# Knowledge Enhanced Multi-modal Fake News Detection

Yi Han, Amila Silva, Ling Luo, Shanika Karunasekera, Christopher Leckie
yi.han<EMAIL_ADDRESS>ling.luo@, karus<EMAIL_ADDRESS>School of Computing and Information Systems
The University of Melbourne

###### Abstract.

Recent years have witnessed the significant damage caused by various types of
fake news. Although considerable effort has been applied to address this issue
and much progress has been made on detecting fake news, most existing
approaches mainly rely on the textual content and/or social context, while
knowledge-level information—entities extracted from the news content and the
relations between them—is much less explored. Within the limited work on
knowledge-based fake news detection, an external knowledge graph is often
required, which may introduce additional problems: it is quite common for
entities and relations, especially with respect to new concepts, to be missing
in existing knowledge graphs, and both entity prediction and link prediction
are open research questions themselves. Therefore, in this work, we
investigate knowledge-based fake news detection that does not require any
external knowledge graph. Specifically, our contributions include: (1)
transforming the problem of detecting fake news into a subgraph classification
task—entities and relations are extracted from each news item to form a single
knowledge graph, where a news item is represented by a subgraph. Then a graph
neural network (GNN) model is trained to classify each subgraph/news item. (2)
Further improving the performance of this model through a simple but effective
multi-modal technique that combines extracted knowledge, textual content and
social context. Experiments on multiple datasets with thousands of labelled
news items demonstrate that our knowledge-based algorithm outperforms existing
counterpart methods, and its performance can be further boosted by the multi-
modal approach.

fake news detection, knowledge graph, graph neural networks, social media

††copyright: acmlicensed††conference: ; ; ††ccs: Computing methodologies
Supervised learning by classification††ccs: Computing methodologies Neural
networks

##  1\. Introduction

The prevalence of fake news111Here we use the definition in (Zhou and
Zafarani, 2018): fake news is intentionally and verifiably false news
published by a news outlet. on social media has serious repercussions on our
society. Especially when equipped with big data analysis, it can accurately
reach specific target audiences to spread fear, aggravate hatred, and cause
riots and violence.

While significant efforts have been made on fake news detection, most existing
work focuses on the textual content and social context, where textual content
includes the news headline and body, and social context means user
interactions over social media. For example, once a news item is published
online, it may be tweeted by multiple users. Each of these tweets and its
retweets form a separate cascade (Vosoughi et al., 2018), and all the cascades
form the propagation pattern/network (we use these two terms interchangeably
in this work) of a news item.

Knowledge-level information, on the other hand, has been much less explored.
Here knowledge refers to the entities in the news content and the relations
between them. Typically, it can be represented in the format of a triple:
(Subject, Predicate, Object), _i.e.,_ SPO triple (noa, 1999). Within the
limited work on knowledge-based fake news detection, an external knowledge
graph is often required. For example, Cui _et al._ (Cui et al., 2020)
incorporate an article-entity bipartite graph and a medical knowledge graph to
better capture the important entities and guide the embedding of news articles
to detect healthcare misinformation. However, it is unlikely for any knowledge
graph to contain all possible entities and relations, and it is not a trivial
task to accurately predict missing ones. To overcome this issue, in this work,
we investigate knowledge-based fake news detection that does not require any
external knowledge graph.

(a) Knowledge-based fake news detection. Information is extracted from all
news items to form a single knowledge graph, where each news item can be
represented by a subgraph, so that the fake news detection problem becomes a
subgraph classification task. A SubGNN model is then trained to classify each
subgraph/news item.

(b) Multi-modal fake news detection that combines extracted knowledge, text
content and propagation network. The propagation-, text- and knowledge-based
models are pre-trained separately, and their generated embeddings are
concatenated to train a MLP.

Figure 1. Overview of the proposed knowledge-based and multi-modal fake news
detection algorithms.

Pan _et al._ (Pan et al., 2018) approach this problem by constructing two
knowledge graphs, with one on real news ($KG_{T}$) and one on fake news
($KG_{F}$). Each of these two knowledge graphs is then used to train a TransE
(Bordes et al., 2013) model, and a news item is verified by comparing the
average or maximum bias of the two models over the extracted relations, where
the bias of a triple $(s,p,o)$ is defined as $||s+p-o||_{2}^{2}$. A news item
is classified as “Real” if the bias of the model trained on $KG_{T}$ is
smaller, and “Fake” otherwise.

In our work, by contrast, we look at the problem from a different perspective:
given the recent development on graph neural networks (GNNs) and their
superior performance on non-Euclidean data, we investigate whether it is
possible to transform fake news detection into a knowledge graph
classification problem.

Initially, we test a method that constructs a separate knowledge graph for
each news item, and then applies GNNs for graph-level classification. However,
our preliminary experiments suggest that this method does not achieve
satisfactory results, with an average accuracy only around 73%. A key reason
is that normally a very limited number of relations/triples can be extracted
from each news item, which means that the constructed knowledge graph is too
small to accurately verify the veracity of a news item.

Therefore, we propose a new approach and transform the problem of fake news
detection into a subgraph classification task. Specifically, our contributions
include:

  * •

Inspired by the work of (Alsentzer et al., 2020), which designs a GNN model
called SubGNN for subgraph classification, we propose to extract entities and
relations from each news item, all of which form a single knowledge graph
where a news item is represented by a subgraph (Fig. 1(a)), and then all
subgraphs with their corresponding news labels (fake/real) are used to train a
SubGNN model, so that the obtained model can classify each subgraph/news item.
As demonstrated by the experimental results in Section 6, this method can
achieve much better performance.

  * •

To further improve the performance of the above model, we develop a simple but
effective multi-modal fake news detection algorithm. In addition to extracted
knowledge, other forms of information, such as the textual content and
propagation network, can also contribute to detecting fake news. Specifically,
as can be seen from Fig. 1(b), in our proposed method three models are first
trained separately: (1) a propagation-based model (Han et al., 2020) that
verifies a news item purely on its propagation pattern; (2) a document
classification model using hierarchical attention networks (Yang et al., 2016)
that operates directly on the news content; and (3) a knowledge-based model
proposed in this work. We demonstrate that by concatenating the generated
embeddings of these three models to train a multilayer perceptron (MLP), this
seemingly simple approach can outperform each individual model by a clear
margin.

The remainder of this paper is organised as follows: Section 2 defines the
research problem; Sections 3 and 4 present our knowledge-based fake news
detection algorithm which does not require any external knowledge graph;
Section 5 introduces an architecture that combines knowledge-, text- and
propagation-based models to facilitate multi-modal fake news detection;
Section 6 provides experimental results to demonstrate the effectiveness of
our proposed methods; Section 7 reviews previous work related to fake news
detection on social media; and finally Section 8 concludes the paper and
offers directions for future work.

##  2\. Problem Definition

Originally, the fake news detection problem can be defined as: given a set of
labelled news items $\mathcal{D}=\\{\left(W_{i},y_{i}\right)|\ i=1,\ 2,\
...\\}$, where $W_{i}\in\mathcal{W}$ is the textual content for news item $i$
(_i.e.,_ a sequence of words), and $y_{i}\in\mathcal{Y}=\\{0\ \text{(Real)},1\
\text{(Fake)}\\}$ is the label of $W_{i}$, the goal is to learn a mapping
$g:\mathcal{W}\to\mathcal{Y}$ that classifies each news item.

In this work, we take a knowledge-based approach and break down the above
formulation into the following two sub-problems:

Knowledge graph construction. The aim of the first step is to extract
information from each news item to construct a knowledge graph.
Conventionally, this involves named entity recognition and relation
extraction, which have been extensively studied. However, we have tested
several existing named entity-based relation extraction techniques, and our
results suggest that when applied to news items, these methods normally
generate a relatively small number of relations, which may lead to substantial
information loss. Therefore, in the first sub-problem we design a new relation
extraction algorithm that extracts a set of relations/triples
$R_{i}=\\{(e_{ij},r_{ij},e_{ij}^{\prime})|j=1,2,...\\}$ from news content
$W_{i}$, where $e_{ij},r_{ij},e_{ij}^{\prime}$ contain one or multiple words
in $W_{i}$. Each of these triples means that $e_{ij}$ and $e_{ij}^{\prime}$
has the relation of $r_{ij}$. For example, a triple (David Warner, troll,
Virat Kohli) can be extracted from the sentence “David Warner trolls Virat
Kohli on Instagram over his grey beard” (Fig. 2). Note that here we do not
pre-define any relation type or named entity.

Subgraph classification. Once all the relations are extracted from news items,
and a single knowledge graph is constructed (in the case where a part of the
graph is isolated from the rest, we only keep the largest connected
component), each news item is assigned to a subgraph based on its extracted
relations. Therefore, the original fake news detection problem is transformed
into a subgraph classification task formulated as follows: given a set of
labelled sub-knowledge graphs $\\{\left(SG_{i},y_{i}\right)|\ i=1,\ 2,\
...\\}$, where $SG_{i}\in\mathcal{SG}$ represents the sub-knowledge graph that
corresponds to news item $i$, and $y_{i}\in\mathcal{Y}=\\{0\ \text{(Real)},1\
\text{(Fake)}\\}$ is the label of $SG_{i}$, then the goal is to learn a
classifier $f:\mathcal{SG}\to\mathcal{Y}$ that labels each subgraph. Note that
here different subgraphs are not necessarily mutually exclusive and may
contain common nodes.

In the following two sections, we explain in detail our solution to the above
two problems.

##  3\. Knowledge Graph Construction

In order to extract relations from news items to build a knowledge graph, we
have first tested the following options: (1) an OpenNRE (Han et al., 2019)
model trained on the Wiki80
dataset222https://github.com/thunlp/OpenNRE/blob/master/benchmark/download_wiki80.sh
with a BERT encoder, (2) an OpenNRE model trained on the Wiki80 dataset with a
CNN encoder, (3) an OpenNRE model trained on the TACRED
dataset333https://nlp.stanford.edu/projects/tacred/ (Zhang et al., 2017) with
a BERT encoder, (4) an ATLOP (Zhou et al., 2021) model trained on the DocRED
dataset (Yao et al., 2019). The first three models are for sentence-level
relation extraction, while the last one is for document-level relation
extraction—hence it can extract more accurate relations than the other three
models in our case.

Our experimental results suggest that all these models extract a relatively
small number of relations from each news item, which may lead to substantial
information loss. In addition, it is also unlikely for all the extracted
relations to form a single graph. A main reason is that the relation types are
pre-defined in these three datasets: Wiki80/TACRED/DocRED contains 80/41/96
relations, which cannot cover all the scenarios. As a result, in many cases
even though a certain type of relation does exist, “no_relation” is returned
instead.

