# Offensive Language Identification in Greek

###### Abstract

As offensive language has become a rising issue for online communities and
social media platforms, researchers have been investigating ways of coping
with abusive content and developing systems to detect its different types:
cyberbullying, hate speech, aggression, etc. With a few notable exceptions,
most research on this topic so far has dealt with English. This is mostly due
to the availability of language resources for English. To address this
shortcoming, this paper presents the first Greek annotated dataset for
offensive language identification: the Offensive Greek Tweet Dataset (OGTD).
OGTD is a manually annotated dataset containing 4,779 posts from Twitter
annotated as offensive and not offensive. Along with a detailed description of
the dataset, we evaluate several computational models trained and tested on
this data.

Keywords: offensive language, hate speech, Greek

Offensive Language Identification in Greek

Zeses Pitenis1, Marcos Zampieri2, Tharindu Ranasinghe1
---
1University of Wolverhampton - Wolverhampton, UK
2Rochester Institute of Technology - Rochester, NY, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

##  1\. Introduction

In the age of social media, offensive content online has become prevalent in
recent years. There are many types of offensive content online such as racist
and sexist posts and insults and threats targeted at individuals or groups. As
such content increasingly occurs online, it has become a growing issue for
online communities. This has come to the attention of social media platforms
and authorities underlining the urgency to moderate and deal with such
content. Several studies in NLP have approached offensive language
identification applying machine learning and deep learning systems on
annotated data to identify such content. Researchers in the field have worked
with different definitions of offensive language with hate speech being the
most studied among these types [Davidson et al. (2017]. [Waseem et al. (2017]
investigate the similarity between these sub-tasks. With a few noteworthy
exceptions, most research so far has dealt with English, due to the
availability of language resources. This gap in the literature recently
started to be addressed with studies on Spanish [Aragón et al. (2018], Hindi
[Kumar et al. (2018], and German [Wiegand et al. (2018], to name a few.

In this paper we contribute in this direction presenting the first Greek
annotated dataset for offensive language identification: the Offensive Greek
Tweet Dataset (OGTD). OGTD uses a working definition of offensive language
inspired by the OLID dataset for English [Zampieri et al. (2019a] used in the
recent OffensEval (SemEval-2019 Task 6) [Zampieri et al. (2019b]. In its
version, 1.0 OGTD contains nearly 4,800 posts collected from Twitter and
manually annotated by a team of volunteers, resulting in a high-quality
annotated dataset. We trained a number of systems on this dataset and our best
results have been obtained from a system using LSTMs and GRU with attention
which achieved 0.89 F1 score.

##  2\. Related Work

The bulk of work on detecting abusive posts online addressed particular types
of such language like textual attacks and hate speech [Malmasi and Zampieri
(2017], aggression [Kumar et al. (2018], and others. OGTD considers a more
general definition of offensiveness inspired by the first layer of the
hierarchical annotation model described in [Zampieri et al. (2019a]. [Zampieri
et al. (2019a] model distinguishes targeted from general profanity, and
considers the target of offensive posts as indicators of potential hate speech
posts (insults targeted at groups) and cyberbulling posts (insults targeted at
individuals).

Offensive Language: Previous work presented a dataset with sentences labelled
as flame (i.e. attacking or containing abusive words) or okay [Razavi et al.
(2010] with a Naïve Bayes hybrid classifier and a user offensiveness
estimation using an offensive lexicon and sentence syntactic structures [Chen
et al. (2012]. A dataset of 3.3M comments from the Yahoo Finance and News
website, labelled as abusive or clean, was utilized in several experiments
using n-grams, linguistic and syntactic features, combined with different
types of word and comment embeddings as distributional semantics features
[Nobata et al. (2016]. The usefulness of character n-grams for abusive
language detection was explored on the same dataset with three different
methods [Mehdad and Tetreault (2016]. The most recent project expanded on
existing ideas for defining offensive language and presented the OLID
(Offensive Language Identification Dataset), a corpus of Twitter posts
hierarchically annotated on three levels, whether they contain offensive
language or not, whether the offense is targeted and finally, the target of
the offense [Zampieri et al. (2019a]. A CNN (Convolutional neural network)
deep learning approach outperformed every model trained, with pre-trained
FastText embeddings and updateable embeddings learned by the model as
features. In OffensEval (SemEval-2019 Task 6), participants had the
opportunity to use the OLID to train their own systems, with the top teams
outperforming the original models trained on the dataset.

Hate Speech: A study dataset of tweets posted after the murder of Drummer Lee
Rigby in the UK, manually annotated as offensive or antagonistic in terms of
race ethnicity or religion for hate speech identification with multiple
classifiers [Burnap and Williams (2015]. A logistic regression classifier
trained with paragraph2vec111https://github.com/thunlp/paragraph2vec word
representations of comments from Yahoo Finance [Djuric et al. (2015]. The
latest approaches in detecting hate speech include a dataset of Twitter posts,
labelled as hateful, offensive or clean, used to train a logistic regression
classifier with part-of-speech and word n-grams and a sentiment lexicon
[Davidson et al. (2017] and a linear SVM trained on character 4-grams, with an
extra RBF SVM meta-classifier that boosts accuracy in hateful language
detection [Malmasi and Zampieri (2018]. Both attempts tried to distinguish
offensive language and hate speech, with the hate class being the hardest to
classify.

###  2.1. Non-English Datasets

Research on other languages includes datasets such as: A Dutch corpus of posts
from the social networking site Ask.fm for the detection of cyberbullying [Van
Hee et al. (2015], a German Twitter corpus exploring the issue of hate speech
targeted to refugees [Ross et al. (2016], another Dutch corpus using data from
two anti-Islamic groups in Facebook [Tulkens et al. (2016], a hate speech
corpus in Italian [Pelosi et al. (2017], an abusive language corpus in Arabic
[Mubarak et al. (2017], a corpus of offensive comments from Facebook and
Reddit in Danish [Sigurbergsson and Derczynski (2020], another Twitter corpus
in German [Wiegand et al. (2018] for GermEval2018, a second Italian corpus
from Facebook and Twitter [Bosco et al. (2018], an aggressive post corpus from
Mexican Twitter in Spanish [Aragón et al. (2018] and finally an aggressive
comments corpus from Facebook in Hindi [Kumar et al. (2018]. SemEval 2019
presented a novel task: Multilingual detection of hate speech specifically
against immigrants and women with a dataset from Twitter, in English and
Spanish [Basile et al. (2019].

##  3\. The OGTD Dataset

The posts in OGTD v1.0 were collected between May and June, 2019. We used the
Twitter API initially collecting tweets from popular and trending hashtags in
Greece, including television programs such as series, reality and
entertainment shows. Due to the municipal, regional as well as the European
Parliament election taking place at the time, many hashtags included tweets
discussing the elections. The intuition behind this approach is that Twitter
as a microblogging service often gathers complaints and profane comments on
widely viewed television and politics, and as such, this period was a good
opportunity for data collection.

Following the methodology described in [Zampieri et al. (2019a] and others,
including a recent comparable Danish dataset [Sigurbergsson and Derczynski
(2020], we collected tweets using keywords such as sensitive or obscene
language. Queries for tweets containing common curse words and expressions
usually found in offensive messages in Greek as keywords (such as the well-
known word for “asshole”, “µαλαϰας” (malakas) or “go to hell”, “ςτο διαολο”
(sto diaolo), etc.) returned a large number of tweets. Aiming to compile a
dataset including offensive tweets of diverse types (sexist, racist, etc.)
targeted at various social groups, the Twitter API was queried with expletives
such as “πουτανα” (poutana, “whore”), “ϰαριολα” (kariola, “bitch”), “πουςτης”
(poustis, “faggot”), etc. and their plural forms, to explore the semantic and
pragmatic differences of the expletives mentioned above in their different
contextual environments. The challenge is to recognize between ironic and
insulting uses of these swear words, a common phenomenon in Greek.

