R}$
and the evolved operator is no longer covered by the theorem.

It turns out that when ${{\mathcal{M}}}_{R}$ is generated by generalized free
fields, much more can be learned and it is possible to generalize the action
of $U(s)$ to all values of $s$ LL :

  1. 1.

The matrix $C_{kk^{\prime}}(s)$ can be determined up to a phase

$C_{kk^{\prime}}(s)={{\delta}}_{{\vec{k}},{\vec{k}}^{\prime}}\sqrt{{\sinh\pi|\omega^{\prime}|\over\sinh\pi|\omega|}}e^{i{{\gamma}}_{\omega}({\vec{k}})-i{{\gamma}}_{\omega^{\prime}}({\vec{k}})}(-s)^{-i(\omega-\omega^{\prime})}{{\Gamma}}(i(\omega-\omega^{\prime}))$
(21)

where the phase $e^{i{{\gamma}}_{\omega}({\vec{k}})}$ depends on the specific
system and the choice of ${{\mathcal{N}}}$.

  2. 2.

For all $s$, we can write

$U^{\dagger}(s)a_{k}^{({\alpha})}U(s)=\sum_{k^{\prime}}\sum_{{\beta}=R,L}\Lambda_{kk^{\prime}}^{{\alpha}{\beta}}(s)a_{k^{\prime}}^{({\beta})},\quad{\alpha}=R,L\
.$ (22)

For ${\alpha}=R$ and $s<0$, equation (22) reduces back to (20) which means
that

$\Lambda_{kk^{\prime}}^{RR}(s)=C_{kk^{\prime}}(s),\quad\Lambda_{kk^{\prime}}^{RL}(s)=0,\quad
s<0\ .$ (23)

$\Lambda^{R{\beta}}_{kk^{\prime}}(s)$ for $s>0$ can also be expressed in terms
of $C_{kk^{\prime}}(s)$,

$\Lambda_{kk^{\prime}}^{RR}(s)={\sinh\pi\omega\over\sinh\pi\omega^{\prime}}C_{kk^{\prime}}(-s),\quad\Lambda_{kk^{\prime}}^{RL}(s)={\sinh\pi(\omega+\omega^{\prime})\over\sinh\pi\omega^{\prime}}C_{k,-k^{\prime}}(-s),\quad
s>0\ .$ (24)

The action of $U(s)$ on ${{\mathcal{O}}}_{L}$, i.e.
$\Lambda^{{\alpha}{\beta}}$ with ${\alpha}=L$, can be obtained from the
relation

$\Lambda_{kk^{\prime}}^{{\alpha}{\beta}}(s)=\Lambda_{-k,-k^{\prime}}^{\bar{\alpha}\bar{\beta}}(-s)$
(25)

which is a consequence of (18) and
${{\mathcal{M}}}^{\prime}=J_{{\mathcal{M}}}{{\mathcal{M}}}J_{{\mathcal{M}}}$.

##  IV Emergence of the bulk Rindler horizon from the boundary

Figure 4: AdS Rindler regions of the bulk spacetime. The vertical lines denote
the boundary and the dashed lines are Rindler horizons.

As a warmup for the black hole story we consider the emergence of the bulk
Rindler horizon from the boundary system using the method outlined in the last
section. Our input is the duality between the AdS Rindler region and the
boundary CFT in the corresponding Rindler patch Hamilton:2006az ; Czech:2012be
; Morrison:2014jha . We will consider as an illustration, a bulk theory in the
Poincare patch of AdS3, dual to a two-dimensional boundary CFT on
${\mathbb{{R}}}^{1,1}$. The bulk spacetime contains two AdS Rindler regions
which respectively have the two Rindler regions in ${\mathbb{{R}}}^{1,1}$ as
their boundaries. The bulk theory in the bulk ${\mathcal{R}}/{{\mathcal{L}}}$
region in Fig. 4 is “reconstructible” from the boundary theory in the
corresponding boundary $R/L$ region in Fig. 2. In this case,
${{\mathcal{M}}}_{R/L}$ is the algebra generated by single-trace operators in
the $R/L$ Rindler regions. We note that going beyond the AdS Rindler horizon
can be achieved by symmetries999See Magan:2020iac  for a discussion. Going
behind the horizon of a black hole in Jackiw-Teitelboim gravity
Maldacena:2018lmt ; Lin:2019qwu  is also similar to the AdS Rindler case, as
it can be done using a symmetry operator. as the usual AdS isometries (which
are dual to boundary conformal symmetries) can take an operator in the
${\mathcal{R}}$ region to the ${{\mathcal{F}}}$ region. But this example
provides a nice illustration of the method of the last section and an
interesting contrast for the discussion of the black hole case where such
symmetries do not exist.