Input: The textual content $W=\\{w_{0},w_{1},...\\}$ of a news record

Output: The list of relations $L$

1 $L\leftarrow\emptyset$;

2 $W\leftarrow coreference\\_resolution(W)$;

3 $\\{D_{0},D_{1},...\\}\leftarrow dependency\\_parsing(W)$;

4 $\\{p_{0},p_{1},...\\}\leftarrow pos\\_tagging(W)$;

5 $\\{l_{0},l_{1},...\\}\leftarrow lemmatization(W)$;

6

7for  _$i$ in $\\{0,1,2,....,|W|-1\\}$_ do

8 if  _$p_{i}$ is a verb_ then

9 $\\{left\\_nodes,right\\_nodes\\}\leftarrow\\{\emptyset,\emptyset\\}$;

10 $edge\\_type\leftarrow l_{i}$;

11 for  _$(w_{k},r,w_{i})$ in $D_{i}$_ do

12 if  _$l_{k}$ is “not”_ then

13 $edge\\_type\leftarrow``not\\_"+edge\\_type$;

14 else if  _$r$ in $\\{nsubj\\}$_ then

15 $left\\_nodes\leftarrow left\\_nodes\cup\\{l_{k}\\}$;

16 else if  _$r$ in $\\{dobj,iobj,attr,xcomp\\}$_ then

17 $right\\_nodes\leftarrow right\\_nodes\cup\\{l_{k}\\}$;

18

19 for  _$left\\_node$ in $left\\_nodes$_ do

20 for  _$right\\_node$ in $right\\_nodes$_ do

21 $L\leftarrow L\cup{(left\\_node,edge\\_type,right\\_node)}$;

22

23

24

25

$return\text{ }L$

Algorithm 1 Relation Extraction Figure 2. An illustration of Algorithm 1.

One option is to create our own dataset from the collected news items with a
larger number of pre-defined relations, and then retrain an OpenNRE or ATLOP
model. However, in order to significantly increase the number of pre-defined
relations, we also need a considerably larger number of relation instances for
training. Considering that DocRED already has a total number of $1,508,320$
instances, it is unlikely to obtain a much larger quantity from the thousands
of collected news items.

Therefore, we design a new technique that can expand the number of extracted
relations. Specifically, we do not pre-define any relation type, and instead
focus on the verbs in each sentence, since verbs often play an important role
in deciding the veracity of a statement. Algorithm 1 summarises our relation
extraction method. As illustrated in Fig. 2, for a news record,

  * •

The title and the body text are concatenated to extract its textual content
$W=\\{w_{0},w_{1},...\\}$.

  * •

Co-reference resolution444https://spacy.io/universe/project/neuralcoref is
performed to replace all the mentions in $W$ that refer to the same real-world
entity with a single token (Line 2). In the given example, “Kohli” is replaced
by “Virat Kohli”.

  * •

The grammatical structure of each sentence in $W$ is extracted using the
dependency parser available in Spacy555https://spacy.io/usage/linguistic-
features#dependency-parse (Line 3). This step returns a set of tuples $D_{i}$
for each word $w_{i}$. Each entry in $D_{i}$ is a tuple $(w_{k},r,w_{i})$,
where $w_{k}$ is another word in the same sentence of $w_{i}$ that is related
to word $w_{i}$ from the relation type $r$, _e.g._ , nominal subject
($nsubj$), direct object ($dobj$), open clausal complement ($xcomp$), as shown
in the third box in Fig. 2. Note that the relation here has different meaning
from the relation between entities mentioned above in the problem definition.

  * •

The Part-of-Speech (POS) tags $\\{p_{0},p_{1},...\\}$ and the base-forms
$\\{l_{0},l_{1},...\\}$ of the words in $W$ are recovered using the POS
tagger666https://spacy.io/usage/linguistic-features#pos-tagging and the
lemmatizer777https://spacy.io/usage/linguistic-features#lemmatization in Spacy
(Line 4-5).

  * •

The verbs in $W$ are identified by looping through POS tags of the words in
$W$ (Line 6-7).

  * •

For each identified verb $w_{i}$, the connected words in the dependency parse
tree $D_{i}$ are analysed. If a negation, _i.e.,_ words that reverse the
meaning of a word, is found to be attached to $w_{i}$, $w_{i}$ is updated as
“not_” $+\ w_{i}$, _e.g._ , “care” becomes “not_care” in the given example
(Line 11-12).

  * •

Then, the connected nodes are categorised as either $left\\_\allowbreak nodes$
or $right\\_nodes$ based on their relation to $w_{i}$. If the relation type of
a connected word is nominal subject ($nsubj$), it is added to $left\\_nodes$
(Line 13-14).

  * •

Otherwise, if the relation to $w_{i}$ belongs to one of the following types:
direct object ($dobj$), indirect object ($iobj$), attribute ($attr$), and open
clausal complement ($xcomp$), it is added to $right\\_nodes$ (Line 15-16). In
the example of Fig. 2, for the verb “troll”, “David Warner”/“Virat Kohli” is
categorised as $left\\_nodes$/$right\\_nodes$, respectively. In addition, even
though “respond” is a verb in the second sentence, it will be extracted as a
possible object of “not_care” by our algorithm due to the xcomp connection
between them.

  * •

In the end, the relation set is constructed by connecting each item in
$left\\_nodes$ with each item in $right\\_nodes$ using the corresponding verb,
_i.e.,_ $w_{i}$ in the above example (Line 17-19).

Our experimental results show that by applying the above approach,
substantially more (over 10 times) number of relations can be extracted from
the news items in total. For example, as shown in Fig. 3, in the left text box
is the snippet of a news
item888https://www.politifact.com/factchecks/2010/feb/04/paul-krugman/krugman-
calls-senate-health-care-bill-similar-law-/, and the relations extracted by
our method are listed in the middle text box. As a comparison, the relations
extracted by the ATLOP model are given on the right.

Figure 3. An example that compares the extracted relations using Algorithm 1
and by an ATLOP model. Left text box: the snippet of a news item. Middle text
box: relations extracted by Algorithm 1. Right text box: relations extracted
by an ATLOP model trained on the dataset of DocRED.

We have compared the cases where the knowledge graph is built using the
relations extracted by (1) Algorithm 1, (2) an ATLOP model trained on the
DocRED dataset (since DocRED is especially collected for document-level
relation extraction), (3) both (1) and (2). Our experiment suggests that the
best result is achieved when only the relations extracted by Algorithm 1 are
used.

Note that we are not proposing a general method for relation extraction, since
it is unlikely for the defined rules to work for all cases, and Algorithm 1 is
only for our knowledge-based fake news detection technique.

After all the relations are extracted from news items, we add them one by one
to form a single knowledge graph. Then each news item can be represented by a
subgraph, based on the relations extracted from it. Therefore, the original
problem of detecting fake news now transforms into classifying each of the
subgraphs, as explained in the next section.

##  4\. Subgraph Classification

In this section, we first give a brief introduction to graph neural networks
and SubGNN, and then explain how a sub-knowledge graph is classified in our
case.

###  4.1. Background on Graph Neural Networks

Given a graph $\mathcal{G}=\allowbreak\mathcal{(V,E)}$, with vertex/node set
$\mathcal{V}$, edge set $\mathcal{E}$, and node feature set
$\mathcal{X}\in\mathbb{R}^{|\mathcal{V}|\times d}$ (_i.e.,_ each node has $d$
features), many GNN models can be formulated as a message passing framework
(Battaglia et al., 2018; Wu et al., 2019) in which information is propagated
from one node to another along edges.

During each message-passing iteration $k$, the embedding for node
$v\in\mathcal{V}$ is updated according to the information aggregated from
$v$’s neighbourhood $\mathcal{N}(v)$, which can be expressed as follows:

$h_{v}^{(k+1)}=UPDATE\left(h_{v}^{(k)},\ AGGREGATE\left(\left\\{h_{u}^{(k)}\
|\ u\in\mathcal{N}(v)\right\\}\right)\right)$,
where $h_{v}^{0}=x_{v}\in\mathcal{X}$. The update step is often omitted by
adding self-loops to the input graph, and the node embedding becomes:

$h_{v}^{(k+1)}=AGGREGATE\left(\left\\{h_{u}^{(k)}\ |\
u\in\mathcal{N}(v)\cup\left\\{v\right\\}\right\\}\right)$.

Take a basic GNN for example (Hamilton, 2020), the model can be defined as:
$H^{(k+1)}=\sigma\left(\left(A+I\right)H^{(k)}M^{(k+1)}\right)$, where (1) $H$
is the matrix of node embeddings/representations; (2) $\sigma$ is a non-linear
activation function, _e.g._ , the rectified linear unit (ReLU) function; (3)
$A\in\\{0,1\\}^{|\mathcal{V}|\times|\mathcal{V}|}$ is the adjacency matrix:
$A_{i,j}=1$ if there is an edge from node $i$ to node $j$, and $A_{i,j}=0$
otherwise; and (4) $M$ is the weight matrix to be learned.

###  4.2. Background on SubGNN

Most existing work on GNN focuses on node- and graph-level prediction tasks,
while subgraphs are much less studied. To bridge this research gap, Alsentzer
_et al._ (Alsentzer et al., 2020) propose SubGNN that learns an embedding
function $\mathcal{F}:\mathcal{SG}\to\mathbb{R}^{d_{S}}$ to map each subgraph
into a lower-dimensional representation.

Specifically, the embedding function $\mathcal{F}$ captures features from
three channels—position, neighbourhood and structure, each of which has two
subchannels—internal and border:

  * •

Position. The internal position of subgraph $SG_{i}$ is defined as the
distance between $SG_{i}$’s components—$SG_{i}$ may contain a single connected
component or multiple isolated components. The border position is defined as
the distance between $SG_{i}$ and rest of $\mathcal{G}$.

  * •

Neighborhood. The internal neighborhood and border neighborhood are defined as
the identity of $SG_{i}$’s internal nodes and border nodes, respectively,
where border nodes refer to nodes within the $k$-hop neighborhood of internal
nodes.

  * •

Structure. The internal structure is defined as the internal connectivity of
$SG_{i}$, while the border structure is defined by edges connecting $SG_{i}$’s
internal nodes to the border neighborhood.

###  4.3. Classify Sub-Knowledge Graphs

Each of the above channels can be used separately or together. We have tested
multiple SubGNN models with different combinations of the three channels, and
find that models that rely only on the structure channel gives the best
result. This is because in our case, the internal and border structures,
_i.e.,_ how nodes are connected with each other within a subgraph and with the
rest of the graph, are more informative for determining whether a news item is
fake or not. Interestingly, the neighborhood channel does not perform as well
as the other two, but this is consistent with the ablation study in the
original work (Alsentzer et al., 2020).