The final query for data collection was for tweets containing “ειςαι” (eisai,
“you are”) as a keyword, inspired by [Zampieri et al. (2019a]. This particular
keyword is considered a stop word as it is quite common and frequent in
languages but was suspected to prove helpful for building the dataset for this
particular project, as offensive language often follows the following
structure: auxiliary verb (be) + noun/adjective. The immediacy of social media
and specifically Twitter provides the opportunity for targeted insults to be
investigated, following data mining of tweets including “you are” as a
keyword. In fact, many tweets present in the dataset showed users verbally
insulting other users or famous people and TV personas, confirming that
“ειςαι” was a facilitating keyword for the task in question.

###  3.1. Pre-processing and annotation

We collected a set of 49,154 tweets. URLs, Emojis and Emoticons were removed,
while usernames and user mentions were filtered as @USER following the same
methodology described in OLID [Zampieri et al. (2019a]. Duplicate punctuation
such as question and exclamation marks was normalized. After removing
duplicate tweets, the dataset was comprised of 46,218 tweets of which 5,000
were randomly sampled for annotation. We used
LightTag222https://www.lighttag.io/ to annotate the dataset due to its simple
and straightforward user interface and limitless annotations, provided by the
software creators.

Based on explicit annotation guidelines written in Greek and our proposal of
the definition of offensive language, a team of three volunteers were asked to
classify each tweet found in the dataset with one of the following tags:
Offensive, Not Offensive and Spam, which was introduced to filter out spam
from the dataset. Inter-annotator agreement was subsequently calculated and
labels with 100% agreement were deemed acceptable annotations. In cases of
disagreement, labels with majority agreement above 66% were selected as the
actual annotations of the tweets in question. For labels with complete
disagreement between annotators, one of the authors of this paper reviewed the
tweets with two extra human judges, to get the desired majority agreement
above 66%. Figure 1 is a confusion matrix that shows the inter-annotator
agreement or reliability, statistically measured by Cohen’s kappa coefficient.
The benchmark annotated dataset produced contained 4,779 tweets, containing
over 29% offensive content. The final distribution of labels in the new
Offensive Greek Tweet Dataset (OGTD), along with the breakdown of the data
into training and testing, is showing in Table 1.

Labels | Training Set | Test Set | Total
---|---|---|---
Offensive | 955 | 446 | 1,401
Not Offensive | 2,390 | 988 | 3,378
All | 3,345 | 1,434 | 4,779
Table 1: Distribution of labels in the OGTD v1.0. Figure 1: Cohen’s Kappa for
each pair of annotators

##  4\. Methods

Before experimenting with OGTD, an unique aspect of Greek which is the
accentuation of characters for correct pronunciation needed to be normalized.
When posting a tweet, many users omit accents due to their haste, resulting in
a mixed dataset containing fully accented tweets, partially-accented tweets,
and non-accented tweets. To achieve data uniformity and to avoid ambiguity,
every word is lower-cased and then normalized to its non-accented equivalent.

Several experiments were conducted with the OGTD, each one utilizing a
different combination from a pool of features (e.g. TF/IDF unigrams, bigrams,
POS and dependency relation tags) to train machine learning models. These
features were selected based on previous methodology used by researchers and
taking the dataset size into consideration. The TF-IDF weighted features are
often used for text classification and are useful for determining how
important a word is to a post in a corpus. The threshold for corpus specific
words was set to 80%, ignoring terms appearing in more than 80% of the
documents while the minimum document frequency was set to 6, and both unigrams
and bigrams were tested. Given the consistent use of linguistic features for
training machine learning models and results from previous work for offensive
language detection, part-of-speech (POS) and dependency relation tags were
considered as additional features. Using the spaCy333https://spacy.io/
pipeline for Greek, POS-tags and dependency relations were extracted for every
token in a tweet, which were then transformed to count matrices. A sentiment
lexicon was considered, but one suitable for this project is as of yet
unavailable for Greek.

| Not Offensive | Offensive | Weighted Average |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1 | F1 Macro
Linear SVM | 0.83 | 0.98 | 0.90 | 0.92 | 0.57 | 0.70 | 0.86 | 0.85 | 0.84 | 0.80
RBF SVM | 0.76 | 0.99 | 0.86 | 0.96 | 0.31 | 0.47 | 0.82 | 0.78 | 0.74 | 0.66
SGDC | 0.84 | 0.96 | 0.90 | 0.86 | 0.61 | 0.71 | 0.85 | 0.85 | 0.84 | 0.80
Multinomial NB | 0.77 | 0.99 | 0.86 | 0.94 | 0.33 | 0.49 | 0.82 | 0.78 | 0.75 | 0.67
Bernoulli NB | 0.83 | 0.89 | 0.86 | 0.71 | 0.61 | 0.66 | 0.80 | 0.80 | 0.80 | 0.76
Table 2: Results for offensive language detection with TF/IDF unigram features. For each model, Precision (P), Recall (R), and F1 are reported on all classes, and weighted averages. Macro-F1 is also listed (best in bold). | Not Offensive | Offensive | Weighted Average |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1 | F1 Macro
Linear SVM | 0.82 | 0.98 | 0.90 | 0.92 | 0.54 | 0.68 | 0.86 | 0.84 | 0.83 | 0.79
RBF SVM | 0.74 | 1.00 | 0.85 | 0.98 | 0.24 | 0.39 | 0.82 | 0.76 | 0.71 | 0.62
SGDC | 0.84 | 0.94 | 0.89 | 0.81 | 0.61 | 0.69 | 0.83 | 0.83 | 0.83 | 0.79
Multinomial NB | 0.77 | 0.99 | 0.87 | 0.93 | 0.32 | 0.48 | 0.82 | 0.79 | 0.75 | 0.67
Bernoulli NB | 0.82 | 0.88 | 0.85 | 0.68 | 0.57 | 0.62 | 0.78 | 0.79 | 0.78 | 0.74
Table 3: Results for offensive language detection with TF/IDF bigram features. For each model, Precision (P), Recall (R), and F1 are reported on all classes, and weighted averages. Macro-F1 is also listed (best in bold). | Not Offensive | Offensive | Weighted Average |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1 | F1 Macro
Linear SVM | 0.84 | 0.96 | 0.90 | 0.88 | 0.58 | 0.70 | 0.85 | 0.85 | 0.83 | 0.80
SGDC | 0.80 | 0.95 | 0.87 | 0.81 | 0.48 | 0.61 | 0.81 | 0.80 | 0.79 | 0.74
Multinomial NB | 0.77 | 0.95 | 0.85 | 0.78 | 0.36 | 0.49 | 0.77 | 0.77 | 0.74 | 0.67
Bernoulli NB | 0.80 | 0.78 | 0.79 | 0.54 | 0.58 | 0.56 | 0.72 | 0.72 | 0.72 | 0.68
Table 4: Results for offensive language detection with TF/IDF unigram features, POS and dependency relation tags. For each model, Precision (P), Recall (R), and F1 are reported on all classes, and weighted averages. Macro-F1 is also listed (best in bold). | Not Offensive | Offensive | Weighted Average |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1 | F1 Macro
Linear SVM | 0.84 | 0.97 | 0.90 | 0.91 | 0.58 | 0.71 | 0.86 | 0.85 | 0.84 | 0.80
SGDC | 0.74 | 0.99 | 0.85 | 0.93 | 0.22 | 0.35 | 0.80 | 0.75 | 0.69 | 0.60
Multinomial NB | 0.77 | 0.99 | 0.86 | 0.93 | 0.33 | 0.49 | 0.82 | 0.78 | 0.75 | 0.68
Bernoulli NB | 0.83 | 0.86 | 0.84 | 0.66 | 0.61 | 0.63 | 0.78 | 0.78 | 0.78 | 0.74
Table 5: Results for offensive language detection with TF/IDF unigram features and POS tags. For each model, Precision (P), Recall (R), and F1 are reported on all classes, and weighted averages. Macro-F1 is also listed (best in bold). | Not Offensive | Offensive | Weighted Average |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1 | F1 Macro
Linear SVM | 0.84 | 0.97 | 0.90 | 0.90 | 0.58 | 0.70 | 0.86 | 0.85 | 0.84 | 0.80
SGDC | 0.87 | 0.66 | 0.75 | 0.51 | 0.78 | 0.61 | 0.76 | 0.70 | 0.71 | 0.68
Multinomial NB | 0.77 | 0.97 | 0.86 | 0.85 | 0.35 | 0.49 | 0.79 | 0.78 | 0.74 | 0.67
Bernoulli NB | 0.82 | 0.81 | 0.81 | 0.58 | 0.60 | 0.59 | 0.74 | 0.74 | 0.74 | 0.70
Table 6: Results for offensive language detection with TF/IDF unigram features
and dependency relation tags. For each model, Precision (P), Recall (R), and
F1 are reported on all classes, and weighted averages. Macro-F1 is also listed
(best in bold).