More explicitly, the metric for an AdS Rindler region can be written as

$ds^{2}=\frac{R^{2}}{w^{2}}\left[-\left(1-w^{2}\right)d\eta^{2}+\left(1-w^{2}\right)^{-1}dw^{2}+d\chi^{2}\right]$
(26)

with the Rindler horizon at $w=1$ and boundary at $w=0$. Consider a bulk
scalar field $\phi$ dual to a boundary operator ${{\mathcal{O}}}$ with
dimension ${{\Delta}}$. The restriction $\phi_{R}(X)$ of $\phi$ to the
${\mathcal{R}}$ region (with $X=(\eta,w,\chi)$) can be expanded in modes as

$\displaystyle\phi_{R}(X)=\int\frac{d^{2}k}{(2\pi)^{2}}\,e^{ik\cdot
x}v_{k}(w)a_{k}^{(R)},\quad k=(\omega,q),\quad k\cdot x=-\omega\eta+q\chi$
(27) $\displaystyle v_{k}(w)=N_{k}f_{k}(w),\quad
q_{\pm}={{\frac{1}{2}}}({{\Delta}}+i(\omega\pm
q)),\quad\bar{q}_{\pm}={{\frac{1}{2}}}({{\Delta}}-i(\omega\pm q))$ (28)
$\displaystyle
N_{k}={\sqrt{\sinh\pi|\omega|}\over\sqrt{2\pi}{{\Gamma}}({{\Delta}})}\left|\Gamma\left(q_{+}\right)\Gamma\left(q_{-}\right)\right|,\quad
f_{k}(w)=w^{\Delta}(1-w^{2})^{-i\omega/2}{}_{2}F_{1}\left(\bar{q}_{-},\bar{q}_{+};\Delta;w^{2}\right)\
.$ (29)

The $a_{k}^{(R)}$ are creation (for $\omega<0$) and annihilation (for
$\omega>0$) operators of the boundary generalized free field theory in the $R$
region, and thus $\phi_{R}(X)$ can be interpreted as an operator in the
boundary theory. There is a similar “bulk reconstruction” equation for
$\phi_{L}$ in terms of $a_{k}^{(L)}$.

Figure 5: Boundary subregions that are used for the construction of a one-
parameter group of unitaries implementing bulk Poincaré $U-$ and
$V-$translations, respectively. $t,x$ are boundary Minkowski coordinates.

Our goal is to use the boundary theory in the $R,L$ regions and (27) to
reconstruct the bulk theory in the full Poincare AdS spacetime, including the
${{\mathcal{F}}}$ and ${{\mathcal{P}}}$ regions in Fig. 4. For this purpose we
take ${{\mathcal{M}}}={{\mathcal{M}}}_{R}$, which is type III1 as it is the
local algebra for a Rindler region. We take ${{\mathcal{N}}}$ to be the
algebra of operators in the region indicated in Fig. 5, corresponding to a
null shift in one of the boundary light cone directions. Clearly
${{\mathcal{N}}}\subset{{\mathcal{M}}}$. For the choice of the left plot of
Fig. 5, the corresponding $C_{kk^{\prime}}$ can be computed explicitly in the
boundary theory. It has the form of (21) with

$e^{i{{\gamma}}_{k}}={{{\Gamma}}(\bar{q}_{+})\over|{{\Gamma}}(\bar{q}_{+})|}\
.$ (30)

From our earlier discussion this fully determines
$\Lambda_{kk^{\prime}}^{{\alpha}{\beta}}(s)$ for all $s$. We can then work out
how a bulk field (27) transforms under the evolution of $U(s)$. From the form
of $v_{k}(w)$ it can be shown that for $s<s_{0}\equiv
e^{-\eta+\chi}\sqrt{1-w^{2}}$

$U^{\dagger}(s)\phi_{R}(X)U(s)=\phi_{R}(X_{s}),$ (31)

where $X_{s}=(\eta_{s},\chi_{s},w_{s})$ with

$w_{s}={w\over\sqrt{1-a_{s}}},\quad
e^{\eta_{s}}={e^{\eta}\sqrt{1-w^{2}}\over\sqrt{1-a_{s}-w^{2}}},\quad
e^{\chi_{s}}=e^{\chi}\sqrt{1-a_{s}},\quad a_{s}\equiv{s(1-w^{2})\over s_{0}}\
.$ (32)

It can be readily checked that the above transformation precisely corresponds
to a shift $U\to U+s$ of the Poincare null coordinate $U$, in terms of which
the AdS metric reads101010The two sets of coordinates are related by
$t=Re^{\chi}\sqrt{1-w^{2}}\sinh\eta,\;x=Re^{\chi}\sqrt{1-w^{2}}\cosh\eta,\;z=Re^{\chi}w$.

$ds^{2}={R^{2}\over z^{2}}\left(-dt^{2}+dx^{2}+dz^{2}\right)={R^{2}\over
z^{2}}\left(-dUdV+dz^{2}\right),\quad U=t-x,\quad V=t+x\ .$ (33)

In particular, $-s_{0}$ is the value of the $U$ coordinate for the point $X$,
and thus $s=s_{0}$ takes $X$ to the Rindler horizon. The transformation (31)
is smooth at $s=s_{0}$, and when $s>s_{0}$ the right hand side also involves
$a_{k}^{(L)}$ with

$U^{\dagger}(s)\phi_{R}(X)U(s)=\phi_{F}(X_{s}),$ (34)

where $X_{s}\in{{\mathcal{F}}}$ is again obtained by a null Poincare shift
$U\to U+s$ and $\phi_{F}(X_{s})$ is the restriction of $\phi$ to the
${{\mathcal{F}}}$ region.

Similarly, choosing ${{\mathcal{N}}}$ as in the right plot of Fig. 5 gives
rise to null Poincare translations $V\to V+s$.

