# Incremental Prototype Tuning for Class Incremental Learning

Jieren Deng  School of Artificial Intelligence, University of Chinese Academy
of Science  Institute of Automation, Chinese Academy of Sciences (CAS)
{dengjieren2019, jianhua.hu, zhanghaojian2014<EMAIL_ADDRESS>Haojian
Zhang  Institute of Automation, Chinese Academy of Sciences (CAS)
{dengjieren2019, jianhua.hu, zhanghaojian2014<EMAIL_ADDRESS>Jianhua
Hu  Institute of Automation, Chinese Academy of Sciences (CAS)
{dengjieren2019, jianhua.hu, zhanghaojian2014<EMAIL_ADDRESS>Yunkuan
Wang Corresponding author Institute of Automation, Chinese Academy of Sciences
(CAS)
{dengjieren2019, jianhua.hu, zhanghaojian2014<EMAIL_ADDRESS>
###### Abstract

Class incremental learning(CIL) has attracted much attention, but most
existing related works focus on fine-tuning the entire representation model,
which inevitably results in much catastrophic forgetting. In the contrast,
with a semantic-rich pre-trained representation model, parameter-additional-
tuning (PAT) only changes very few parameters to learn new visual concepts.
Recent studies have proved that PAT-based CIL can naturally avoid fighting
against forgetting by replaying or distilling like most of the existing
methods. However, we find that PAT-based CIL still faces serious semantic
drift, the high-level forgetting problem caused by classifier learning bias at
different learning phases, which significantly reduces the performance of PAT-
based CIL. To address this problem, we propose Incremental Prototype Tuning
(IPT), a simple but effective method that tunes category prototypes for
classification and learning example prototypes to compensate for semantic
drift. Extensive experiments demonstrate that our method can effectively
compensate for semantic drift. Combined with well-pre-trained Vit backbones
and other PAT methods, IPT surpasses the state-of-the-art baselines on
mainstream incremental learning benchmarks.

##  1 Introduction

Open learning ability is the key to improve the level of artificial
intelligence. Class incremental learning (CIL) [26] is exactly the natural way
to raise the openness of AI systems, but it faces a notorious problem called
catastrophic forgetting [22] – directly end-to-end training the neural network
learned from the old task on the new task leads to a sharp decline in the
performance of the old task.

Most class incremental approaches resist catastrophic forgetting by replaying
the seen data [26, 19, 35] or distilling the previous models [17, 7, 4].
However, when just feed a few new visual concepts, the representation model
learned from last phase is still needed to be fine-tuned in the next phase
endlessly, affecting the whole body of the model and bringing in lots of
forgetting, so that it limits the performance of class incremental learning.

Different from the fine-tuning methods that tune the entire model, parameter-
additional-tuning methods (PAT) such as linear probing, prompt tuning, and
adapter, only allow a very small number of additional parameters to be
trained. Therefore PAT can effectively weaken forgetting while retaining the
ability to learn new knowledge, as well as reducing the overfitting of each
small task in CIL with the support of rich prior knowledge stored in the pre-
trained model. In addition, PAT converges quickly because of its little
learnable parameter scale. A recent study L2P[32] has proved that PAT with a
strong pre-trained representation model can even help class incremental
learning to obtain competitive results without fine-tuning the entire
representation model, replaying old data, or distilling from the previous
model. We believe that PAT is a very promising method to apply CIL to
practical applications.

Although PAT with strong pre-trained representation can greatly improve the
performance of class incremental learning systems while avoiding replaying
original samples, the performance is not satisfactory enough. We find a major
reason is that the categories at different phases are not trained together,
and the classifier may overlap the category prototypes learned at different
phases in the representation space, as shown in the left subfigure of Figure
1, which is called semantic drift [36].

Figure 1: T-SNE[20] of CIFAR-10 [14] to show semantic drift and its
compensation. Note that the black arrows reflect the semantic drift. IPT
without example prototypes may cause serious semantic drift as shown in the
left picture, in which the category prototypes may be far away from the
density center of the target class and even close to another class. Only one
example prototype can even greatly alleviates this problem as shown in the
right picture.

To address semantic drift, we propose a novel PAT method. In detail, extract
several learnable prototypes for each category (namely example prototypes).
These example prototypes are classified by all learned category prototypes to
constrain the classifier to isolate the category prototypes when learning new
categories from different phases. Example prototypes aim to imitate each
class’ distribution in representation space, so they should not be
homogenized. In order to ensure the diversity of example prototypes, we
initialize them by K-means clustering and design a new maximum similarity loss
to optimize them. Summarizing the above steps, our method called incremental
prototype tuning (IPT) can effectively reduce semantic drift as shown in the
right subfigure of Figure 1. Unlike replaying methods storing the original
samples, IPT only stores several prototypes of each category thus saving a lot
of storage resources.

By combining IPT and L2P, our method can outperform baselines and others
state-of-the-art models on CIFAR-100 [14], ImageNet-100 [26] and ImageNet-1000
[28] benchmarks. IPT is indeed simple but surprisingly enables an AI system to
learn new concepts incrementally without replaying original data or
distillation from previous models.

In summary, we provide the following contributions:

1\. We argue that PAT is a promising direction for CIL, and point out that
semantic drift is one of the major factor that affects the performance of PAT-
based CIL. To quantitatively analyze this effect, we also propose a new metric
called category drift rate (CDR) to measure semantic drift.

2\. We propose a novel PAT-based method named incremental prototype tuning
(IPT), which alleviates semantic drift by classifying the learnable example
prototypes. A new maximum similarity loss (MSL) is designed to maintain the
diversity of example prototypes to ensure the effectiveness of the
classification constraints of the example prototypes.

3\. IPT achieves the new state-of-the-art on standard incremental learning
benchmarks without replaying original data and distilling from previous
models, which reduces the requirement of memory storage and improves the
feasibility of PAT-based CIL in practical application.

##  2 Related work

###  2.1 Class Incremental Learning

Deep learning models overspecialize at individual tasks while lacking the
ability to learn openly to deal with other tasks, which is in sharp contrast
to human beings. To address this problem, incremental learning attempts to
develop an artificial intelligence system that can continuously learn and
process new tasks from new data, while retaining the knowledge learned from
previous tasks. Incremental learning can be divided into three categories
[31]: task incremental learning, domain incremental learning, and class
incremental learning (CIL) [26], where CIL is more challenging and closer to
the needs of practical applications which the task identity is not informed
when classifying test samples. Recent popular CIL methods can be categorized
into three classes: replay-based, regularization-based, and parameter-
isolation-based [2, 24, 18]. Replay-based methods preserve a small amount of
data in previous tasks in memory or a generative model in order to replay
these data when training the model on new data to overcome catastrophic
forgetting [26, 19, 35, 10, 33]. The regularization-based methods provide
terms in the final loss to restrict the model with prior or knowledge
distillation to change too much to forget previous knowledge [12, 15, 37, 17,
7, 4]. For parameter-isolation-based methods, different subsets of the model
parameters are dedicated to each task to prevent the previous tasks from
getting any possible forgetting [21, 30, 29, 1].

###  2.2 Parameter-additional-tuning with pre-trained model

Parameter-additional-tuning (PAT) aims to alias the pre-trained
representations and features of downstream tasks by adjusting the additional
task-specific parameters while remaining the pre-trained model fixed.
Recently, the most popular PAT can be categorized into two kinds of tuning
methods: adapter-based, and prompt-based. The adapter-based approach injects
small-scale neural network modules (adapters) connected to the transformer
layer, and only adjusts these adapters for model adaptation [8, 11, 27]. The
adapter module contains a down-projection layer and an up-projection layer and
achieves impressive performance while only owning about 0.5% 8% parameters of
the whole model. Instead of injecting neural modules to the Transformer model,
the prompt-based approach learns new input tokens for transformer backbone [6,
16, 9], has achieved promising performance in various NLP tasks. L2P [32] is
the first work that introduces the idea of prompt-tuning into incremental
learning. We choose to use it as one of our baseline methods.

##  3 Incremental prototype tuning

###  3.1 Problem setup

Given a data stream of a labeled sample super set
$\\{X^{1},X^{2},\cdots,X^{t}\\}$, where
$X^{y}=\\{x_{1}^{y},x_{2}^{y},\cdots,x_{n_{y}}^{y}\\}$ is a set containing all
samples belonging to class $y$, class incremental learning models learn from a
dataset stream {$D_{0},D_{1},\cdots,D_{t},\cdots$}. At each phase t, only
$D_{t}=\\{X^{s_{t}+1},X^{s_{t}+2},\cdots,X^{s_{t+1}}\\}$ is accessible for
training the model, while the seen data ${X^{1},\cdots,X^{s_{t}}}$ is no
longer available. Here, $s_{t}$ represents the sum of the number of categories
learned before phase t. During the test, the trained model is expected to
classify all seen categories. It is needed to note that joint training whose
accuracy is regarded as the upper bound of class incremental learning puts all
samples of all categories together for training.

###  3.2 Baseline

L2P is one of the baseline methods of our work. As shown in Figure 2. It
learns a pool of prompt queries as one part of the input for the pre-trained
Vit [3] backbone and mitigates catastrophic forgetting by a key-value-pair-
based prompt selection strategy. Though both IPT and L2P are PAT methods, the
difference between IPT and L2P is that IPT chooses to learn category
prototypes as the input for the classifier and servers a plug-in module for
all PAT methods. Since semantic drift is a higher-level forgetting problem, it
is less obvious to see the semantic drift when the representation model is not
frozen because it is mixed with a lower level of forgetting (representation
forgetting). Therefore, another baseline of our work is the method that
naively freezes the backbone as the representation model to avoid the
interference of representation forgetting.

Figure 2: Structure of L2P.“C" means concatenation.

###  3.3 Overview

An overview of incremental prototype tuning (IPT) for class incremental
learning is presented in Figure 3. The IPT consists of a pre-trained model
with additional parameters, category prototypes, and examples prototypes.

Figure 3: Overview of IPT. IPT consists of category prototypes and example
prototypes that are optimized by three kinds of loss including maximum similar
loss, loss of sample classification, and loss of example classification. The
backbone can be frozen or tuned by any PAT method such L2P.

Three Stage Training strategy. The training strategy of our model for each
phase involves three stages, the initialization stage, the prototype training
stage, and the additional parameter tuning stage.