For the first six deep learning models we used Greek word embeddings trained
on a large Greek web corpus [Outsios et al. (2018]. Each Greek word can be
represented with a 300 dimensional vector using the trained model. The vector
then can be used to feed in to the deep learning models which will be
described in section 4.1.2.. For the last deep learning architecture we wanted
to use a BERT [Devlin et al. (2019] model trained on Greek. However there was
no BERT model available for Greek language. The model that came closest our
requirement was multilingual BERT model 444https://github.com/google-
research/bert trained on 108 languages [Devlin et al. (2019] including Greek.
Since training BERT is a very computationaly expensive task we used the
available multilingual BERT cased model for the sixth deep learning
architecture.

###  4.1. Models

####  4.1.1. Classical Machine Learning Models

Every classical model was considered on the condition it could take matrices
as input for fitting and was trained with the default settings because of the
size of the dataset. Five models were trained: Two SVMs, one with linear
kernel and the other with a radial basis function kernel (RBF), both with a
value of 1 in the penalty parameter C of the error term. The gamma value of
the RBF SVM which indicates how much influence a single training example has,
was set to 2. The third classifier trained was another linear classifier with
Stochastic Gradient Descent (SGDC) learning. The gradient of the loss is
estimated each sample at a time and the SGDC is updated along the way with a
decreasing learning rate. The parameters for maximum epochs and the stopping
criterion were defined using the default values in scikit-learn. The final
classifier was two models based on the Bayes theorem: Multinomial Naïve Bayes,
which works with occurrence counts, and Bernoulli Naïve Bayes, which is
designed for binary features.

####  4.1.2. Deep Learning Models

Six different deep learning models were considered. All of these models have
been used in an aggression detection task. The models are Pooled GRU [Plum et
al. (2019], Stacked LSTM with Attention [Plum et al. (2019], LSTM and GRU with
Attention [Plum et al. (2019], 2D Convolution with Pooling [Ranasinghe et al.
(2019], GRU with Capsule [Hettiarachchi and Ranasinghe (2019], LSTM with
Capsule and Attention [Ranasinghe et al. (2019] and BERT [Devlin et al.
(2019]. These models has been used in HASOC 2019 and achieved a third place
finish in English task and a eighth place finish in German and Hindi subtasks
[Ranasinghe et al. (2019]. Parameters described in [Ranasinghe et al. (2019]
were used as the default parameters in order to ease the training process. The
code for the deep learning has been made available on Github
555https://github.com/tharindudr/aggression-detection-greek.

| Not Offensive | Offensive | Weighted Average |
---|---|---|---|---
Model | P | R | F1 | P | R | F1 | P | R | F1 | F1 Macro
Pooled GRU | 0.90 | 0.99 | 0.95 | 0.94 | 0.65 | 0.75 | 0.91 | 0.86 | 0.86 | 0.87
Stacked LSTM with Attention | 0.91 | 0.99 | 0.96 | 0.95 | 0.66 | 0.76 | 0.92 | 0.87 | 0.87 | 0.88
LSTM and GRU with Attention | 0.92 | 0.99 | 0.96 | 0.96 | 0.68 | 0.77 | 0.93 | 0.88 | 0.88 | 0.89
2D Convolution with Pooling | 0.91 | 0.98 | 0.96 | 0.95 | 0.64 | 0.74 | 0.90 | 0.86 | 0.85 | 0.88
GRU with Capsule | 0.92 | 0.99 | 0.95 | 0.94 | 0.64 | 0.75 | 0.91 | 0.86 | 0.85 | 0.88
LSTM with Capsule and Attention | 0.91 | 0.98 | 0.95 | 0.94 | 0.66 | 0.75 | 0.90 | 0.86 | 0.86 | 0.87
BERT-Base Multilingual Cased | 0.85 | 0.84 | 0.84 | 0.65 | 0.60 | 0.58 | 0.77 | 0.76 | 0.75 | 0.73
Table 7: Results for offensive language detection for Deep Learning models
with Greek word embeddings. For each model, Precision (P), Recall (R), and F1
are reported on all classes, and weighted averages. Macro-F1 is also listed
(best in bold).

###  4.2. Results

The performance of individual classifiers for offensive language
identification with TF/IDF unigram features is demonstrated in table 2 below.
We can see that both linear classifiers (SVM and SGDC) outperform the other
classifiers in terms of macro-F1, which does not take label imbalance into
account. The Linear SVM and SGDC perform almost identically, with the Linear
SVM performing slightly better in recall score for the Not Offensive class and
SGDC in recall score for the Offensive class. Bernoulli Naïve Bayes performs
better than all classifiers in recall score for the Offensive class but yields
the lowest precision score of all classifiers. While the RBF SVM and
Multinomial Naïve Bayes yield better recall score for the Not Offensive class,
their recall scores for the Offensive class are really low. For a binary text
classification task like offensive language detection, a high recall score for
both classes, especially for the Offensive class, is important for a model to
be considered successful. Thus, the Linear SVM can be considered the
marginally best model trained with OGTD, as its weighted average precision and
recall scores are higher.

Models trained with TF/IDF bigram features performed worse, with scores of all
evaluation metrics dropping with the exception of Multinomial Naïve Bayes
which improved in F1-score for the Not Offensive class. The full results are
reported in table 3 below. Three other approaches were opted for training the
models with the implementation of POS and dependency relation tags via a
transformation pipeline, also including TF/IDF unigram features, performing
better than the addition of bigrams.

Experiments with linguistic features were conducted, to inspect their
efficiency for this task. For these experiments, the RBF SVM was not used due
to data handling problems by the model in the scikit-learn library. In the
first experiment, TF/IDF unigram features were combined with POS and
dependency relation tags. The results of implementing all three features are
shown in table 4 below. While the Linear SVM model improved the recall score
on the previous model trained with bigrams, the other models show a
significant drop in their performance.

In the next experiment, POS tags were used in conjunction with TF/IDF unigram
features. Surprisingly, the addition of POS tags in the Linear SVM yields the
same F1-score as the first model trained on TF/IDF unigram features, yielding
lower precision scores for both classes, while the recall score for the
Offensive class improved marginally. The Naïve Bayes models show a marginal
decrease in their performance. On the other hand, the performance of SGDC
significantly decreases with POS tags only and, interestingly enough, its
recall score for the Offensive class is the worst among classifiers. The
complete results are presented in table 5 below.

The experiment with linguistic features was the combination of dependency
relation tags with TF/IDF unigrams. This experimented yielded the same
F1-score of 80% as the other Linear SVM classifiers, performing almost
identically with the previous model trained with POS tags, only bested in
precision for the Offensive class. While the recall score for Offensive
instances improves on the first model trained only on TF/IDF unigrams by
0.01%, the recall score for Not Offensive instances drops by the same amount.
The recall score for the Not Offensive class was already high, so this
increase in recall score could slightly facilitate the offensive language
detection task. Without improving upon the first SGDC presented, the SGDC
rised in performance overall and as for the Naïve Bayes representatives, the
both the Multinomial and Bernoulli approaches performed better than in the
second experiment. The complete results are shown in table 6 below.