The above construction explicitly demonstrates the emergence of the AdS
Rindler horizons and the associated bulk causal structure from the boundary
theory.

##  V Emergence of the black hole event horizon and Kruskal time

We now consider the emergence of the interior of the black hole from the
boundary theory. The bulk theory in the eternal black hole geometry is now
dual to two copies of the boundary theory on ${\mathbb{{R}}}\times S^{d-1}$ in
the state ${\lvert{\rm TFD}\rangle}$. For illustration we again take $d=2$,
i.e. a BTZ black hole Banados:1992wn , whose metric has exactly the same form
as (26) but now with $\chi$ compact.111111In units of (26), the inverse
Hawking temperature is ${\beta}=2\pi$ while the size of $\chi$ is a free
parameter. The bulk reconstruction formula for a scalar field $\phi$ in the
$R$ region has the same form as (27) except that the integration over $q$ is
replaced by a discrete sum. The corresponding boundary manifold, instead of
being a Rindler patch of Minkowski spacetime, is given by
${\mathbb{{R}}}\times S^{1}$. We take ${{\mathcal{M}}}={{\mathcal{M}}}_{R}$,
the algebra generated by single-trace operators in CFTR. The corresponding
algebra of bulk fields in the $R$ region is $\widetilde{{\mathcal{M}}}_{R}$.

Figure 6: The left plot gives the boundary subregion that is used for the
construction of a one-parameter group of unitaries that carries an observer
from the $R$ region across the future horizon of the black hole into the $F$
region. The boundary is ${\mathbb{{R}}}\times S^{1}$ (vertical boundaries in
the figure are identified). The relevant subalgebra is that generated by
single-trace operators supported up to some maximum value of the boundary time
(which we will take to be $\eta=0$). The right plot is the boundary subregion
used to construct the unitaries to carry an observer from the $R$ region
across the past horizon of the black hole. The relevant subalgebra is that
generated by single-trace operators supported at boundary times greater than
some minimum value (which we will again take to be $\eta=0$).  Figure 7: In a
generalized free field theory, the algebras ${{\mathcal{A}}}_{1}$ and
${{\mathcal{A}}}_{2}$ of the two different slices shown are inequivalent, even
though they share a causal diamond.

In this case the constructions of last section no longer apply since when
$\chi$ is compact, a boundary null shift as in Fig. 5 is not well defined. We
will instead take ${{\mathcal{N}}}$ to be the operator algebra associated with
the region indicated in the left plot of Fig. 6. To see that this is a
sensible choice, it should be emphasized that generalized free fields do not
satisfy any Heisenberg equations, and thus the algebras generated by them are
not defined by causal diamonds. For example, the algebras associated with the
two spacetime regions in Fig. 7 are inequivalent, even though they share the
same causal diamond. The state ${\lvert{\rm TFD}\rangle}$ is clearly
separating with respect to ${{\mathcal{N}}}$. While we do not have a rigorous
mathematical proof, we will assume that it is also cyclic with respect to
${{\mathcal{N}}}$.

Figure 8: The respective proposed bulk entanglement wedges for the boundary
subregions indicated in Fig. 6.

Finding $U(s)$ for this choice of ${{\mathcal{N}}}$ is now difficult. We will
find its explicit form by proposing a candidate for the entanglement wedge of
${{\mathcal{N}}}$ in the bulk. We propose that the operator algebra
${{\mathcal{N}}}$ is dual to the algebra of bulk fields in the region
$\widetilde{{\mathcal{N}}}$ shown in Fig. 8. This proposal is natural as this
is the causal wedge associated with that part of the boundary, and we
conjecture it is also the entanglement wedge. We emphasize that the
entanglement wedge here is not defined by an extremal surface. We will provide
further support for this identification below.

Using the bulk theory it is then possible to explicitly construct $U(s)$. For
a bulk scalar field (27) the corresponding matrix $C_{kk^{\prime}}$ again has
the form (21) with phase ${{\gamma}}$ given by

$\displaystyle
e^{i{{\gamma}}_{k}}=e^{i{{\delta}}_{k}}{|{{\Gamma}}(i\omega)|\over{{\Gamma}}(i\omega)}=e^{-i\omega\log
2}{{{\Gamma}}(\bar{q}_{-}){{\Gamma}}(\bar{q}_{+})\over|{{\Gamma}}(\bar{q}_{-}){{\Gamma}}(\bar{q}_{+})|}$
(35)

where $e^{i{{\delta}}_{k}}$ is the phase shift for the scalar field at the
horizon, and in the second equality we have given the explicit expression for
the case of a BTZ black hole ($\bar{q}_{\pm}$ was defined earlier in (28)).
More explicitly, ${{\delta}}_{k}$ can be read from the behavior of bulk mode
function $v_{k}(w)$ near $w=1$

$v_{k}(w)={1\over\sqrt{2|\omega|}}\left(e^{-i\omega\xi+i\delta_{k}}+e^{i\omega\xi-i\delta_{k}}\right),\quad
w\to 1,\quad\xi={{\frac{1}{2}}}\log{1-w\over 1+w}$ (36)

where $\xi$ is the tortoise coordinate. With the explicit form of
$C_{kk^{\prime}}$ we can then work out the action of $U(s)$ on $\phi_{R}(X)$.
The resulting operator $\Phi(X,s)\equiv U^{\dagger}(s)\phi_{R}(X)U(s)$ has the
following properties:

  1. 1.

