# Introducing Self-Attention to Target Attentive Graph Neural Networks

1st Sai Mitheran  National Institute of Technology,
Tiruchirappalli
<EMAIL_ADDRESS>2nd Abhinav Java  Delhi Technological University
<EMAIL_ADDRESS>3rd Surya Kant Sahu  The Learning Machines
<EMAIL_ADDRESS>4th Arshad Shaikh  BYJU’s
<EMAIL_ADDRESS>
###### Abstract

Session-based recommendation systems suggest relevant items to users by
modeling user behavior and preferences using short-term anonymous sessions.
Existing methods leverage Graph Neural Networks (GNNs) that propagate and
aggregate information from neighboring nodes i.e., local message passing. Such
graph-based architectures have representational limits, as a single sub-graph
is susceptible to overfit the sequential dependencies instead of accounting
for complex transitions between items in different sessions. We propose a new
technique that leverages a Transformer in combination with a target attentive
GNN. This allows richer representations to be learnt, which translates to
empirical performance gains in comparison to a vanilla target attentive GNN.
Our experimental results and ablation show that our proposed method is
competitive with the existing methods on real-world benchmark datasets,
improving on graph-based hypotheses. Code is available at
https://github.com/The-Learning-Machines/SBR.

###### Index Terms:

Session-based recommendation, Graph neural networks, Transformers

##  I Introduction

Traditional recommendation systems use user-item interactions across multiple
sessions to model user preferences. However, in session-based recommendation
(SBR) systems, the users are anonymous; hence inter-session data cannot be
used. The goal here is to predict the items with which the user is likely to
interact, given previous item interactions within a single session. In
general, the number of user-item interactions is limited, and as a result,
modeling user intent is a challenging task. Nevertheless, session-based
recommendation is gaining momentum due to increasing privacy concerns. Recent
advancements in session-based recommendation systems have focused on modeling
the user-item interaction as a directed graph and hence leverages graph-based
architectures and related multi-level feature extraction techniques. However,
these methods are disposed to representational limits [1] as a sub-graph tends
to overfit the sequential dependencies, while the essence of extracting the
representations that connote complex transitions between multi-session items
are lost.

Figure 1: Session-based Recommendation Systems

In this work we present TAGNN++, incorporating Transformers as universal
function approximators to enhance capturing complex transitions that address
the limitations of GNNs in learning rich representations. We model item
interactions using a GNN and both global and local user interaction with a
Transformer. We show that our model is competitive with the existing state-of-
the-art techniques on the Diginetica benchmark and on the Yoochoose benchmark.
We also explore Adaptive Gradient Clipping [2] for Transformer-based
architectures specific to our task and present an ablation study to analyze
the performance of the model.

##  II Related works

###  II-A Deep Learning based SBRs

GNNs offer an intuitive approach to session-based recommendation since each
session can be mapped into a graph’s chain. Each node of the graph represents
an item, and each edge represents an interaction. The natural compatibility
between data modeled in such a manner and GNNs allow this method to perform
well and was introduced in [3]. Despite the convenient representation of
sessions offered by GNNs, it lacks the ability to model long-range
dependencies and intricate interactions as substantiated in Graph-
Contextualised Self-Attention model [4]. Qiu et al. [5] proposed using a
weighted graph attention layer for focusing on the essential parts of the item
embeddings. Extending the idea of using attention with GNNs, Yu et al.[6]
proposed using both item embeddings with a GNN and target embedding with an
attention mechanism to achieve significant performance gain. However, a GNN
applied on a single sub-graph is susceptible to overfit on sequential
dependencies instead of accounting for complex transitions between items in
different sessions. Using a dual-channel GNN capable of complex item
transition modeling addresses this issue [7].

###  II-B Use of Adaptive Gradient Clipping (AGC)

Clipping the gradient is a commonly used approach to improving gradient
descent [8], but manual selection of the clipping threshold increases the
number of hyperparameters. However, the hyperparameter for gradient clipping
needs to be tuned carefully, as it is susceptible to the loss function and
architecture [9]. The chosen threshold is vital as if it is set too high, then
the gradient norm will always be smaller than that, and clipping is never
applied. If too low, then the network’s step size may be too small and cause
convergence issues leading to unstable learning. We can clip gradients based
on the unit-wise ratio of gradient norms to parameter norms [2]. This helps
ensure stable training across different batch sizes, allowing larger learning
rates to ensure quick convergence, and is effective to tackle poorly
conditioned loss landscapes. By incorporating this, we verify that performance
is marginally increased for Transformer architectures specific to our task.

##  III Method

###  III-A Problem Formulation and Preliminaries

The SBR problem contains three types of entities : Users, Items, and User-item
interactions. Here, we formally introduce the SBR problem and associated
notations. Let $S=[S_{0},S_{1},S_{2},...,S_{N-1}]$ be the set of $N$ sessions
and $V=[V_{0},V_{1},V_{2},...,V_{M-1}]$ be the set of $M$ unique items in the
dataset. Let $S_{k}=[x_{0},x_{1},x_{2},...,x_{t}]$ be the item sequence within
a session $S_{k}$, where $x_{t}\in V$ is the item clicked by a user at time
step $t=0,1,2...$. The goal is to estimate the parameter set $\theta$ such
that $p(x_{t+1}|x_{0},x_{1},x_{2},...,x_{t};\theta)$ is the maximum-likelihood
estimator of $x_{t+1}$, where $k\in[0,N-1]$.

The proposed TAGNN++ leverages two key components, namely - _Graph Neural
Networks and Self-Attention_. We use the same background strategy for
constructing the session graph from the given data as proposed in [6] and [3].
Each session is stored in memory as an adjacency matrix representing a
directed graph. In this graph $\mathds{G_{s}}$, the $i^{th}$ node in the node
set $\mathds{V_{s}}$ represents an item such that $n_{s,i}\in\mathds{V_{s}}$.
Each edge connecting nodes $i$ and $j$ denotes subsequent item selections by
the user in the given session $s$ such that
$(n_{s,i},n_{s,j})\in\mathds{E_{s}}$ [6]. Lastly, each node has an incoming
weight and an outgoing weight associated to it. We effectively capture item
transitions and connections using a gated GNN [10], by learning both item and
session embeddings as in [6].

###  III-B Proposed method - TAGNN++

We observe that a simple attention model is unable to capture both local and
global context [11], hence indicating the need for a more robust way to
represent sequences. Furthermore, GNNs build a representation of data by
message passing or neighborhood aggregation, using only local information.
Garg et al. show that even simple graph structures are indistinguishable by
GNNs relying only on local information, making it hard for a GNN to compute
several graph properties [1]. Additionally, they perform a thorough study of
generalization bounds for message passing, which accentuates the need for
better function approximators.

Figure 2: Proposed Architecture : TAGNN++

To that end, we propose a multi-headed Transformer-based design for target-
aware predictions. The proposed design enables us to leverage both GNNs and
powerful attention models; in essence, model item interaction in the form of a
graph and a user’s long and short-term interaction through the Transformer. We
hypothesize that our design is a superior function approximator for the task
of next-item prediction in session-based recommendation, providing empirical
evidence and a carefully crafted ablative study. Recently, Transformers have
been successfully used in Natural Language Processing and Computer Vision by
parallelizing the self-attention [11] mechanism over multiple heads achieving
state of the art results. [12] also gives a compact support based proof of why
Transformers have high representation capacity and are universal approximators
of permutation equivariant sequence to sequence functions.

At the $l^{th}$ layer the following update takes place in a single head:

$h_{i}^{l+1}=\sum_{\forall j\in
S}\sigma(\frac{Q^{l}h_{i}^{l}\cdot(K^{l}h_{j}^{l})^{T}}{\sqrt{d_{k}}})V^{l}$
(1)

where $h_{j}^{l}$ is the hidden state of the $j^{th}$ item in sequence $S$ at
layer $l$ such that $h_{j}^{l}\forall j\in S$. $Q^{l}$, $K^{l}$, $V^{l}$ are
Queries, Keys and Values respectively, and $\sigma$ denotes the softmax
operation. $Q^{l}$, $K^{l}$, $V^{l}$ are learnable parameters that are updated
in parallel as opposed to sequentially, and $d_{k}$ is a hyperparameter
indicating the dimension of the linear layer in the Transformer. In our
approach, we allow the Transformer to concatenate several representations or
transformations of the interaction between the hidden states of the target and
sequence. This is done using multiple such attention heads and the
concatenation of the outputs before normalizing them across the layer, before
being passed on to a Feed-forward module. At the feature level, it is possible
that the values in the attention matrix cover a large domain, which is not
desirable since the network finds it hard to learn the optimal parameter
across different scales quickly. This is where Layer Normalization comes into
play by normalizing across the feature space.

Finally, the token-wise Feed-forward Layer (FFL) transforms the normalized
context vector to the output sequence. The composition of these FFLs
implicitly implement a scalar quantization map such that each input is mapped
to the output [12]. We stress that the proposed augmentations to the TAGNN
architecture make it more robust to different kinds of data streams. The
various operations in the Transformer that include Self-attention, Layer Norm,
and the Feed-forward Layer with skip connections play a vital role in enabling
Transformers to be universal approximators of sequence-to-sequence functions.

TABLE I: Comparison of several baseline methods with the proposed method. Method | Yoochoose 1/64 | Diginetica
---|---|---
| HR@20 | MRR@20 | HR@20 | MRR@20
GRU4REC [13] | 60.64 | 22.89 | 29.45 | 8.33
NARM [14] | 68.32 | 28.63 | 49.70 | 16.17
RepeatNet [15] | 70.71 | 31.03 | 47.79 | 17.66
CSRM [16] | 71.45 | 30.36 | 50.55 | 16.38
SR-GNN [3] | 70.87 | 30.94 | 50.73 | 17.59
GC-SAN [4] | 70.66 | 30.04 | 51.70 | 17.61
TAGNN [6] | 71.02 | 31.12 | 51.31 | 18.03
LESSR [17] | 70.64 | 30.97 | 51.71 | 18.15
| TAGNN++
---
71.91 | 31.57 | 51.86 | 17.93

##  IV Setup and Experimental Results

Evaluation Metrics: We use two metrics from previous studies, i.e., Hit
Rate@N, and MRR@N, where N $=20$. Hit Rate calculates the number of ”hits” in
an N-sized list of ranked items, where a ”hit” refers to something that the
user has clicked on, purchased, or ”saved for later”, based on context. Mean
reciprocal rank (MRR) is used to judge a system where the order/positions of
the retrieved items are important.