The performance of the deep learning models is presented in table 7. As we can
see LSTM and GRU with Attention outperformed all the other models in-terms of
macro-f1. Notably it outperformed all other classifical models and deep
learning models in precision, recall and f1 for Offensive class as well as the
Not Offensive class. However, fine tuning BERT-Base Multilingual Cased model
did not achieve good results. For this task monolingual Greek word embeddings
perform significantly better than the multilingual bert embeddings. LSTM and
GRU with Attention can be considered as the best model trained for OGTD.

###  4.3. Discussion

The data annotated in OGTD proved to be facilitating in offensive language
detection with a significant success for Greek, taking into consideration its
size and label distribution, with the best model (LSTM and GRU with Attention)
achieving a F1-macro of 0.89. Among the classical machine learning approaches,
the linear SVM model achieved the best results, 0.80, whereas the the
Stochastic Gradient Descent (SGD) learning classifier yielded the best recall
score for the Offensive class, at 0.61. In terms of features used, TF/IDF
matrices of word unigrams proved to work work well with multiple classical ML
classifiers. Overall, it is clear that deep learning models with word
embedding feature provide better results than the classical machine learning
models.

Of the linguistic features, POS tags improved the performance of the Linear
SVM marginally in terms of recall for the Offensive class, other classifiers
deteriorated in their performance.It is not yet clear whether this is due to
the accuracy of the Greek model available for spaCy in producing such tags or
the tags themselves as features and is a subject that can be explored with
further improvements of spaCy or other NLP tools developed for Greek. The
dataset itself contains many instances with neologisms, creative uses of
language or and even rare slang words, therefore training the existing model
with such instances could improve both spaCy’s accuracy for POS and dependency
relation tags and the Linear SVM’s performance in text classification for
Greek.

##  5\. Conclusion

This paper presented the Offensive Greek Tweet Dataset (OGTD), a manually
annotated dataset for offensive language identification and the first Greek
dataset of its kind. The OGTD v1.0 contains a total of 4,779 tweets,
encompassing posts related to an array of topics popular among Greek people
(e.g. political elections, TV shows, etc.). Tweets were manually annotated by
a team volunteers through an annotation platform. We used the same guidelines
used in the annotation of the English OLID dataset [Zampieri et al. (2019a].
Finally, we run several machine learning and deep learning classifiers and the
best results were achieved by a LSTM and GRU with Attention model.

###  5.1. Ongoing - OGTD v2.0 and OffensEval 2020

We have recently released OGTD v2.0 as training data for OffensEval 2020
(SemEval-2020 Task 12) [Zampieri et al.
(2020].666https://sites.google.com/site/offensevalsharedtask/home The
reasoning behind the expansion of the dataset was to have a larger Greek
dataset for the competition. New posts were collected in November 2019
following the same approach we used to compile v1.0 described in this paper.
This second batch of tweets included tweets with trending hashtags, shows and
topics from Greece at the time. Additionally, keywords that proved to retrieve
interesting tweets in the first version were once again used in the search,
along with new keywords like pejorative terms. When the collection was
finished, 5,508 tweets were randomly sampled to be then annotated by a team of
volunteers. The annotation guidelines were the same ones we used for v1.0.
OGTD v2.0 combines the existing with the newly annotated tweets in a larger
dataset of 10,287 instances.

Labels | Training Set | Test Set | Total
---|---|---|---
Offensive | 2,486 | 425 | 2,911
Not Offensive | 6,257 | 1119 | 7,376
All | 8,743 | 1,544 | 10,287
Table 8: Distribution of labels in the OGTD v2.0.

Finally, both OGTD v1.0 and v2.0 provide the opportunity for researchers to
test cross-lingual learning methods as it can be used in conjunction with the
English OLID and other datasets annotated using the same guidelines such as
the one by ?) for Danish and by ?) for Turkish while simultaneously
facilitating the development of language resources for NLP in Greek.

## Acknowledgements

We would like to acknowledge Maria, Raphael and Anastasia, the team of
volunteer annotators that provided their free time and efforts to help us
produce v1.0 of the dataset of Greek tweets for offensive language detection,
as well as Fotini and that helped review tweets with ambivalent labels.
Additionally, we would like to express our sincere gratitude to the LightTag
team and especially to Tal Perry for granting us free use for their annotation
platform.

## Bibliographical References

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# Normal modes for two-dimensional gravitating kinks

Yuan Zhong<EMAIL_ADDRESS>School of Physics, Xi’an Jiaotong
University, Xi’an 710049, China  Institute of Theoretical Physics, Xi’an
Jiaotong University, Xi’an 710049, China

###### Abstract

We study small perturbations around an arbitrary static kink solution of a
two-dimensional (2D) gravity-scalar system, where the gravity part is
described by a subclass of 2D dilaton gravity theory, and the scalar matter
field has generalized dynamics. We expand the action around an arbitrary
static solution and keep terms up to the second order of the perturbations.
After variation the linear-order action leads to background field equations,
as expected. The quadratic action of the normal modes are obtained after
fixing the gauge and using the constraint equation. The linear perturbation
equations obtained from the quadratic action are consistent with those
obtained by linearizing the field equations under the dilaton gauge. All the
calculations are assisted by a Mathematica code, which is also provided as a
supplementary material.

###### keywords:

2D dilaton gravity , K-field , Kinks , Normal modes

††journal: Journal of LaTeX Templates

##  1 Introduction

Two-dimensional (2D) gravity theories are simpler mathematically, but still
reflect part of the features of their higher-dimensional counterparts [1, 2].
For this reason, 2D gravity theories have been applied to discuss quantization
of gravity [3, 4], gravitational collapse [5, 6], black hole evaporation [7,
8, 9, 10, 11], and many other difficult issues, see [12, 13, 14, 15] for
comprehensive reviews on early works. Recently, the study of the Sachdev-Ye-
Kitaev (SYK) model [16, 17] displays a potential application of 2D gravity in
the study of gauge/gravity duality [18, 19, 20, 21], see [22, 23, 24] for
pedagogical introductions.

In parallel to the study of 2D black holes, it is natural to consider 2D
gravity models with nonlinear scalar fields, and regarding smooth gravitating
kink solutions as 2D versions of some 5D thick brane solutions. The later have
been extensively studied in the past two decades [25, 26, 27, 28, 29] as they
offer some regular extensions of the Randall-Sundrum thin brane world models
[30, 31, 32, 33].

Although the study of 2D self-gravitating kinks can be traced back to 1995
[34, 35, 36, 37, 38], most of these solutions contain some space-time
singularities. To describe 2D thick brane, we require some gravitating kink
solutions without any singularity.

To the author’s knowledge, the first solution of this type was reported by
Stötzel [35], who considered the 2D gravity model of Mann, Morsink, Sikkema
and Steele (MMSS) [39]. Stötzel’s solution has no singularity, and the
corresponding metric is asymptotic AdS2, and therefore, is an excellent
example of 2D thick brane.

In a recent work [40], the present author pointed out that the procedure used
by Stötzel for deriving his solution is nothing but the superpotential method,
which has been used repeatedly in the study of 5D thick branes [25, 26, 41,
42, 43, 44, 45, 46]. This observation makes it an easy task to construct
analytical solutions of 2D gravitating kink, some examples can be found in
Ref. [40] for the case with canonical scalar field, and [47] for the case with
noncanonical scalar field. All these solutions are regular, and can be
regarded as 2D thick branes.

Before using these solutions for further explorations, one should carefully
discuss the stability of these solutions under small field perturbations. This
issue was first analyzed in Ref. [40], where the author conducted a direct
linearization of the field equations. After gauging out the perturbation of
the dilaton field, and using the constraint equations, a Schrödinger-like
equation is obtained for the physical perturbation. In particular, the
Hamiltonian operator of the Schrödinger-like equation can be factorized as the
form $\hat{H}=\hat{\mathcal{A}}\hat{\mathcal{A}}^{\dagger}$, which indicates
that the eigenvalues are positive-semidefinite, and therefore, the solutions
are stable. Similar results were also obtained for the case with noncanonical
scalar field [47]. Remarkablly, the perturbation equations derived in Refs.
[40, 47] take similar form as the scalar perturbation equations of some 5D
thick brane model with Einstein gravity [48, 49, 50]. This verified the
conjecture that the MMSS model may be the closest thing there is to general
relativity in two dimensions [51].