At the initialization stage, new category prototypes and new examples
prototypes are initialized and integrated into the model The new category
prototypes can simply be initialized as the sample embeddings of target
categories or the embeddings of the class name provided by the text encoder
pre-trained together with the representation model, such as CLIP [25]. The new
example prototypes are initialized by a clustering algorithm.

At the prototype training stage, the loss at phase t is computed by the simple
addition:

$\displaystyle L^{t}=L_{CES}^{t}+L_{CEP}^{t}+L_{MS}^{t}.$ (1)

At this stage, only category prototypes, example prototype of phase t are
trainable.

At the additional parameter tuning stage, we add the parameter-additional-
tuning loss term $L_{PAT}^{t}$ to the previous loss:

$\displaystyle L^{t}=L_{PAT}^{t}+L_{CES}^{t}+L_{CEP}^{t}+L_{MS}^{t}.$ (2)

At this stage, only additional parameters are trainable. If there is no
additional parameter for tuning, the training of this stage can be canceled.

Inference. The inference architecture consists of the pre-trained model,
additional parameters, and all category prototypes. Samples are classified
according to which category prototypes their embeddings, extracted by the pre-
trained representation model with additional parameters, are most similar
with.

###  3.4 Loss of sample classification

At the prototype training stage during phase $t$, each sample is extracted as
a representation $\Phi(x,\theta)$ by the pre-trained representation model
$\theta$. For each sample x, suppose the similarity between $\Phi(x,\theta)$
and category prototype $c_{i}$ is represented as:

$\displaystyle s(x,c_{i})=<\Phi(x,\theta),c_{i}>,$ (3)

where $i\in\\{s_{t},\cdots,s_{t+1}-1\\}$ and $x\in D_{t}.$ We get the
classification result of samples after applying a softmax function:

$\displaystyle\begin{split}[y_{s_{t}}(x),\cdots,y_{s_{t+1}-1}(x)]^{T}=softmax([s(x,c_{s_{t}})\\\
,\cdots,s(x,c_{s_{t+1}-1})]^{T}).\end{split}$ (4)

Finally, cross-entropy loss during phase t is computed as:

$\displaystyle
L_{CES}^{t}(x)=\sum_{i=s_{t}}^{s_{t+1}-1}\hat{y}_{i}(x)\log{y_{i}(x)},$ (5)

where $\hat{y}_{i}(x)$ is the ground truth of probability of sample $x$
belongs to class $i$.

At the additional parameters tuning stage, the parameter-additional-tuning
loss $L_{PAT}^{t}$ is computed by the similar computing way of $L_{CES}^{t}$,
the minor difference is that each sample is extracted as a representation
$\Phi(x,\theta,\gamma)$ by the pre-trained representation model $\theta$ with
additional parameters $\gamma$. Sometimes $L_{PAT}^{t}$ may also contains an
additional loss such as prompt key select loss in L2P [32], which is dependent
by which parameter-additional-tuning method is used.

###  3.5 Learning example prototypes

Motivation. As shown in Figure 1, when we map category prototypes and sample
embeddings into low dimension space for visualization, we find that category
prototypes are not always located in the distribution of the target class
sample embeddings, instead, it may be located in the distribution of other
class embeddings, which is called semantic drift [36]. Since the samples at
the former and latter phases are not available simultaneously, as long as the
category prototype can correctly classify the samples at the current phase,
the loss function does not necessarily continue to encourage the category
prototypes to enter the sample representation density center of the target
class. The category prototypes may even occupy the sample representation
density center of the categories at other phase, resulting in classification
confusion that greatly harms the performance of PAT. So that we propose to
optimizing the classification loss of several example prototypes to prevent
semantic drift.

Maximum similarity loss (MSL). We define the example prototypes of class $i$
as $e_{i}^{j},i=1,\cdots,N_{c},j=1,\cdots,N_{e}$, where $N_{c}$ is the total
number of all classes, and $N_{e}$ is the number of example prototypes.
Example prototypes are supposed to imitate the distribution of representation
embeddings of every category, they are trained by the maximum similarity loss
(MSL) computed as:

$\displaystyle
L_{MS}^{t}(x)=1-\max\limits_{j}(<\Phi(x,\theta),e^{j}_{g(x)}>),$ (6)

where $j=1,2,\cdots,N_{e}$, $x\in D_{t}$, and $g(x)$ is ground truth class id
label of sample $x$. Note that we compute $\Phi(x,\theta)$ without additional
parameters.Maximum similarity does not require each sample prototype to be
close to each sample of the target category. In this way, it is possible to
find some difficult samples that deviate from the density center of the
sample.

Initialization of example prototypes. The optimization direction of example
prototypes is mainly dominated by maximum similarity loss. When example
prototypes are far from the target category embedding center at the
initialization stage, only a few example prototypes can eventually enter the
sample distribution space of the target category, resulting in the example
prototype’s underfitting of target class distribution. Therefore, the example
prototype needs a better initialization method to shorten the initial distance
between all example prototypes and the target category embedding center. Note
that this distance should not be too small, otherwise example prototype will
lose the ability to mine difficult samples. So we use k-means as our
initialization method. For each new class $C$, we extract the representation
of all its training samples, then use the k-means algorithm to split them into
$N_{e}$ clusters, and then use these cluster centers as the initialization
values of new example prototypes. This initialization method not only allows
example prototypes to cover the sample representation distribution of the
corresponding class, but also retains the diversity of example prototypes.

###  3.6 Loss of example classification

Semantic drift can be prevented by classification loss of example prototypes.
During phase $t$, classification loss of example prototypes is computed as:

$\displaystyle
L^{t}_{CEP}(e_{i}^{j})=-\sum_{k=1}^{s_{t+1}-1}\hat{y_{k}}(e_{i}^{j})\log{y_{k}(e_{i}^{j})},$
(7)

where $i\in\\{1,2,\cdots,s_{t+1}-1\\}$,

$\displaystyle[y_{k}(e_{i}^{j})]^{T}=softmax([<e_{i}^{j},c_{k}>]^{T}),$ (8)

where $k\in\\{1,2,\cdots,s_{t+1}-1\\}$.

Discussion. Classification loss of example prototypes forces $c_{k}$ to be far
away from $e_{i}^{j}$ for each $i\neq k$, which helps to isolate category
prototypes from different phases. Moreover, since it can also adjust those
category prototypes born before phase $t$, accuracy of phase $t^{b}(t^{b}<t)$
can even be continually improved, achieving positive backward transfer.
Another advantage of example prototypes is that they naturally avoid the
class-imbalanced problem, since the number of example prototypes used in
equation 7 for each category is equal which makes $L^{t}_{CEP}$ a class-
balanced loss.

##  4 Experiment

###  4.1 Experiment setup

Benchmark rotocols. We validate our method with two widely used benchmark
protocols including the protocol of half initialization (PHI) proposed and the
protocol of no initialization (PNI). The PHI starts training the models on
half the classes (i.e., $50$ for CIFAR-100 and ImageNet-100, $500$ for
ImageNet-1000), then equally divides the remaining classes and incrementally
learns them at the future phase. The PNI simply divides the total classes
equally among phases and learns them phase by phase. Note that an $n$ phases
PHI has actually $n+1$ phases with one phase for half initialization, while
PNI has exactly $n$ phases.

Metrics. We use four indicators to evaluate our method: average incremental
accuracy, final accuracy, forgetting rate, and CDR. Rebuffi et al. [26]
introduce average incremental accuracy (AIA), which averages the accuracy of
all phases. Forgetting rate (FR), proposed by Liu et al. [19], measures the
degree of performance degradation in the first initialization phase after
incremental training. Besides, in order to see more clearly the influence of
semantic drift, we propose a new metric named category drift rate (CDR) to
measure semantic drift as followed:

$\displaystyle CDR=N_{d}/N_{c},$ (9)

where $N_{d}$ is the number of categories that misclassify more than 50 % of
all the samples that are classified as that category, and $N_{c}$ is the total
number of categories. The CDR is computed after all training phased on the
test dataset.

Figure 4: Relationship curves between IPT-based class incremental learning
performance and the number of example prototypes. In order to avoid infinity
and display better, the horizontal axis adopts the logarithm of the number of
example prototypes plus one. The legend content “k-means” or “random” means
whether to use k-means to initialize the example prototypes and "freeze" or
"free" means whether to freeze those category prototypes before the current
learning phase.

Implementation details. We implement our method in PyTorch with an RTX 3090.
We train 20 epochs at the prototype training stage and 10 epochs at the
additional parameters tuning stage with batch size 32 for all datasets. SGD is
used for optimization with a base learning rate of 0.01. Momentum and weight
decay parameters are set to 0.9 and 0.0005, respectively. We multiply the
learning rate by 0.1 at the beginning of the 5th, 10th, and 15th epoch. We use
a clipping gradient in the range of 5 to 100 to accelerate training. We choose
cosine distance to uniformly measure the distance among sample embeddings,
category prototypes, and example prototypes. Code will be available after
review.

###  4.2 Main results

Metric | Method | CIFAR100 | Standford Car | Oxford 102 Flower
---|---|---|---|---
phase=5 | 10 | 25 | 7 | 14 | 49 | 3 | 17
AIA(%) $\uparrow$ | Fixed ViTB/32 | 78.05 | 75.61 | 68.37 | 72.97 | 69.37 | 60.64 | 93.69 | 88.29
Fixed ViTB/32 + IPT | 80.54 | 80.08 | 79.22 | 80.81 | 80.23 | 80.01 | 93.74 | 94.42
L2P ViTB/16 | 79.56 | 76.72 | 73.88 | 78.35 | 76.60 | 67.43 | 93.33 | 78.23
L2P ViTB/16 + IPT | 81.47 | 79.73 | 79.42 | 83.34 | 82.41 | 83.32 | 92.21 | 93.20
Final Acc(%) $\uparrow$ | Fixed ViTB/32 | 72.14 | 68.28 | 57.96 | 63.25 | 56.22 | 43.19 | 90.19 | 81.40
Fixed ViTB/32 + IPT | 75.45 | 75.11 | 74.75 | 75.76 | 75.14 | 74.18 | 90.96 | 92.39
L2P ViTB/16 | 74.24 | 70.93 | 65.33 | 68.59 | 65.78 | 52.41 | 88.86 | 57.20
L2P ViTB/16 + IPT | 76.70 | 74.64 | 74.36 | 78.08 | 76.68 | 76.51 | 88.81 | 90.10
CDR(%) $\downarrow$ | Fixed ViTB/32 | 7 | 12 | 26 | 17 | 22 | 37 | 2 | 6
Fixed ViTB/32 + IPT | 7 | 6 | 7 | 4 | 6 | 8 | 1 | 1
L2P ViTB/16 | 7 | 11 | 20 | 19 | 20 | 32 | 3 | 34
L2P ViTB/16 + IPT | 6 | 7 | 7 | 3 | 4 | 4 | 2 | 1
Upper Bound | Fixed ViTB/32 | 77.89 | 80.39 | 94.96
L2P ViTB/16 | 83.91 | 83.18 | 96.01

Table 1: Comparison with baselines on CIFAR100, Standford Car, and Oxford 102
Flower under PHI protocol.