In order to facilitate subgraph-level message passing for the structure
channel, a number of connected components (each subgraph may contain single or
multiple components) are randomly sampled via triangular random walks (Boldi
and Rosa, 2012)—they are called anchor patches
$\mathcal{A}=\\{A_{i}|i=1,2,...,n_{A}\\}$. The embedding of a subgraph
component ($SC$) is then represented in the following form:
$h^{(k+1)}_{SC}=\sigma\left(M^{(k+1)}\cdot\left[h^{(k)}_{SC};AGG\left(\left\\{\gamma\left(SC,A_{i}\right)\cdot
a_{i}|i=1,...,n_{A}\right\\}\right)\right]\right)$,
where $\gamma$ is a pre-defined similarity function, $a_{i}$ is the learned
representation of $A_{i}$, $AGG$ is an aggregation function, _e.g._ , the sum
operator, and $M$ is a layer-wise trainable weight matrix.

If a subgraph contains multiple isolated connected components, its embedding
is generated by concatenating the embeddings of all components.

##  5\. Knowledge Enhanced Multi-modal Fake News Detection

In addition to the extracted knowledge, the textual content itself and how the
news item propagates through the social network also provide valuable
information for detecting fake news. Therefore, in this section we study how
to combine our knowledge-based approach with existing content- and
propagation-based methods for more accurate detection.

Formally, given a set of labelled news items $\\{(SG_{i},W_{i},P_{i},y_{i})\
|\ i=1,\ 2,\ ...\\}$, where $SG_{i}\in\mathcal{SG}$ is the sub-knowledge
graph, $W_{i}\in\mathcal{W}$ is the textual content, $P_{i}\in\mathcal{P}$ is
the propagation network, and $y_{i}\in\mathcal{Y}=\\{0\ \text{(Real)},1\
\text{(Fake)}\\}$ is the label of the corresponding news, the goal is to learn
a classifier $f:\mathcal{SG}\times\mathcal{W}\times\mathcal{P}\to\mathcal{Y}$
that can label each news item.

As shown in Fig. 1(b), we first train three knowledge-, text- and propagation-
based models separately on the same training dataset. Then for each training
instance, we feed its $SG_{i},W_{i},\text{and}\ P_{i}$ into the obtained
models to generate three separate embeddings $h_{i_{K}},\
h_{i_{T}},\allowbreak h_{i_{P}}$, all of which are concatenated to form the
final embedding $h_{i}=h_{i_{K}}\oplus h_{i_{T}}\oplus h_{i_{P}}$. In the end,
an MLP is trained on the set of embeddings $\\{h{{}_{i}}|i=1,2,...\\}$.

Our experimental results in Section 6 demonstrate that this seemingly simple
method outperforms each individual model by a clear margin. Specifically, in
this work we choose the following two as the text- and propagation-based
models:

  * •

A document classification model using hierarchical attention networks (Yang et
al., 2016). The overall architecture of this model consists of four
components: (1) a word encoder where words are embedded with a GRU-based
sequence encoder; (2) a word-level attention layer where the importance of a
word is measured by its similarity with a word-level context vector, which is
jointly learned during the training process; (3) a sentence encoder that is
also based on bidirectional GRU; and (4) a sentence-level attention layer that
calculate the weight of a sentence in a similar way to (2). The reason why we
choose this model is that previous work has shown that it performs better than
other text-based models for fake news detection (Shu et al., 2019a), such as
LIWC (Pennebaker et al., 2015), text-CNN (Kim, 2014).

  * •

A propagation-based algorithm (Han et al., 2020) that applies the GNN model of
DiffPool (Ying et al., 2018) (built on top of GraphSage (Hamilton et al.,
2017)) to verify a news item purely on its propagation pattern (as explained
in the introduction) and the features of the corresponding Twitter users,
including (1) whether the user is verified, (2) the timestamp when the user
was created, (3) the number of followers, (4) the number of friends, (5) the
number of lists, (6) the number of favourites; (7) the number of statuses, (8)
the timestamp of the tweet. The adjacency matrix corresponding to the
propagation pattern of a news item and the node feature matrix are fed as
input for graph-level classification. We choose this model due to its
simplicity and efficiency.

Note that (1) although the experimental results in the next section suggest
that our knowledge-based model works well together with the above two models,
especially the propagation-based model, they can be replaced by other options
too; (2) in addition to text content and propagation network, there are other
useful forms of information as well, including user replies, images and
external knowledge graphs, especially domain-specific knowledge graphs. We
leave more sophisticated fusion techniques for future work.

##  6\. Experimental Verification

In this section, we empirically verify the effectiveness of our proposed
knowledge-based and multi-modal fake news detection algorithms over two
datasets with thousands of labelled news items.

###  6.1. Datasets

While there are a number of public datasets on fake news detection covering
different domains, we choose the dataset of FakeNewsNet (Shu et al., 2018) in
our work. FakeNewsNet contains labelled news from two websites:
politifact.com999https://www.politifact.com/ and
gossipcop.com101010https://www.gossipcop.com/—we call them PolitiFact and
GossipCop hereafter. For each news item, the dataset provides both linguistic
and visual information, all the tweets and retweets, as well as the
information of the corresponding Twitter users. For more details please refer
to (Shu et al., 2018).

The reasons why we choose PolitiFact and GossipCop over other options include:
(1) they align with the our definition of fake news—fake news is intentionally
and verifiably false news published by a news outlet. Some public datasets on
fake news detection, _e.g._ , Twitter16 (Ma et al., 2017), Weibo (Ma et al.,
2016), Pheme (Kochkina et al., 2017) are intended for detecting rumours,
satires, misinformation, etc. (2) They provide accurate ground truth labels,
which are collected using fact-checking websites. As a result, they are more
accurate than those datasets where news items are weakly labelled by applying
distant supervision techniques. For example, the datasets of ReCOVery (Zhou et
al., 2020b) and CoAID (Cui and Lee, 2020) label news records based on the
reliability of the source. (3) They provide social context data, which is
missing in datasets such as BuzzFeedNews (Santia and Williams, 2018), Ma-
Twitter (Ma et al., 2016) and LIAR (Wang, 2017). This type of information is
required by the chosen propagation-based approach to construct the propagation
network for each news item.

Table 1. Sstatistics of the PolitiFact and GossipCop datasets. Dataset | No. of Fake News | No. of Real News
---|---|---
PolitiFact | 185 | 225
GossipCop | 4942 | 2520

The statistics of the dataset are listed in Table 1. Note that the values are
smaller than those reported in (Shu et al., 2018) because (1) many news items,
especially fake news, have already been removed, (2) some tweets and retweets
of a news items are no longer retrievable, and we cannot build the
corresponding propagation pattern. Therefore, those news items are also
excluded from our experiments.

Table 2. Performance comparison of fake news detection on the dataset of
PolitiFact

K1 |  T1 | P1 | Accuracy | Precision | Recall | F1
---|---|---|---|---|---|---
✓ |  |  | 0.837 | 0.844 | 0.801 | 0.836
| ✓ |  | 0.829 | 0.838 | 0.824 | 0.826
|  | ✓ | 0.854 | 0.855 | 0.852 | 0.852
✓ | ✓ |  | 0.858 | 0.853 | 0.842 | 0.857
| ✓ | ✓ | 0.884 | 0.922 | 0.825 | 0.883
✓ |  | ✓ | 0.876 | 0.862 | 0.877 | 0.876
✓ | ✓ | ✓ | 0.919 | 0.913 | 0.913 | 0.919
Avg bias (Pan et al., 2018) | 0.752 | 0.725 | 0.776 | 0.729
Max bias (Pan et al., 2018) | 0.662 | 0.617 | 0.638 | 0.618

  * 1

K: knowledge-based, T: text-based (Yang et al., 2016), P: propagation-based
(Han et al., 2020)

Table 3. Performance comparison of fake news detection on the dataset of
GossipCop

K1 |  T1 | P1 | Accuracy | Precision | Recall | F1
---|---|---|---|---|---|---
✓ |  |  | 0.783 | 0.813 | 0.870 | 0.750
| ✓ |  | 0.788 | 0.767 | 0.747 | 0.754
|  | ✓ | 0.872 | 0.855 | 0.860 | 0.858
✓ | ✓ |  | 0.800 | 0.823 | 0.888 | 0.768
| ✓ | ✓ | 0.861 | 0.882 | 0.910 | 0.843
✓ |  | ✓ | 0.879 | 0.916 | 0.899 | 0.866
✓ | ✓ | ✓ | 0.871 | 0.903 | 0.900 | 0.856
Avg bias (Pan et al., 2018) | 0.749 | 0.646 | 0.621 | 0.629
Max bias (Pan et al., 2018) | 0.723 | 0.625 | 0.627 | 0.626

  * 1

K: knowledge-based, T: text-based (Yang et al., 2016), P: propagation-based
(Han et al., 2020)

###  6.2. Baselines

Three baselines are considered: in addition to the two models chosen in
Section 5, _i.e.,_ test-based (Yang et al., 2016) and propagation-based (Han
et al., 2020), we also implement the knowledge-based method (Pan et al., 2018)
described in the introduction.

In order to compare these models, the datasets are split as follows: 70% of
the data are used for training, 15% are for validation, and the remaining 15%
are for test. In addition, all models are evaluated with the following
commonly used metrics: accuracy, precision, recall and F1 score.

Model hyper-parameters. (1) For the text-based model, both the word embedding
dimension and the GRU dimension are set to 100, the learning rate is 0.003,
and the batch size is 32 for PolitiFact and 128 for GossipCop. (2) For the
propagation-based model, the hyper-parameters for the DiffPool algorithm are:
2 pooling layers, 64 hidden dimensions and 64 embedding dimensions. In
addition, the learning rate is 0.001, and the batch size is 128. (3) For our
knowledge-based model, the hyper-parameters for SubGNN are selected from the
following ranges: batch size $\in[64,128]$, learning rate
$\in[3e{\text{-}}5,\allowbreak 1e{\text{-}}3]$, number of layers $\in[1,4]$,
number of structure anchor patches $|A_{S}|\in[15,45]$, and feed forward
hidden dimension sizes $\in[32,64]$ with dropout $\in[0.0,0.4]$. (4) For our
multi-modal approach, number of feed forward layers $\in[2,4]$ with hidden
dimension sizes $\in[8,64]$ and dropout $\in[0.0,0.2]$. (5) For the baseline
of (Pan et al., 2018), the embedding dimensions of $KG_{F}$ and $KG_{T}$ are
in the range of $[30,50]$ for PolitiFact, and $[50,100]$ for GossipCop.

###  6.3. Performance Comparison of Fake News Detection

Tables 2 and 3 show the performance comparison on the two datasets of
PolitiFact and GossipCop (results in bold correspond to the best values or
values less than $0.01$ below the best values). The results suggest that:

  * •

Our knowledge-based model outperforms the baseline of (Pan et al., 2018) on
both datasets.

  * •

In terms of each single model of $K,\ T\ \text{and}\ P$, the difference among
their performances on the dataset of PolitiFact is not significant, but the
propagation-based model achieves the best performance on the dataset of
GossipCop, while the other two perform similarly again.

  * •

In terms of the multi-modal approach, the performance of any combination of
two models is almost always better than each individual model.