In the present work, we notice that there are two different approaches for
deriving the linear perturbation equations. In the first approach, one
directly linearizes the field equations, as has been done in Refs. [40, 47].
In the other, one first expands the action around a background solution, and
keep only the quadratic terms of the perturbations. After canonicalizing this
quadratic action, one can easily read out the master equations for the normal
modes of the perturbations. Naively, one may expect that these two approaches
leads to equivalent results. However, counterexamples have already been
reported in some 5D thick brane models [48, 49, 50].

In brief, Refs. [48, 49, 50] tell us that one must fix the gauge properly if
he adopt the equation of motion approach. Otherwise, the Schrödinger-like
equation might be different from the one obtained from the quadratic approach.
For example, if one starts with the widely used longitude gauge, he would
obtain a factorizable Hamiltonian operator
$\hat{H}_{-}=\hat{\mathcal{A}}^{\dagger}\hat{\mathcal{A}}$ for the physical
perturbation modes. While the Hamiltonian operator derived from the quadratic
action approach is $\hat{H}=\hat{\mathcal{A}}\hat{\mathcal{A}}^{\dagger}$. One
may recognize that $\hat{H}$ and $\hat{H}_{-}$ are a pair of superpartners in
the sense of supersymmetric quantum mechanics [52, 53].

Usually, a pair of superpartner Hamiltonians share the same spectra, except
their zero modes [53]. So the properties of $\hat{H}$ can be learnt from
$\hat{H}_{-}$, and vise versa. The situation changes, however, if the
potential term of either $\hat{H}$ or $\hat{H}_{-}$ diverges. In this case,
the supersymmetry might be explicitly broken, and the eigenvalues of $\hat{H}$
or $\hat{H}_{-}$ are only partial degenerated, if the potential is symmetric,
or completely non-degenerated, if the potential is asymmetric [53, 54, 55, 56,
57]. Even worse, one of the superpartner Hamiltonians might has a eigenstate
with negative eigenvalue, a sign for instability! Therefore, when singular
potential appears, one must adopt the quadratic approach, as it gives the
correct Hamiltonian for the normal perturbation modes.

Unfortunately, for 2D gravitating kink solutions considered in Refs. [40, 47],
singular potentials in the Schrödinger-like equation seems unavoidable within
some parameter ranges. For this reason, it is necessary to conduct an
independent derivation of the linear perturbation equations from the quadratic
action approach, and see if they are equivalent with those obtained in Refs.
[40, 47].

In the next section, we briefly review the main results of gravitational
perturbation theory for an arbitrary background metric solution of the
following model [47]:

$\displaystyle S=\frac{1}{\kappa}\int d^{2}x\sqrt{-g}\left[\mathcal{X}+\varphi
R+\kappa\mathcal{L}(\phi,X)\right],$ (1)

where $\kappa$ is the gravitational coupling constant, $g$ is the determinant
of the metric,
$\mathcal{X}\equiv-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\mu}\varphi$
and $X\equiv-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\mu}\phi$ are the
kinetic terms of the dilaton field and the scalar field, respectively. In Sec.
3, we come back to the case with static solutions. After taking the dilaton
gauge, we derive the quadratic action of the normal perturbation modes. All
our calculations are conducted with a Mathematica code, which can be found in
the supplementary material. The results are summarized in Sec. 4.

##  2 Gravitational perturbation theory: a brief review

Consider a small perturbation $\delta g_{\mu\nu}$ around an arbitrary
background metric solution $g_{\mu\nu}$. If we denote the perturbation of the
inverse metric as $\delta g^{\nu\sigma}$, and require the orthogonal relation

$\displaystyle(g_{\mu\nu}+\delta g_{\mu\nu})(g^{\nu\sigma}+\delta
g^{\nu\sigma})=\delta^{\sigma}_{\mu},$ (2)

then we would obtain the following equation:

$\displaystyle g_{\mu\nu}\delta g^{\nu\sigma}+g^{\nu\sigma}\delta
g_{\mu\nu}+\delta g_{\mu\nu}\delta g^{\nu\sigma}=0.$ (3)

Obviously, the last two term are not of the same order, therefore, this
equation can only be satisfied order by order, which means that $\delta
g^{\nu\sigma}=\delta^{(1)}g^{\nu\sigma}+\delta^{(2)}g^{\nu\sigma}+\cdots$ is a
summation of perturbations with different orders.

To the first and second orders, one obntains

$\displaystyle\delta^{(1)}g^{\mu\nu}$ $\displaystyle=$
$\displaystyle-g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma},$ (4)
$\displaystyle\delta^{(2)}g^{\mu\nu}$ $\displaystyle=$
$\displaystyle-g^{\mu\rho}\delta g_{\rho\sigma}\delta^{(1)}g^{\sigma\nu},$ (5)

respectively. The Christoffel symbol is completely defined by the metric, and
its first and second-order perturbations take the following forms:

$\displaystyle\delta^{(1)}\Gamma_{\mu\nu}^{\lambda}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\delta^{(1)}g^{\lambda\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)$
(6) $\displaystyle+$
$\displaystyle\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}\delta
g_{\sigma\nu}+\partial_{\nu}\delta g_{\sigma\mu}-\partial_{\sigma}\delta
g_{\mu\nu}\right),$ $\displaystyle\delta^{(2)}\Gamma_{\mu\nu}^{\lambda}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\delta^{(2)}g^{\lambda\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)$
(7) $\displaystyle+$
$\displaystyle\frac{1}{2}\delta^{(1)}g^{\lambda\sigma}\left(\partial_{\mu}\delta
g_{\sigma\nu}+\partial_{\nu}\delta g_{\sigma\mu}-\partial_{\sigma}\delta
g_{\mu\nu}\right).$

Similarly, one can write out the perturbations of the Ricci tensor

$\displaystyle\delta^{(1)}R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\rho}\delta^{(1)}\Gamma_{\mu\nu}^{\rho}-\partial_{\nu}\delta^{(1)}\Gamma_{\mu\rho}^{\rho}+\delta^{(1)}\Gamma_{\lambda\rho}^{\rho}\Gamma_{\mu\nu}^{\lambda}$
(8) $\displaystyle+$
$\displaystyle\Gamma_{\lambda\rho}^{\rho}\delta^{(1)}\Gamma_{\mu\nu}^{\lambda}-\delta^{(1)}\Gamma_{\lambda\nu}^{\rho}\Gamma_{\mu\rho}^{\lambda}$
$\displaystyle-$
$\displaystyle\Gamma_{\lambda\nu}^{\rho}\delta^{(1)}\Gamma_{\mu\rho}^{\lambda},$
$\displaystyle\delta^{(2)}R_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\rho}\delta^{(2)}\Gamma_{\mu\nu}^{\rho}-\partial_{\nu}\delta^{(2)}\Gamma_{\mu\rho}^{\rho}+\delta^{(2)}\Gamma_{\lambda\rho}^{\rho}\Gamma_{\mu\nu}^{\lambda}$
(9) $\displaystyle+$
$\displaystyle\delta^{(1)}\Gamma_{\lambda\rho}^{\rho}\delta^{(1)}\Gamma_{\mu\nu}^{\lambda}+\Gamma_{\lambda\rho}^{\rho}\delta^{(2)}\Gamma_{\mu\nu}^{\lambda}$
$\displaystyle-$
$\displaystyle\delta^{(2)}\Gamma_{\lambda\nu}^{\rho}\Gamma_{\mu\rho}^{\lambda}-\delta^{(1)}\Gamma_{\lambda\nu}^{\rho}\delta^{(1)}\Gamma_{\mu\rho}^{\lambda}$
$\displaystyle-$
$\displaystyle\Gamma_{\lambda\nu}^{\rho}\delta^{(2)}\Gamma_{\mu\rho}^{\lambda},$

and the scalar curvature

$\displaystyle\delta^{(1)}R$ $\displaystyle=$
$\displaystyle\delta^{(1)}g^{\mu\nu}R_{\mu\nu}+g^{\mu\nu}\delta^{(1)}R_{\mu\nu},$
(10) $\displaystyle\delta^{(2)}R$ $\displaystyle=$
$\displaystyle\delta^{(2)}g^{\mu\nu}R_{\mu\nu}+\delta^{(1)}g^{\mu\nu}\delta^{(1)}R_{\mu\nu}$
(11) $\displaystyle+$ $\displaystyle g^{\mu\nu}\delta^{(2)}R_{\mu\nu}.$