It is not a local operator, but may be understood as $\phi$ smeared over a
certain spacetime region. Writing $X=(\eta_{0},\chi_{0},w_{0})$ in terms of
Kruskal coordinates, $X=(U_{0},V_{0},\chi_{0})$ (see Appendix A for the
transformation between the coordinates of (26) and the Kruskal coordinates),
we find $\Phi(X;s)$ is supported only for $U<U_{0}+s$. In particular, for
$s<s_{0}\equiv-U_{0}$, $\Phi(X;s)\in\widetilde{{\mathcal{M}}}_{R}$, while for
$s>s_{0}$, $a_{k}^{(L)}$ is now also involved.

  2. 2.

For $X$ near the horizon, i.e. $-U_{0}\ll 1$, $U(s)$ acts as a point-wise $U$
translation, $\Phi(X;s)=\phi(X_{s})$ with $X_{s}=(U_{0}+s,V_{0},\chi_{0})$, to
leading order in $-U_{0}$.

  3. 3.

For general points $X_{1}\in R$ and $X_{2}\in L$, we find that

$[U^{\dagger}(s)\phi_{R}(X_{1})U(s),\phi_{L}(X_{2})]=0$ (37)

for $s<s_{12}\equiv-U_{1}+U_{2}$, but the commutator becomes nonzero when
$s>s_{12}$, precisely reproducing the casual structure expected from the black
hole geometry. See Fig. 9. From the boundary theory perspective, this means
that the two boundary systems are casually connectable in the sense of (8).

  4. 4.

Acting on an operator ${{\mathcal{O}}}_{R}$ at boundary point
$(\eta_{0},\chi_{0})$ it is a nonlocal transformation with support only for
$\eta<-\log(e^{-\eta_{0}}-s)$. This agrees with (17)121212Note the boundary
time $\eta$ is related to modular time $t$ of (17) by $\eta={\beta}t=2\pi t$.,
and provides a nontrivial consistency check of the identification of the
shaded region $\widetilde{{\mathcal{N}}}$ in Fig. 8 as the entanglement wedge
of the boundary region ${{\mathcal{N}}}$.

  5. 5.

In the large $\Delta$ limit, with $q=0$ (i.e. if we dimensionally reduce both
the bulk and boundary theories on the circle $\chi$), the transformation is
point-wise

$\Phi(X;s)=\lambda_{X}\phi(X_{s}),\quad\lambda_{X}=\sqrt{1-s\sqrt{1-w_{0}^{2}}e^{\eta_{0}}}$
(38)

with $X_{s}$ given by

$\displaystyle e^{2\eta_{s}}={e^{2\eta_{0}}\over
1-{2se^{\eta_{0}}\over\sqrt{1-w^{2}_{0}}}+s^{2}e^{2\eta_{0}}},\quad
w_{s}={w_{0}\over 1-s\sqrt{1-w^{2}_{0}}e^{\eta_{0}}}\ .$ (39)

The above transformation can be expressed in terms of Kruskal coordinates as

$U_{s}=U_{0}+s,\qquad V_{s}={V_{0}\over 1-sV_{0}}\ .$ (40)

The trajectories following from (40) are shown in Fig. 10.

Figure 9: When a bulk field $\phi_{R}(X_{1})$ with $X_{1}\in R$ is transported
by a null Kruskal coordinate distance $-U_{1}+U_{2}$ (since $U_{1}<0$), it
enters the lightcone of $\phi_{L}(X_{2})$. The shaded region is a cartoon for
the spread of $\Phi(X_{1};s)$. The orange dashed lines are event horizons, and
the purple dashed lines give the light cones of $X_{2}$. The boundaries and
singularities suppressed in the figure.

Figure 10: The left plot gives trajectories of (40). The right plot gives
constant $s$ surfaces evolved from the $\eta=0$ slice. The orange dashed lines
are the event horizons, black solid lines are the boundaries, while the red
solid lines are the singularities.

Figure 11: The counterparts of Fig. 10 when using ${{\mathcal{N}}}$ as in the
right plot of Fig. 6.

By choosing ${{\mathcal{N}}}$ to be the algebra associated with region in the
right plot of Fig. 6, we can similarly construct unitary evolutions as above
but with the roles of Kruskal $U$ and $V$ swapped. See Fig. 11 for the
corresponding flow trajectories.

##  VI Discussion

###  VI.1 Possible choices of $U(s)$ and more general states

In Sec. V we discussed two possible choices of ${{\mathcal{N}}}$ and the
corresponding $U(s)$. These are only the simplest choices. There are an
infinite number of others. For example, for both plots in Fig. 6 instead of
letting the region describing ${{\mathcal{N}}}$ be bounded by the $\eta=0$
slice on the boundary, we can choose a slice $\eta=f(\chi)$ where $\chi$ is
the boundary spatial coordinate and $f$ an arbitrary periodic function.
Alternatively, instead of taking the cyclic and separating vector
$\lvert{\Omega}\rangle$ to be the “vacuum” $\lvert{\Omega_{0}}\rangle$ of the
GNS Hilbert space dual to the bulk Hartle-Hawking vacuum we can also choose
other $\lvert{\Omega}\rangle$. The simplest possibilities are obtained by
acting on $\lvert{\Omega_{0}}\rangle$ unitaries from ${{\mathcal{M}}}_{R}$ and
${{\mathcal{M}}}_{L}$, i.e.