Datasets: For testing our hypothesis, we employ two widely used real-world
datasets, Yoochoose22footnotemark: 2 and Diginetica11footnotemark: 1. Table
III provides information on the dataset contents. We use only the recent 1/64
fraction of the Yoochoose dataset, denoted as Yoochoose 1/64, which comprises
various sessions that specify the clicks by a given user. It is a collection
of records in a file containing a Session ID, Timestamp, Item ID, and
Category. Diginetica contains anonymized search and browsing logs, product
data, anonymized transactions, and an extensive collection of product images.

Preprocessing: For simplicity, we apply the same preprocessing as [6], [3]. We
drop all unit length sessions and remove items that appear less than five
times for both datasets as same as previous studies. For generating training
and test sets, sessions of last days are used as the test set for Yoochoose
1/64, and sessions of last weeks as the test set for Diginetica. For an
existing session, we generate a series of input session sequences and
corresponding labels. We filter out items from the test set which do not
appear in the training set.

Hyperparameters: We retain most hyperparameter settings from previous
baselines to display the advantage of learning better representations using
Transformers. We keep $10\%$ of our datasets for validation and use a batch
size of $50$, for $15$ epochs. We use the Adam [18] optimizer with an initial
learning rate of $10^{-4}$, with momentum parameters $\beta_{1}$ = $0.9$,
$\beta_{2}$ = $0.999$, and decay it by a factor of $0.1$ every $3$ epochs.
Additionally, we set the $L2$ penalty to $10^{-6}$. For Multi-Head Attention
in the Transformer, we set the number of heads as $2,8$ and dropout of $0.1$,
with the embedding dimension as $100,120$ for Yoochoose 1/64 and Diginetica
respectively.

Baselines: We compare our method with baseline GNN and Attention-based methods
for session-based recommendation, as shown in Table I. The RNN-based methods
[13] [14] [16], and further [15] that takes repeat consumption into account
were then outperformed by graph-based methods [3], those involving the notion
of attention [4] [6].

TABLE II: Ablation Study Architecture | Yoochoose 1/64 | Diginetica
---|---|---
| HR@20 | MRR@20 | HR@20 | MRR@20
TAGNN++ | 71.91 | 31.57 | 51.86 | 17.93
\- AGC | 71.80 | 31.41 | 51.57 | 17.65
\- GNN | 71.75 | 31.62 | 51.48 | 17.59
\- PE | 71.69 | 31.48 | 51.64 | 17.66
\- Transformer | 71.03 | 30.69 | 51.42 | 17.84
TABLE III: Dataset Stats Dataset |  | Clicks
---
| Train
---
Sessions
| Test
---
Sessions
| Total
---
Items
| Avg.
---
Length
Diginetica | 982961 | 719470 | 68977 | 43074 | 5.13
Yoochoose 1/64 | 557248 | 375043 | 55898 | 16339 | 6.11

Inferences: We can infer from the values of HR@20 and MRR@20 that our method
is competitive with the previous state-of-the-art on the given benchmarks,
whilst outperforming the graph-based TAGNN model [6]. Hence, it is evident
that Deep learning-based methods are more capable of learning a better user-
item interaction representation by _capturing complex data distributions and
transitions_ in the dataset. Incorporating Transformers to overcome the
representational limitations in a GNN improves the performance of our
recommendation system. The improvement observed verifies that learning better
representations as proposed is helpful to model complex patterns, as shown in
Fig. 3. To verify the efficacy of our architecture, we also perform an
ablation study as shown in Table II. It can be observed that the performance
is only marginally affected by the removal of secondary techniques such as AGC
(Adaptive Gradient Clipping) and PE (Positional Embedding). Further, the
removal of the Transformer decreases the overall performance, verifying that
GNNs fail at learning accurate representations in some cases.

Fig. 3 indicates better performance on Yoochoose 1/64 than Diginetica, due to
a greater average session length in the former - as more historical data would
enhance the Transformer’s attention mechanism to leverage global context. Both
datasets consist of short, medium, and long sessions [19]. When the user
wishes to build a session-based recommender consisting of long sessions
(Yoochoose 1/64) with better rank/order as the main objective (tweets,
webpages, music), we show that our model outperforms existing methods by a
considerable margin. For short/medium length sessions (Diginetica), where both
order and the number of desired items in the top N of the ranked list are
important (shopping items, movies/videos), our model is competitive with the
existing methods.

Figure 3: Scatter plot comparison of models on the evaluation metrics

##  V Conclusion

This paper proposes an alternative method for learning richer representations
in session-based recommendation models, to overcome the limitations posed by
message-passing GNNs. We perform intelligent feature extraction using
Transformers with target attentive GNNs. We leverage Multi-head Attention to
capture both local and global context. Our method is competitive with the
previous state-of-the-art on the aforementioned benchmarks. We show that our
method is suitable for rank-based retrieval in long sessions (Yoochoose), and
also establish competitive performance for short/medium length sessions
(Diginetica). Motivating research in this direction would enable making
informative and practical choices in streaming, business operations, and
E-Commerce.

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#  Spin-$2$ operators in AdS2/CFT1

###### Abstract

We study spin-2 fluctuations around an infinite family of warped backgrounds
of the form
$\text{AdS}_{2}\times\text{S}^{2}\times\text{T}^{4}\times\text{S}^{1}\times\mathcal{I}_{\rho}$
in the type IIB theory, which possesses $\mathcal{N}=4$ supersymmetry. We find
that there exists a special set of solutions which is independent of the
background data; the minimal universal class. This particular class of
solutions corresponds to operators with scaling dimension $\Delta=\ell+1$,
with $\ell$ being the quantum number of the angular-momentum on the
$\text{S}^{2}$. Using the mode solutions of these spin-2 states, we compute
the central charge of the dual field theories. We comment on the relation of
our results to the seed
$\text{AdS}_{3}\times\text{S}^{2}\times\text{T}^{4}\times\mathcal{I}_{\rho}$
backgrounds.

###### Contents

  1. 1 Prolegomena
  2. 2 The supergravity backgrounds and the dual field theories
  3. 3 Spin-2 fluctuations in the type IIB backgrounds
  4. 4 Unitarity bound and universal class of solutions
  5. 5 The holographic central charge
  6. 6 On the SU(1,1$|$2) multiplets
  7. 7 Non-universal classes of solutions
  8. 8 From $\text{AdS}_{2}/\text{CFT}_{1}$ to $\text{AdS}_{3}/\text{CFT}_{2}$
  9. 9 Epilogue

##  1 Prolegomena

The AdS/CFT duality [1, 2, 3] represents a primo realization of the
holographic principle. The correspondence postulates that specific string
theories with an AdS factor in their geometry are equivalent descriptions of
field theories on the boundary of the space. Since its discovery, there’s been
a renewed interest in the systematic study of superconformal field theories
across different dimensions and with various amounts of supersymmetry and the
construction of the dual AdS descriptions.

The space of consistent and non-trivial half-maximal superconformal field
theories is very rich in structures, in addition to being well under control
due to the high amount of symmetry. It has received a lot of attention with
various studies having been performed using different approaches and
techniques.

For the four-dimensional $\mathcal{N}=2$ theories first suggested by Gaiotto
[4], the duals were derived in [5, 6], with further studies in this context to
have been pursued in [7, 8, 9, 10, 11]. The case of five-dimensional field
theories with $\mathcal{N}=1$ supersymmetry has been thoroughly studied both
from the field theory and the supergavity points of view [12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Progress has, also, been
made in the context of six-dimensional $\mathcal{N}=(1,0)$ theories [30, 31],
which have known supergravity duals in the context of massive type IIA [32,
33, 34, 35, 36, 37, 38].

There has been intense reasearch and progress not only for the case of four
and higher dimensional field theories, but also for lower dimensional ones.
Three-dimensional $\mathcal{N}=4$ theories have been studied [28, 39, 40, 41,
42, 43, 44, 45] and the very appealing case of two-dimensional superconformal
field theories and their $\text{AdS}_{3}$ dual backgrounds has gained its own
attention and led to new, interesting and exciting results [46, 47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59].

In the above beautiful picture, there is an enigmatic corner. The case being
made for the $\text{AdS}_{2}/\text{CFT}_{1}$ duality which is the less
understood example in the context of holography due to some subtleties that
arise in that case and are not present in the higher-dimensional holographic
duals [60, 61, 62, 63]. When considering the effect of backreaction for
excitations of finite-energy, one obtains a divergence at one of the two
asymptotic boundaries. Moreover, since global $\text{AdS}_{2}$ possesses two
disconnected boundaries, it seems that the putative holographic description
should be given in terms of two copies of a one-dimensional theory. This is
contradiction to the calculations of black hole entropy in the context of
string theory.

Be that as it may, owing to the similarities between the $d=1$ and $d=2$
superconformal algebras, or the geometric connections between $\text{AdS}_{2}$
and $\text{AdS}_{3}$ spaces, there is, by now, some progress towards the
better understanding of the picture [64, 65, 66, 67, 68, 69, 70, 71, 72, 73,
74, 75, 76, 77]. Studies pertaining to $\text{AdS}_{2}$ geometries that have
been motivated either black holes physics, or a purely geometric/field
theoretic viewpoint, or both [78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
90].

An interesting antithesis was presented in [91] where a broad class of
geometries that are regular was constructed. This class asymptotes to
$\text{AdS}_{2}\times\text{S}^{2}$. In these wormhole-like solutions, the
presence of two boundaries was a feature rather than a bug of the
construction. The expansion of these solutions around the
$\text{AdS}_{2}\times\text{S}^{2}$ background results to perturbations similar
in spirit to the ones we will consider in this work, although in a different
string theory embedding.