  * •

If we compare the different combinations, $K+T<T+P\approx K+P\leq K+T+P$: (1)
since the knowledge- and text-based models rely on the same source of
information—the news content, the combination of these two performs the worst.
(2) When the knowledge- or text-based model is combined with the propagation-
based model, they perform similarly and both better than (1). (3) The
combination of all three models performs the best overall.

The above results suggest that while our knowledge-based approach that does
not require any external knowledge graph is effective, combining it with other
sources of information can further boost its performance.

###  6.4. Time sensitivity analysis

In addition to the metrics of accuracy, precision, recall and F1 score, it is
important to understand how the performance of a model changes over time,
since in real cases a deployed fake news detection system is likely to face
highly dynamic and volatile data.

(a) Accuracy

(b) Precision

(c) Recall

(d) F1

Figure 4. Time sensitivity analysis for the dataset of GossipCop. The x-axis
is the number of weeks from the test item to the last training item.

We run time sensitivity analysis over a period of 26 weeks (half a year) on
the dataset of GossipCop (the dataset of PolitiFact contains only around 400
news items spread over a number of months, which are insufficient to run the
analysis): all news items are sorted by their timestamp, and the first 70% are
used for training while the last 30% are for testing. During test time the
data are divided into separate groups based on the number of weeks from the
test news item to the last training item, then the four metrics are calculated
over each group, _i.e.,_ the model performance is measured weekly. Fig. 4
shows how different models perform over time.

We can see that (1) the multi-modal approach performs the best in most cases.
(2) In the first few weeks, the performance drop of the text-based and the
knowledge-based approaches is more obvious than that of the multi-modal and
the propagation-based approaches. (3) All four models are relatively stable
between Week 5 and Week 25. The results further confirm the effectiveness of
the multi-modal approach that combines extracted knowledge, text content and
propagation network.

##  7\. Related Work

In this section, we provide a brief review of the related work on fake news
detection, which has become a popular research problem over recent years.
Specifically, following a similar taxonomy in (Shu et al., 2017; Pierri and
Ceri, 2019), we classify existing work into three categories: content-based
approaches, context-based approaches and mixed approaches, which mainly rely
on news content, social context, and a mix of both for detection,
respectively. In addition, we also summarise prior work on fake news early
detection and explainability.

###  7.1. Content-based Approaches

The most straightforward content-based approach is to consider fake news
detection as a text classification problem, and apply techniques such as RST
(Rubin et al., 2015), LIWC (Pennebaker et al., 2015) and text-CNN (Kim, 2014)
to identify fake news. These algorithms often serve as baselines.

In addition to the knowledge-based detection methods discussed in this paper,
another line of research studies fake news from a style-based perspective.

####  7.1.1. Style-based Detection

Since the purpose of fake news is to mislead the public, they should exhibit
unique writing styles that are rarely seen in real news. This is supported by
forensic psychological studies (Undeutsch, 1967), which have shown that
statements based on factual experiences differ from those derived from
fabrication or fiction in both content and quality.

Therefore, style-based methods aim to identify the different content style,
which can be represented by quantifiable features, including attribute-based
language features or structure-based language features. For example, Perez-
Rosas _et al._ (Pérez-Rosas et al., 2018) train linear SVMs on the following
linguistic features to detect fake news: unigrams, bigrams, punctuation,
psycholinguistic, readability and syntax features. Other methods that fall
into this category include (Horne and Adali, 2017; Volkova et al., 2017; Wang,
2017; Potthast et al., 2018).

In addition to textual information, images posted in social media have also
been investigated to facilitate the detection of fake news (Jin et al., 2017;
Yang et al., 2018; Wang et al., 2018; Zhou et al., 2020c).

###  7.2. Context-based Approaches

Social context here refers to the interactions between users, including tweet,
retweet, reply, mention and follow. As mentioned in the introduction, these
engagements can form the propagation pattern for a news item, and a number of
studies have used various types of models to identify the difference in the
propagation pattern between real and fake news: Wu _et al._ (Wu et al., 2015)
use a hybrid SVM; Ma _et al._ (Ma et al., 2017) use Propagation Tree Kernel;
Wu _et al._ (Wu and Liu, 2018) incorporate LSTM cells into the RNN model; Liu
_et al._ (Liu and Wu, 2018) use both RNNs and CNNs; Shu _et al._ (Shu et al.,
2019b) and Zhou _et al._ (Zhou and Zafarani, 2019) propose different types of
features and compare multiple commonly used machine learning models; Monti _et
al._ (Monti et al., 2019), Lu _et al._ (Lu and Li, 2020) and Bian _et al._
(Bian et al., 2020) apply GNNs to study propagation patterns.

In addition to the propagation network, other types of graphs can also be
built from social context. For example, Jin _et al._ (Jin et al., 2016) build
a stance network where the weight of an edge represents how much each pair of
posts support or deny each other. Then the credibility of all the posts
related to a news item is estimated to decide whether the news is fake or
real, which can be formalised as a graph optimisation problem.

In another example, Tacchini _et al._ (Tacchini et al., 2017) propose to
detect fake news based on users who liked them on Facebook. They tested
logistic regression-based and harmonic Boolean label crowdsourcing-based
methods, and their results suggest that both methods can achieve high
accuracy.

While all the above methods are supervised, an unsupervised approach is
proposed by Yang _et al._ (Yang et al., 2019), which builds a Bayesian
probability graphical model to capture the generative process among the
validity of news, user opinions and user credibility.

###  7.3. Mixed Approaches

Since both news content and social context can provide valuable evidence,
mixed approaches use these two sources of information to differentiate between
fake news and real news.

Ruchansky _et al._ (Ruchansky et al., 2017) design a three-module architecture
that combines the text of a news article, the received user response and the
source of the news: (1) in order to capture temporal representations of
articles, the first module trains a RNN that takes the user response, news
content and user features as input; (2) the second module generates for each
user a score and a low-dimensional representation based on user features; (3)
the third module takes the output of the first two modules and trains a neural
network to label the news item.

A pre-extracted word set is used in (Zhang et al., 2018) to construct explicit
features from the news content, user profile and news subject description.
Meanwhile, RNNs are applied to learn latent features, such as news article
content information inconsistency and profile latent patterns. Once the
features are obtained, a deep diffusive network is built to learn the
representations of news articles, creators and subjects.

Shu _et al._ (Shu et al., 2019c) propose to use the tri-relationship among
publishers, news articles and users to detect fake news. Specifically, the
latent representations for news content and users are learned with non-
negative matrix factorization, and the problem is formalised as an
optimisation over the linear combination of each relation. They compare a
number of machine learning algorithms to solve the optimisation problem in
their experiments.

###  7.4. Explainability

In addition to the above work, a few recent papers have started to work on
explainability, which provides evidence to support why their model labels
certain news items as fake/real (Popat et al., 2018; Shu et al., 2019a; Lu and
Li, 2020).

For example, Lu _et al._ (Lu and Li, 2020) propose a novel attention mechanism
for fake news detection which jointly considers news textual content, retweet
sequences in propagation networks, and user co-occurrence networks.

###  7.5. Fake News Early Detection

Considering that it is difficult to correct people’s perception towards an
issue, even if the previous impression is inaccurate (keersmaecker and Roets,
2017), it is more crucial to detect fake news at an early stage before it
becomes widespread. Therefore, another line of research works on early
detection of fake news (Liu and Wu, 2018; Shu et al., 2020; Zhou et al.,
2020a). Shu _et al._ (Shu et al., 2020) study multiple weak signals of user
sentiment, bias and credibility, and then combine weakly labelled data with a
small amount of manually labelled data to train a fake news detection model.

##  8\. Conclusions and Future Work

A series of incidents over recent years have demonstrated the significant
damage that fake news can cause to society. In this work, we investigate
knowledge-enhanced multi-modal techniques for fake news detection.
Specifically, we transform the problem of detecting fake news into a subgraph
classification task, and design a knowledge-based algorithm that does not
require any external knowledge graph. In addition, we propose a multi-modal
detection algorithm that combines extracted knowledge, textual content and
social context. Experimental results on two datasets with thousands of
labelled news items demonstrate the effectiveness of our approaches.

For future work, we will further explore more sophisticated methods rather
than simple concatenation to combine different sources of information.
Moreover, in addition to news content and propagation networks, we intend to
exploit other modalities as well, including images and external knowledge
graphs.

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11institutetext:  Centre of Advanced Studies, Department of Physics, The
University of Burdwan,
Burdwan 713 104, India

String and brane phenomenology Quantum cosmology Particle-theory and field-
theory models of the early Universe (including cosmic pancakes, cosmic
strings, chaotic phenomena, inflationary universe, etc.)

# Single field slow-roll effective potential from Kähler moduli stabilizations
in type IIB/F-theory

Abhijit Let  Arunoday Sarkar  Chitrak Sarkar  Buddhadeb Ghosh

###### Abstract

We derive a single field slow-roll inflaton potential in three intersecting
$D7$ branes configuration under type IIB/F-theory compactification. Among
three resulting Kähler moduli corresponding to three orthogonal directions,
two are stabilized via perturbative corrections in Kähler potential arising
from large volume scenario ($\alpha^{\prime 3}$) and four graviton scattering
amplitude upto one loop level and the remaining Kähler modulus is stabilized
by KKLT-type non-perturbative correction in superpotential. The symmetric
combination of two canonically normalized and perturbatively stabilized Kähler
moduli gives the inflaton field and the anti-symmetric combination manifests
as an auxiliary field.