Comparison with baselines. We evaluate the proposed IPT on CIFAR100 [14],
Standford-Cars [13], and Oxford-102-Flower [23]. We use two baseline methods
including L2P [32] and a naive fixed backbone with category prototypes for
comparison. The models are pre-trained on CLIP400M [25], a dataset consists of
$400$ million image-text pairs. The main results of comparison with the
baseline methods is shown in Table 1. An intuitive impression of these results
is that high CDR will lead to low accuracy, while IPT can significantly reduce
CDR and improve accuracy, which verifies the hypothesis that semantic drift
seriously reduces the performance of PAT-based CIL and the effectiveness of
IPT for mitigating semantic drift.

Protocol | PHI | PNI
---|---|---
Phases | 5 | 10 | 5 | 10 | 50
Metrics | AIA(%)$\uparrow$ | FR(%)$\downarrow$ | AIA(%)$\uparrow$ | FR(%)$\downarrow$ | AIA(%)$\uparrow$ | AIA(%)$\uparrow$ | AIA(%)$\uparrow$
iCaRL-CLIP-VitB/16 | 58.18 ± 0.57 | 31.88 | 52.78 ± 0.96 | 33.1 | 71.45 ± 0.44 | 66.01 ± 0.92 | 57.08 ± 0.84
UCIR-CLIP-VitB/16 | 62.98 ± 1.02 | 18.3 | 60.56 ± 0.71 | 22.34 | 63.77 ± 0.84 | 58.65 ± 0.73 | 56.76 ± 3.54
BiC-CLIP-VitB/16 | 57.86 ± 0.36 | 31.22 | 53.28 ± 0.91 | 31.5 | 73.30 ± 0.45 | 68.89 ± 1.10 | 62.19 ± 0.75
DyTox-CLIP-VitB/16 | 76 ± 1.12 | 6.77 | 75.82 ± 1.21 | 8.30 | 77.42 ± 0.62 | 77.03 ± 1.24 | 75.12 ± 1.13
L2P-CLIP-VitB/16 | 79.56 | 4.86 | 76.72 | 6.42 | 83.42 | 80.17 | 70.20
L2P-CLIP-VitL/14 | 86.90 | 2.86 | 85.54 | 2.48 | 88.60 | 84.28 | 77.80
Fixed CLIP-VitB/16 + IPT | 82.32 ± 0.86 | 0.047 | 82.1 ± 1.07 | 0.187 | 83.12 ± 2.00 | 83.30 ± 2.40 | 84.25 ± 2.40
Fixed CLIP-VitL/14 + IPT | 87.73 ± 0.55 | -0.4 | 87.45 ± 0.83 | -0.46 | 88.62 ± 1.22 | 87.83 ± 1.35 | 89.09 ± 1.98

Table 2: Comparison with state-of-the-art on CIFAR-100 benchmarks

Datasets | ImageNet-100 | ImageNet-1000
---|---|---
Protocols | PHI | PHI
Phases | 5 | 10 | 5 | 10
Metrics | AIA(%)$\uparrow$ | FR(%)$\downarrow$ | AIA(%)$\uparrow$ | FR(%)$\downarrow$ | AIA(%)$\uparrow$ | FR(%)$\downarrow$ | AIA(%)$\uparrow$ | FR(%)$\downarrow$
iCaRL-CLIP-VitB/16 | 66.56 | 42.4 | 62.0 | 42.36 | 53.34 | 25.03 | 42.78 | 31.56
UCIR-CLIP-VitB/16 | 72.15 | 30.12 | 68.82 | 31.56 | 66.45 | 22.18 | 62.34 | 25.41
BiC-CLIP-VitB/16 | 70.98 | 25.14 | 66.34 | 30.14 | 43.45 | 29.14 | 42.29 | 33.71
DyTox-CLIP-VitB/16 | 78.53 | 7.56 | 76.44 | 8.34 | 71.80 | 4.30 | 69.54 | 6.37
L2P-CLIP-VitB/16 | 76.63 | 0.44 | 75.92 | 2.44 | 77.13 | 1.41 | 75.51 | 2.09
L2P-CLIP-VitL/14 | 75.62 | 0.88 | 71.49 | 2.6 | 79.64 | 0.37 | 75.31 | 3.42
Fixed CLIP-VitB/16 + IPT | 82.75 | 0.56 | 83.22 | 0.73 | 79.43 | -0.90 | 79.12 | -0.74
Fixed CLIP-VitL/14 + IPT | 85.64 | 0.16 | 85.98 | 0.04 | 83.37 | -0.41 | 83.10 |  0.31

Table 3: Comparison with state-of-the-art on ImageNet-100 and ImageNet-1000 benchmarks Number of class | 1 | 10 | 100
---|---|---|---
NE = 0 | 0.00% | 0.01% | 0.13%
NE = 1 | 0.00% | 0.03% | 0.26%
NE = 5 | 0.01% | 0.08% | 0.79%
NE = 10 | 0.01% | 0.15% | 1.45%
NE = 40 | 0.05% | 0.54% | 5.41%
Entire Model Size(MB) | 614.2
Table 4: Extra memory proportion of different number of example prototypes
under PNI settings. Extra memory proportion refers to the proportion of
additional parameter memory in the total model memory. “NE" means the number
of example prototypes. All the results in this table are based on a fixed
VitB/32 on CIFAR-100.

Comparison with state-of-the-art. We also compare our methods with the state-
of-the-art results of several non-PAT-based methods including DyTox [5], BiC
[34], UCIR [7], iCaRL [26] and the recent state-of-the-art PAT-based method
L2P [32]. All the results are based on the models including Vit-B/16, Vit-L/14
[3] pre-trained by CLIP [25] on CLIP400M. $10$ example prototypes are used for
IPT. We run experiments for different protocols and phase settings on
CIFAR100, ImageNet100 [26] and ImageNet1000 [28]. As shown in Table 2 and 3,
our method performs better than other methods on different incremental phase
settings, and the average incremental accuracy does not decrease significantly
along the increase of phases. In particular, the FR of our method is very
close to zero and even negative, suggesting that the model has achieved
positive backward transfer. On the contrary, due to too many parameters being
fine-tuned, the non-PAT-based methods are easy to cause overfitting on split
incremental tasks and suffer from huge forgetting, making themself unable to
take advantage of large-scale pre-trained models.

###  4.3 Ablation of prototypes.

Settings. We conduct a further study about the contribution of example
prototypes and category prototypes for our method. In detail, we employ
different numbers of example prototypes (including number zero which means
there is no example prototype used) under the PHI and PNI setting for both 10
and 50 phases on CIFAR-100. We also compare random and k-means initialization
for the example prototypes. The results of whether to freeze those category
prototypes born before phase $n$ while learning at phase $n+t$ to prevent
those knowledge learned in new phases to be transferred to old phases are also
submitted. VitB/16 and VitB/32 pre-trained on CLIP400M are used as the fixed
representation models in this section.

Results. Figure 4 exhibits the result of the prototype ablation study. The
methods of using example prototypes are better than the methods of not using,
particularly under the more challenging settings such as the 50-phase setting.
Although one example prototype is enough to achieve obvious improvement under
10 phase setting, the performances of IPT-based methods under more difficult
settings still need more example prototypes to be ensured. However, there is
an obvious marginal effect in the number of used example prototypes. Also, the
more example prototypes are used, the more IPT depends on initialization
methods, and of course, the more storage space is required. It is also worth
noting that the methods that do not freeze the categories prototypes born
before phase $n$ while learning at phase $n+t$ outperform those methods that
freeze. In other words, a positive backward transfer is actually achieved
behind the tuning of category prototypes.

###  4.4 Further study

Extra memory cost. IPP only adds category prototypes and example prototypes to
the model at each phase. Since all the additional prototypes are vectors only,
the incremental memory requirement rises slowly, as shown in Table 4. The
concern that using too many example prototypes will bring a large memory
footprint can be addressed by just setting the number of example prototypes to
$1$. As shown in Figure 4, one example prototype can successfully solve
semantic drift while only maintaining the same memory consumption as the
classifier weight (category prototypes).

Ablation of maximum similarity loss. To see the effectiveness of the maximum
similarity loss (MSL) of our method, we also replace the equation 6 by
adopting average similarity loss (ASL) as the followed equation:

$\displaystyle
L_{AS}^{t}(x)=1-\frac{\sum\limits_{j}(<\Phi(x,\theta),e^{j}_{g(x)}>)}{N_{e}}.$
(10)

By comparing the performances of MSL and ASL, Table 5 demonstrates that IPT
does benefit from MSL. The more phases the incremental learning split the
dataset into, the more effective MSL is.

Setting | 10 Phase PNI | 50 Phase PNI
---|---|---
Metrics | AIA(%)$\uparrow$ | Final Acc(%)$\uparrow$ | AIA(%)$\uparrow$ | Final Acc(%)$\uparrow$
MSL | 82.07 | 73.59 | 82.19 | 73.49
ASL | 80.14 | 72.00 | 78.46 | 68.21

Table 5: Ablation results of maximum similarity loss. All the results are
produced based on a frozen VitB/16 pre-trained CLIP400M.Ten example prototypes
are used for this experiment.

##  5 Conclusions

In this work, we emphasize that semantic drift is a serious problem that
limits the performance of PAT-based CIL, and propose IPT to alleviate semantic
drift. At each new phase t, IPT freezes the examples prototypes born before
phase t, generates new category prototypes and examples prototypes, and
optimize them with two classification losses and maximum similarity loss.
Exhaustive experiments show that our method can significantly reduce semantic
drift and improve the performances of PAT-based CIL methods without replaying
original samples and distilling from previous models. The superior performance
of our method further verifies that PAT-based CIL is promising.