Recently the classification of the $\text{AdS}_{2}$ string backgrounds and the
associated dual field theory picture got enlarged [92, 93, 94, 95, 96]. More
specifically, the authors in [92] derived $\text{AdS}_{2}$ string geometries
that possess one $\text{SU}(2)$ isometry. This leads to the study of
$\mathcal{N}=4$ superconformal quantum mechanics. The background solutions,
which are constructed in the type IIB theory, are trustworthy as dual
descriptions of the $d=1$ conformal field theories, so long as the number of
nodes of the quiver and the ranks of each gauge group are large.

Since $\text{AdS}_{3}$ and $\text{AdS}_{2}$ share structural similarities,
this suggests that some of the holographic studies in the $\text{AdS}_{3}$
backgrounds can naturally be extended to the case of $\text{AdS}_{2}$
solutions. It also suggests, that we should be able to use
$\text{AdS}_{3}/\text{CFT}_{2}$ results, in order to obtain information on
$\text{AdS}_{2}/\text{CFT}_{1}$ and of course vice versa.

Related to a better understanding of any superconformal field theory, a very
important aspect in the study of the spectrum of operators, and the
understanding of how they organise themselves into the different
representations of the underlying algebra.

By virtue of the AdS/CFT, in order to describe the spectrum of the gauge-
invariant field theory operators we need to consider the linearised
fluctuations of the bulk fields around a given supergravity solution.
Computing all the linearised fluctuations for the full supergravity background
is a daunting task, see the works of [97, 98] for the computations of the full
Kaluza-Klein spectrum in $\text{AdS}_{5}\times\text{S}^{5}$ and
$\text{AdS}_{3}\times\text{S}^{3}$ respectively.

Be that as it may, the work of [99] proved that if we consider only the
spectrum of spin-2 operators and the situation simplifies dramatically. In
essence the authors showed that spin-2 fluctuations, which are perturbations
of the geometry, satisfy an equation that depends only on data that we can
extract by examining the underlying geometry of the background supergravity
configuration. While that paper was in the context of an $\text{AdS}_{4}$
solution, generalizing the logic is quite straightforward. This paved the way
for many new interesting results on the spectrum of holographic spin-2
operators in the contexts of $\text{AdS}_{7}/\text{CFT}_{6}$ [100],
$\text{AdS}_{6}/\text{CFT}_{5}$ [101, 102], $\text{AdS}_{5}/\text{CFT}_{4}$
[103, 104], $\text{AdS}_{4}/\text{CFT}_{3}$ [105, 106] and
$\text{AdS}_{3}/\text{CFT}_{2}$ [107, 108].

Taking that path, we will be extending the catalogue of the previous spin-2
results by studying some of the linearised fluctuations that arise in the
$\text{AdS}_{2}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$
supergravity background recently constructed in [92]. The dual picture is
described by $\mathcal{N}=4$ superconformal quantum mechanics. Our motivation
is to holographically construct part of the spectrum of operators of those
theories and to examine how much information we can obtain for the seed
$\text{AdS}_{3}$ from studying the $\text{AdS}_{2}$ solutions. As byproducts
of our studies, we derive a bound on the mass of the modes and the central
charge of the associated theories.

The structure of this work is the following: in section 2 we review and
discuss the basic aspects of the type IIB $\text{AdS}_{2}$ solutions that were
derived in [92] as well as some aspects of the dual field theory
interpretation. Then we proceed to section 3 to derive the equation that
governs the dynamics of the spin-2 fluctuations of the background metric.
These are the transverse-traceless perturbations along the non-compact part of
the geometry under consideration and they correspond to massive rank-2
tensors. In section 4 we derive a bound for the mass and we associate it to
the unitarity bound of conformal dimension of the associated operators. We
are, also, able to obtain a special class of solutions that saturates the
bound and is independent of functions tha define the background, hence this
class of solutions is present in all quivers related to these type IIB
solutions. The modes derived by this special class of solutions are dubbed
minimal universal class. Moving along, in section 5 we use the action of
spin-2 fluctuations to compute the holographic central charge of the
associated theories, and we find agreement with an independent computation.
Section 6 contains discussion on the field theory implications of our studies.
In section 7 we provide an example of a non-universal solution, while section
8 contains discuss on the relation of our results to preciously obtained
studies in the context of the $\tfrac{1}{4}$-BPS $\text{AdS}_{3}$ seed
solutions. We summarize our findings and conclude in section 9.

##  2 The supergravity backgrounds and the dual field theories

We would like to begin by reviewing the basic characteristics of the
geometries derived in [92] and their associated dual field theories. These are
solutions in the type IIB theory with the form
$\text{AdS}_{2}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$,
where $\mathcal{I}_{\rho}$ is a finite interval. They were originally derived
by considering a T-duality transformation along an $AdS$ direction of the
massive type IIA solutions that have the form
$\text{AdS}_{3}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\rho}$
originally constructed in [52].

#### The supergravity description

The AdS2 solution derived in [92] is described by the geometry

$ds^{2}=\frac{u}{\sqrt{\hat{h}_{4}h_{8}}}\left(\frac{1}{4}ds^{2}_{\text{AdS}_{2}}+\frac{\hat{h}_{4}h_{8}}{4\hat{h}_{4}h_{8}+(u^{\prime})^{2}}ds^{2}_{\text{S}^{2}}\right)+\sqrt{\frac{\hat{h}_{4}}{h_{8}}}ds^{2}_{\text{CY}_{2}}+\frac{\sqrt{\hat{h}_{4}h_{8}}}{u}(d\rho^{2}+d\psi^{2})\,,$
(2.1)

and is equipped with the following fields in the NS-NS sector

$\begin{split}e^{-2\Phi}&=\frac{h_{8}}{4\hat{h}_{4}}\left(4\hat{h}_{4}h_{8}+(u^{\prime})^{2}\right)\,,\\\
H_{3}&=\frac{1}{2}d\left(-\rho+\frac{uu^{\prime}}{4\hat{h}_{4}h_{8}+(u^{\prime})^{2}}\right)\wedge\mathop{\rm
vol}\nolimits_{S^{2}}+\frac{1}{2}\mathop{\rm
vol}\nolimits_{\text{AdS}_{2}}\wedge d\psi\,,\end{split}$ (2.2)

with $\psi$ being the T-dual coordinate that takes values in the range
$[0,2\pi]$. Here $\Phi$ is the dilaton, $H_{3}$ the NSNS three-form and the
metric is written in the string frame. The functions $u$, $\hat{h}_{4}$,
$h_{8}$ are functions only of the $\rho$ coordinate, and we have used a prime
to denote a derivative with respect to $\rho$-coordinate.

A more complicated variant of the above description is when $\hat{h}_{4}$ has
an explicit dependence on $(\rho,\text{CY}_{2})$. This more general form of
backgrounds is briefly described [92, appendix A]. For us, it is sufficient to
say that the case $\hat{h}_{4}=\hat{h}_{4}(\rho,\text{CY}_{2})$ is associated
to a function in the seed $\text{AdS}_{3}$ backgrounds, which was named
$H_{2}$. Upon considering that $H_{2}$ vanishes, we obtain the solutions shown
above, which will be the case of interest for us here.

The RR sector our solutions reads

$\begin{split}F_{1}&=h_{8}^{\prime}d\psi\,,\\\
F_{3}&=-\frac{1}{2}\left(h_{8}-\frac{h_{8}^{\prime}u^{\prime}u}{4h_{8}\hat{h}_{4}+(u^{\prime})^{2}}\right)\mathop{\rm
vol}\nolimits_{\text{S}^{2}}\wedge
d\psi+\frac{1}{4}\left(d\left(\frac{u^{\prime}u}{2\hat{h}_{4}}\right)+2h_{8}d\rho\right)\wedge\mathop{\rm
vol}\nolimits_{\text{AdS}_{2}}\,,\\\
F_{5}&=-(1+\star)\,\hat{h}_{4}^{\prime}\,\mathop{\rm
vol}\nolimits_{\text{CY}_{2}}\wedge d\psi\,,\\\
&=-\hat{h}_{4}^{\prime}\,\mathop{\rm vol}\nolimits_{\text{CY}_{2}}\wedge
d\psi+\frac{\hat{h}_{4}^{\prime}h_{8}u^{2}}{4\hat{h}_{4}(4\hat{h}_{4}h_{8}+(u^{\prime})^{2})}\mathop{\rm
vol}\nolimits_{\text{AdS}_{2}}\wedge\mathop{\rm
vol}\nolimits_{\text{S}^{2}}\wedge d\rho\,,\\\ \end{split}$ (2.3)

and the higher-form RR fields are given by $F_{7}=-\star F_{3}$ and
$F_{9}=\star F_{1}$.

The Type IIB equations of motion are satisfied by imposing the BPS equations
and Bianchi identities:

$u^{\prime\prime}=0$ (2.4)

and

$\hat{h}_{4}^{\prime\prime}=0\,,\qquad h_{8}^{\prime\prime}=0\,.$ (2.5)

Hence, it is easy to see that the three functions $u$, $\hat{h}_{4}$ and
$h_{8}$ that are the warp factors and fully determine the $AdS_{2}$ solutions
are at most linear in the $\rho$-coordinate. This holds true away from
explicit brane sources. In the presence of branes, there is a violation of the
Bianchi identities as the right-hand sides of equation 2.5 receive infinite
contributions in the form of delta functions. This leads to $\hat{h}_{4}$ and
$h_{8}$ being piecewise linear functions.

The above background can be depicted pictorially by the brane intersection
presented in table 1

| $x^{0}$ | $x^{1}$ | $x^{2}$ | $x^{3}$ | $x^{4}$ | $x^{5}$ | $x^{6}$ | $x^{7}$ | $x^{8}$ | $x^{9}$
---|---|---|---|---|---|---|---|---|---|---
D1 | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
D3 | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | — | — | — | $\bullet$
D5 | — | — | — | — | — | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$
D7 | — | — | — | — | — | $\bullet$ | — | — | — | $\bullet$
NS5 | — | — | — | — | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | —
F1 | — | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | $\bullet$ | —
Table 1: A pictorial depiction of the brane-scan that describes the
backgrounds given by equations 2.1, 2.2 and 2.3. In our notation (—) stands
for the brane extending along that particular direction, while $\bullet$
denotes a coordinate is transverse to the brane. In the above, $x^{0}$ is the
time direction of the ten-dimensional spacetime. The directions
$(x^{1},\dots,x^{4})$ span the Calabi-Yau two-fold, $x^{5}$ is the direction
associated with $\rho$, $(x^{6},x^{7},x^{8})$ are the transverse directions
realising the $\text{SO}(3)$-symmetry of the $\text{S}^{2}$, and $x^{9}$ is
the $\psi$ direction.