###### pacs:

11.25.Wx

###### pacs:

98.80.Qc

###### pacs:

98.80.Cq

In recent years, major efforts have been made [1, 2, 3, 4, 5, 6] to obtain
cosmological inflation and inflaton potential(s) via Kähler moduli
stabilization in type-IIB/F-theory since the inflationary picture in large
scale limit is experimentally connected to the CMBR anisotropy and
polarisation data [7, 8, 9]. In fact, Planck-2018 [7] has confirmed the
efficacy of the single field slow-roll type potentials e.g. the
$\alpha$-attractors in explaining the observational bounds with significant
precision. Quite obviously, moduli stabilization is essential in order to
connect the string compactification to low energy effective theories, such as
inflation. Therefore, obtaining a plateau-type inflaton potential via moduli
stabilization is an important task in the interface of string theory and
cosmology. We have already shown in an earlier publication [10] that the mode-
dependent behaviours of cosmological parameters of $\alpha$-attractor
potentials conform to the Planck-2018 data to a great extent. This class of
potentials originates from the geometry of Kähler manifold in $\mathcal{N}=1$
minimal supergravity. Motivated by this success, we felt it is pertinent to
extract such type of experimentally favoured potential from more fundamental
theory viz., the superstring theory through stabilizing the Kähler moduli
fields by quantum corrections in the topology of the internal compact manifold
which is a Calabi-Yau threefold. In the string frame, one is concerned with
two potentials: i) the Kähler potential [11] $\mathcal{K}$ which generates the
metric of the moduli space of the internal manifold, ii) the superpotential
[12] $\mathcal{W}$ which is generated by world volume fluxes of branes.
Usually, $\mathcal{K}$ and $\mathcal{W}$ get contributions from perturbative
[5, 11, 13, 14, 1, 15, 16] and non-perturbative [5, 17, 18, 19, 20] effects,
respectively. In the process of compactification in string theory, many moduli
fields (massless scalars in four dimension) appear which are related to
$\mathcal{K}$ and $\mathcal{W}$. The number of moduli fields may be reduced by
fluxes [21, 22], $D$ brane compactification [23] and orientifold projection
[24]. A single-field inflation is driven by a potential $V(\phi)$, where
$\phi$ is the inflaton field. In order to connect string theory to inflation
one has to i) derive $V(\phi)$ from a potential $V(\mathcal{K},\mathcal{W})$,
which is called the $F$-term potential, and ii) make a transition from $AdS$
space to $dS$ space, the latter having a positive cosmological constant, which
is required for inflationary expansion of the universe. Moreover, the
potential $V(\phi)$ has to be a slow-roll one. The main motivation of the
present work is to obtain a slow-roll potential, $V(\phi)$, incorporating the
perturbative ($\alpha^{\prime 3}$, four-graviton scattering upto genus one and
3 intersecting D7 branes wrapping over 4-cycles), non-perturbative (one
instanton/gaugino condensation) corrections in $V(\mathcal{K},\mathcal{W})$.
Obtaining a stable $dS$ vacuum from superstring theory is a challenging task
because of the recently proposed swampland conjecture [25, 26] in the context
of quantum gravity. But still, efforts have been made to find an effective
potential from the stringy perspective, which includes both perturbative[11,
14, 15] and non-perturbative[18, 19, 20] elements in the topology of
compactified extra dimensions of space-time. Another aspect of this scenario
is to uplift the single field effective potential from the $AdS$ to the $dS$
space maintaining a slow-roll plateau. In this paper we have proposed a scheme
for deriving a slow-roll $dS$-potential for the inflaton field by stabilizing
all the Kähler moduli and suitably uplifting the $AdS$ minimum to the $dS$
one. Our calculational framework is based on the type IIB superstring theory
compactified on a 6d $T^{6}/\mathbb{Z}_{N}$ orbifold limit of Calabi-Yau
3-fold (CY3)111or an elliptically fibered Calabi-Yau 4-fold in $F$-theory
[27], which will be designated as $\mathcal{X}_{6}$ such that the target space
is $\mathcal{M}_{4}\times\mathcal{X}_{6}$, where $\mathcal{M}_{4}$ is the 4d
Minkowski space. The internal space $\mathcal{X}_{6}$ is equipped with some
non-perturbative objects like $D7$ branes and $O7$ planes. These $O$-planes
are necessary to project out the 12d $F$-theory in 10d type IIB theory [28,
29]. Furthermore, the simplest configuration of three magnetised non-
interacting and intersecting $D7$ branes is considered which wrap around the
4-cycles and warp the metric topology of $\mathcal{X}_{6}$ such that
$h^{1,1}=3$. In this set-up, the complexified Kähler moduli which control the
volume of $\mathcal{X}_{6}$ are expressed as,

$\small\rho_{k}=b_{k}+i\tau_{k},\quad k=1,2,3.$ (1)

$b_{k}$ is connected with RR $C_{4}$ potential and $\tau_{k}$ is identified as
4-cycle volume transverse to the 7-branes. The internal volume of
$\mathcal{X}_{6}$ can be expressed in terms of $\tau_{1,2,3}$ as,

$\small\mathcal{V}=\sqrt{\tau_{1}\tau_{2}\tau_{3}}=\sqrt{\frac{(\rho_{1}-\bar{\rho_{1}})(\rho_{2}-\bar{\rho_{2}})(\rho_{3}-\bar{\rho_{3}})}{(2i)^{3}}},$
(2)

where we have assumed that the intersection number of the branes is one.
Branes, wrapping the $\mathcal{X}_{6}$, produce generalised fluxes [23, 21]
threading the 4-cycles of the internal manifold arising due to some potentials
$C_{p}$ and the associated form fields $F_{p}=dC_{p-1}$, $p=0,2,4$,
complexified axion-dilaton $S=C_{0}+ie^{-\varphi}$, where $\varphi$ is the
dilaton field which is related to the string coupling constant as
$\langle\varphi\rangle=\ln{g_{s}}$; $F_{3}=dC_{2}$, the Kalb-Ramond field
$B_{2}$, $H_{3}=dB_{2}$ and $G_{3}=F_{3}-SH_{3}$. These fields depend on
$h^{2,1}$ complex structure moduli $z_{a}$ which dictate the shape of
$\mathcal{X}_{6}$. The CY3 being a compact Kähler manifold (which means it is
an orientable Riemann surface having finite volume) with vanishing first Chern
class and Ricci flatness [30] admits a non-zero closed holomorphic (3,0) form
$\Omega(z_{a})$ [31] (i.e. $d\Omega(z_{a})=0$) everywhere, which is a non-
trivial element of Hodge-de Rham cohomology group $H^{3,0}$. The 3-form field
$G_{3}$ and $\Omega(z_{a})$ together define the compatibility of CY3 with
supersymmetry [30, 21] through a flux-generated tree level superpotential
[12],

$\small\mathcal{W}_{0}(S,z_{a})=\int_{\mathcal{X}_{6}}G_{3}(S,z_{a})\wedge\Omega(z_{a}),$
(3)

which is a holomorphic function of $z_{a}$ and $S$. Such type of potential is
also described in the $\mathcal{N}=1$ supergravity [32] except that, in that
case it will be a function of superfields. The supersymmetric constraints
require that $z_{a}$ and $S$ should be supersymmetrically stabilized acquiring
large masses at supersymmetric minimum. Therefore, the covariant derivative of
$\mathcal{W}_{0}$ w.r.t. $z_{a}$ and $S$ must vanish, i.e.

$\small\mathcal{D}_{S}\mathcal{W}_{0}=\mathcal{D}_{z_{a}}\mathcal{W}_{0}=0$
(4)

where,
$\mathcal{D}\mathcal{W}_{0}=\partial\mathcal{W}_{0}+\mathcal{W}_{0}\partial\mathcal{K}$.
$\partial\mathcal{K}$ is the connection on the moduli space of CY3. This
stabilization ensures that those moduli fields can never appear in large four
dimensions. Now, so far as the Kähler structure (which is the complex
structure with a Riemannian metric) of $\mathcal{X}_{6}$ is concerned, the
metric of the moduli space
$\mathcal{M}(\mathcal{X}_{6})=\mathcal{M}^{2,1}(\mathcal{X}_{6})\times\mathcal{M}^{1,1}(\mathcal{X}_{6})$,
$\mathcal{K}_{I\bar{J}}$ being an exact 2-form i.e. it is derivable (at least
locally) from a scalar potential called Kähler potential $\mathcal{K}_{0}$,

$\small K_{I\bar{J}}=\partial_{I}\partial_{\bar{J}}\mathcal{K}_{0}.$ (5)

Here $\mathcal{K}_{0}$ is the classical version of the Kähler potential and it
depends on three moduli viz., $S$, $z_{a}$ and $\rho_{k}$ as [31, 21],

$\small\begin{split}\mathcal{K}_{0}(S,z_{a},\rho_{k})=&-\sum_{k=1}^{3}\ln(-i(\rho_{k}-\bar{\rho}_{k}))-\ln(-i(S-\bar{S}))\\\
&-\ln(-i\int\Omega\wedge\bar{\Omega}).\end{split}$ (6)

Using Eq. (2) we get,

$\small\mathcal{K}_{0}(S,z_{a},\rho_{k})=-2\ln\mathcal{V}-\mathcal{K}_{0}(z_{a},S),$
(7)

where a constant factor $\ln 8$ has been absorbed in
$\mathcal{K}_{0}(z_{a},S)$. $\mathcal{K}_{0}(S,z_{a},\rho_{k})$ satisfies an
interesting condition,

$\small\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{h^{1,1}=3}{\mathcal{K}_{0}}^{{\rho_{k}}\bar{\rho}_{k^{\prime}}}\partial_{\rho_{k}}\mathcal{K}_{0}\partial_{\bar{\rho}_{k^{\prime}}}\mathcal{K}_{0}=3$
(8)

called the ‘no-scale’ structure, which is necessary to maintain the
supersymmetry [32]. The superpotential $\mathcal{W}_{0}$ and the Kähler
potential $\mathcal{K}_{0}(S,z_{a},\rho_{k})$ together provide a 4d effective
potential called the $F$-term potential,

$\small\begin{split}V_{F}&=e^{\mathcal{K}_{0}}\sum_{I,J\in\mathcal{M}({\mathcal{X}_{6}})}\left({\mathcal{K}_{0}}^{I\bar{J}}\mathcal{D}_{I}\mathcal{W}_{0}\mathcal{D}_{\bar{J}}\mathcal{\bar{W}}_{0}-3\mathcal{W}_{0}\mathcal{\bar{W}}_{0}\right)\\\
&=e^{\mathcal{K}_{0}}{\mathcal{K}_{0}}^{S\bar{S}}\mathcal{D}_{S}\mathcal{W}_{0}\mathcal{D}_{\bar{S}}\bar{\mathcal{W}_{0}}\\\
&+e^{\mathcal{K}_{0}}\sum_{a,b\in\mathcal{M}^{2,1}({\mathcal{X}_{6}})}^{h^{2,1}}{\mathcal{K}_{0}}^{z_{a}\bar{z}_{b}}\mathcal{D}_{z_{a}}\mathcal{W}_{0}\mathcal{D}_{\bar{z}_{b}}\bar{\mathcal{W}_{0}}\\\
&+e^{\mathcal{K}_{0}}(\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{h^{1,1}=3}{\mathcal{K}_{0}}^{\rho_{k}\bar{\rho}_{k^{\prime}}}\mathcal{D}_{\rho_{k}}\mathcal{W}_{0}\mathcal{D}_{\bar{\rho}_{k^{\prime}}}\bar{\mathcal{W}_{0}}\\\
&-3\mathcal{W}_{0}\bar{\mathcal{W}_{0}})\\\ \end{split}$ (9)

which vanishes at classical level due to Eqs. (4) and (8). Therefore, the
Kähler moduli are not fixed at tree level leading to the ‘moduli stabilization
problem’. In order to avoid this problem we have to come out of the classical
description and turn on quantum corrections to break the supersymmetric no-
scale structure of the Kähler potential which will give a non-zero $F$-term
potential. Let us first focus on the non-perturbative contributions of the
Kähler moduli to $\mathcal{W}_{0}$ as [17]