IPT reduces the high-level forgetting (semantic drift) of the PAT-based CIL
method but still remains low-level forgetting (representation forgetting)
unsolved. Although fixing the representation can prevent representation
forgetting, it hinders learning better feature extraction ability. This is an
open problem about balancing plasticity and stability.

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††institutetext: 1Centre for Ocean, River, Atmosphere and Land Sciences,
Indian Institute of Technology Kharagpur, Kharagpur 721 302,
India††institutetext: 2Department of Physics,Indian Institute of Technology
Kharagpur, Kharagpur 721 302, India††institutetext: 3Rayagada Autonomous
College, Rayagada, 765001, India

# Pulsating string solution and stability in two parameter $\chi$-deformed
background

Rashmi R Nayak1 Nibedita Padhi2 Manoranjan Samal3<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate pulsating string solutions within the context of
a two-parameter $\chi$-deformed $\mathcal{R}\times S^{2}$ background. We
derive the energy and oscillation number relation for the deformed
$\mathcal{R}\times S^{2}$ in a short string limit. Additionally, we derive the
first-order perturbation equation for a pulsating string in this specific
background. As a special case of the two-parameter deformed background, we
examine the stability of the pulsating string solution in one-parameter
deformed $\mathcal{R}\times S^{2}$ and $AdS_{3}$ background by setting one of
the deformation parameters to zero.

###### Keywords:

AdS/CFT Correspondence, Pulsating string, Perturbation

##  1 Introduction

A remarkable example of AdS/CFT correspondence lies in the complex connection
between the spectrum of close superstring in $AdS_{5}\times S^{5}$ space time
and gauge invariant operators in 4d $\mathcal{N}=4$ Supersymmetric Yang-Mills
theory Maldacena:1997re . A precise match between the string states and the
dual operators of both the sides is hard to achieve. However, it has been
shown that in the plane wave limit it is possible to identify the energy of
the string states with the anomalous dimension of the gauge invariant
operators considering large R-charge Berenstein:2002jq ; Santambrogio:2002sb .
There have been a lot of studies in various settings over the years using a
variety of classical strings as probes, such as folded strings Gubser:2002tv ,
giant magnons Hofman:2006xt , spiky strings Kruczenski:2004wg  along with
their detailed dual gauge theories. Pulsating strings are distinct from all
other string solutions due to their time-dependent nature and better
stability. In Ref. Minahan:2002rc  the dimension of the dual operator has been
predicted by computing the energy of circular string pulsating in
$AdS_{5}\times S^{5}$ with semiclassical limit. Similar analysis has also led
to extensive work on pulsating strings in different backgrounds, where the
energy of the string states can be expressed as complicated functions of
string tension and various charges Khan:2003sm ; Dimov:2004xi ;
Smedback:2004udl ; Kruczenski:2004cn ; Chen:2008qq ; Dimov:2009rd ;
Bobev:2004id ; Arnaudov:2010by ; Park:2005kt ; Pradhan:2013sja ;
Beccaria:2010zn ; Giardino:2011jy ; Banerjee:2014bca ; Panigrahi:2014sia ;
Rotating2 ; Barik1 ; Banerjee:2014rza ; HN3 ; Banerjee:2016xbb ;
Chakraborty:2022eeq . Recently, we have studied a pulsating string in NS5
brane and I-brane background Biswas:2023uuq , by expressing its energy in
terms of the adiabatic invariant $\mathcal{N}$. In this article, we will
consider a multiwrapped pulsating string moving in a two-parameter deformed
$\mathcal{R}\times S^{2}$ background.

Understanding the stability properties of such string solutions on curved
space backgrounds is also a topic of great interest. One effective approach to
comprehend the stability properties is by analysing the worldsheet
fluctuations of string in curved backgrounds. The authors of Ref.
Larsen:1993iva  have developed a covariant approach to study the dynamics of
the small disturbances. The procedure involves performing a perturbative
series expansion around the precisely known solution for a generic string in
the specified curved spacetime. Following the covariant approach, it has been
investigated the emergence of string instabilities in D-dimensional black hole
spacetimes and de-Sitter spacetime. The perturbation of spiky strings in flat
and AdS space has been studied in Bhattacharya:2016ixc ; Bhattacharya:2018unr
; Bhattacharya:2021xfc , a circular string in a power law expanding universe
was shown in Larsen:1994jt , the planetoid strings in Ellis geometry and in
(2+1) BTZ black hole is discussed in Kar:1997zi  and a pulsating string in
$AdS_{3}\times S^{3}$ was studied in Barik:2017opb . Following this method
numerous studies have been conducted in various gravitational and cosmological
backgrounds, one can check Garriga:1991ts ; Guven:1993ew ; Guven:1993ex ;
Larsen:1994ah ; Larsen:1993mx ; Khan:2005fc ; Larsen:2003ma ; Larsen:1996yb ;
Mahapatra:1996ap ; Larsen:1995af ; Li:2018jxy ; Norma:2016soj ; Hioki:2009hp ;
Brandenberger:2007by . The covariant formalism of perturbation has also been
used to understand the stability properties of moving branes. A higher order
perturbation equation for branes has been formulated in Capovilla:2021xfy  and
first order perturbation equation for a macroscopic string and relativistic
membranes moving in an arbitrary curved space-time was derived following the
covariant formalism in Larsen:2000sg ; deVega:1988ch ; Kiosses:2014tua . In
addition to the above-mentioned works, semiclassical quantization makes use of
the study of tiny perturbations surrounding classical string solutions.
Numerous studies have also been made in this direction. Originally Vega and
Sanchez developed the method to study string solutions in the presence of a
strong curvature regime deVega:1987veo . This method computes first and second
order quantum fluctuations, which result in the determination of physical
quantities such as scattering amplitudes and mass spectrum. For additional
details on this one can check deVega:1987veo ; deVega:1988jh . The purpose of
the present paper is to study the stability of small perturbations on
pulsating strings moving in a single and two parameter $\chi$ deformed
$\mathcal{R}\times S^{2}$ background.

Regarding the deformed backgrounds, the construction of one parameter
deformation of the Matsaev- Tseytlin action in the context of supercoset sigma
models describe dynamics of a system in terms of fields, leading to
classically integrable model Delduc:2013qra . This construction involves
q-deformed symmetries, spacetime metric deformations and the development of a
R-matrix with associate S-matrix and dispersion relation for certain string
theories in specific spacetime backgrounds Beisert:2008tw ; Hoare:2023zti ;
Hoare:2022asa . On the other hand, a two parameter deformed sigma model on the
world sheet of superstrings on $AdS_{3}\times S^{3}\times M^{4}$ was first
constructed by Hoare in Hoare:2014oua . The deformed metric involves two
parameters $\chi_{+}$ and $\chi_{-}$ and due to the presence of a
$\textbf{Z}_{2}$ symmetry of the geometry $\chi_{+}$ $\leftrightarrow$
$\chi_{-}$, we can take $\chi_{+}$ $\geq$ $\chi_{-}$ without any loss of
generality. Despite the deformation, the classical integrability of the
original model is preserved. The metric reduces to one-parameter deformed case
Hoare:2014pna  taking $\chi_{+}\rightarrow 0$ and to a known squashed $S^{3}$
case Cherednik:1981df  for $\chi_{+}$ $=$ $\chi_{-}$. The formula for Ramond-
Ramond fluxes in the two parameter deformed background has been formulated in
Seibold:2019dvf  and the authors of Seibold:2021lju  has computed the massive
tree level scattering matrices for the two parameter deformed background using
perturbation theory. A quantum spin chain model on this background has been
studied in Wen:2019mgv . The authors have also found the complicated
dispersion relation for giant magnon solutions and various hanging string
solutions treating the deformation parameters perturbatively.

In order to understand the interesting aspect of $\chi$-deformed sigma models
we have considered the pulsating string solutions with two parameter
deformation in the background of $\mathcal{R}\times S^{2}$ in section 2. In
the first case, we study the string motion which seems to be equivalent to the
motion of a particle under the influence of a potential $V(\theta)$ for
different values of $\chi_{+}$ and $\chi_{-}$. Then we have used the Bohr-
Sommerfeld like quantisation to find the oscillation number $\mathcal{N}$ and
its relation to the string energy, whereas the one parameter case has been
studied in Banerjee:2016xbb . In section 3 we have studied the propagation of
a perturbation along the pulsating string and derived the equation governing
perturbation. We have also checked the stability of such string solutions in
the deformed background where one of the deformation parameter is set to zero.
Finally, in section 4 we present our conclusion.

##  2 Pulsating string in two parameter deformed $\mathcal{R}\times S^{2}$

In this section, we will delve into the semiclassical quantisation of a closed
pulsating string in two parameter $\chi$ deformed $\mathcal{R}\times S^{2}$.
The generic metric for a two parameter deformed $AdS_{3}\times S^{3}$
background is given by Hoare:2014oua ,

$\begin{split}ds^{2}=&\frac{1}{F(\rho)}\left[-(1+\rho^{2})dt^{2}+\frac{d\rho^{2}}{1+\rho^{2}}+\rho^{2}d\psi^{2}-\left(\chi_{-}(1+\rho^{2})dt-\chi_{+}\rho^{2}d\psi\right)^{2}\right]\\\
&+\frac{1}{\tilde{F}(r)}\left[(1-r^{2})d\zeta^{2}+\frac{dr^{2}}{1-r^{2}}+r^{2}d\phi^{2}-\left(\chi_{-}(1-r^{2})d\zeta+\chi_{+}r^{2}d\phi\right)^{2}\right],\end{split}$
(2.1)

with

$F(\rho)=1+\chi_{-}^{2}(1+\rho^{2})-\chi_{+}^{2}\rho^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{F}(r)=1+\chi_{-}^{2}(1-r^{2})+\chi_{+}^{2}r^{2}.$

The metric is not supported by any B- field as it is a total derivative. We
now consider $\rho=0$ and $r=\sin{\theta}$ in equation (2.1) which reduces it
to the deformed $\mathcal{R}\times S^{2}$ background