$\lvert{\Omega}\rangle=V_{L}W_{R}\lvert{\Omega_{0}}\rangle,\quad
V_{L}\in{{\mathcal{M}}}_{L},\quad W_{R}\in{{\mathcal{M}}}_{R}$ (41)

which results in a $U(s)=V_{L}W_{R}U_{0}(s)W_{R}^{\dagger}V_{L}^{\dagger}$
with $U_{0}$ denoting the evolution operator corresponding to
$\lvert{\Omega_{0}}\rangle$.

Our discussion can also be generalized to more general entangled states of
CFTR and CFTL. A simple variant is to act on ${\lvert{\rm TFD}\rangle}$ by a
left unitary $U_{L}$ which does not change the reduced density matrix
$\rho_{\beta}$ of the CFTR, i.e.

$\lvert{\Psi}\rangle=U_{L}{\lvert{\rm TFD}\rangle}\ .$ (42)

The story depends on whether $\lvert{\Psi}\rangle$ lies in the GNS Hilbert
space built from ${\lvert{\rm TFD}\rangle}$. If $\lvert{\Psi}\rangle$ lies in
the GNS Hilbert space, the bulk geometry is still described by the eternal
black hole, now with some small excitations on the left due to insertion of
$U_{L}$. The construction of $U(s)$ is the same as that for ${\lvert{\rm
TFD}\rangle}$. In particular, there are an infinite number of choices of
$\lvert{\Omega}\rangle$ as in (41). When $\lvert{\Psi}\rangle$ does not lie in
the GNS Hilbert space, for example if $U_{L}$ changes the energy of the system
by an amount which scales with $N$, the story is different. We need to work
with the GNS space ${{\mathcal{H}}}_{\Psi}^{\rm GNS}$ associated with
$\lvert{\Psi}\rangle$, which does not overlap with that associated with
${\lvert{\rm TFD}\rangle}$, and the corresponding representations
${{\mathcal{M}}}_{L,R}$ of single-trace operator algebras are also different
from those of associated with ${\lvert{\rm TFD}\rangle}$.131313The appearance
of a different representation in this case is also required by the duality
since the bulk geometry should also be modified. In this case there is no
simple relation between $U(s)$ for $\lvert{\Psi}\rangle$ with those for
${\lvert{\rm TFD}\rangle}$ as they act on different GNS Hilbert spaces.

###  VI.2 Interpretation of the black hole singularity

From a generic bulk point $X\in R$, the flow (40) reaches the future
singularity for a finite value of $s$. Since we expect in general that the
wave function of a bulk field should become singular at the singularity, the
presence of the singularity in the black hole geometry should imply that the
emergent evolution $U(s)$ breaks down at finite values of $s$.

It is instructive to contrast the nature of the $U(s)$ of Sec. V with those of
Sec. IV. In the discussion of Sec. IV, while we also used generalized free
fields, the evolution operator $U(s)$ from the choices of ${{\mathcal{N}}}$ in
Fig. 5 can be defined for the full theory at finite $N$. Thus $U(s)$ there
should be well defined for $s\in(-\infty,+\infty)$. But the $U(s)$ of Sec. V
only exists in the large $N$ limit, so the associated sharp horizon and
interior (i.e. causal connectability from the boundary perspective) only exist
in this limit. We can thus interpret the black hole singularity as a
limitation on this emergent causal connectability; the connection of left and
right observers cannot be extended indefinitely,141414See Nomura:2018kia  for
a related perspective. unlike the case of AdS Rindler.

###  VI.3 The nature of UV divergences in gravity and factorization of modular
operator

The entanglement entropy, $S_{R}$, of CFTR in ${\lvert{\rm TFD}\rangle}$ is
given by the generalized entropy on the gravity side

$S_{R}={A\over 4G_{N}}+{\mathcal{S}}_{R}$ (43)

where $A$ is the horizon area and ${\mathcal{S}}_{R}$ denotes entropy of
matter fields in the $R$ region of the black hole geometry. There is also a
relation between the corresponding modular operators Jafferis:2015del

$H_{R}={\hat{A}\over 4G_{N}}+K_{R}$ (44)

where $H_{R}$ is the Hamiltonian of CFTR, $K_{R}$ is the modular operator for
the bulk field algebra $\widetilde{{\mathcal{M}}}_{R}$ and $\hat{A}$ is the
horizon area operator. $K_{R}$ suffers from bulk UV divergences as does
${\mathcal{S}}_{R}$.151515Strictly speaking, $K_{R}$ cannot be mathematically
defined due to UV divergences. But the left hand sides of (43) and (44) are
well defined. For these expressions to make sense, the UV divergences of
$K_{R}$ and ${\mathcal{S}}_{R}$ must exactly be canceled by those in $G_{N}$
(understood as the bare coupling) to all orders in $G_{N}$ expansion.