$\small\mathcal{W}(S,z_{a},\rho_{k})=\mathcal{W}_{0}(S,z_{a})+\sum_{k=1}^{3}A_{k}(z_{a})e^{i\alpha_{k}\rho_{k}}$
(10)

arising from various non-perturbative effects like gaugino condensation [19]
and instanton correction [20] where $\alpha_{k}$’s are small constants.
According to the large volume scenario (LVS) [14], in non-compact limit, all
4-cycles do not expand to infinity, rather, at least one of the 4-cycles must
be smaller than others. Also Ref. [33] says that due to certain choices of
world volume fluxes in the presence of $E3$ magnetised branes, some of the
Kähler moduli will have non-vanishing contributions to superpotential, which
help in applying perturbative string loop effects in Kähler potential as
explained in [15]. In our framework we consider $\tau_{1}$ to be smallest
among $\tau_{1,2,3}$ and thus suppressing the effects of larger $\rho$’s we
get from Eq. (10)

$\small\begin{split}\mathcal{W}(S,z_{a},\rho_{k})&=\mathcal{W}_{0}(S,z_{a})+A(z_{a})e^{i\alpha\rho_{1}}\\\
&=\mathcal{W}_{0}(S,z_{a})+A^{\prime}(z_{a})e^{-\alpha\tau_{1}},\end{split}$
(11)

where $A^{\prime}(z_{a})=A(z_{a})e^{i\alpha b_{1}}$. The non-renormalization
theorem [34] forbids us to modify the superpotential by perturbative
corrections. The tree level Kähler potential is perturbatively corrected
through the stabilization of the volume term in Eq. (7) by the classical
$\alpha^{\prime 3}$ correction [11] and quantum string loop effects due to
multi-graviton scattering [15]. The second type of correction is related to
the 4d localized Einstein-Hilbert term (second part of Eq. (12)), originating
through the process of compactification from the 10d effective action in
gravitational sector given in Ref. [15], containing higher derivative
objects222Higher derivative terms like $\nabla^{4}R^{4}$ are neglected as
their effects are very small in four-graviton scattering amplitude [35, 36].
like $R^{4}$ which is also proportional to the non-zero Euler number ($\chi$)
of the internal manifold. In this way the 10d effective action reduces to (see
[15, 5] for details),

$\small\begin{split}S_{\mathrm{grav}}&=\frac{1}{(2\pi)^{7}\alpha^{\prime
4}}\int_{\mathcal{M}_{4}\times\mathcal{X}_{6}}e^{-2\varphi}R_{(10)}\\\
&+(\alpha^{\prime})^{3}\frac{\chi}{(2\pi\alpha^{\prime})^{4}}\int_{\mathcal{M}_{4}}(2\zeta(3)e^{-2\varphi}+4\zeta(2))R_{(4)},\end{split}$
(12)

where $R_{(10)}=R\wedge e^{8}$ and $R_{(4)}=R\wedge e^{2}$. Here $R$ is the
Ricci curvature 2-form with $e$ being some generalized basis vector over
$\mathcal{X}_{6}$ (also called vielbein), and

$\small\chi=\frac{3!}{(2\pi)^{3}}\int_{\mathcal{X}_{6}}R\wedge R\wedge R.$
(13)

The terms associated with $\zeta(3)$ and $\zeta(2)$ come from genus zero
scattering amplitude which is actually analogous to $\alpha^{\prime 3}$
correction in large volume limit and genus one amplitude which arises from one
loop correction, respectively. This computation is done in
$T^{6}/\mathbb{Z}_{N}$ orbifold limit of CY3 (see [37, 38] for similar
calculations), where there are some points which remain invariant under
discrete $\mathbb{Z}_{N}$ group of transformations, called orbifold fixed
points or EH vertices, where $\chi\neq 0$ 333Although Ref. [15] shows that for
orbifolds tree level contribution vanishes, which creates a little paradoxical
situation.. They act as the sources of emission of massless gravitons and
massive Kaluza-Klein (KK) modes in the internal space. The one loop term in
the action of Eq. (12) is modified by a correction arising from the effect of
exchange of massless as well as massive closed string excitations between EH
vertices and $D7$ branes and $O7$ planes, viz., logarithmic correction [15],
which takes the form

$\small\begin{split}S_{\mathrm{grav}}&=\frac{1}{(2\pi)^{7}\alpha^{\prime
4}}\int_{\mathcal{M}_{4}\times\mathcal{X}_{6}}e^{-2\varphi}R_{(10)}\\\
&+(\alpha^{\prime})^{3}\frac{\chi}{(2\pi\alpha^{\prime})^{4}}\int_{\mathcal{M}_{4}}(2\zeta(3)e^{-2\varphi}\\\
&+4\zeta(2)(1-\sum_{k}e^{2\varphi}T_{k}\ln(R_{\perp}^{k}/w)))R_{(4)},\end{split}$
(14)

where $T_{k}$ is the tension of $k$th $D7$ brane, $R_{\perp}$ stands for size
of the transverse 2-cycle volume and $w$ is the width of effective UV cut-off
for graviton/KK modes [38]. Considering all these corrections, the volume term
can be re-written as [39, 5]

$\small\mathcal{V^{\prime}}=\mathcal{V}+\xi+\sum_{k=1}^{3}\eta_{k}\ln(\tau_{k}).$
(15)

For the sake of simplicity, it is assumed that all branes are identical so
that they all have same tension of constant magnitude viz.,
$e^{-\varphi}T_{0}$. Therefore,

$\small\small\eta_{k}=\eta=-\frac{1}{2}e^{\varphi}T_{0}\xi,$ (16)

where $\eta<0$, $\xi,T_{0},e^{\varphi}>0$ and $|\eta|<<\xi$ [15]. Then,

$\small\mathcal{V^{\prime}}=\mathcal{V}+\xi+\eta\sum_{k=1}^{3}\ln(\tau_{k})=\mathcal{V}+\xi+\eta\ln(\mathcal{V}),$
(17)

where a factor $2$ is temporarily absorbed in $\eta$.
$\xi=-\frac{\chi}{2}\zeta(2)e^{2\varphi}$ for orbifold and
$-\frac{\chi}{4}\zeta(3)$ for smooth CY3 [15], that means $\chi<0$ so that
$\xi>0$ which will be used later. Now, we can finally write the modified
version of Eq. (7),

$\small\mathcal{K}(S,z_{a},\rho_{k})=-2\ln(\mathcal{V}+\xi+\eta\ln\mathcal{V})-\mathcal{K}_{0}(S,z_{a})$
(18)

where the 6-cycle CY-volume satisfies Eq. (2). As we are interested only in
Kähler moduli stabilization, afterwards we will safely ignore the
$\mathcal{K}_{0}(S,z_{a})$ term (which is also clear from the third term of
Eq. (9)) and will proceed with only the Kähler moduli dependent term

$\small\mathcal{K}(\rho_{k})=-2\ln(\mathcal{V}+\xi+\eta\ln\mathcal{V}).$ (19)

This Kähler potential breaks the supersymmetric no-scale structure i.e., now,

$\small\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{3}{\mathcal{K}}^{{\rho_{k}}\bar{\rho}_{k^{\prime}}}\partial_{\rho_{k}}\mathcal{K}\partial_{\bar{\rho}_{k^{\prime}}}\mathcal{K}\neq
3$ (20)

leading to a non-vanishing $F$-term potential

$\small
V_{F}=e^{\mathcal{K}}\left(\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{3}{\mathcal{K}}^{\rho_{k}\bar{\rho}_{k^{\prime}}}\mathcal{D}_{\rho_{k}}\mathcal{W}\mathcal{D}_{\bar{\rho}_{k^{\prime}}}\bar{\mathcal{W}}-3\mathcal{W}\bar{\mathcal{W}}\right),$
(21)

where $\mathcal{W}$ is the corrected fluxed superpotential of Eq. (11). Thus,
the Kähler moduli sector of $\mathcal{X}_{6}$ is stabilized by perturbative
and non-perturbative quantum corrections in classical $F$-term potential. We
assume that the non-supersymmetrically stabilized $\tau_{2,3}$ will be just
massive enough to be able to sneak out to $\mathcal{M}_{4}$, for which
cosmological inflation will be possible. Now we split the Eq. (21) into
$V_{1}$, $V_{2}$ and $V_{3}$ such that $V_{F}=V_{1}+V_{2}+V_{3}$:

$\small
V_{1}=e^{\mathcal{K}}\left(\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{3}{\mathcal{K}}^{\rho_{k}\bar{\rho}_{k^{\prime}}}\partial_{\rho_{k}}\mathcal{K}\partial_{\bar{\rho}_{k^{\prime}}}\mathcal{K}-3\right)\mathcal{W}\bar{\mathcal{W}},$
(22) $\small
V_{2}=e^{\mathcal{K}}\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{3}{\mathcal{K}}^{\rho_{k}\bar{\rho}_{k^{\prime}}}\partial_{\rho_{k}}\mathcal{W}\partial_{\bar{\rho}_{k^{\prime}}}\bar{\mathcal{W}},$
(23) $\small\begin{split}&V_{3}=\\\
&e^{\mathcal{K}}\sum_{k,k^{\prime}\in\mathcal{M}^{1,1}({\mathcal{X}_{6}})}^{3}{\mathcal{K}}^{\rho_{k}\bar{\rho}_{k^{\prime}}}\left(\bar{\mathcal{W}}\partial_{\rho_{k}}\mathcal{W}\partial_{\bar{\rho}_{k^{\prime}}}\mathcal{K}+\mathcal{W}\partial_{\rho_{k}}\mathcal{K}\partial_{\bar{\rho}_{k^{\prime}}}\bar{\mathcal{W}}\right).\end{split}$
(24)

The term $e^{\mathcal{K}}$ can be approximated using Eq. (19) as

$\small
e^{\mathcal{K}}\approx\frac{1}{\mathcal{V}^{2}}-\frac{2(\xi+\eta\ln\mathcal{V})}{\mathcal{V}^{3}}+\frac{6\xi\eta}{\mathcal{V}^{4}},$
(25)

where the terms $\mathcal{O}(\xi^{2})$, $\mathcal{O}(\eta^{2})$ and
$\mathcal{O}(\frac{1}{\mathcal{V}^{5}})$ are neglected in large volume limit.
Similarly, we compute the three terms $V_{1}$, $V_{2}$ and $V_{3}$ using Eqs.
(1), (2), (11) and (19) in WOLFRAM MATHEMATICA 12 and write the results,

$\small\begin{split}V_{1}&=3e^{\mathcal{K}}(\tilde{A}+\mathcal{W}_{0})^{2}\\\
&\frac{(-8\eta+2\xi+2\eta\ln\mathcal{V})\mathcal{V}^{2}-2\eta\mathcal{V}(\eta+2\xi+2\eta\ln\mathcal{V})}{(16\eta-2\xi-2\eta\ln\mathcal{V}+4\mathcal{V})\mathcal{V}^{2}+2\eta\mathcal{V}(3\eta+2\xi+2\eta\ln\mathcal{V})},\end{split}$
(26)