The perturbations of $\sqrt{-g}$ are also useful

$\displaystyle\delta^{(1)}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{-g}g^{\mu\nu}\delta g_{\mu\nu},$ (12)
$\displaystyle\delta^{(2)}\sqrt{-g}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sqrt{-g}\left[(g^{\mu\nu}\delta
g_{\mu\nu})^{2}+2\delta g_{\mu\nu}\delta^{(1)}g^{\mu\nu}\right].$ (13)

In addition to the metric perturbation, we also have perturbations of the
dilaton and the scalar fields, which are denoted as $\delta\varphi$ and
$\delta\phi$, respectively. One can testifies that the perturbations of the
dilaton kinetic term are

$\displaystyle\delta^{(1)}\mathcal{X}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\delta^{(1)}g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi-g^{\mu\nu}\nabla_{\mu}\delta\varphi\nabla_{\nu}\varphi,$
(14) $\displaystyle\delta^{(2)}\mathcal{X}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\delta^{(2)}g^{\mu\nu}\nabla_{\mu}\varphi\nabla_{\nu}\varphi-\delta^{(1)}g^{\mu\nu}\nabla_{\mu}\delta\varphi\nabla_{\nu}\varphi$
(15) $\displaystyle-$
$\displaystyle\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\delta\varphi\nabla_{\nu}\delta\varphi,$

which after replacing $\varphi\to\phi$ give us the expressions of
$\delta^{(1)}X$ and $\delta^{(2)}X$.

Finally, the perturbations of the scalar Lagrangian density are

$\displaystyle\delta^{(1)}\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\phi}\delta\phi+\mathcal{L}_{X}\delta^{(1)}X,$ (16)
$\displaystyle\delta^{(2)}\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X\phi}\delta\phi\delta^{(1)}X+\frac{1}{2}\mathcal{L}_{\phi\phi}(\delta\phi)^{2}$
(17) $\displaystyle+$
$\displaystyle\frac{1}{2}\mathcal{L}_{XX}(\delta^{(1)}X)^{2}+\mathcal{L}_{X}\delta^{(2)}X.$

In this work, we defined
$\mathcal{L}_{\phi}\equiv\frac{\partial\mathcal{L}}{\partial\phi}$,
$\mathcal{L}_{X}\equiv\frac{\partial\mathcal{L}}{\partial X}$, and so on.

With all these preparation, we can now write out the perturbations of the
action (1):

$\displaystyle\delta^{(1)}S=\frac{1}{\kappa}\int
d^{2}x\\{\delta^{(1)}\sqrt{-g}[\mathcal{X}+\varphi R+\kappa\mathcal{L}]$ (18)
$\displaystyle+$ $\displaystyle\sqrt{-g}[\delta^{(1)}\mathcal{X}+\delta\varphi
R+\varphi\delta^{(1)}R+\kappa\delta^{(1)}\mathcal{L}]\\},$

and

$\displaystyle\delta^{(2)}S=\frac{1}{\kappa}\int
d^{2}x\\{\delta^{(2)}\sqrt{-g}[\mathcal{X}+\varphi R+\kappa\mathcal{L}]$ (19)
$\displaystyle+$
$\displaystyle\delta^{(1)}\sqrt{-g}[\delta^{(1)}\mathcal{X}+\delta\varphi
R+\varphi\delta^{(1)}R+\kappa\delta^{(1)}\mathcal{L}]$ $\displaystyle+$
$\displaystyle\sqrt{-g}[\delta^{(2)}\mathcal{X}+\delta\varphi\delta^{(1)}R+\varphi\delta^{(2)}R+\kappa\delta^{(2)}\mathcal{L}]\\}.$

No doubt, it would be very complicate to calculate the perturbations of the
action manually. Fortunately, in two dimension these calculations can be
accomplished by writing Mathematica codes, see the supplementary material.

##  3 The case of static solution

Now, let us turn to the special case with static solutions, where
$g_{\mu\nu}=g_{\mu\nu}(x)$, $\varphi=\varphi(x)$, $\phi=\phi(x)$. In two
dimensions one can always express a static metric into the following form:

$\displaystyle ds^{2}=-e^{2A(x)}dt^{2}+dx^{2}.$ (20)

This coordinate system is very convenient for constructing first-order
formalism, from which analytical gravitating kinks can be derived, see Refs.
[40, 47] for details.

However, to discuss the perturbation issue, it is more convenient to introduce
a spatial transformation

$\displaystyle r\equiv\int e^{-A(x)}dx,$ (21)

with which the metric can be written as a conformally flat form

$\displaystyle g_{\mu\nu}(r)=e^{2A(r)}\eta_{\mu\nu}.$ (22)

Following Refs. [40, 47], we define the metric perturbation as

$\displaystyle\delta g_{\mu\nu}(t,r)$ $\displaystyle\equiv$ $\displaystyle
e^{2A(r)}h_{\mu\nu}(t,r)$ (25) $\displaystyle=$ $\displaystyle
e^{2A(r)}\left(\begin{array}[]{cc}h_{00}(t,r)&\Phi(t,r)\\\
\Phi(t,r)&h_{rr}(t,r)\\\ \end{array}\right).$

Note that although the background solution is static, the field perturbations
$\delta g_{\mu\nu}(t,r)$, $\delta\varphi(t,r)$, $\delta\phi(t,r)$ are all
functions of both $t$ and $r$.

Using Eqs. (22) and (25), one can proof that

$\displaystyle\delta^{(1)}S=\frac{1}{\kappa}\int d^{2}x\big{\\{}-2\varphi
A^{\prime}\dot{\Phi}-2\varphi\dot{\Phi}^{\prime}$ (26) $\displaystyle-$
$\displaystyle\varphi^{\prime}\delta\varphi^{\prime}-2A^{\prime\prime}\delta\varphi+\kappa
e^{2A}\mathcal{L}_{\phi}\delta\phi-\kappa\mathcal{L}_{X}\phi^{\prime}\delta\phi^{\prime}$
$\displaystyle+$ $\displaystyle\frac{1}{4}h_{00}\left(4\varphi
A^{\prime\prime}-2e^{2A}\kappa\mathcal{L}+{\varphi^{\prime}}^{2}\right)$
$\displaystyle+$ $\displaystyle\varphi A^{\prime}h_{00}^{\prime}+\varphi
h_{00}^{\prime\prime}+\varphi A^{\prime}h_{rr}^{\prime}+\varphi\ddot{h}_{rr}$
$\displaystyle+$ $\displaystyle\frac{1}{4}h_{rr}\left(4\varphi
A^{\prime\prime}+2e^{2A}\kappa\mathcal{L}+2\kappa\mathcal{L}_{X}{\phi^{\prime}}^{2}+{\varphi^{\prime}}^{2}\right)\big{\\}}.$

Here we used over dots and primes to denote the derivatives with respect to
$t$ and $r$, respectively. Obviously, the $\Phi$ terms can be written as a
total derivative term (as $\dot{A}=0=\dot{\varphi}$), and does not lead to
field equation.