$\begin{split}ds^{2}=\frac{1}{\tilde{F}(\theta)}&\bigg{[}-\tilde{F}(\theta)dt^{2}+\cos^{2}{\theta}(1+{\chi_{-}}^{2}\cos^{2}{\theta})d\zeta^{2}+d\theta^{2}+\sin^{2}{\theta}(1+\chi_{+}^{2}\sin^{2}{\theta})d\phi^{2}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+2\chi_{-}\chi_{+}\cos^{2}{\theta}\sin^{2}{\theta}d\phi d\zeta\bigg{]},\\\
&\text{where}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\tilde{F}(\theta)=1+\chi_{-}^{2}cos^{2}{\theta}+\chi_{+}^{2}sin^{2}{\theta}.\end{split}$
(2.2)

As our interest lies in a fundamental string motion, we consider the Polyakov
action

$\displaystyle S_{P}=-\frac{\sqrt{\hat{\lambda}}}{4\pi}\int d\tau
d\sigma\sqrt{-h}h^{\alpha\beta}G_{\alpha\beta}.$ (2.3)

where $\hat{\lambda}$ is the modified ’t Hooft coupling for the deformed
model, $h^{\alpha\beta}$ is the internal metric and $\gamma^{\alpha\beta}$ is
the worldsheet metric. The induced metric $G_{\alpha\beta}$ on the string
worldsheet is defined by

$G_{\alpha\beta}=g_{\mu\nu}\frac{\partial
X^{\mu}}{\partial\sigma^{\alpha}}\frac{\partial
X^{\nu}}{\partial\sigma^{\beta}},$ (2.4)

where $\sigma^{\beta}$ are worldsheet coordinates and $X^{\mu}$ are space time
coordinates. The equations of motion derived from the action (2.3) along with
the conformal gauge constraint are given by

$\ddot{X^{\mu}}-X^{\prime\prime\mu}+\Gamma^{\mu}_{\rho\sigma}\Big{(}\dot{X^{\rho}}\dot{X^{\sigma}}-X^{\prime}{\rho}X^{\prime}{\sigma}\Big{)}=0,$
(2.5)

and,

$G_{\alpha\beta}-\frac{1}{2}h_{\alpha\beta}G^{\gamma}_{\leavevmode\nobreak\
\gamma}=0.$ (2.6)

We now take the ansatz for a pulsating string in the deformed
$\mathcal{R}\times S^{2}$ background (2.1) as

$t=t(\tau),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\theta=\theta(\tau),\leavevmode\nobreak\ \leavevmode\nobreak\
\zeta=0,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\phi=m\sigma.$ (2.7)

With the help of the ansatz the resulting action is given by

$S=-\frac{\sqrt{\hat{\lambda}}}{4\pi}\bigg{[}\dot{t}^{2}+\frac{1}{\tilde{F}(\theta)}\left(m^{2}sin^{2}{\theta}(1+{\chi_{+}}^{2}sin^{2}{\theta})-\dot{\theta}^{2}\right)\bigg{]}.$
(2.8)

The equations of motion for coordinate $t$ and $\theta$ give

$\displaystyle\dot{t}=\mathcal{E},$ (2.9) $\displaystyle
2\tilde{F}^{2}(\theta)\ddot{\theta}+\dot{\theta}^{2}+m^{2}\sin{2\theta}(1+2\chi_{+}^{2}\sin^{2}{\theta})\tilde{F}(\theta)-m^{2}\sin^{2}{\theta}(1+\chi_{+}^{2}\sin^{2}{\theta})=0,$
(2.10)

and the conformal gauge constraint equation (2.6) gives

$\dot{\theta}^{2}=(1+\chi_{-}^{2}\cos^{2}{\theta}+\chi_{+}^{2}\sin^{2}{\theta})\mathcal{E}^{2}-m^{2}\sin^{2}{\theta}(1+\chi_{+}^{2}\sin^{2}{\theta}).$
(2.11)

Equation (2.11) can also be written in the following form,

$\dot{z}^{2}=m^{2}\chi_{+}^{2}(1-z^{2})(z^{2}-Z_{1})(Z_{2}-z^{2}),$ (2.12)

where $z=\sin{\theta}$ and $Z_{1}$ and $Z_{2}$ are given by

$\displaystyle
Z_{1}=\frac{1}{2}\left(-\frac{1}{{\chi_{+}}^{2}}+\frac{\mathcal{E}^{2}\left({\chi_{+}}^{2}-{\chi_{-}}^{2}\right)}{m^{2}{\chi_{+}}^{2}}-\sqrt{\left(\frac{1}{{\chi_{+}}^{2}}-\frac{\mathcal{E}^{2}\left({\chi_{+}}^{2}-{\chi_{-}}^{2}\right)}{m^{2}{\chi_{+}}^{2}}\right)^{2}+\frac{4\mathcal{E}^{2}\left({\chi_{-}}^{2}+1\right)}{m^{2}{\chi_{+}}^{2}}}\right),$
$\displaystyle
Z_{2}=\frac{1}{2}\left(-\frac{1}{{\chi_{+}}^{2}}+\frac{\mathcal{E}^{2}\left({\chi_{+}}^{2}-{\chi_{-}}^{2}\right)}{m^{2}{\chi_{+}}^{2}}+\sqrt{\left(\frac{1}{{\chi_{+}}^{2}}-\frac{\mathcal{E}^{2}\left({\chi_{+}}^{2}-{\chi_{-}}^{2}\right)}{m^{2}{\chi_{+}}^{2}}\right)^{2}+\frac{4\mathcal{E}^{2}\left({\chi_{-}}^{2}+1\right)}{m^{2}{\chi_{+}}^{2}}}\right).$

Now, solving the differential equation (2.12) for $\dot{z}$ yields the
solution of string motion in terms of Jacobi functions111$cn(x)$ is the
solution to differential equations
$y^{\prime}(x)^{2}-(1-y(x)^{2})(1-m+my(x)^{2})=0$ , and $sn(x)$ is solution to
the differential equations  $y^{\prime}(x)^{2}-(1-y(x)^{2})(1-my(x)^{2})=0$ as

$\sin{\theta}=z(\tau)=\cfrac{\sqrt{Z_{1}Z_{2}}\leavevmode\nobreak\
sn\left(m\chi_{+}\sqrt{Z_{2}-Z_{1}}\tau\leavevmode\nobreak\
|\leavevmode\nobreak\
\frac{\left(Z_{1}-1\right)Z_{2}}{Z_{1}-Z_{2}}\right)}{\sqrt{Z_{1}-Z_{2}\leavevmode\nobreak\
cn\left(m\chi_{+}\sqrt{Z_{2}-Z_{1}}\tau\leavevmode\nobreak\
|\leavevmode\nobreak\
\frac{\left(Z_{1}-1\right)Z_{2}}{Z_{1}-Z_{2}}\right)^{2}}}.$ (2.13)

Equation (2.11) can also be written in terms of an effective potential
$V(\theta)$ such as

$\Pi_{\theta}^{2}+V(\theta)=0,$

where

$\displaystyle\Pi_{\theta}=\frac{\dot{\theta}}{(1+\chi_{-}^{2}cos^{2}{\theta}+\chi_{+}^{2}sin^{2}{\theta})},$
(2.14) $\displaystyle
V(\theta)=\frac{m^{2}\sin^{2}\theta\left(1+\chi_{+}^{2}\sin^{2}\theta\right)}{\left(1+\chi_{+}^{2}\sin^{2}\theta+\chi_{-}^{2}\cos^{2}\theta\right)^{2}}-\frac{\mathcal{E}^{2}}{1+\chi_{+}^{2}\sin^{2}\theta+\chi_{-}^{2}\cos^{2}\theta}.$
(2.15)

and $\Pi_{\theta}$ is the canonical momentum corresponding to coordinate
$\theta$. In figure 1 and 2 we show the nature of the potential for different
values of the deformation parameters.

Figure 1: Potential $V(\theta)$ plot for $\chi_{+}$=$\chi_{-}$$=\chi$ (left)
and for different values of $\chi_{+}$ fixing $\chi_{-}=1$,(right), for both
the plots $m=2$ and $\mathcal{E}=0.1$

Figure 2: Potential $V(\theta)$ plot for different values of $\chi_{-}$ at
fixed $\chi_{+}=1$,$m=2$ and $\mathcal{E}=0.1$ (left) and for different values
of $m$ (right) where the other parameters are fixed at
$\chi_{+}=7$,$\chi_{-}=2$ and $\mathcal{E}=0.1$.

In order to get a deeper understanding of the string dynamics we compute the
string oscillation number and its relation to energy. The oscillation number
associated with the periodic angular movement can be expressed as

$\mathcal{N}=\frac{1}{2\pi}\oint d\theta\leavevmode\nobreak\ \Pi_{\theta}.$
(2.16)

Now substituting equation (2.11) in equation (2.16) and $\sin^{2}{\theta}=x$,
the oscillation number can be expressed as following

$\mathcal{N}=\frac{1}{\pi}\int_{0}^{{Z_{2}}}\frac{dx}{((1+\chi_{-}^{2})+(\chi_{+}^{2}-\chi_{-}^{2})x)}\leavevmode\nobreak\
\sqrt{\frac{-\mathcal{E}^{2}((1+\chi_{-}^{2})+(\chi_{+}^{2}-\chi_{-}^{2})x)+m^{2}x(1+\chi_{+}^{2}x)}{x(x-1)}}.$
(2.17)

The above integral can be readily evaluated considering a short string limit
where $\mathcal{E}\rightarrow 0$ and we obtain the oscillation number
expressed as a series in $\mathcal{E}$

$\begin{split}\mathcal{N}&=\frac{\mathcal{E}^{2}}{2m}-\Big{[}\left(3\chi_{+}^{2}+1\right)\chi_{-}^{2}+\chi_{+}^{2}-1\Big{]}\frac{\mathcal{E}^{4}}{16m^{3}}+\Big{[}\left(35\chi_{+}^{4}+22\chi_{+}^{2}+3\right)\chi_{-}^{4}\\\
&+2\left(11\chi_{+}^{4}-6\chi_{+}^{2}-1\right)\chi_{-}^{2}+3\chi_{+}^{4}-2\chi_{+}^{2}+11\Big{]}\frac{\mathcal{E}^{6}}{128m^{5}}+O\left(\mathcal{E}^{7}\right).\end{split}$
(2.18)

Performing an inversion of the aforementioned series, we can deduce the string
energy in terms of the oscillation number as follows,

$\begin{split}\mathcal{E}&=\sqrt{2m\mathcal{N}}\Bigg{[}1+\Big{[}\left(3\chi_{+}^{2}+1\right)\chi_{-}^{2}+\chi_{+}^{2}-1\Big{]}\frac{\mathcal{N}}{8m}-\Big{[}\left(77\chi_{+}^{4}+46\chi_{+}^{2}+5\right)\chi_{-}^{4}\\\
&+\left(46\chi_{+}^{4}-20\chi_{+}^{2}+6\right)\chi_{-}^{2}+(5\chi_{+}^{4}+6\chi_{+}^{2}+37)\Big{]}\frac{\mathcal{N}^{2}}{128m^{2}}+O\left(\mathcal{N}^{3}\right)\Bigg{]}.\end{split}$
(2.19)

One can check for $\chi_{-}$ $\rightarrow$ $0$ and $\chi_{+}$ $\rightarrow$
$0$, the dispersion relation matches with the undeformed case studied in
Beccaria:2010zn . The above result can also be verified by performing the
quantisation starting with a Nambu-Goto action.