$\small\begin{split}V_{2}&=4e^{\mathcal{K}}\tilde{A}^{2}\alpha^{2}\tau_{1}^{2}\mathcal{V}\\\
&\frac{(2\eta\mathcal{V}(\eta+\xi+\eta\ln\mathcal{V})+2\mathcal{V}^{2}(3\eta+\mathcal{V}))(2\eta\ln\mathcal{V}+2(\xi+\mathcal{V}))}{(\mathcal{V}^{2}+\eta\mathcal{V})(2\eta\mathcal{V}(3\eta+2\xi+2\eta\ln\mathcal{V})+\mathcal{V}^{2}(16\eta-2\xi-2\eta\ln\mathcal{V}+4\mathcal{V}))},\end{split}$
(27)
$\small\begin{split}V_{3}&=8e^{\mathcal{K}}\alpha\tau_{1}\tilde{A}(\tilde{A}+\mathcal{W}_{0})\\\
&\frac{(\mathcal{V}^{2}+\eta\mathcal{V})(2\eta\ln\mathcal{V}+2\xi+2\mathcal{V})}{2\eta\mathcal{V}(3\eta+2\xi+2\eta\ln\mathcal{V})+\mathcal{V}^{2}(16\eta-2\xi-2\eta\ln\mathcal{V}+4\mathcal{V})},\end{split}$
(28)

where $\tilde{A}=Ae^{-\alpha\tau_{1}}$. Using Eq. (25) we binomially expand
$V_{1,2,3}$ in large volume limit and release the $2$ factor in $\eta$ (which
was absorbed in Eq. (17)) to obtain,

$\small\begin{split}V_{1}&\approx\frac{3}{2}\mathcal{W}_{0}^{2}\frac{\xi-2\eta(4-\ln\mathcal{V})}{\mathcal{V}^{3}}-9\mathcal{W}_{0}^{2}\frac{\xi\eta\ln\mathcal{V}}{\mathcal{V}^{4}}\\\
&+(2\mathcal{W}_{0}\tilde{A}+\tilde{A}^{2})\left(\frac{3(\xi-2\eta(4-\ln\mathcal{V}))}{2\mathcal{V}^{3}}-\frac{9\xi\eta\ln\mathcal{V}}{\mathcal{V}^{4}}\right)\\\
&=V_{1\mathrm{p}}+V_{1\mathrm{m}},\end{split}$ (29)
$\small\begin{split}V_{2}&\approx\frac{4\alpha\tau_{1}\tilde{A}(\alpha\tau_{1}\tilde{A})}{\mathcal{V}^{2}}\\\
&-2\alpha\tau_{1}\tilde{A}(\alpha\tau_{1}\tilde{A})\left(\frac{\xi+2\eta(4+\ln\mathcal{V})}{\mathcal{V}^{3}}+\frac{2\xi\eta(2-3\ln\mathcal{V})}{\mathcal{V}^{4}}\right)\\\
&=V_{2\mathrm{np}}+V_{2\mathrm{m}},\end{split}$ (30)
$\small\begin{split}V_{3}&\approx\frac{4\alpha\tau_{1}\tilde{A}(\tilde{A}+\mathcal{W}_{0})}{\mathcal{V}^{2}}-2\alpha\tau_{1}\tilde{A}(\tilde{A}+\mathcal{W}_{0})\\\
&\left(\frac{\xi+2\eta(6+\ln\mathcal{V})}{\mathcal{V}^{3}}+\frac{6\xi\eta(1-\ln\mathcal{V})}{\mathcal{V}^{4}}\right)\\\
&=V_{3\mathrm{np}}+V_{3\mathrm{m}},\end{split}$ (31)

where the indices ‘p’, ‘np’ and ‘m’ respectively refer to perturbative, non-
perturbative and mixed terms in $V_{1,2,3}$. Assembling all terms we finally
obtain the perturbative, non-perturbative and mixed parts of $V_{F}$ as,

$\small
V_{F_{1}}=V_{1\mathrm{p}}=\frac{3}{2}\mathcal{W}_{0}^{2}\frac{\xi-2\eta(4-\ln\mathcal{V})}{\mathcal{V}^{3}}-9\mathcal{W}_{0}^{2}\frac{\xi\eta\ln\mathcal{V}}{\mathcal{V}^{4}},$
(32) $\small
V_{F_{2}}=V_{2\mathrm{np}}+V_{3\mathrm{np}}=\frac{4\alpha\tau_{1}}{\mathcal{V}^{2}}\tilde{A}(\tilde{A}+\alpha\tau_{1}\tilde{A}+\mathcal{W}_{0})$
(33)

and

$\small
V_{F_{3}}=V_{1\mathrm{m}}+V_{2\mathrm{m}}+V_{3\mathrm{m}}=\tilde{A}(\tilde{A}f+\mathcal{W}_{0}g)$
(34)

where,

$\small\begin{split}f=&(3\xi-8\eta(2\alpha\tau_{1}(2\alpha\tau_{1}+3)+3)-4\xi\alpha\tau_{1}(\alpha\tau_{1}+1)\\\
&-2\eta(2\alpha\tau_{1}-1)(2\alpha\tau_{1}+3)\ln\mathcal{V})/(2\mathcal{V}^{3})\\\
&+\frac{\eta\xi(2\alpha\tau_{1}+3)((6\alpha\tau_{1}-3)\ln\mathcal{V}-4\alpha\tau_{1})}{\mathcal{V}^{4}},\end{split}$
(35)

$\small\begin{split}g=&\frac{(3-2\alpha\tau_{1})(\xi+2\eta\ln\mathcal{V})-24\eta(1+\alpha\tau_{1})}{\mathcal{V}^{3}}\\\
&-6\eta\xi\frac{(3-2\alpha\tau_{1})\ln\mathcal{V}+2\alpha\tau_{1}}{\mathcal{V}^{4}}.\end{split}$
(36)

To stabilize the smallest Kähler modulus $\tau_{1}$ supersymmetrically we
first approximate the Kähler potential as,

$\small\mathcal{K}\approx-2\ln\mathcal{V}$ (37)

by considering the perturbative corrections $\xi$ and $\eta$ to have
negligible contributions, which will make calculations simpler. Then equating
the covariant derivative of $\mathcal{W}$ of Eq. (11) w.r.t $\rho_{1}$ to zero
at $\rho_{1}=i\tau_{1}$ we get,

$\small\begin{split}&D_{\rho_{1}}\mathcal{W}|_{\rho_{1}=i\tau_{1}}=ie^{-\alpha\tau_{1}}\left(\alpha
A+\frac{A+\mathcal{W}_{0}e^{\alpha\tau_{1}}}{2\tau_{1}}\right)=0\\\
&\mathrm{or,}\quad(-\alpha\tau_{1}-\frac{1}{2})e^{(-\alpha\tau_{1}-\frac{1}{2})}=\frac{\mathcal{W}_{0}}{2A\sqrt{e}}\in\mathbb{R}.\end{split}$
(38)

The physical solution of this equation is called the ‘Lambert W-function’,
$w=W_{0}(\frac{\mathcal{W}_{0}}{2A\sqrt{e}})$ corresponding to the $0$-branch
[5]. Therefore,

$\small-\alpha\tau_{1}-\frac{1}{2}=w\\\
\longrightarrow\tau_{1}=-\frac{1+2w}{2\alpha}.$ (39)

Here $\tau_{1}$ will be chosen as 40 as in [5]. Also from Eq. (38) we get,

$\small\tilde{A}=Ae^{-\alpha\tau_{1}}=-\frac{\mathcal{W}_{0}}{1+2\alpha\tau_{1}}=\frac{\mathcal{W}_{0}}{2w}.$
(40)

Let, $\epsilon=\frac{1+2w}{w}\quad(\approx 1$) [5], then from Eqs. (39) and
(40) we obtain,

$\small 2\alpha\tau_{1}=-\frac{\epsilon}{\epsilon-2}$ (41)

and

$\small\tilde{A}=\frac{\epsilon-2}{2}\mathcal{W}_{0}.$ (42)

Using Eqs. (41) and (42) we can rewrite Eqs. (32), (33) and (34) as,

$\small
V_{F_{1}}=\frac{3}{2}\mathcal{W}_{0}^{2}\frac{\xi-2\eta(4-\ln\mathcal{V})}{\mathcal{V}^{3}}-9\mathcal{W}_{0}^{2}\frac{\xi\eta\ln\mathcal{V}}{\mathcal{V}^{4}},$
(43) $\small
V_{F_{2}}=-(\epsilon\mathcal{W}_{0})^{2}\frac{\mathcal{V}}{4\mathcal{V}^{3}}$
(44)

and

$\small\begin{split}V_{F_{3}}&=(\epsilon\mathcal{W}_{0})^{2}\left(\frac{2\xi+4\eta(\ln\mathcal{V}-1)}{4\mathcal{V}^{3}}-\eta\xi\frac{3\ln\mathcal{V}-1}{\mathcal{V}^{4}}\right)\\\
&-\left(\frac{3}{2}\mathcal{W}_{0}^{2}\frac{\xi-2\eta(4-\ln\mathcal{V})}{\mathcal{V}^{3}}-9\mathcal{W}_{0}^{2}\frac{\xi\eta\ln\mathcal{V}}{\mathcal{V}^{4}}\right).\end{split}$
(45)

Now we can finally write the $F$-term potential as,

$\small\begin{split}V_{F}&=V_{F_{1}}+V_{F_{2}}+V_{F_{3}}\\\
&=-(\epsilon\mathcal{W}_{0})^{2}\left(\frac{\mathcal{V}-2\xi+4\eta(1-\ln\mathcal{V})}{4\mathcal{V}^{3}}-\eta\xi\frac{1-3\ln\mathcal{V}}{\mathcal{V}^{4}}\right)\end{split}$
(46)

which matches with the corresponding expression in [5]. This $F$-term
potential arises because of deviation from the no-scale structure of the tree
level Kähler potential vis-à-vis the supersymmetry breaking through non-
perturbative quantum correction of fluxed superpotential by the smallest
Kähler modulus $\tau_{1}$, which ensures that $\tau_{1}$ is stabilized
supersymmetrically and lies in compactified dimensions. The other two moduli
$\tau_{2,3}$ are perturbatively stabilized through overall volume correction
of the internal space $\mathcal{X}_{6}$. Being just massive enough, they can
come out of the compactified dimensions and appear in non-compact 4-manifold
$\mathcal{M}_{4}$. These relatively larger moduli $\tau_{2,3}$ play very
significant role in cosmology. Certain symmetric combination of them will give
rise to the inflaton field $\phi$ which drives the inflation by providing
required large vacuum energy, whereas their anti-symmetric combination is
another field which will remain in the background. Now, it is quite clear from
Eq. (46) that this $V_{F}$ can not alone be responsible for cosmological
inflation because its minima is at $AdS$ space. It requires some uplifting
agent to obtain a slow-roll like $dS$ vacuum. There are actually so many
uplifting mechanisms which include applying $\bar{D3}$ brane [40], Fayet-
Iliopoulos (FI) $D$-term [41] and nilpotent superfield [42]. Here we use
$D$-term contributions associated with $U(1)$ factors in the models of
intersecting $D7$ branes [3, 1, 43, 19, 44] and the corresponding $D$-term
potential takes the form [1, 5]