Following [92], we will be interested in the case of a finite interval
$\mathcal{I}_{\rho}$, where both the $\hat{h}_{4}$ and $h_{8}$ vanish at the
endpoints of the interval. Therefore, to start fixing conventions, let us call
$\mathcal{I}_{\rho}=[0,\rho^{*}]$ and
$\hat{h}_{4}(\bar{\rho})=h_{8}(\bar{\rho})=0$, when $\bar{\rho}$ is equal to
$0$ or $\rho^{*}$. It is convenient [92] to set $\rho^{*}=2\pi(P+1)$, with $P$
being a large integer. The remaining function, $u$, vanishes only at
$\rho=0$111there exists, also, the possibility that $u=\texttt{constant}$
everywhere, which we will not discuss in this work.. The general form for
$\hat{h}_{4}$, $h_{8}$ and $u$ is given by the following expressions:

$\hat{h}_{4}(\rho)=\Upsilon\begin{cases*}\frac{\beta_{0}}{2\pi}\rho&$0\leq\rho\leq
2\pi$\\\ \beta_{0}+\dots+\beta_{k-1}+\frac{\beta_{k}}{2\pi}(\rho-2\pi k)&$2\pi
k<\rho\leq 2\pi(k+1)\,,\quad k=1,\cdots,P-1$\\\
\alpha_{P}-\frac{\alpha_{P}}{2\pi}(\rho-2\pi P)&$2\pi P<\rho\leq
2\pi(P+1)\,,$\end{cases*}$ (2.6)
$h_{8}(\rho)=\begin{cases*}\frac{\nu_{0}}{2\pi}\rho&$0\leq\rho\leq 2\pi$\\\
\nu_{0}+\dots+\nu_{k-1}+\frac{\nu_{k}}{2\pi}(\rho-2\pi k)&$2\pi k<\rho\leq
2\pi(k+1)\,,\quad k=1,\cdots,P-1$\\\ \mu_{P}-\frac{\mu_{P}}{2\pi}(\rho-2\pi
P)&$2\pi P<\rho\leq 2\pi(P+1)\,.$\end{cases*}$ (2.7)

In the above, the quantities $(\alpha_{k},\beta_{k},\mu_{k},\nu_{k})$ are
integration constants, and continuity of $\hat{h}_{4}$ and $h_{8}$ determines
that:

$\alpha_{k}=\sum_{j=0}^{k-1}\beta_{j}\,,\qquad\mu_{k}=\sum_{j=0}^{k-1}\nu_{j}\,.$
(2.8)

Finally, the function $u(\rho)$ is given by:

$u=\frac{b_{0}}{2\pi}\rho\,.$ (2.9)

By performing the asymptotic expansions at the endpoints of the
$\mathcal{I}_{\rho}$ we find that as $\rho=0$ the background is regular. At
the other endpoint, denoted by $\rho=2\pi(P+1)-x$, we perform the expansion
for small $x$, and we obtain:

$ds^{2}\sim\frac{1}{x}ds^{2}_{\text{AdS}_{2}}+x(dx^{2}+d\psi^{2}+ds^{2}_{\text{S}^{2}})+ds^{2}_{\text{CY}_{2}}\,,\qquad
e^{-2\Phi}\sim 1\,.$ (2.10)

The above describes the superposition of O1- and O5-planes that extend along
$\text{AdS}_{2}$ and are smeared on the $\text{CY}_{2}$ and
$\text{AdS}_{2}\times\text{CY}_{2}$ respectively.

#### The dual field theories

Here, we wish to discuss some basic aspects of the $\mathcal{N}=4$
superconformal quantum mechanical theories which were proposed to be the duals
to the backgrounds that we consider in this work equations 2.1, 2.2 and 2.3.
We have already mentioned that there deep relations between the
$\text{AdS}_{3}$ and $\text{AdS}_{2}$ solutions, and as expected this is,
also, reflected in the field theory side. The authors of [92] argued that the
type IIB solutions the obtained equations 2.1, 2.2 and 2.3, with the defining
functions being given by equations 2.6, 2.7 and 2.9 are dual to the IR limit
of the quiver depicted in figure 1. It is worthwhile stressing that the shape
of this quiver diagram is directly inherited by the seed $\text{AdS}_{3}$
backgrounds [92].

The proposal that was put forth in [92] is that the way to think about the
dynamics of the quantum mechanical quiver is to view it as the dimensional
reduction along the spatial direction of the two-dimensional
$\mathcal{N}=(4,0)$ quiver theories that were described in [53]. Since the
shape of the quiver is inherited by the seed two-dimensional theory the same
holds true for the matter content.

We can read off the ranks of colour and flavour groups from the Page charges.
These were computed in [92, equation 3.9]. Let us discuss the $k^{th}$ entry
of the quiver diagram that is in correspondence with the $[2\pi k,2\pi(k+1)]$
interval of the geometry. We have $\text{U}(\alpha_{k})$ and
$\text{U}(\mu_{k})$ for the colour groups and

$\alpha_{k}=\sum_{j=0}^{k-1}\beta_{j}\,,\qquad\mu_{k}=\sum_{j=0}^{k-1}\nu_{j}\,.$
(2.11)

These are coupled via bifundamental hypermultiplets and Fermi multiplets with
the adjacent nodes. The connections with the $k^{th}$ flavour groups of ranks
$\text{SU}(\nu_{k-1}-\nu_{k})$ and $\text{SU}(\beta_{k-1}-\beta_{k})$ is
mediated by Fermi fields and by bifundamental hypermultiplets.

$F_{k}$$\alpha_{k}$$\mu_{k}$$\tilde{F}_{k}$$F_{2}$$\alpha_{2}$$\mu_{2}$$\tilde{F}_{2}$$F_{1}$$\alpha_{1}$$\mu_{1}$$\tilde{F}_{1}$$\bf{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots}$$\bf{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots}$$\bf{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots}$$\bf{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots}$$\bf{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots}$$\bf{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\ldots}$
Figure 1: The general quiver with an IR limit that is captured by the
$\text{AdS}_{2}$ background given by equations 2.1, 2.2 and 2.3 and defined by
equations 2.6 and 2.7. Each gauge node has vector multiplets as degrees of
freedom. The straight solid and wavy lines represent the bifundamental
hypermultiplets, while the dashed lines denote Fermi multiplets. In the seed
2d CFT the straight lines are (4,4) hypermultiplets, while the wavy lines
(4,0).

Finally, the various $F$ and $\tilde{F}$ are not independent of the other
numbers of the quiver. It was noticed in [53] that the two-dimensional theory
is chiral and might be afflicted by gauge anomalies. Hence, the $F$ and
$\tilde{F}$ have to be chosen in such a way that gauge anomalies cancel out at
each gauge node of the quiver. The exact relation was shown to be [53]

$F_{k-1}=\nu_{k-1}-\nu_{k}\,,\quad\tilde{F}_{k-1}=\beta_{k-1}-\beta_{k}\,.$
(2.12)

#### R-symmetry

The R-symmetry group of supersymmetric $\mathcal{N}=4$ quantum mechanics is
$\text{SO}(4)=\text{SU}(2)\times\text{SU}(2)$. There are two possibilities for
the superalgebra: the $\mathfrak{d}(2,1;\alpha)$, with two $\mathfrak{su}(2)$
R-symmetries, and the $\mathfrak{su}(1,1|2)$, with only one
$\mathfrak{su}(2)$.

In the $\mathfrak{d}(2,1;\alpha)$, which is usually named the large
superconformal algebra, the parameter $\alpha$ is the one that parametrises
the relative strength of the two Kac-Moody levels, $k_{-}$ and $k_{+}$
pertaining to the R-symmetries. Taking into consideration that we have only
one $SU(2)$, the geometric realization being the one $\text{S}^{2}$ that
appears in the backgrounds under consideration, we deduce that the
superalgebra is the $\mathfrak{su}(1,1|2)$. Another piece fo evidence comes
from examining the seed $\text{AdS}_{3}$ backgrounds and the fact that the
supercharges were in the $(\mathbf{1,2;2})$ of
$\text{SL}(2,\mathbb{R})\times\text{SL}(2,\mathbb{R})\times\text{SU}(2)_{R}$.

We would, also, like to stress that the superalgebras in $d=1$ and $d=2$
dimensions are closely related to one another. More specifically, each chiral
sector of a two-dimensional superconformal field theory provides a
superalgebra and it is realisation for a $d=1$ superconformal quantum
mechanics. This is what makes it possible to identify central charges in the
cases of $d=1$ and $d=2$ [80] and it will important for the discussion of our
results in section 8 where we compare the central charge that we compute in
the type IIB $\text{AdS}_{2}$ solutions to the central charge of the seed
$\text{AdS}_{3}$ theories.