##  3 Perturbation of pulsating string in two parameter deformed
$\mathcal{R}\times S^{2}$

In this section, we use the covariant approach developed by Vega and Sanchez
deVega:1987veo  to study the perturbation of bosonic string moving in an
arbitrary curved space-time. The equations governing the perturbations are the
well-known Jacobi equations and are derived from the second variation of the
action (2.3). The variation of the embedding coordinates can be expressed in
terms of normal ($N_{i}^{\mu}$) and tangent vectors ($X^{\mu}_{,\alpha}$) as
$\delta X^{\mu}=\Phi^{i}N_{i}^{\mu}+\psi^{\alpha}X^{\mu}_{,\alpha}$. Here
$\Phi^{i}$ and $\psi^{\alpha}$ are the perturbations along normal and tangent
directions respectively. Here $i$ denotes the number index. The normal vectors
satisfy the orthogonality condition

$g_{\mu\nu}N^{\mu}_{i}N^{\nu}_{j}=\delta_{ij}\leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
g_{\mu\nu}N^{\mu}_{i}\partial_{\alpha}X^{\nu}=0.$ (3.1)

The extrinsic curvature and normal fundamental forms are defined by

$K_{i,\alpha\beta}=g_{\mu\nu}N^{\mu}_{i}X^{\rho}_{,\alpha}\nabla_{\rho}X^{\nu}_{,\beta},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\mu_{ij,\alpha}=g_{\mu\nu}N^{\mu}_{i}\partial_{\alpha}X^{\rho}\nabla_{\rho}N^{\nu}_{j},$
(3.2)

where $\nabla_{\rho}$ is the covariant derivative w.r.t space time
coordinates. Now considering all this the first order perturbation equation
satisfied by $\varphi^{i}$ is given by Larsen:1993mx ,Khan:2005fc

$\Big{(}\delta^{kl}h^{\alpha\beta}D_{ik\alpha}D_{lj\beta}+\frac{2}{G^{c}_{c}}K^{\alpha\beta}_{i}K_{j,\alpha\beta}-h^{\alpha\beta}R_{\mu\rho\sigma\nu}N^{\rho}_{i}N^{\sigma}_{j}X^{\mu}_{,\alpha}X^{\nu}_{,\beta}\Big{)}\varphi^{j}=0,$
(3.3)

where $D_{ij\alpha}=\delta_{ij}D_{\alpha}+\mu_{ij\alpha}$. The background
metric is the two parameter deformed $\mathcal{R}\times S^{2}$ given in
section 2. From equation (2.7) and equation (2.4) the induced metric is given
by

$\displaystyle
G_{\alpha\beta}=\frac{m^{2}\sin^{2}{\theta}(1+\chi_{+}^{2}\sin^{2}{\theta})}{1+\chi_{-}^{2}cos^{2}{\theta}+\chi_{+}^{2}sin^{2}{\theta}}\left(-d\tau^{2}+d\sigma^{2}\right).$
(3.4)

The tangent and normal vectors to the worldsheet are given by

$\dot{X}=(\mathcal{E},\dot{\theta},0)\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
X^{\prime}=(0,0,m)$ (3.5) $\displaystyle
N^{\mu}=\bigg{\\{}\frac{\sqrt{\tilde{F}(\theta)\mathcal{E}^{2}-m^{2}\sin^{2}{\theta}(1+\chi_{+}^{2}\sin^{2}{\theta})}}{m\sin{\theta}\sqrt{1+\chi_{+}^{2}\sin^{2}{\theta}}}\,,\frac{\mathcal{E\tilde{F}(\theta)}}{m\sin{\theta}\sqrt{1+\chi_{+}^{2}\sin^{2}{\theta}}}\,,0\bigg{\\}}.$
(3.6)

Here all the normal fundamental forms vanish. The extrinsic curvature tensor
of the worldsheet is found to be

$\displaystyle
K_{\tau\tau}=-\frac{m\mathcal{E}\cos{\theta}\left(\left(\chi_{+}^{2}\sin^{2}{\theta}+1\right)\left(\left(\chi_{-}^{2}+\chi_{+}^{2}\right)\sin^{2}{\theta}+1\right)+\chi_{-}^{2}\cos^{2}{\theta}\left(2\chi_{+}^{2}\sin^{2}{\theta}+1\right)\right)}{\sqrt{\chi_{+}^{2}\sin^{2}{\theta}+1}\left(\chi_{+}^{2}\sin^{2}{\theta}+\chi_{-}^{2}\cos^{2}{\theta}+1\right)},$
$\displaystyle
K_{\sigma\sigma}=-\frac{m\mathcal{E}\cos{\theta}\left(\left(\chi_{+}^{2}\sin^{2}{\theta}+1\right)\left(\left(\chi_{-}^{2}+\chi_{+}^{2}\right)\sin^{2}{\theta}+1\right)+\chi_{-}^{2}\cos^{2}{\theta}\left(2\chi_{+}^{2}\sin^{2}{\theta}+1\right)\right)}{\sqrt{\chi_{+}^{2}\sin^{2}{\theta}+1}\left(\chi_{+}^{2}\sin^{2}{\theta}+\chi_{-}^{2}\cos^{2}{\theta}+1\right)},$
$\displaystyle K_{\tau\sigma}=K_{\sigma\tau}=0.$ (3.7)

Using all the quantities mentioned above, we now proceed to derive the
perturbation equation and obtain the following equation satisfied by the
scalar $\varphi$

$\Phi^{\prime\prime}(\tau,\sigma)-\ddot{\Phi}(\tau,\sigma)+\left[\frac{2\left(\chi_{-}^{2}+1\right)\mathcal{E}^{2}}{\sin^{2}{\theta}}+\frac{2\chi_{-}^{2}\left(2\chi_{+}^{4}+5\chi_{+}^{2}+3\right)\mathcal{E}^{2}}{\chi_{+}^{2}(1+\chi_{+}^{2}\sin^{2}{\theta})}\right.\\\
\left.-\frac{3\chi_{-}^{2}\left(\chi_{+}^{2}+1\right)^{2}\mathcal{E}^{2}}{\chi_{+}^{2}\left(1+\chi_{+}^{2}\sin^{2}{\theta}\right)^{2}}-\frac{\chi_{-}^{2}\left(5\chi_{+}^{2}+3\right)\mathcal{E}^{2}}{\chi_{+}^{2}}+(\chi_{+}^{2}-1)\mathcal{E}^{2}\right]\Phi(\tau,\sigma)=0.$
(3.8)

Let us take a fourier expansion
$\Phi(\tau,\sigma)=\sum_{n}\epsilon_{0}\,\alpha_{n}(\tau)e^{in\sigma}$ under
which equation (3.8) reduces to

$\displaystyle\ddot{\alpha_{n}}(\tau)+\Bigg{[}n^{2}-\mathcal{E}^{2}\left(\chi_{+}^{2}-1\right)-\frac{2\mathcal{E}^{2}\left(\chi_{-}^{2}+1\right)}{\sin^{2}{\theta}}-\frac{2\chi_{-}^{2}\left(2\chi_{+}^{4}+5\chi_{+}^{2}+3\right)\mathcal{E}^{2}}{\chi_{+}^{2}(1+\chi_{+}^{2}\sin^{2}{\theta})}$
$\displaystyle+\frac{3\chi_{-}^{2}\left(\chi_{+}^{2}+1\right){}^{2}\mathcal{E}^{2}}{\chi_{+}^{2}\left(1+\chi_{+}^{2}\sin^{2}{\theta}\right){}^{2}}+\frac{\chi_{-}^{2}\left(5\chi_{+}^{2}+3\right)\mathcal{E}^{2}}{\chi_{+}^{2}}\Bigg{]}\alpha_{n}(\tau)=0.$
(3.9)

where $\epsilon_{0}<<1$ can be considered as the amplitude of perturbation.
One can rearrange the equation (3.9) to get a second order differential
equation of the form

$\frac{d^{2}Y}{d\tau^{2}}+\\{n^{2}-V(\tau)\\}Y=0,$ (3.10)

where $V(\tau)$ is given by,

$\displaystyle V(\tau)$ $\displaystyle=$
$\displaystyle\mathcal{E}^{2}\Bigg{[}(\chi_{+}^{2}-1)-\frac{\left(5\chi_{+}^{2}+3\right)\chi_{-}^{2}}{\chi_{+}^{2}}+\frac{2\left(\chi_{-}^{2}+1\right)}{z(\tau)^{2}}+\cfrac{2\chi_{-}^{2}\left(2\chi_{+}^{4}+5\chi_{+}^{2}+3\right)}{\chi_{+}^{2}\left(1+z(\tau)^{2}\right)}$
(3.11) $\displaystyle\hskip
241.84842pt-\cfrac{3\chi_{-}^{2}\left(\chi_{+}^{2}+1\right)}{\chi_{+}^{2}\left(1+z(\tau)^{2}\right)^{2}}\Bigg{]}$

Figure 3 shows the nature of potential $V(\tau)$ for $\mathcal{E}$=0.03, m=2,
$\chi_{+}$=0.5, $\chi_{-}$=0.3.

Figure 3: V($\tau$) vs $\tau$ plot for $\mathcal{E}$=0.03, m=2,
$\chi_{+}$=0.5, $\chi_{-}$=0.3 .