From the identification of ${{\mathcal{M}}}_{R}$ and
$\tilde{{\mathcal{M}}}_{R}$, $K_{R}$ can be identified with the modular
operator of ${{\mathcal{M}}}_{R}$, and its divergences must then originate
from the emergent type III1 structure. This provides a different perspective
on the bulk UV divergences and renormalization of the Newton constant
$G_{N}$.161616Recall that in the usual AdS/CFT dictionary, the bulk UV
divergence is understood from the boundary theory as coming from a truncation
of operators dual to stringy modes in the bulk. The divergence in $K_{R}$ is a
reflection that, in a type III1 algebra, the modular operator for
${{\mathcal{M}}}_{R}$ cannot be factorized into those for the $R$ and $L$
systems. Since the algebra for the full CFT is type I, the corresponding
modular operator is factorizable, and thus the area term in (44) must
“restore” the algebra from type III1 to type I.

###  VI.4 Some future directions

There are many more questions to be understood and we mention a few here. It
is clearly of great interest to understand better the emergence of the type
III1 structure in the large $N$ limit,171717The emergence of a type III1
structure, and the associated symmetries and continuous spectrum are also
closely related to the discussion of Goheer:2003tx  of incompatibility of an
exact SL(2,R) symmetry with a finite number of states, and the factorization
of Wilson line problem discussed in Harlow:2015lma .  and what becomes of the
in-falling evolution operators at finite $N$. In particular, it is important
to understand more precisely the emergence of singularities in the boundary
theory. The discussion here should also be generalizable to single-sided black
holes including evaporating ones. We expect such constructions can shed new
light on the information loss problem. We also expect that the manner in which
an in-falling time emerges from the boundary theory here should teach us
valuable lessons about holography for asymptotically flat and cosmological
spacetimes. This should be especially helpful for understanding time in
cosmological spacetimes including de Sitter.

Acknowledgements

We would like to thank Netta Engelhardt, Hao Geng, Daniel Harlow, Gary
Horowitz, Daniel Jafferis, Lampros Lamprou, Juan Maldacena, Donald Marolf,
Leonard Susskind, Aron Wall, and Edward Witten for discussions, and Horacio
Casini for communications. This work is supported by the Office of High Energy
Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567
and and DE-SC0019127. SL acknowledges the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC).

##  Appendix A Kruskal coordinates for the BTZ spacetime

For the BTZ metric (26), the tortoise coordinate is given by

$\xi=-\int{dw\over 1-w^{2}}={{\frac{1}{2}}}\log{1-w\over 1+w}\ .$ (45)

The Kruskal coordinates in the right exterior region are

$\displaystyle U=-e^{\xi-\eta}=-\sqrt{1-w\over 1+w}e^{-\eta},\quad
V=e^{\xi+\eta}=\sqrt{1-w\over 1+w}e^{\eta},$ (46)
$\displaystyle-e^{2\xi}=UV={w-1\over w+1},\quad w={1+UV\over 1-UV},\quad
e^{2\eta}=-{V\over U}\ .$ (47)

The singularity lies at $UV=1$ and the boundary at $UV=-1$. The event horizons
are at $U,V=0$.

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clearly bounded, open, and also it has the Lipschitz boundaries which are the
two spheres $S_{r}$ and $S_{R}$), then with $\bm{\mu}(\mathbf{x})=\mathbf{x}$,
we have

$\bm{\mu}\cdot\mathbf{n}=\|\mathbf{x}\|=\left\\{\begin{array}[]{cc}R\textrm{
on }S_{R},\\\ r\textrm{ on }S_{r}\end{array}\right\\}\geq r=:\delta.$

Also, if we take $\epsilon=1/4$, then

$\|f\|_{L^{2}(S_{r})}^{2}\leq\|f\|_{L^{2}(\partial A)}^{2}\leq
3\frac{\|\bm{\mu}\|_{C^{1}(\overline{A})}}{r}\|f\|_{H^{1}(A)}^{2}.$

As

$\|\bm{\mu}\|_{C^{1}(\overline{A})}=\max\limits_{\overline{A}}\|\bm{\mu}\|+\max\limits_{\overline{A}}|\nabla\cdot\bm{\mu}|=R+n,$

we obtain

$\|f\|_{L^{2}(S_{r})}^{2}\leq 3\frac{R+n}{r}\|f\|_{H^{1}(A)}^{2}.$

Now we will construct $(r_{k_{m}})_{m}$ and $(\delta_{m})_{m}$.

We choose $k_{1}$ such that $r_{k_{1}}>n$. Let $\delta_{1}$ be such that
$0<\delta_{1}<r_{k_{1}}-n$. Then for the annulus

$A_{1}:=\\{\mathbf{x}:r_{k_{1}}<\|\mathbf{x}\|<r_{k_{1}}+\delta_{1}\\},$

we have

$\|f\|^{2}_{H^{1}(A_{1})}\geq\frac{r_{k_{1}}/3}{(r_{k_{1}}+\delta_{1})+n}\|f\|^{2}_{L^{2}(S_{r_{k_{1}}})}\geq\frac{1/3}{1+\frac{\delta_{1}+n}{r_{k_{1}}}}\epsilon>\frac{1/3}{1+1}\epsilon=\frac{\epsilon}{6}=:\widetilde{\epsilon}.$

Now suppose $r_{k_{1}},\cdots,r_{k_{m}},\delta_{1},\cdots,\delta_{m}$
possessing the desired properties have been constructed. Choose $k_{m+1}$ such
that $r_{k_{m+1}}>r_{k_{m}}+\delta_{m}$. Let $\delta_{m+1}$ be such that
$0<\delta_{m+1}<r_{k_{m+1}}-n$.