After partial integrations and deminding the coefficient of $\delta\varphi$
equals to zero, one immediately obtains the dilaton equation

$\displaystyle\varphi=2A.$ (27)

Similarly, for $\delta\phi$, $h_{00}$ and $h_{rr}$ terms, the corresponding
field equations are:

$\displaystyle(\mathcal{L}_{X}\phi^{\prime})^{\prime}+e^{2A}\mathcal{L}_{\phi}=0,$
(28) $\displaystyle 4A^{\prime\prime}-2A^{\prime 2}=\kappa e^{2A}\mathcal{L},$
(29)

and

$\displaystyle 2{A^{\prime}}^{2}=\kappa
e^{2A}\mathcal{L}+\kappa\mathcal{L}_{X}{\phi^{\prime}}^{2},$ (30)

where we have used the dilaton equation (27). The combination of Eqs. (29) and
(30) also gives another useful equation

$\displaystyle-4A^{\prime\prime}+4{A^{\prime}}^{2}=\kappa\mathcal{L}_{X}{\phi^{\prime}}^{2}.$
(31)

The above field equations are consistent with those obtained in Ref. [47].

The quadratic action of the perturbations are much more complicated. To
simplify the calculation, we first note that the general invariance of action
(1) under the general coordinate transformation

$\displaystyle x^{\mu}\to\tilde{x}^{\mu}=x^{\mu}+\xi^{\mu}(t,r),$ (32)

induces an invariance of $\delta^{(2)}S$ under the following gauge
transformations [40]:

$\displaystyle\Delta h_{00}$ $\displaystyle=$ $\displaystyle
2\partial_{t}\xi^{0}+2\frac{a^{\prime}}{a}\xi^{1},$ (33)
$\displaystyle\Delta\Phi$ $\displaystyle=$
$\displaystyle-\partial_{t}\xi^{1}+\partial_{r}\xi^{0},$ (34)
$\displaystyle\Delta h_{11}$ $\displaystyle=$
$\displaystyle-2\partial_{r}\xi^{1}-2\frac{a^{\prime}}{a}\xi^{1},$ (35)
$\displaystyle\Delta\delta\phi$ $\displaystyle=$
$\displaystyle-\phi^{\prime}\xi^{1},$ (36) $\displaystyle\Delta\delta\varphi$
$\displaystyle=$ $\displaystyle-\varphi^{\prime}\xi^{1},$ (37)

where $\Delta h_{00}\equiv\widetilde{h}_{00}-h_{00}$, and so on.

To fix the gauge degrees of freedom, we first define a new variable $\Xi\equiv
2\dot{\Phi}-h_{00}^{\prime}$, whose gauge transformation only depends on
$\xi^{1}$:

$\displaystyle\Delta\Xi=-2\left[\ddot{\xi}^{1}+\left(A^{\prime}\xi^{1}\right)^{\prime}\right].$
(38)

Thus, the introducing of $\Xi$ eliminates the the degree of freedom of
$\xi^{0}$. To fix the other, we simply chose the gauge condition such that
$\delta\varphi=0$ [40].

After this gauge fixing process, we can separate $\delta^{(2)}S$ into three
parts

$\displaystyle\delta^{(2)}S=\delta^{(2)}S_{\Xi}+\delta^{(2)}S_{h_{rr}}+\delta^{(2)}S_{\delta\phi},$
(39)

where

  * 1.

$\delta^{(2)}S_{\Xi}\equiv$ all the terms with $\Xi$ and its derivatives.

  * 2.

$\delta^{(2)}S_{h_{rr}}\equiv$ all the terms with $h_{rr}$ and its
derivatives, but without $\Xi$ and its derivatives.

  * 3.

$\delta^{(2)}S_{\delta\phi}\equiv$ all the terms with only $\delta\phi$ and
its derivatives, in other words, the quadratic terms of $\delta\phi$.

The above definitions prevent us from overcounting the cross terms.

By conducting partial integrations many times and discarding all the boundary
terms, we obtain

$\displaystyle\delta^{(2)}S_{\Xi}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\int
d^{2}x\left\\{\frac{1}{2}\kappa\phi^{\prime}\mathcal{L}_{X}\delta\phi-A^{\prime}h_{rr}\right\\}\Xi,$
(40) $\displaystyle\delta^{(2)}S_{h_{rr}}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\int
d^{2}x\bigg{\\{}\frac{1}{2}\gamma\left(A^{\prime\prime}-A^{\prime
2}\right)h_{rr}^{2}$ (41) $\displaystyle+$ $\displaystyle\frac{1}{2}A^{\prime
2}h_{rr}^{2}+2\gamma\left(A^{\prime
2}-A^{\prime\prime}\right)\frac{\delta\phi^{\prime}}{\phi^{\prime}}h_{rr}$
$\displaystyle+$ $\displaystyle 2A^{\prime}\left(A^{\prime\prime}-A^{\prime
2}\right)\frac{\delta\phi}{\phi^{\prime}}h_{rr}$ $\displaystyle+$
$\displaystyle\gamma\left(A^{\prime\prime}-A^{\prime
2}\right)\frac{X^{\prime}}{X}\frac{\delta\phi}{\phi^{\prime}}h_{rr}\bigg{\\}},$

where

$\displaystyle\gamma\equiv 1+2\frac{\mathcal{L}_{XX}X}{\mathcal{L}_{X}},$ (42)

and

$\displaystyle\delta^{(2)}S_{\delta\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa}\int d^{2}x\bigg{\\{}\frac{1}{2}\kappa
e^{-2A}\mathcal{L}_{XX}\phi^{\prime 2}\delta\phi^{\prime 2}$ (43)
$\displaystyle+$ $\displaystyle\frac{1}{2}\kappa
e^{2A}\mathcal{L}_{\phi\phi}\delta\phi^{2}-\frac{1}{2}\kappa\mathcal{L}_{X}\delta\phi^{\prime
2}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\kappa\mathcal{L}_{X}\dot{\delta\phi}^{2}-\kappa\mathcal{L}_{X\phi}\phi^{\prime}\delta\phi\delta\phi^{\prime}\bigg{\\}}.$

Obviously, $\Xi$ is a Lagrangian multiplier, whose variation leads to a
constraint equation:

$\displaystyle
2A^{\prime}{h}_{rr}=\kappa\mathcal{L}_{X}\phi^{\prime}{\delta\phi}.$ (44)

The same equation was also derived in Ref. [47] from the equation of motion
approach.

After eliminating $h_{rr}$, $\delta^{(2)}S$ becomes a quadratic action of
$\delta\phi$. The simplification of $\delta^{(2)}S$ is the most intricate part
of the present work. A key observation is that all the coefficients of can be
rearranged as functions of only $A^{\prime},\phi^{\prime}$, $\gamma$ (or
equivalently $\mathcal{L}_{XX}$), $X$, $\mathcal{L}_{X}$, and the derivatives
of $\gamma$, $X$, and $\mathcal{L}_{X}$.

To see this clearly, we first note that Eq. (31) allows us to express
$A^{\prime\prime}$ as a function of $A^{\prime},\phi^{\prime}$, and
$\mathcal{L}_{X}$:

$\displaystyle A^{\prime\prime}=A^{\prime
2}-\frac{1}{4}\kappa\mathcal{L}_{X}\phi^{\prime 2}.$ (45)

Similarly, by taking the derivative of $X$, we can express
$\phi^{\prime\prime}$ in terms of $A^{\prime},\phi^{\prime}$, $X$ and
$X^{\prime}$

$\displaystyle\phi^{\prime\prime}=A^{\prime}\phi^{\prime}+\frac{1}{2}\frac{X^{\prime}}{X}\phi^{\prime}.$
(46)

Then, by taking the derivative of $\mathcal{L}_{X}$ we obtain

$\displaystyle\mathcal{L}_{X\phi}=\frac{1}{\phi^{\prime}}\left(\mathcal{L}_{X}^{\prime}-\mathcal{L}_{XX}X^{\prime}\right).$
(47)