The potential corresponding to the perturbation equation is a well-like
potential and hence, there will be bound state solutions corresponding to the
normal mode fluctuations. Now considering the complicated expression of
$V(\tau)$, it is a difficult task to solve the second order differential
equation (3.10) analytically. We next try to simplify the complexity of the
problem by setting one of the deformation parameter to zero.

###  3.1 Perturbation of pulsating string $\mathcal{R}\times S_{\chi}^{2}$

In this subsection, we want to analyse the perturbation of string in a
simplified scenario where one of the deformation parameters from the two-
parameter case is set to zero. By fixing one parameter to zero, we reduce the
complexity of the background deformation, allowing us to do an analytic study
on the perturbation equation. Now setting $\chi_{+}=0$ and $\chi_{-}=\chi$,
the metric equation (2.2) reduces to

$ds^{2}=-dt^{2}+\frac{d\theta^{2}}{1+\chi^{2}\cos^{2}{\theta}}+\frac{\sin^{2}{\theta}d\phi^{2}}{1+\chi^{2}\cos^{2}{\theta}}.$
(3.12)

Considering the pulsating string ansatz as given in equation (2.7) and from
equation (2.11) the Virasoro equation reduces to

$\dot{\theta}^{2}=\mathcal{E}^{2}(1+\chi^{2}\cos^{2}{\theta})-m^{2}\sin^{2}{\theta},$
(3.13)

and the solution to the theta equation is given by,

$\sin{\theta}=\sqrt{\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}}\leavevmode\nobreak\
\leavevmode\nobreak\
\textbf{sn}\left(\sqrt{m^{2}+\mathcal{E}^{2}\chi^{2}}\tau\leavevmode\nobreak\
|\leavevmode\nobreak\
\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}\right),$
(3.14)

Following a similar analysis as the previous section we get the reduced
perturbation equation as

$\Phi^{\prime\prime}(\tau,\sigma)-\ddot{\Phi}(\tau,\sigma)+\mathcal{E}^{2}\left(\chi^{2}+1\right)(1+2\cot^{2}{\theta})\Phi(\tau,\sigma)=0.$
(3.15)

To solve the perturbation equation analytically we first choose to consider a
short string limit. In this limit neglecting the higher order terms in
$\frac{\mathcal{E}}{m}$, equation (3.14) and equation (3.15) correspondingly
take the form,

$\sin{\theta}=\frac{\mathcal{E}\sqrt{(1+\chi^{2})}}{m}\leavevmode\nobreak\
\leavevmode\nobreak\ \sin{m\tau},$ (3.16)
$\ddot{\alpha_{n}}(\tau)+\Big{(}n^{2}-\frac{2m^{2}}{\sin^{2}{m\tau}}+\mathcal{E}^{2}\left(\chi^{2}+1\right)\Big{)}\alpha_{n}(\tau)=0.$
(3.17)

where we have used
$\Phi(\tau,\sigma)=\sum_{n}\epsilon_{0}\,\alpha_{n}(\tau)e^{in\sigma}$. From
equation (3.17) it is clearly evident that the differential equation is a
special case of the well-known Poschl-Teller equation and a general solution
for such a differential equation is given in the form

$\alpha_{n}(\tau)=C_{1}P_{1}(\tau)+C_{2}P_{2}(\tau)$ (3.18)

where,

$\displaystyle P_{1}(\tau)=\sin^{2}{m\tau}\leavevmode\nobreak\
_{2}F_{1}\left(1+\frac{\sqrt{n^{2}+\mathcal{E}^{2}(1+\chi^{2})}}{2m},1-\frac{\sqrt{n^{2}+\mathcal{E}^{2}(1+\chi^{2})}}{2m},\frac{5}{2};\sin^{2}{m\tau}\right)$
$\displaystyle P_{2}(\tau)=\frac{1}{\sin{m\tau}}\leavevmode\nobreak\
_{2}F_{1}\left(\frac{\sqrt{n^{2}+\mathcal{E}^{2}(1+\chi^{2})}-m}{2m},\frac{-\sqrt{n^{2}+\mathcal{E}^{2}(1+\chi^{2})}-m}{2m},-\frac{1}{2};\sin^{2}{m\tau}\right)$
(3.19)

and $C_{1}$,$C_{2}$ are constants.

Now the scalar function can be written as

$\Phi(\tau,\sigma)=\sum_{n}\epsilon_{0}\left(C_{1}P_{1}(\tau)+C_{2}P_{2}(\tau)\right)e^{in\sigma}$
(3.20)

Now to understand the nature of the solution to the perturbation equation we
have plotted $P_{1}(\tau)$ and $P_{2}(\tau)$. Figure (4) shows $P_{1}(\tau)$
for different values of $n$ (left) and $\chi$ (right). It is observed that the
solution $P_{1}(\tau)$ is oscillatory and the amplitude of perturbation
decreases with increasing values of both $n$ and $\chi$. Similarly, the
solution $P_{2}(\tau)$ is shown in figure (5). Unlike the first solution, it
is diverging periodically. Hence, $P_{2}(\tau)$ does not contribute to the
physical perturbations.

Figure 4: (a)Plot showing nature of solution $P_{1}(\tau)$ for for different
values of $n=$(10(blue),30(green),50(black),100(red)), $m=5$,
$\mathcal{E}=0.03,\leavevmode\nobreak\ \chi=0.09$, and (b)behavior of
$P_{1}(\tau)$ for for different values of
$\chi=$(100(blue),200(red),500(green),1000(purple)) ,$m=5$,
$\mathcal{E}=0.03$, $n=5$. From both the plots we observe that the amplitude
of oscillations decreases for higher values of $n$ and $\chi$.

Figure 5: (a)Behavior of solution $P_{2}(\tau)$ for different values of $n=$
(10(blue),30(black),35(green),50(red)) and m=5,
$\mathcal{E}=0.03,\leavevmode\nobreak\ \chi=100$,(b)Behavior of solution
$P_{2}(\tau)$ for different values of $\chi=$
(100(blue),250(orange),500(green),1000(red)) and m=5, $\mathcal{E}=0.03$,
$n=5$.

Now, the physical perturbations for short string solution can be written as

$\delta
t=N^{t}\Phi(\tau,\sigma)=\epsilon_{0}\sum_{n}\sin{m\tau}\sqrt{1-\left(1+\frac{\mathcal{E}^{2}\chi^{2}}{m^{2}}\right)\sin^{2}{m\tau}}\leavevmode\nobreak\
P_{1}(\tau)\cos{n\sigma}$ (3.21)
$\delta\theta=N^{\theta}\Phi(\tau,\sigma)=\epsilon_{0}\sum_{n}\sin{m\tau}\frac{1+\chi^{2}\left(1-\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}}\sin^{2}{m\tau}\right)}{\sqrt{1+\chi^{2}}}\leavevmode\nobreak\
P_{1}(\tau)\cos{n\sigma}$ (3.22) $\delta\phi=N^{\phi}\Phi(\tau,\sigma)=0$
(3.23)

Let’s discuss the perturbation without considering the short string limit.
Then substituting equation (3.14) in the perturbation equation (3.15) and
using $\Phi(\tau,\sigma)=\sum_{n}\alpha_{n}(\tau)e^{in\sigma}$, we get

$\ddot{\alpha_{n}}(\tau)+\Bigg{(}n^{2}-\cfrac{2(m^{2}+\mathcal{E}^{2}\chi^{2})}{sn^{2}\left(\sqrt{m^{2}+\mathcal{E}^{2}\chi^{2}}\tau\leavevmode\nobreak\
|\leavevmode\nobreak\
\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}\right)}+\mathcal{E}^{2}\left(\chi^{2}+1\right)\Bigg{)}\alpha_{n}(\tau)=0.$
(3.24)

Let’s take
$sn^{2}\left(\sqrt{m^{2}+\mathcal{E}^{2}\chi^{2}}\tau|\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}\right))=u$
in equation (3.24) and with a substitution $\alpha_{n}=us$ we have,

$\displaystyle\frac{d^{2}s}{du^{2}}+\frac{1}{2}\Bigg{[}\frac{5}{u}+\frac{1}{u-1}+\cfrac{1}{u-\cfrac{\mathcal{E}^{2}\chi^{2}+m^{2}}{\mathcal{E}^{2}(1+\chi^{2})}}\leavevmode\nobreak\
\leavevmode\nobreak\
\Bigg{]}\cfrac{ds}{du}+\frac{\Bigg{[}\frac{3}{2}u+\frac{n^{2}-4(\mathcal{E}^{2}\chi^{2}+m^{2}))}{4\mathcal{E}^{2}(1+\chi^{2})}-\frac{3}{4}\Bigg{]}}{u(u-1)\bigg{(}u-\cfrac{\mathcal{E}^{2}\chi^{2}+m^{2}}{\mathcal{E}^{2}(1+\chi^{2})}\bigg{)}}s=0.$
(3.25)

This is exactly the canonical form of Heun’s general equation and the solution
to the above differential equation can be written as in terms of Heun’s
functions.