##  3 Spin-2 fluctuations in the type IIB backgrounds

In order to begin our analysis we consider the background geometry given by
equation 2.1 in the Einstein frame. To do so, we multiply the string-frame
metric, given by equation 2.1, by $e^{-\Phi/2}$, with $\Phi$ being the dilaton
of our background. We can write it in the following manner

$ds^{2}=f_{1}e^{-\Phi/2}ds^{2}_{\text{AdS}_{2}}+\hat{g}_{ab}dz^{a}dz^{b}\,,$
(3.1)

with the warp factor and the internal part of the geometry being given by the
following expressions:

$\displaystyle f_{1}$
$\displaystyle=\frac{1}{4}\frac{u}{\sqrt{\hat{h}_{4}h_{8}}}\,,$ (3.2)
$\displaystyle\hat{g}_{ab}dz^{a}dz^{b}$
$\displaystyle=e^{-\Phi/2}\left(\frac{u\sqrt{\hat{h}_{4}h_{8}}}{4\hat{h}_{4}h_{8}+(u^{\prime})^{2}}ds^{2}_{\text{S}^{2}}+\sqrt{\frac{\hat{h}_{4}}{h_{8}}}ds^{2}_{\text{CY}_{2}}+\frac{\sqrt{\hat{h}_{4}h_{8}}}{u}(d\psi^{2}+d\rho^{2})\right)\,.$

In what follows we will consider the Calabi-Yau to be a four-torus
$\text{T}^{4}$ parametrised by four angles which we name
$\theta_{i},(i=1,\dots,4)$, and, of course, $\theta_{i}\cong\theta_{i}+2\pi$.
We can consider a metric fluctuation, denoted by $h$, along the
$\text{AdS}_{2}$ directions in the ten-dimensional background metric in the
following manner:

$ds^{2}=f_{1}e^{-\Phi/2}(ds^{2}_{\text{AdS}_{2}}+h_{\mu\nu}dx^{\mu}dx^{\nu})+\hat{g}_{ab}dz^{a}dz^{b}\,.$
(3.3)

We can decompose $h$ into a transverse-traceless mode on $\text{AdS}_{2}$ and
a scalar one along the internal manifold like so:

$h_{\mu\nu}(x,z)=h^{(tt)}_{\mu\nu}(x)\psi(z)\,.$ (3.4)

Following the logic of [99], the transverse-traceless fluctuation
$h^{(tt)}_{\mu\nu}(x)$ satisfies the following equation of motion on $AdS_{2}$

$\mathop{\rm\Box}\nolimits_{\text{AdS}_{2}}^{(2)}h^{(tt)}_{\mu\nu}(x)=(M^{2}-2)h^{(tt)}_{\mu\nu}(x)\,,$
(3.5)

where $\mathop{\rm\Box}\nolimits_{\text{AdS}_{2}}^{(2)}$ is the Laplacian
acting on a massive rank-two tensor in $\text{AdS}_{2}$, see [109]. In [99],
the authors showed that the linearised Einstein equations can be reduced to
the ten-dimensional Laplace equation given by:

$\frac{1}{\sqrt{-g}}\partial_{M}\sqrt{-g}g^{MN}\partial_{N}h_{\mu\nu}=0\,.$
(3.6)

For the background metric in equation 3.1 and with the transverse-traceless
$\text{AdS}_{2}$ mode, $h_{\mu\nu}^{(tt)}$, satisfying the equation equation
3.5, it is a matter of some straightforward algebra to obtain the following
equation for the internal part of the fluctuation, $\psi(z)$,

$\frac{1}{\sqrt{\hat{g}}}\partial_{a}\left[f_{1}e^{-\Phi/2}\sqrt{\hat{g}}\hat{g}^{ab}\partial_{b}\right]\psi=-M^{2}\psi\,,$
(3.7)

where we have suppressed the explicit dependence on $z$ for convenience.

We expand explicitly equation 3.7 to obtain

$\left[\frac{1}{\hat{h}_{4}h_{8}}\frac{d}{d\rho}\left(u^{2}\frac{d}{d\rho}\right)+\right(4+\frac{(u^{\prime})^{2}}{\hat{h}_{4}h_{8}}\left)\nabla^{2}_{S^{2}}+\frac{u}{\hat{h}_{4}}\sum^{4}_{i=1}\partial^{2}_{\theta_{i}}+\frac{u^{2}}{\hat{h}_{4}h_{8}}\partial^{2}_{\psi}+M^{2}\right]\psi=0\,.$
(3.8)

In order to proceed, it is convenient to decompose the mode $\psi$ into
eigenstates along the different parts of the internal geometry. This can be
done by considering the spherical harmonics on $\text{S}^{2}$, denoted by
$Y_{\ell,m}$, and the plane-waves on the $\text{T}^{4}$ and
$\mathcal{I}_{\psi}$ in the following manner:

$\psi=\sum\Psi_{\ell mnp}Y_{\ell,m}e^{in\cdot\theta}e^{ip\cdot\psi}\,.$ (3.9)

In the above we have used $n$ as a short-hand for the four integers
$(n_{1},n_{2},n_{3},n_{4})$ on the four-torus and
$n\cdot\theta=n_{1}\theta_{1}+n_{2}\theta_{2}+n_{3}\theta_{3}+n_{4}\theta_{4}$.
Substituting equation 3.9 into equation 3.8 we get the following equation for
$\Psi$

$\displaystyle\frac{d}{d\rho}\left(u^{2}\frac{d\Psi_{\ell mnp}}{d\rho}\right)$
$\displaystyle-4l(l+1)h_{4}h_{8}\Psi_{\ell
mnp}-(u^{\prime})^{2}l(l+1)\Psi_{\ell mnp}$ (3.10)
$\displaystyle-\frac{u}{h_{4}}h_{4}h_{8}n^{2}\Psi_{\ell
mnp}-u^{2}p^{2}\Psi_{\ell mnp}+4h_{4}h_{8}M^{2}\Psi_{\ell mnp}=0\,.$

It is convenient to consider $\Psi_{\ell mnp}=u^{\ell}\varphi$ in the above
equation. Of course, $\varphi$ also depends on all the quantum numbers and we
should formally write $\varphi=\varphi_{\ell mnp}$, however, we suppress the
subscripts in what follows. In terms of $\varphi$ we obtain:

$\displaystyle u^{\ell+2}\frac{d\varphi}{d\rho}$
$\displaystyle+2\ell(\ell+1)\frac{d\varphi}{d\rho}\frac{du}{d\rho}$ (3.11)
$\displaystyle-4\ell(\ell+1)h_{4}h_{8}u^{\ell}\varphi-\frac{u}{h_{4}}h_{4}h_{8}n^{2}u^{\ell}\varphi-u^{2}p^{2}u^{\ell}\varphi+4h_{4}h_{8}M^{2}u^{\ell}\varphi=0\,.$

In deriving the above, we used $u^{\prime\prime}=0$ which holds globally true.

We can re-express equation 3.11 in a more suggestive and intuitive form,
namely we can write it as a Sturm-Liouville problem222the integrating factor
in this case turns out to be given by $\mu(\rho)=u^{\ell}$

$S(\rho)\varphi+Q(\rho)\varphi=-\lambda W(\rho)\varphi\,,$ (3.12)

with

$\displaystyle\begin{aligned}
S(\rho)&=\frac{d}{d\rho}\left(P(\rho)\frac{d}{d\rho}\right)\,,&Q(\rho)&=-(n^{2}h_{8}u+p^{2}u^{2})u^{2\ell}\varphi\,,\\\
P(\rho)&=u^{2(\ell+1)}\,,&W(\rho)&=h_{4}h_{8}u^{2\ell}\,,\end{aligned}$ (3.13)

and the eigenvalue $\lambda$ being given by

$\lambda=4(M^{2}-\ell(\ell+1))\,.$ (3.14)

The variable $\rho$ takes values in the $\mathcal{I}_{\rho}$ interval;
$[0,2\pi(P+1)]$. The function $u$ vanishes at the $\rho=0$ endpoint, and hence
we have a singular Sturm-Liouville problem.

##  4 Unitarity bound and universal class of solutions

We would like to proceed by obtaining a bound for $M^{2}$ that emerges from
the Sturm-Liouville problem derived in the previous section, equations 3.12,
3.13 and 3.14. When this bound is saturated we are able to find a specific
class of solutions which we will call minimal universal class. This class of
solutions is independent of the defining functions $\hat{h}_{4}$ and $h_{8}$.
We discuss the regularity conditions for the scalar mode $\Psi$ as well.

### The unitarity bound

We begin by multiplying the Sturm-Liouville problem defined by equations 3.12,
3.13 and 3.14 through by $\varphi$ and then we carry out the integration over
$\mathcal{I}_{\rho}$ interval. That is, we obtain

$\int_{\mathcal{I}_{\rho}}d\rho\,\varphi\frac{d}{d\rho}\left(u^{2(\ell+1)}\frac{d\varphi}{d\rho}\right)-\left(n^{2}h_{8}u+p^{2}u^{2}\right)u^{2\ell}\varphi^{2}+4\left(M^{2}-\ell(\ell+1)\right)\hat{h}_{4}h_{8}u^{2\ell}\varphi^{2}=0\,,$
(4.1)

We focus on the first term of the above and we perform an integration by
parts. This gives

$\begin{split}\int_{\mathcal{I}_{\rho}}d\rho\left(-\varphi^{\prime
2}u^{2(\ell+1)}-\left(n^{2}h_{8}u+p^{2}u^{2}\right)u^{2\ell}\varphi^{2}+4\left(M^{2}-\ell(\ell+1)\right)\hat{h}_{4}h_{8}u^{2l}\varphi^{2}\right)=\\\
-\varphi\varphi^{\prime}u^{2(\ell+1)}\Big{\rvert}_{0}^{\rho^{*}}\,.\end{split}$
(4.2)

We note that the right-hand side of equation 4.2,
$\varphi\varphi^{\prime}u^{2(\ell+1)}$, vanishes when we evaluate it at the
endpoint $\rho=0$, since $u$ vanishes at $\rho=0$ \- recall that $u$ is linear
in $\rho$ and its exact definition is given by equation 2.9 \- as long as
$\varphi$ and $\varphi^{\prime}$ are regular there, while it does not vanish
when evaluated at the other endpoint; $\rho=\rho^{*}$. In what follows, we
will focus on functions $\varphi$ such that which
$\varphi\varphi^{\prime}u^{2(\ell+1)}$ vanishes at $\rho=\rho^{*}$ as well.
Therefore, equation 4.2 reduces to the following:

$\int_{\mathcal{I}_{\rho}}d\rho\left(\varphi^{\prime
2}u^{2(\ell+1)}+\left(n^{2}h_{8}u+p^{2}u^{2}\right)u^{2\ell}\varphi^{2}\right)=4(M^{2}-\ell(\ell+1))\int_{\mathcal{I}_{\rho}}d\rho\,\hat{h}_{4}h_{8}u^{2\ell}\varphi^{2}\,,$
(4.3)