Finally, by taking the derivative of Eq. (28), and using Eqs. (45)-(47), we
get

$\displaystyle\mathcal{L}_{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{L}_{X}X^{\prime\prime}}{\phi^{\prime
2}}+\frac{X^{\prime 2}\mathcal{L}_{{XX}}}{\phi^{\prime
2}}-\frac{\mathcal{L}_{X}X^{\prime 2}}{2X\phi^{\prime 2}}$ (48)
$\displaystyle+$
$\displaystyle\frac{X^{\prime}\mathcal{L}_{X}^{\prime}}{\phi^{\prime
2}}-\frac{1}{2}\kappa
X\mathcal{L}_{X}^{2}+\frac{2X\mathcal{L}_{X}^{\prime\prime}}{\phi^{\prime
2}}.$

With Eqs. (45)-(48), a direct calculation shows that $\delta^{(2)}S$ takes the
following form:

$\displaystyle\delta^{(2)}S=\frac{1}{2}\int
d^{2}x\mathcal{L}_{X}\left\\{-\ddot{\delta\phi}+\gamma\delta\phi^{\prime\prime}+\gamma
U\delta\phi\right\\}\delta\phi,$ (49)

where

$\displaystyle U$ $\displaystyle\equiv$
$\displaystyle-\frac{\kappa\mathcal{L}_{X}X^{\prime}\phi^{\prime
2}}{2XA^{\prime}}-\frac{\kappa\gamma^{\prime}\mathcal{L}_{X}\phi^{\prime
2}}{4\gamma A^{\prime}}-\frac{\kappa^{2}\mathcal{L}_{X}^{2}\phi^{\prime
4}}{8A^{\prime 2}}$ (50) $\displaystyle-$
$\displaystyle\frac{\kappa\phi^{\prime
2}\mathcal{L}_{X}^{\prime}}{2A^{\prime}}-\frac{\gamma^{\prime\prime}}{2\gamma}-\frac{X^{\prime\prime}}{2X}-\frac{\gamma^{\prime}X^{\prime}}{2\gamma
X}+\frac{X^{\prime 2}}{4X^{2}}$ $\displaystyle-$
$\displaystyle\frac{X^{\prime}\mathcal{L}_{X}^{\prime}}{2X\mathcal{L}_{X}}-\frac{\gamma^{\prime}\mathcal{L}_{X}^{\prime}}{\gamma\mathcal{L}_{X}}-\frac{1}{4}\kappa\mathcal{L}_{X}\phi^{\prime
2}-\frac{\mathcal{L}_{X}^{\prime\prime}}{2\mathcal{L}_{X}}.$

To canonicalize the quadratic action, we conduct [47] a field transformation

$\displaystyle G(t,r)\equiv\mathcal{L}_{X}^{1/2}\gamma^{1/4}\delta\phi,$ (51)

as well as a spatial coordinated transformation

$\displaystyle r\to y\equiv\int dr\gamma^{-1/2},$ (52)

after simplification, we obtain

$\displaystyle\delta^{(2)}S=\frac{1}{2}\int
dtdy\left\\{-\partial_{t}^{2}G+\partial_{y}^{2}G-V_{\textrm{eff}}(y)G\right\\}G,$
(53)

where

$\displaystyle V_{\textrm{eff}}(y)\equiv\frac{\partial_{y}^{2}f}{f},\quad
f\equiv\mathcal{L}_{X}^{1/2}\gamma^{1/4}\frac{\partial_{y}\phi}{\partial_{y}A}.$
(54)

Note that in order the definitions in Eqs. (51) and (52) are meaningful, we
demand $\mathcal{L}_{X}>0$ and $\gamma>0$.

Obviously, the equation for the normal mode $G$ is

$\displaystyle-\partial_{t}^{2}G+\partial_{y}^{2}G-V_{\textrm{eff}}(y)G=0.$
(55)

After conducting the mode expansion

$\displaystyle G(t,y)=\sum_{n}e^{i\omega_{n}t}\psi_{n}(y),$ (56)

one immediately sees that $\psi_{n}(y)$ satisfy a Schrödinger-like equation:

$\displaystyle\hat{H}\psi_{n}=-\frac{d^{2}\psi_{n}}{dy^{2}}+V_{\textrm{eff}}(y)\psi_{n}=\omega_{n}^{2}\psi_{n}.$
(57)

This equation is exactly the same one derived in Ref. [47].

One can check that the Hamiltonian operator in Eq. (57) can be factorized as
follows111The definitions for $\hat{\mathcal{A}}$ and
$\hat{\mathcal{A}}^{\dagger}$ here are different from those of Ref. [47]. The
present definitions are consistent with those used in literatures of
supersymmetric quantum mechanics [53].

$\displaystyle\hat{H}=\hat{\mathcal{A}}\hat{\mathcal{A}}^{\dagger},$ (58)

where

$\displaystyle\hat{\mathcal{A}}=\frac{d}{dy}+\frac{\partial_{y}f}{f},\quad\hat{\mathcal{A}}^{\dagger}=-\frac{d}{dy}+\frac{\partial_{y}f}{f}.$
(59)

Usually, the factorization of the Hamiltonian operator indicate the spectrum
is positive-semidefinite [58], and therefore the background kink solution is
stable. The ground state is the zero mode with the following wave function:

$\displaystyle\psi_{0}\propto f.$ (60)

Factorizable Hamiltonian operators are also closely related to the study of
supersymmetric quantum mechanics [52, 53]. For example, the superpotential is
simply

$\displaystyle W(y)=\frac{\partial_{y}f}{f},$ (61)

in terms of which, the effective potential $V_{\textrm{eff}}$ can be written
as

$\displaystyle V_{\textrm{eff}}(y)=W^{2}+\partial_{y}W.$ (62)

Also, one can define a superpartner Hamiltonian for $\hat{H}$:

$\displaystyle\hat{H}_{-}=\hat{\mathcal{A}}^{\dagger}\hat{\mathcal{A}},$ (63)

the corresponding potential is

$\displaystyle V_{-}(y)=f\partial_{y}^{2}(f^{-1})=W^{2}-\partial_{y}W.$ (64)

When both $V_{\textrm{eff}}$ and $V_{-}$ are nonsingular, the spectra of
$\hat{H}$ and $\hat{H}_{-}$ are identical, except their zero modes. But if any
one of the potentials is singular at some points, the supersymmetry might be
broken, and the spectra are at most partially identical [53, 54, 55, 56, 57].
Even worse, bound states with negative eigenvalues might appear in the
spectrum of $\hat{H}_{-}$.

As shown in Refs. [40, 47], the effective potential $V_{\textrm{eff}}$ for 2D
symmetric gravitating kinks are typically divergent at $x=0=y$ (because
$\partial_{y}A=0=\partial_{x}A$ here), and one must be careful with the
stability issue.

##  4 Conclusions

In this work, we noticed the fact that the effective potentials
$V_{\textrm{eff}}$ for symmetric gravitating kinks are typically divergent at
the origin, which means that the supersymmetry between $\hat{H}$ and
$\hat{H}_{-}$ might be explicitly broken. If this is the case, then not just
the symmetry between the spectra of $\hat{H}$ and $\hat{H}_{-}$ are broken,
the spectrum of $\hat{H}_{-}$ are not positive-semidefinite any more. On the
other hand, the study of 5D thick branes has shown that if one starts with the
equation of motion approach, but chooses an improper gauge condition (the
longitude gauge, for instance), he might obtain $\hat{H}_{-}$ rather than
$\hat{H}$. Only by starting with the quadratic action approach, could one
obtain the correct Hamiltonian for the normal modes of perturbation.

For this reason, we reconsidered the stability issue of 2D gravitating kink
solutions of Refs. [40, 47] from the quadratic action approach. Our results
indicate that $\delta\varphi=0$ is a proper gauge condition for the present 2D
gravitating kink model, in the sense that the Schrödinger-like equations
obtained from the equation of motion approach and from the quadratic action
approach are equivalent. We also showed that the present 2D model is so simple
that all the calculations can be conducted with the help of some Mathematica
codes. The result of the present work can also be applied to other static
solutions.

## Declaration of competing interest

The authors declare that they have no known competing finan-cial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Grant numbers 12175169, 11847211, 11605127), the Fundamental Research Funds
for the Central Universities (Grant number xzy012019052).

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