Then for the annulus

$A_{m+1}:=\\{\mathbf{x}:r_{k_{m+1}}<\|\mathbf{x}\|<r_{k_{m+1}}+\delta_{m+1}\\},$

we have

$\|f\|_{H^{1}(A_{m+1})}^{2}>\frac{r_{k_{m+1}}/3}{(r_{k_{m+1}}+\delta_{m+1})+n}\|f\|_{L^{2}(S_{r_{k_{m+1}}})}\geq\frac{1/3}{1+\frac{\delta_{m+1}+n}{r_{k_{m+1}}}}\epsilon>\frac{\epsilon}{6}=\widetilde{\epsilon}.$

This completes the induction step.

So we have arrived at the contradiction that

$+\infty>\|f\|_{H^{1}(\mathbb{R}^{n})}^{2}\geq\sum_{m}\|f\|^{2}_{H^{1}(A_{m})}\geq\sum_{m}\widetilde{\epsilon}=+\infty.$

This shows that our original assumption was incorrect, and so

$\lim_{r\rightarrow\infty}\int_{S_{r}}|f|^{2}d\sigma_{r}=0,$

completing the proof of our lemma. ∎

An analogous result also holds for the cylinder $\mathbb{R}\times S^{n-1}$.
This was used in the proof of our Theorem 4.2.

######  Lemma 7.3.

If $n\geq 3$ and $f,g\in H^{1}(\mathbb{R}\times S^{n-1})$, then

$\displaystyle\lim_{t\rightarrow+\infty}\int_{S^{n-1}}fgd\Omega=0=\lim_{t\rightarrow-\infty}\int_{S^{n-1}}fgd\Omega.$

###### Proof.

(Sketch.) The proof is based on the same idea as the above, but is somewhat
simpler, since the radius of $S^{n-1}$ doesn’t change, and the constants one
has in the trace theorem for a ‘cylindrical band’ of the form $(a,b)\times
S^{n-1}$ already work, as opposed to having to keep careful track, via Theorem
7.2, of the constants in the earlier case when the radii of the $S_{r}^{n-1}$
were changing. Procceding in the same way as in the previous lemma, we assume
that

$\neg\left(\lim_{t\rightarrow+\infty}\int_{S^{n-1}}|f|^{2}d\Omega=0\right),$

and so there exists an $\epsilon>0$ and a sequence $(t_{k})_{k\in\mathbb{N}}$
such that $\lim\limits_{k\rightarrow\infty}t_{k}=+\infty$, and

$\lim_{k\rightarrow+\infty}\int_{S^{n-1}}|f(t_{k},\cdot)|^{2}d\Omega>\epsilon.$

In order to fatten the ‘circle’ $\\{t_{k}\\}\times S^{n-1}$ to a cylindrical
band of the form $I=(t_{k},t_{k}+\delta)\times S^{n-1}$, while keeping the
$L^{2}$-norm of $f$ on the band uniformly (in $k$) bigger than a fixed
positive quantity, one can use the inequality

$\|f(t_{k},\cdot)\|_{L^{2}(S^{n-1})}\leq C\|f\|_{H^{1}(I\times S^{n-1})}.$

This follows from [24, Prop. 4.5, p.287], by taking
$\Omega=[t_{k},t_{k}+\delta]\times S^{n-1}$. The rest of the proof is the
along the same lines. ∎

##  8\. Appendix C: Sharpness of bound when $|m|=\frac{n}{2}$ in Theorem 3.1

In this appendix, we will show the sharpness of the bound from Theorem 3.1 we
had obtained for the decay of the solution to the Klein-Gordon equation in the
de Sitter universe in flat FLRW form, when $|m|=\frac{n}{2}$. Let us recall
this bound:

$\forall t\geq
t_{0},\;\;\|\phi(t,\cdot)\|_{L^{\infty}(\mathbb{R}^{n})}\lesssim
a^{-\frac{n}{2}}\log a.$

If $|m|=\frac{n}{2}$, then with $\psi:=a^{\frac{n}{2}}\phi$, we had seen that

$\ddot{\psi}-\frac{1}{a^{2}}\Delta\psi=0.$

We will now construct a solution $\psi$ that satisfies

$\|\psi(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}\sim A+Bt\;\textrm{ as
}t\rightarrow\infty,$

showing that

$\|\phi(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}\sim(A+Bt)a^{-\frac{n}{2}}\;\textrm{
as }t\rightarrow\infty,$

and so the bound

$\|\phi(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}\lesssim(A+Bt)a^{-\frac{n}{2}}\;\textrm{
for all large }t$

cannot be improved.