Taking into consideration that the defining functions of our backgrounds $u$,
$\hat{h}_{4}$ and $h_{8}$ are non-negative, and that the integrals are finite,
we obtain from equation 4.3 the following lower bound for $M^{2}$

$M^{2}\geq\ell(\ell+1)\,.$ (4.4)

### Minimal universal class of solutions

Let us consider here the case where $M$ saturates its bound;
$M^{2}=\ell(\ell+1)$. We, also, consider the case $n=p=0$, or in other words
the modes are not excited along the directions of the $\text{T}^{4}$ and the
$\mathcal{I}_{\psi}$. After those considerations, the Sturm-Liouville problem
defined by equations 3.12, 3.13 and 3.14 reduces to

$\frac{d}{d\rho}\left(u^{2(\ell+1)}\frac{d\varphi}{d\rho}\right)=0\,.$ (4.5)

We can integrate equation 4.5 to obtain

$\varphi^{\prime}=\frac{\texttt{constant}}{u^{2(\ell+1)}}\,.$ (4.6)

However, for the class of geometries considered in this work, $u$ vanishes at
$\rho=0$, and therefore $\varphi^{\prime}$ is not finite at $\rho=0$. This
suggests that both $\varphi$ and $\Psi_{\ell m}=u^{\ell}\varphi$ are not
finite at $\rho=0$. However, we are interested in finding fluctuations for the
modes that remain finite anywhere in the allowed region of the geometries that
we consider, and hence the only solution to equation 4.5 that satisfies this
condition is the constant solution. This, in turn, implies that

$\varphi=\texttt{constant}\,,\qquad\Psi=\texttt{constant}\cdot
u^{\ell}\,,\qquad M^{2}=\ell(\ell+1)\,.$ (4.7)

This class of solutions does not depend on the form that the functions $u$,
$\hat{h}_{4}$ and $h_{8}$ have and in this sense it is a universal class of
solutions. Furthermore, they are the solutions obtained for the minimal value
of $M$ for a given $\ell$.

More specifically, the spin-2 fluctuations that we considered above are dual
to field theory operators that carry dimension $\Delta$. The conformal
dimensions of those operators is related to the mass via
$M^{2}=\Delta(\Delta-1)$ using the standard AdS/CFT formula. The inequality
derived earlier, equation 4.4, implies for the conformal dimension of the
field theory operators is bounded from below

$\Delta\geq\Delta_{\texttt{min}}\,,\qquad\Delta_{\texttt{min}}=\ell+1\,.$
(4.8)

For the case of non-universal solutions, i.e. solutions for which
$M^{2}>\ell(\ell+1)$ holds, or non-minimal solutions, such that the modes are
excited on $\text{T}^{4}$ or along the $\mathcal{I}_{\psi}$, it is necessary
to specify what the three functions $u$, $\hat{h}_{4}$ and $h_{8}$ are.

##  5 The holographic central charge

In this section we will compute the central charge for the theories described
by equations 2.1, 2.3 and 2.2. Before we proceed with our computation, some
commentary is in order. Trying to define the central charge in conformal
quantum mechanics is a very subtle issue. The reasoning is as follows: let us
consider a one-dimensional conformal theory. Then, we have only one component
of the stress-energy tensor, $T_{\mu\nu}$, the trace of which must vanish, and
hence this implies that $T_{tt}=0$. The quantity that we name as holographic
central charge should be interpreted as the number of vacuum states in the
associated superconformal quantum mechanics picture [92]. Our reasoning behind
choosing that particular name is to stay consistent with the existing
literature.

We will be following the approach described [102]. The central charge has
been, also, computed in [92] using the algorithmic approach developed first in
[110] and revisited in [111] to account for more general warp factors..

Our starting point is the action for the type IIB theory, which we only need
schematically. In the Einstein frame it reads

$S_{\text{IIB}}=\frac{1}{2\kappa_{10}^{2}}\int
d^{10}x\sqrt{-g}\left(R+\cdots\right)\,.$ (5.1)

Following the logic outlined in [102], we expand the supergravity action to
second order and we are led to an action for the $h_{\mu\nu}$

$\delta^{2}S=\frac{1}{\kappa_{10}^{2}}\int
d^{10}x\,{h}^{\mu\nu}\partial_{M}\sqrt{-g}g^{MN}\partial_{N}h_{\mu\nu}+\texttt{boundary
terms}\,.$ (5.2)

We can evaluate equation 5.2 for the bulk and internal pieces and drop the
boundary terms to obtain

$S[h]=\frac{1}{\kappa_{10}^{2}}\int
d^{10}x\sqrt{-g_{AdS_{2}}}\sqrt{\hat{g}}\,{h}^{\mu\nu}\left\\{\mathop{\rm\Box}\nolimits_{AdS_{2}}^{(2)}+2+\hat{\mathop{\rm\Box}\nolimits}\right\\}h_{\mu\nu}\,,$
(5.3)

with $\hat{\mathop{\rm\Box}\nolimits}$ in the above being the operator on the
left-hand side of equation 3.7. We decompose the metric perturbations by using
the ansatz

$h_{\mu\nu}=(h_{lmnp}^{(tt)})_{\mu\nu}Y_{lm}\psi_{lmnp}e^{in\cdot\theta}e^{in\cdot\theta}e^{ip\cdot\psi}$
(5.4)

and we find

$\delta^{2}S=\sum_{lmnp}C_{lmnp}\int
d^{2}x\sqrt{-g_{AdS_{2}}}\,{(h^{(tt)}_{lmnp})}^{\mu\nu}\left\\{\mathop{\rm\Box}\nolimits_{AdS_{2}}^{(2)}+2-M^{2}\right\\}{(h^{(tt)}_{lmnp})}_{\mu\nu}$
(5.5)

where the coefficients $C_{lmnp}$ are given by

$C_{lmnp}=\frac{1}{\kappa_{10}}\mathop{\rm vol}\nolimits_{T^{4}}\mathop{\rm
vol}\nolimits_{S^{2}}\mathop{\rm
vol}\nolimits_{\psi}\int_{\mathcal{I}_{\rho}}d\rho\sqrt{\hat{g}}\,|\psi_{lmnp}|^{2}\,,$
(5.6)

and we have used the standard normalisation $\int
Y_{lm}Y_{l^{\prime}m^{\prime}}=\delta_{ll^{\prime}}\delta_{mm^{\prime}}$ for
the spherical harmonics.

The integral in equation 5.6 is finite for the all those solutions discussed
described by fluctuations that are finite everywhere. In particular, by
specialising to the minimal universal solution class of solution, equation
4.7, and by setting all the quantum numbers to zero, $l=m=n=p=0$ such that
$\psi_{lmnp}=1$, equation 4.7 evaluated on equation 2.1 defined in the
Einstein frame yields

$C_{\\{0\\}}=\frac{1}{4\kappa_{10}^{2}}\mathop{\rm
vol}\nolimits_{T^{4}}\mathop{\rm vol}\nolimits_{S^{2}}\mathop{\rm
vol}\nolimits_{\psi}\int_{\mathcal{I}_{\rho}}d\rho\,\hat{h}_{4}h_{8}\,.$ (5.7)

For the volume-forms in the above we use

$\mathop{\rm vol}\nolimits_{T^{4}}=16\pi^{4}\,,\qquad\mathop{\rm
vol}\nolimits_{S^{2}}=4\pi\,,\qquad\mathop{\rm vol}\nolimits_{\psi}=2\pi\,,$
(5.8)

and also

$2\kappa^{2}=(2\pi)^{7}\,,$ (5.9)

in units such that $\alpha^{\prime}=g_{s}=1$ in order to follow closely [92].
Then, the final result is

$C_{\\{0\\}}=\frac{1}{2\pi}\int_{\mathcal{I}_{\rho}}d\rho\,\hat{h}_{4}h_{8}\,,$
(5.10)

which agrees with [92, equation 4.4] modulo an irrelevant numerical factor.

In fact, if we are persistent on fixing the final numerical factor we can
consider the integral along the AdS directions. The integrand is zero, which
in turn implies that we have the multiplication of equation 5.10 by a
constant. That constant can be fixed by matching directly to the holographic
central charge computed in [92, equation 4.4].

##  6 On the SU(1,1$|$2) multiplets

For each one of the mode solutions that are derived from the fluctuation of
the metric that we have considered in this work, there exists an operator in
the dual field theory picture. We should attempt to understand to what kind of
operators these metric perturbations correspond. With that in mind, it will be
useful to try and identify the associated field theory multiplets. We will
denote the states that correspond to the metric fluctuations using the
relevant conformal dimension and the angular quantum number due to the
$\text{S}^{2}$ isometries. Hence we have

$[\Delta,\ell]\,.$ (6.1)

The unitary, irreducible representations of the $\text{SU}(1,1|2)$ have been
constructed long ago in [112], see also [113]. The short multiplets are
labelled by a half-integer

$(k,k)\oplus 2(k+\tfrac{1}{2},k-\tfrac{1}{2})\oplus(k+1,k-1)\,,$ (6.2)

and we have in total four states, the $\text{SL}(2,\mathbb{R})$ primaries,
except for the lowest case, $k=\tfrac{1}{2}$, where the last one is absent.

As we pointed out earlier, the mass of a spin-2 field in the bulk and the
conformal dimension of the dual operator are related by means of
$M^{2}=\Delta(\Delta-1)$. Therefore, the minimal universal solution derived
earlier, equation 4.7, for which we have $M^{2}=\ell(\ell+1)$, corresponds to
operators with scaling dimension $\Delta=\ell+1$. Therefore we can re-write
the state as

$[\ell+1,\ell]\,.$ (6.3)

Starting from the lowest value of the $\text{S}^{2}$ harmonics $\ell=0$, we
that the state is $[1,0]$. Taking into consideration that this should
correspond to the $k=\tfrac{1}{2}$ value we can relate the values as
$k=\tfrac{1}{2}+\ell$.

##  7 Non-universal classes of solutions

Here, we wish to consider two particular classes of solutions to the Sturm-
Liouville problem, which are distinct from the universal ones. In other words,
the solutions we discuss here do not saturate the bound $M^{2}=\ell(\ell+1)$.