$s=H\bigg{(}\cfrac{\mathcal{E}^{2}\chi^{2}+m^{2}}{\mathcal{E}^{2}(1+\chi^{2})},\frac{3}{4}-\frac{n^{2}-4(\mathcal{E}^{2}\chi^{2}+m^{2}))}{4\mathcal{E}^{2}(1+\chi^{2})};1,\frac{3}{2},\frac{5}{2},1;sn^{2}\left(\sqrt{m^{2}+\mathcal{E}^{2}\chi^{2}}\tau\leavevmode\nobreak\
|\leavevmode\nobreak\
\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}\right)\bigg{)}$

and

$\displaystyle\alpha_{n}(\tau)=sn^{2}\left(\sqrt{m^{2}+\mathcal{E}^{2}\chi^{2}}\tau\leavevmode\nobreak\
\leavevmode\nobreak\ |\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}\right)$
$\displaystyle
H\bigg{(}\cfrac{\mathcal{E}^{2}\chi^{2}+m^{2}}{\mathcal{E}^{2}(1+\chi^{2})},\frac{3}{4}-\frac{n^{2}-4(\mathcal{E}^{2}\chi^{2}+m^{2}))}{4\mathcal{E}^{2}(1+\chi^{2})};1,\frac{3}{2},\frac{5}{2},1;sn^{2}\left(\sqrt{m^{2}+\mathcal{E}^{2}\chi^{2}}\tau\leavevmode\nobreak\
\leavevmode\nobreak\ |\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{\mathcal{E}^{2}(1+\chi^{2})}{m^{2}+\mathcal{E}^{2}\chi^{2}}\right)\bigg{)}$

###  3.2 Perturbation of pulsating string $AdS_{3_{\chi}}$

In this subsection, we study the worldsheet perturbation of the pulsating
string in the deformed $AdS_{3}$ background where $\chi_{+}=0$,
$\chi_{-}=\chi$ and the spherical coordinates are not in use. We begin by
considering the metric

$ds^{2}=-\frac{\cosh^{2}{\rho}}{1-\chi^{2}\sinh^{2}{\rho}}dt^{2}+\frac{d\rho^{2}}{1-\chi^{2}\sinh^{2}{\rho}}+\sinh^{2}{\rho}d\phi^{2}.$
(3.26)

To avoid the singularity surface present in this coordinate system as
explained in Banerjee:2016xbb ,Kameyama:2014vma  we use a coordinate
transformation following Kameyama:2014vma

$\cosh{\Theta}=\frac{\cosh{\rho}}{\sqrt{1-\Theta^{2}\sinh^{2}{\rho}}},$

under which the metric transforms to

$\displaystyle
ds^{2}=-\cosh^{2}{\Theta}dt^{2}+\frac{d\Theta^{2}}{1+\chi^{2}\cosh^{2}{\Theta}}+\frac{\sinh^{2}{\Theta}}{1+\chi^{2}\cosh^{2}{\Theta}}d\phi^{2}.$
(3.27)

Now considering a pulsating string ansatz,

$t=t(\tau),\leavevmode\nobreak\ \Theta=\Theta(\tau),\leavevmode\nobreak\
\phi=m\sigma,$ (3.28)

we obtain the equation of motion for coordinate $t$ and $\Theta$ as follows,

$\displaystyle\dot{t}=\frac{\mathcal{E}}{\cosh^{2}{\Theta}},$ (3.29)
$\displaystyle\dot{\Theta}^{2}=\frac{\mathcal{E}^{2}(1+\chi^{2}\cosh^{2}{\Theta})}{\cosh^{2}{\Theta}}-m^{2}\sinh^{2}{\Theta}.$
(3.30)

In this new coordinate system solving the above differential equation (3.30),
we get the solution

$\sinh{\Theta}=\sqrt{\frac{-R_{-}R_{+}}{R_{+}-R_{-}}}\textbf{sd}\Big{(}m\sqrt{R_{+}-R_{-}}\tau\leavevmode\nobreak\
\leavevmode\nobreak\ |\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{R_{+}}{R_{+}-R_{-}}\Big{)},$ (3.31)

where,

$R_{\pm}=\frac{(\mathcal{E}^{2}\chi^{2}-m^{2})\pm\sqrt{(\mathcal{E}^{2}\chi^{2}-m^{2})^{2}+4m^{2}\mathcal{E}^{2}(1+\chi^{2})}}{2m^{2}}.$

For more details on the solution one can check Banerjee:2016xbb . Now, the
induced metric on the worldsheet is given by

$ds^{2}=\frac{m^{2}\sinh^{2}{\Theta}}{1+\chi^{2}\cosh^{2}{\Theta}}\left(-d\tau^{2}+d\sigma^{2}\right).$
(3.32)

The tangent vectors are given by

$\dot{X}^{\mu}=\left(\frac{\mathcal{E}}{\cosh^{2}{\Theta}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\dot{\Theta},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
0\right),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
X^{\prime\mu}=\left(0,\leavevmode\nobreak\ \leavevmode\nobreak\
0,\leavevmode\nobreak\ \leavevmode\nobreak\ m\right),$ (3.33)

and normal vector is given by

$\displaystyle N^{\mu}=$
$\displaystyle\Bigg{(}\frac{\sqrt{\mathcal{E}^{2}\left(\text{sech}^{2}{\Theta}+\chi^{2}\right)-m^{2}\sinh^{2}(\Theta)}}{m\sinh{\Theta}\cosh{\Theta}}\leavevmode\nobreak\
,\leavevmode\nobreak\
\frac{\mathcal{E}\left(\chi^{2}\cosh^{2}{\Theta}+1\right)}{m\sinh{\Theta}\cosh{\Theta}}\leavevmode\nobreak\
,\leavevmode\nobreak\ 0\Bigg{)}.$ (3.34)

Now the components of the extrinsic curvature are given by

$K_{\tau\tau}=-\frac{m\mathcal{E}\left(\chi^{2}+1\right)}{1+\chi^{2}\cosh^{2}{\Theta}},\quad
K_{\tau\sigma}=0,\quad K_{\sigma\tau}=0,\quad
K_{\sigma\sigma}=-\frac{m\mathcal{E}\left(\chi^{2}+1\right)}{1+\chi^{2}\cosh^{2}{\Theta}}\leavevmode\nobreak\
.$ (3.35)

Here all the normal fundamental forms vanishes. Hence the first order
perturbation equation takes the form

$\Phi^{\prime\prime}(\tau,\sigma)-\ddot{\Phi}(\tau,\sigma)+\left(m^{2}-m^{2}\cosh{2\Theta}+2\mathcal{E}^{2}(1+\chi^{2})\csch^{2}{\Theta}\right)\Phi(\tau,\sigma)=0.$
(3.36)

Now taking a Fourier expansion,
$\Phi(\tau,\sigma)=\sum_{n}\epsilon_{0}\alpha_{n}(\tau)e^{in\sigma}$, the
above differential equation takes the form

$\ddot{\alpha}_{n}(\tau)+\Big{(}n^{2}+2m^{2}\sinh^{2}{\Theta}-\frac{2\mathcal{E}^{2}(1+\chi^{2})}{\sinh^{2}{\Theta}}\Big{)}\alpha_{n}(\tau)=0.$
(3.37)

Now substituting the $\sinh{\Theta}$ value from equation (3.31) , we have

$\displaystyle\ddot{\alpha}_{n}(\tau)+\Bigg{[}n^{2}-2m^{2}\frac{R_{-}R_{+}}{R_{+}-R_{-}}\textbf{sd}^{2}\Big{(}m\sqrt{R_{+}-R_{-}}\leavevmode\nobreak\
\leavevmode\nobreak\ |\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{R_{+}}{R_{+}-R_{-}}\Big{)}$
$\displaystyle+\frac{2\mathcal{E}^{2}(1+\chi^{2})}{\frac{R_{-}R_{+}}{R_{+}-R_{-}}\textbf{sd}^{2}\Big{(}m\sqrt{R_{+}-R_{-}}\leavevmode\nobreak\
\leavevmode\nobreak\ |\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{R_{+}}{R_{+}-R_{-}}\Big{)}}\Bigg{]}\alpha_{n}(\tau)=0.$ (3.38)

For a short string limit neglecting the higher order terms in
$\frac{\mathcal{E}}{m}$, we have equation (3.38) reduces to

$\ddot{\alpha}_{n}(\tau)+\Big{(}n^{2}-\frac{2m^{2}}{\sin^{2}{m\tau}}\Big{)}\alpha_{n}(\tau)=0.$
(3.39)

Let us consider $y=\sin^{2}{m\tau}$. Equation (3.39) takes the form

$y(1-y)\alpha_{n}^{{}^{\prime\prime}}+(\frac{1}{2}-y)\alpha_{n}^{{}^{\prime}}+\frac{1}{4}\left(\frac{n^{2}}{m^{2}}-\frac{2}{y}\right)\alpha_{n}=0.$

From the above equation, it is clear that the perturbation equation in the
deformed background in a short string limit is independent of the deformation
parameter. The solution to the above differential equation has a similar form
as the undeformed case.

$\displaystyle\alpha_{n}(\tau)=C_{1}\sin^{2}{m\tau}P_{1}(\tau)+\frac{C_{2}}{\sin{m\tau}}P_{2}(\tau)$
(3.40) $\displaystyle
P_{1}(\tau)=\,_{2}F_{1}\left(1-\frac{n}{2m},\frac{n}{2m}+1;\frac{5}{2};\sin^{2}(m\tau)\right)$
(3.41) $\displaystyle
P_{2}(\tau)=i\,_{2}F_{1}\left(-\frac{n}{2m}-\frac{1}{2},\frac{n}{2m}-\frac{1}{2};-\frac{1}{2};\sin^{2}(m\tau)\right)$
(3.42)

Analysing the behavior of these two solutions $P_{1}(\tau)$ and $P_{2}(\tau)$
for a short string limit, we conclude that one of the solutions $P_{1}(\tau)$
is oscillatory and hence defines the physical stable perturbation, whereas the
other solution is divergent and hence is not stable.

##  4 Summary and Conclusion

Let us first summarise the results obtained in this article. Firstly, we have
considered the motion of a pulsating string in the novel two parameter
deformed $\mathcal{R}\times S^{2}$ background. With the help of the Virasoro
constraint, we found the string motion is equivalent to the motion of a
particle under the influence of a potential. The nature of the potential is
studied. We have derived the conserved quantities and shown a relation between
the string energy and it’s oscillation number in order to get a better
understanding in the string dynamics. Next, we write down the fundamental
equations describing the fluctuations of a fundamental string with pulsating
nature in the two parameter deformed $\mathcal{R}\times S^{2}$ background.
Because of the complicated form of the perturbation equation we have not
solved the perturbation equation. We have simplified our analysis by
considering one of the deformation parameters to be zero and study the
perturbation equation in both $\chi$\- deformed $\mathcal{R}\times S^{2}$ and
$AdS_{3}$ background. Finally, we comment on the stability nature of the
string solution by looking at the solutions governing perturbations. By
examining the solutions in both an undeformed and a single-parameter deformed
background, we anticipate similar behavior of the solutions in a two-parameter
deformed background, which may include both stable and unstable solutions.

We have shown our result is a generalization of the solutions obtained in the
one parameter deformed background. It would be an interesting direction to
verify our result by reducing the current string sigma model into the well-
known 1D integrable Neumann-Rosochatius model using a pulsating string ansatz.
Exploring the dual gauge theory of such string background can be quite
fascinating for future work. An extension to our work can be finding the
dispersion relation in a three parameter deformed $\mathcal{R}\times S^{2}$
and background constructed in Garcia:2021iox  in the presence of a mixed NS-NS
and RR- field for any generic string solution.

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