We want

$\ddot{\psi}-\frac{1}{e^{2t}}\Delta\psi=0.$ (24)

Taking the Fourier transform with respect to only the (spatial)
$\mathbf{x}$-variable, and denoting

$\widehat{\psi}(t,\bm{\xi}):=\int_{\mathbb{R}^{n}}\psi(t,\mathbf{x})e^{i\langle\bm{\xi},\mathbf{x}\rangle}d^{n}\mathbf{x},$

(24) becomes

$\frac{\partial^{2}}{\partial
t^{2}}\widehat{\psi}(t,\bm{\xi})+\frac{\|\bm{\xi}\|^{2}}{e^{2t}}\widehat{\psi}(t,\bm{\xi})=0,$
(25)

which is a family of ordinary differential equations in $t$, parameterised by
$\bm{\xi}\in\mathbb{R}^{n}$. For a fixed $\bm{\xi}\in\mathbb{R}^{n}$, the
general solution to the ODE (25) is given by

$\widehat{\psi}(t,\bm{\xi})=C_{1}(\bm{\xi})\cdot
J_{0}(\|\bm{\xi}\|e^{-t})+C_{2}(\bm{\xi})\cdot Y_{0}(\|\bm{\xi}\|e^{-t}),$

where

  * •

$J_{0}$ is the Bessel function of first kind and of order $0$, and

  * •

$Y_{0}$ is the Bessel function of second kind and of order $0$.

In order to construct our $\psi$, we will make special choices of $C_{1}$ and
$C_{2}$.

We recall [1, (9.1.7-8)] that

$\displaystyle J_{0}(z)$ $\displaystyle\sim$ $\displaystyle 1,$ $\displaystyle
Y_{0}(z)$ $\displaystyle\sim$ $\displaystyle\frac{2}{\pi}\log z$

as $z\searrow 0$ ($z\in\mathbb{R}$).

Now as $t\rightarrow\infty$, $e^{-t}\searrow 0$, and so from the above
limiting behaviour of $J_{0}$ and $Y_{0}$, we obtain that as
$t\rightarrow\infty$,

$\displaystyle\widehat{\psi}(t,\bm{\xi})$ $\displaystyle\sim$ $\displaystyle
C_{1}(\bm{\xi})\cdot
1+C_{2}(\bm{\xi})\cdot\left(\frac{2}{\pi}\log(\|\bm{\xi}\|e^{-t})\right)$
$\displaystyle=$ $\displaystyle
C_{1}(\bm{\xi})+\frac{2}{\pi}C_{2}(\bm{\xi})\log\|\bm{\xi}\|-\frac{2}{\pi}t\cdot
C_{2}(\bm{\xi}).$

By Plancherel’s identity (see e.g. [24, Prop. 3.2])

$\|\widehat{\psi}(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}=\|\psi(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}.$

Since we want the linear behaviour in $t$ of
$\|\psi(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}$, we keep $C_{2}$ nonzero, but may
take $C_{1}\equiv 0$. Then as $t\rightarrow\infty$,

$\widehat{\psi}(t,\bm{\xi})=C_{2}(\bm{\xi})\cdot Y_{0}(\|\bm{\xi}\|e^{-t}).$

In order to have $\widehat{\psi}(t,\cdot)$ (and so also $\psi(t,\cdot)$) in
$L^{2}(\mathbb{R}^{n})$ for all $t$, we choose $C_{2}$ to have a sufficiently
fast decay.

We recall [1, §9.2.2] that

$Y_{0}(z)=\sqrt{\frac{2}{\pi
z}}\left(\sin\Big{(}z-\frac{\pi}{4}\Big{)}+O\Big{(}\frac{1}{|z|}\Big{)}\right)$

as $z\rightarrow\infty$ ($z\in\mathbb{R}$). So we have

$Y_{0}(\|\bm{\xi}\|e^{-t})=\sqrt{\frac{2}{\pi\|\bm{\xi}\|e^{-t}}}\left(\sin\Big{(}\|\bm{\xi}\|e^{-t}-\frac{\pi}{4}\Big{)}+O\Big{(}\frac{1}{\|\bm{\xi}\|e^{-t}}\Big{)}\right)$

as $\|\bm{\xi}\|\rightarrow+\infty$ (and $t$ is kept fixed). So to arrange
$\widehat{\psi}(t,\cdot)\in L^{2}(\mathbb{R}^{n})$ for all $t$, we may take

$C_{2}(\bm{\xi}):=\frac{\|\bm{\xi}\|}{(\|\bm{\xi}\|^{2}+1)^{1+\frac{n}{4}}}.$

(Also this choice makes

$\bm{\xi}\mapsto C_{2}(\bm{\xi})\log\|\bm{\xi}\|\in L^{2}(\mathbb{R}^{n}),$

which will be needed below.)

Then $\widehat{\psi}(t,\cdot)\in L^{2}(\mathbb{R}^{n})$ for all $t$. Also, as
$t\rightarrow\infty$,

$\widehat{\psi}(t,\bm{\xi})\sim\frac{2}{\pi}\bigg{(}\underbrace{\frac{\|\bm{\xi}\|}{(\|\bm{\xi}\|^{2}+1)^{1+\frac{n}{4}}}\log\|\bm{\xi}\|}_{=:f\in
L^{2}(\mathbb{R}^{n})}-t\underbrace{\frac{\|\bm{\xi}\|}{(\|\bm{\xi}\|^{2}+1)^{1+\frac{n}{4}}}}_{=:g\in
L^{2}(\mathbb{R}^{n})}\bigg{)},$

and

$\|\widehat{\psi}(t,\cdot)\|_{L^{2}(\mathbb{R}^{n})}\geq\frac{2}{\pi}\Big{(}t\underbrace{\|g\|_{L^{2}(\mathbb{R}^{n})}}_{\neq
0}-\|f\|_{L^{2}(\mathbb{R}^{n})}\Big{)}\geq 0$

for large $t$.

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