### Case I: Solutions with $n=p=0$

We start by considering $n=p=0$ still. This means that the bulk modes are not
excited along the $\text{T}^{4}$ directions and on the $S^{1}$. The difference
with the minimal universal class is only that the mass does not saturate the
unitarity bound. In order to solve the Sturm-Liouville problem, we will have
to choose some particular $u$, $\hat{h}_{4}$ and $h_{8}$. This choice, in
turn, corresponds to a particular background.

Let us start off by considering the case of

$\hat{h}_{4}=h_{8}=\beta_{0}\begin{cases}\rho/2\pi&0\leq\rho\leq\pi P\\\
P-\rho/2\pi&\pi P<\rho\leq 2\pi P\end{cases}\,,\quad\quad
u=\frac{\beta_{0}}{2\pi}\rho\,.$ (7.1)

A solution to equations 3.12, 3.13 and 3.14 splits into two solutions along
the two intervals $\mathcal{I}_{1}=\left[0,\pi P\right]$ and
$\mathcal{I}_{2}=\left(\pi P,2\pi P\right]$, since both $\hat{h}_{4}$ and
$h_{8}$ are only piecewise continuous. Moreover, since we are interested in a
smooth solution for the fluctuations, we will impose that the solution and its
derivative are continuous at $\rho=\pi P$.

For the particular choice of $u$, $\hat{h}_{4}$, $h_{8}$ we made here and for
$n=p=0$ the Sturm-Liouville problem equations 3.12, 3.13 and 3.14 becomes in
$\mathcal{I}_{1}$

$\varphi^{\prime\prime}(\rho)+\frac{2\ell+2}{\rho}\varphi^{\prime}(\rho)+\lambda~{}\varphi(\rho)=0\,,$
(7.2)

while in $\mathcal{I}_{2}$ we have

$\varphi^{\prime\prime}(\rho)+\frac{2\ell+2}{\rho}\varphi^{\prime}(\rho)+\lambda~{}\frac{(P-\rho/2\pi)^{2}}{\rho^{2}}\varphi(\rho)=0\,,$
(7.3)

with $\lambda$ being given by equation 3.14. The general solution of equation
7.2 reads

$\varphi=\rho^{-\ell-\frac{1}{2}}\left(c_{1}~{}J_{\ell+\frac{1}{2}}\big{(}\sqrt{\lambda}\rho\big{)}+c_{2}~{}Y_{\ell+\frac{1}{2}}\big{(}\sqrt{\lambda}\rho\big{)}\right)\,,$
(7.4)

with $J_{n}(z)$ and $Y_{n}(z)$ being the Bessel functions of the first and
second kind, respectively, and $c_{1,2}$ are just the integration constants.
However, since we are interested in the solution and its derivative anywhere
within $\mathcal{I}_{1}$ we must set $c_{2}=0$ and hence the final solution
reads:

$\varphi=c_{1}~{}\rho^{-\ell-\frac{1}{2}}~{}J_{\ell+\frac{1}{2}}\big{(}\sqrt{\lambda}\rho\big{)}\,.$
(7.5)

We can, also, solve equation 7.3 whose general solution is given by:

$\varphi=e^{-\sqrt{\lambda}\rho}\rho^{-(\ell+\frac{1}{2}-\frac{\gamma}{2})}\left(c_{3}~{}U(\alpha,1+\gamma,2i\sqrt{\lambda}\rho)+c_{4}~{}L^{\gamma}_{-\alpha}(2i\sqrt{\lambda}\rho)\right)\,,$
(7.6)

where in the above $U(a,b,z)$ is the confluent hypergeometric function,
$L^{a}_{b}(z)$ is the generalized Laguerre polynomial, $c_{3,4}$ are
integration constants and we have defined the auxiliary parameters
$\alpha,\gamma$ as follows:

$\displaystyle\alpha$ $\displaystyle=\frac{1}{2}-2i\pi
P\sqrt{\lambda}+\frac{1}{2}\gamma\,,$ (7.7) $\displaystyle\gamma$
$\displaystyle=\sqrt{(1+2\ell)^{2}-16P^{2}\pi^{2}\lambda}\,.$

The above solution is well-defined everywhere along the $\mathcal{I}_{2}$
interval and hence there is no need to tune any of the constants to zero.

We can now proceed to match the solutions derived above and their derivatives
at $\rho=\pi P$. This leads to two conditions. A third condition is coming
from imposing that the solution we derived in the $\mathcal{I}_{2}$ interval
or its derivative should vanish at $\rho=2\pi P$; see the discussion below
equation 4.2. We choose to make the function vanish. After we take these into
consideration, we have three equations and we need to determine three unknown
parameters, namely $c_{1,3,4}$. This system admits a non-trivial solution if
and only if the determinant of the following matrix

$\begin{pmatrix}\varphi_{1}(\rho=\pi P)&-\varphi_{3}(\rho=\pi
P)&-\varphi_{4}(\rho=\pi P)\\\ \varphi^{\prime}_{1}(\rho=\pi
P)&-\varphi^{\prime}_{3}(\rho=\pi P)&-\varphi^{\prime}_{4}(\rho=\pi P)\\\
0&\varphi_{3}(\rho=2\pi P)&\varphi_{4}(\rho=2\pi P)\end{pmatrix}\,,$ (7.8)

is equal to zero.

It is at this point that we would have to resort to a careful numerical
analysis for different $M^{2}$ such that the above holds true. The expectation
is that the consistent solution for $M^{2}$ is of the schematic form

$M^{2}=\ell(\ell+1)+p_{1}f(p_{1},p_{2})\,,$ (7.9)

with $p_{1}$ would be a positive integer number and $f$ a positive function
such that $f(0,p_{2})$ is a regular solution.

It is possible realize and explain the form of equation 7.9 analytically by
examining carefully the results and conclusions of section 4. We know from the
holographic dictionary that $M^{2}=\Delta(\Delta-1)$ in the case of AdS2/CFT1.
We, also, know that the conformal dimension is bounded from below owing to the
unitarity of the CFT, in other words $\Delta\geq\Delta_{\texttt{min}}$ with
$\Delta_{\texttt{min}}=\ell+1$. The value of $\Delta_{\texttt{min}}$ is
carried by the universal spin-2 fluctuations that were studied in section 4
and are present in all the CFT duals of these AdS2 backgrounds, regardless of
the form of the defining functions $u,\hat{h}_{4}$ and $h_{8}$. These
universal fluctuations are, of course, present in the specific examples that
we study in this section. The field theory operators dual to the modes
described by equations 7.5 and 7.6 necessarily carry a conformal dimension
bigger than that of $\ell+1$. This is reflected in their mass spectrum by the
addition of $p_{1}f(p_{1},p_{2})$333in simpler terms this quantity denotes a
numerical interpolating function. to the minimum $M^{2}$ that was derived in
equation 4.7.

### Case II: Solutions with $n=0$

Here, we allow the modes to have some excitation on the $\text{S}^{1}$ while
keeping them at $n=0$, i.e no excitation along the $\text{T}^{4}$. We choose
to examine, as before, the backgrounds described by defining functions given
by equation 7.1. Similarly to our previous consideration, the Sturm-Liouville
problem equations 3.12, 3.13 and 3.14 is split into two part. In
$\mathcal{I}_{1}$ interval the equation that governs the spin-2 dynamics is
given by:

$\varphi^{\prime\prime}(\rho)+\frac{2\ell+2}{\rho}\varphi^{\prime}(\rho)+(\lambda-p^{2})~{}\varphi(\rho)=0\,,$
(7.10)

while in $\mathcal{I}_{2}$ we have

$\varphi^{\prime\prime}(\rho)+\frac{2\ell+2}{\rho}\varphi^{\prime}(\rho)+\left(-p^{2}+\lambda~{}\frac{(P-\rho/2\pi)^{2}}{\rho^{2}}\right)\varphi(\rho)=0\,,$
(7.11)

with $\lambda$ being given by equation 3.14.

It is obvious that the resulting equations in this case, equations 7.10 and
7.11, are very similar to the ones obtained when we did not consider any
excitation on the $\text{S}^{1}$, see equations 7.2 and 7.3. The solutions to
the above equations are the ones we expect from the previous case. Namely, the
general solution of equation 7.10 reads

$\varphi=\rho^{-\ell-\frac{1}{2}}\left(c_{1}~{}J_{\ell+\frac{1}{2}}\big{(}-i\sqrt{p^{2}-\lambda}\rho\big{)}+c_{2}~{}Y_{\ell+\frac{1}{2}}\big{(}-i\sqrt{p^{2}-\lambda}\rho\big{)}\right)\,,$
(7.12)

with $J_{n}(z)$ and $Y_{n}(z)$ being the Bessel functions of the first and
second kind, respectively, and $c_{1,2}$ are just the integration constants.
However, since we are interested in the solution and its derivative anywhere
within $\mathcal{I}_{1}$ we must set $c_{2}=0$ and hence the final solution
reads:

$\varphi=c_{1}~{}\rho^{-\ell-\frac{1}{2}}~{}J_{\ell+\frac{1}{2}}\big{(}-i\sqrt{p^{2}-\lambda}\rho\big{)}\,.$
(7.13)

We can, also, solve equation 7.3 whose general solution is given by:

$\varphi=e^{-\sqrt{p^{2}-\lambda}\rho}\rho^{-(\ell+\frac{1}{2}-\frac{\gamma}{2})}\left(c_{3}~{}U(\alpha,1+\gamma,2\sqrt{p^{2}-\lambda}\rho)+c_{4}~{}L^{\gamma}_{-\alpha}(2\sqrt{p^{2}-\lambda}\rho)\right)\,,$
(7.14)

where in the above $U(a,b,z)$ is the confluent hypergeometric function,
$L^{a}_{b}(z)$ is the generalized Laguerre polynomial, $c_{3,4}$ are
integration constants and we have defined the auxiliary parameters
$\alpha,\gamma$ as follows:

$\displaystyle\alpha$ $\displaystyle=\frac{1}{2}+\frac{2\pi
P\lambda}{\sqrt{p^{2}-\lambda}}+\frac{1}{2}\gamma\,,$ (7.15)
$\displaystyle\gamma$
$\displaystyle=\sqrt{(1+2\ell)^{2}-16P^{2}\pi^{2}\lambda}\,.$

The above solution is well-defined everywhere along the $\mathcal{I}_{2}$
interval and hence there is no need to tune any of the constants to zero.

Of course, one would have to proceed as before in order to set up a system of
equations that would need to analyse numerically, with very minor changes from
our previous treatment. More specifically, we again proceed to match the
solutions derived above and their derivatives at the point $\rho=\pi P$. This
leads to two conditions. A third condition is coming from imposing that the
solution we derived in the $\mathcal{I}_{2}$ interval or its derivative should
vanish at $\rho=2\pi P$; see the discussion below equation 4.2. We choose to
make the function vanish. After we take these into consideration, we have
three equations and we need to determine three unknown parameters, namely
$c_{1,3,4}$. This system admits a non-trivial solution if and only if the
determinant of the following matrix

$\begin{pmatrix}\varphi_{1}(\rho=\pi P)&-\varphi_{3}(\rho=\pi
P)&-\varphi_{4}(\rho=\pi P)\\\ \varphi^{\prime}_{1}(\rho=\pi
P)&-\varphi^{\prime}_{3}(\rho=\pi P)&-\varphi^{\prime}_{4}(\rho=\pi P)\\\
0&\varphi_{3}(\rho=2\pi P)&\varphi_{4}(\rho=2\pi P)\end{pmatrix}\,,$ (7.16)

is equal to zero.

The mass spectrum in this case, also, assumes the generic form

$M^{2}=\ell(\ell+1)+r_{1}g(r_{1},r_{2})\,,$ (7.17)

with $r_{1}$ would be a positive integer number and $g$ a positive function
such that $g(0,r_{2})$ is a regular solution.

It is possible realize and explain the form of equation 7.17 analytically by
examining carefully the results and conclusions of section 4. We know from the
holographic dictionary that $M^{2}=\Delta(\Delta-1)$ in the case of AdS2/CFT1.
We, also, know that the conformal dimension is bounded from below owing to the
unitarity of the CFT, in other words $\Delta\geq\Delta_{\texttt{min}}$ with
$\Delta_{\texttt{min}}=\ell+1$. The value of $\Delta_{\texttt{min}}$ is
carried by the universal spin-2 fluctuations that were studied in section 4
and are present in all the CFT duals of these AdS2 backgrounds, regardless of
the form of the defining functions $u,\hat{h}_{4}$ and $h_{8}$. These
universal fluctuations are, of course, present in the specific examples that
we study in this section. The field theory operators dual to the modes
described by equations 7.5 and 7.6 necessarily carry a conformal dimension
bigger than that of $\ell+1$. This is reflected in their mass spectrum by the
addition of $r_{1}g(r_{1},r_{2})$444in this case, as in the previous one, it
is just a numerical interpolating function. to the minimum $M^{2}$ that was
derived in equation 4.7.

##  8 From $\text{AdS}_{2}/\text{CFT}_{1}$ to $\text{AdS}_{3}/\text{CFT}_{2}$

In this section we discuss how we can use the $\text{AdS}_{2}/\text{CFT}_{1}$
picture developed in [92] to extract information about the seed
$\text{AdS}_{3}/\text{CFT}_{2}$ theories [52].

### The spin-2 dynamics

Let us recall the basic equation that governs the dynamics of the spin-2 modes
before decomposing along the various submanifolds

$\left[\frac{1}{\hat{h}_{4}h_{8}}\frac{d}{d\rho}\left(u^{2}\frac{d}{d\rho}\right)+\right(4+\frac{(u^{\prime})^{2}}{\hat{h}_{4}h_{8}}\left)\nabla^{2}_{S^{2}}+\frac{u}{\hat{h}_{4}}\sum^{4}_{i=1}\partial^{2}_{\theta_{i}}+\frac{u^{2}}{\hat{h}_{4}h_{8}}\partial^{2}_{\psi}+M^{2}\right]\psi=0\,.$
(8.1)

The difference between the above and [107, equation (16)] is merely the
$\partial^{2}_{\psi}$-term which is associated with the $\text{S}^{1}$.
However, keeping in mind that the $\text{S}^{1}$ was derived by a T-duality
transformation along the spatial direction of the seed $\text{AdS}_{3}$, one
can just disregard it if the interest is to extract information about the seed
backgrounds from the lower-dimensional ones.

### The minimal universal solutions

The minimal universal solutions described by equation 4.7 are exactly the same
as those obtained in the seed $\text{AdS}_{3}$ backgrounds, see [107, equation
(28)]. This is perhaps not too surprising, since there are many structural
similarities between the two different backgrounds as they are T-duals. Of
course, the unitarity bounds are different in the two cases, however, by
adopting the line of reasoning described above and solving the modified
$\text{AdS}_{2}$ Sturm-Liouville problem one would be able to start from an
$\text{AdS}_{2}$ study and derive a result pertaining to the two-dimensional
superconformal field theory dual to the $\text{AdS}_{3}$ background solutions.

### The non-universal solutions

The reason that we chose the defining functions to have the particular form
given by equation 7.1 was to make a direct comparison [107, appendix A].
Specifically, the non-universal Class I mode solutions are the same both in
the $\text{AdS}_{2}$ backgrounds and the $\text{AdS}_{3}$ solutions. We are
referring to our solutions given by equations 7.4 and 7.6 that can be compared
directly to [107, equations (39) and (40)]. A more interesting question would
be, perhaps, to try and understand how the Class II non-universal solutions
are related to the $\text{AdS}_{3}$ backgrounds. Recall, that these are the
solutions obtained by considering some non-trivial excitation on the
$\text{S}^{1}$ which is obtained by a T-duality transformation along an AdS
isometry in the seed massive type IIA theories.

### The holographic central charge

A final comment needs to be made before we close this section. From the
R-symmetry discussion in section 2 we expect that there is a match between the
central charge we computed here, and the central charge of the
$\mathcal{N}=(4,0)$ $\text{AdS}_{3}$ seed solutions. In the latter context,
the effective three-dimensional gravitational coupling $\kappa_{3}$ is related
to $C_{\\{0\\}}$ by $C_{\\{0\\}}=1/\kappa_{3}^{2}$. Also, the expanded
quadratic action for the $h_{\mu\nu}$ fluctuations computes the two-point
function of the dual stress-energy tensor. The coefficient of the latter is
proportional to the central charge of the dual two-dimensional superconformal
field theory. Equation 5.10 is, as expected, equal to the central charge
derived in [53, equation 2.27] up to an unimportant numerical factor, which
can be fixed as we described earlier in section 5.

A, perhaps, more natural and straightforward comparison for the expression of
the central charge can be made when considering the result we derived earlier
in equation 5.10 and the result for the central charge computed in [107]. In
the latter, the same method that was adopted here was used to compute the
central charge of the seed field theory solutions, namely the spin-2
fluctuations were studied in the seed $\text{AdS}_{3}$ massive type IIA
solutions. Equation 5.10 is exactly the same as [107, equation 37] as
expected.

##  9 Epilogue

In this work we have investigated some aspects of spin-2 fluctuations around a
warped type IIB solution of the form
$\text{AdS}_{2}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$
of [92], which is related via T-duality to the
$\text{AdS}_{3}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\rho}$
background of [52]. We have derived the equation that governs the dynamics of
these spin-2 modes following the logic of [99]. We have seen that they fall
into two big classes, the universal and non-universal. The former are
independent of the background supergravity data and are present in all the
$d=1$ field theories that are associated with the backgrounds considered here.

As we have seen in the main body of our work, when the universal class of
fluctuations saturates the bound on the mass, it is dual to operators with
scaling dimension $\Delta=\ell+1$, where $\ell$ is the angular-momentum-charge
on the $\text{S}^{2}$. This is the holographic manifestation of the
$SU(2)_{R}$ symmetry of the dual field theory.

The non universal solutions are more difficult to analyse as they depend on
background data. In other words, one has to make specific choices for
functions $u$, $\hat{h}_{4}$ and $h_{8}$ that fully determine the
$\text{AdS}_{2}$ solutions.

Finally, we have computed the central charge for the one-dimensional dual
quiver field theory using the action for the spin-2 fluctuations,
$h_{\mu\nu}$.

As we have, already, seen these $\text{AdS}_{2}$ backgrounds share certain
similarities in their structures with their higher-dimensional seed solutions.
We have seen in detail the relations of the minimal universal class of modes
to the corresponding ones in the $\text{AdS}_{3}$ solutions and how the
central captures the correct behaviour of the two-dimensional seed conformal
field theory. We have, also, discussed these similarities in some particular
non-universal solutions.

It would be interesting to take inspiration from the various $\text{AdS}_{3}$
studies and compute additional holographic observables to understand the
$\text{AdS}_{2}\times\text{S}^{2}\times\text{CY}_{2}\times\mathcal{I}_{\psi}\times\mathcal{I}_{\rho}$
more thoroughly. It would also be interesting to consider metric fluctuations
in the $\text{T}^{4}$ part of the geometry. Since, none of the defining
functions of the background depend on those directions these fluctuations
would, also, satisfy a massless Klein-Gordon equation in ten dimensions, and
would capture part of the spin-0 spectrum of the dual $\mathcal{N}=4$ quantum
mechanical theory555we are grateful to Christoph Uhlemann for suggesting this
point and related comments.. It would be interesting to study these in our
setup as well as in the seed $\text{AdS}_{3}$ and spot any similarities or
differences. Finally, having already a map that takes us from $\text{AdS}_{2}$
to the $\text{AdS}_{3}$ solutions, it would be interesting to study the
backgrounds that are described by $u=\texttt{constant}$ in $\text{AdS}_{2}$
and use the results in relation to the $\text{AdS}_{3}$ theories.

## Acknowledgments

I have greatly benefited from discussions and suggestions with Carlos Nunez. I
am grateful to Nick J. Evans, Jose Manuel Penin and Stefano Speziali for
related conversations. I would, also, like to thank Yolanda Lozano, Oleg
Lunin, Carlos Nunez and Christoph Uhlemann for reading this manuscript and
offering valuable comments.

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