$\small
V_{D}=\sum_{i=1}^{3}\frac{g_{i}^{2}}{2}\left(\sqrt{-1}Q_{i}\partial_{\rho_{i}}\mathcal{K}+\sum_{j}q_{j}|\langle\Phi_{j}\rangle|^{2}\right)^{2}\approx\sum_{i=1}^{3}\frac{d_{i}}{\tau_{i}^{3}},$
(47)

where $g_{i}$’s (${g_{i}}^{-2}=\tau_{i}$ \+ flux dependent and curvature
corrections involving dilaton [19]) are the $U(1)$ gauge couplings and
$d_{i}={Q_{i}}^{2}/8>0$ ($i=1,2,3$) correspond to the charges carried out by
the 7-branes. $q_{j}$’s are the charges of the matter fields $\Phi_{j}$’s
whose VEVs are considered to be zero. This is a valid approximation because we
are considering only the gravitational sector (see Eq. (12)) and also this is
a safe and simple assumption for obtaining a non-vanishing $V_{D}$ (see [19]
and [44] for more details). $\frac{d_{1}}{\tau_{1}^{3}}$ acts as constant
uplifting factor similar to FI $D$-term. Remaining two terms will manifest
through inflaton and an auxiliary field. The effective potential can now be
expressed using Eqs. (46) and (47) as

$\small\begin{split}V_{\mathrm{eff}}&=V_{F}+V_{D}=\\\
&-(\epsilon\mathcal{W}_{0})^{2}\left(\frac{\mathcal{V}-2\xi+4\eta(1-\ln\mathcal{V})}{4\mathcal{V}^{3}}-\eta\xi\frac{1-3\ln\mathcal{V}}{\mathcal{V}^{4}}\right)\\\
&+\sum_{i=1}^{3}\frac{d_{i}}{\tau_{i}^{3}}.\end{split}$ (48)

Let us transform $\tau_{2,3}$ into two canonically normalized fields $t_{2,3}$
as

$\small t_{2}=\frac{1}{\sqrt{2}}\ln(\sqrt{\tau_{1}}\tau_{2}),\quad
t_{3}=\frac{1}{\sqrt{2}}\ln(\sqrt{\tau_{1}}\tau_{3})$ (49)

where $\tau_{1}$ is considered to be constant according to Eq. (39) as it is
supersymmetrically stabilized. A symmetric combination of $t_{2,3}$ yields

$\small\phi=\frac{1}{\sqrt{2}}(t_{2}+t_{3})=\frac{1}{2}\ln(\tau_{1}\tau_{2}\tau_{3})=\ln\mathcal{V}.$
(50)

Thus, the inflaton field manifests as logarithm of the CY3 volume. Also from
Eq. (50) we can write

$\small\tau_{2}\tau_{3}=\frac{e^{2\phi}}{\tau_{1}}.$ (51)

By antisymmetrizing $t_{2,3}$ we obtain

$\small
u=\frac{1}{\sqrt{2}}(t_{2}-t_{3})=\frac{1}{2}\ln\left(\frac{\tau_{2}}{\tau_{3}}\right)\longrightarrow\quad\frac{\tau_{2}}{\tau_{3}}=e^{2u}.$
(52)

These $\phi$ and $u$ fields are just two new avatars of $\tau_{2,3}$ and
therefore we can express $\phi$ and $u$ in terms of $\tau_{2,3}$ by inverting
Eqs. (51) and (52) as

$\small\tau_{2}=\frac{e^{(\phi+u)}}{\sqrt{\tau_{1}}},\quad\tau_{3}=\frac{e^{(\phi-u)}}{\sqrt{\tau}_{1}}.$
(53)

Now, we can transform the effective potential of Eq. (48) as a 2-field
potential (see Figure 3)

$\small\begin{split}&V_{\mathrm{eff}}(\phi,u)=-(\epsilon\mathcal{W}_{0})^{2}\left(\frac{e^{\phi}-2\xi+4\eta(1-\phi)}{4e^{3\phi}}-\eta\xi\frac{1-3\phi}{e^{4\phi}}\right)\\\
&+\frac{d_{1}}{\tau_{1}^{3}}+\tau_{1}^{3/2}(d_{2}e^{-3(\phi+u)}+d_{3}e^{-3(\phi-u)}).\end{split}$
(54)

Now, in order to stabilize the $u$ field we set,

$\small\left(\frac{\partial V_{\mathrm{eff}}(\phi,u)}{\partial
u}\right)_{u=u_{0}}=0\longrightarrow
u_{0}=\frac{1}{6}\ln\left(\frac{d_{2}}{d_{3}}\right)$ (55)

and

$\small\left(\frac{\partial^{2}V_{\mathrm{eff}}(\phi,u)}{\partial
u^{2}}\right)_{u=u_{0}}=9d\tau_{1}^{3/2}e^{-3\phi}>0$ (56)

where $d=2\sqrt{d_{2}d_{3}}$. Now, we finally obtain the single field slow-
roll inflaton potential with a stable $dS$ vacuum,

$\small\begin{split}&V(\phi,u_{0})\equiv
V(\phi)=\eta(\epsilon\mathcal{W}_{0})^{2}e^{-3\phi}\\\
&\left[\phi+\left(\frac{\xi}{2\eta}-1+\frac{d\tau_{1}^{3/2}}{\eta\epsilon^{2}\mathcal{W}_{0}^{2}}\right)-\frac{e^{\phi}}{4\eta}+\xi
e^{-\phi}(1-3\phi)\right]+\frac{d_{1}}{\tau_{1}^{3}}.\end{split}$ (57)

We can compress this equation by considering

$\small\alpha=\eta(\epsilon\mathcal{W}_{0})^{2},\beta=\left(\frac{\xi}{2\eta}-1+\frac{d\tau_{1}^{3/2}}{\eta\epsilon^{2}\mathcal{W}_{0}^{2}}\right),\\\
\gamma=\frac{1}{4\eta},\lambda=\frac{d_{1}}{\tau_{1}^{3}}$ (58)

as

$\small\boxed{V(\phi)=\alpha e^{-3\phi}\left[\phi+\beta-\gamma e^{\phi}+\xi
e^{-\phi}(1-3\phi)\right]+\lambda}.$ (59)

Our derived inflaton potential $V(\phi)$ (Eq. (59)) crucially depends on four
parameters $\alpha$, $\beta$, $\gamma$ and $\lambda$ which in turn depend on
perturbative and non-perturbative string theoretic parameters: $\epsilon$,
$\mathcal{W}_{0}$, $\xi$, $\eta$, $g_{s}$, $T_{0}$, $\chi$, $\tau_{1}$,
$d_{1}$ and $d$ according to the Eq. (58). In our framework we choose these
parameters as follows:

$\epsilon$ | $\mathcal{W}_{0}$ | $\xi$ | $\eta$ | $g_{s}$ | $T_{0}$ | $\chi$ | $\tau_{1}$
---|---|---|---|---|---|---|---
1 | 1.59 | 23 | -0.71 | 0.6 | 0.103 | -77 | 40
1 | 1.59 | 52 | -0.71 | 0.6 | 0.045 | -173 | 40
1 | 1.59 | 65 | -0.71 | 0.6 | 0.037 | -217 | 40
1 | 1.59 | 65 | -0.71 | 0.6 | 0.037 | -217 | 40
1 | 1.59 | 65 | -0.71 | 0.6 | 0.037 | -217 | 40
1 | 1.59 | 80 | -0.71 | 0.6 | 0.029 | -267 | 40
Table 1: In our first parameter space we have chosen $\epsilon$, $\mathcal{W}_{0}$, $\eta$, $g_{s}$ and $\tau_{1}$ to be fixed parameters while $\xi$, $T_{0}$ and $\chi$ are varied. The parameters are consistent with the constraints given in [5, 15]. $T_{0}$ and $\chi$ are parameterized in such a way so that $\eta$ remains almost fixed.  $\alpha$ | $\beta$ | $\gamma$ | $\lambda$ | $d_{1}$ | $d$
---|---|---|---|---|---
-1.805 | -70 | -0.35 | $5.47\times 10^{-6}$ | 0.35 | 0.45
-1.805 | -80 | -0.35 | $5.47\times 10^{-6}$ | 0.35 | 0.45
-1.805 | -90 | -0.35 | $5.47\times 10^{-6}$ | 0.35 | 0.17
-1.805 | -100 | -0.35 | $5.47\times 10^{-6}$ | 0.35 | 0.31
-1.805 | -110 | -0.35 | $5.47\times 10^{-6}$ | 0.35 | 0.45
-1.805 | -120 | -0.35 | $5.47\times 10^{-6}$ | 0.35 | 0.45
Table 2: In our second parameter space we have treated $\alpha$, $\gamma$,
$\lambda$ and $d_{1}$ as constants and $\beta$, $d$ to be variables. The
$\alpha$, $\beta$, $\gamma$ and $\lambda$ are obtained from the Eq. (58) using
the parameters in Table 1 and $d_{1}$, $d$ are suitably fixed to yield the
slow-roll structure of the potential. Although $d_{1}$ satisfies the condition
given in [5].

In Figure 1 we have shown the inflaton potential $V(\phi)$ against the
inflaton field $\phi$ from Eq. (59) without the uplifting term $\lambda$. This
potential has a plateau-type slow-roll feature with an $AdS$ minimum at
$\phi\approx 6.02$, which can not drive the inflationary expansion. In Figures
2 we have described the actual inflaton potentials in $dS$ space for two sets
of the parameter $\beta$ (see Eq. (58)): one is $\beta=-70,-90,-110$ for three
values of the non-perturbative parameter $d=0.17,0.31,0.45$ respectively
keeping the perturbative parameter fixed at $\xi=65$ (see upper figure) and
the other is $\beta=-80,-100,-120$ by varying $\xi=23,52,80$ respectively for
a particular value of $d=0.45$ (see lower figure). Both the figures highlight
an uplifting and a slight shift in $\phi$ direction of the $dS$ vacuum by the
increase of $\beta$ maintaining the same slow-roll plateau as found in $AdS$
space. We find that, in these two figures, uplifting the $dS$ vacuum does not
disturb the flat direction, necessary for inflation. It is observed that the
two perturbative parameters $\xi$ and $\eta$ play a major role for shaping the
inflaton potential as slow-roll one, the non-perturbative parameters $d$ and
$d_{1}$ are responsible for the uplifting and the $\alpha$ fixes the overall
energy scale of inflation which is $\sim 10^{-6}$ in our case. The smallness
of this energy scale firmly indicates the microscopic origin of our inflaton
potential viz., the moduli stabilization in type IIB/F-theory compactification
with string, brane, orientifold and fluxes- which is certainly a prime
motivation of our approach.

[width=0.5]New_plots/AdS_pot.eps

Figure 1: Inflaton potential with $AdS$ minima at $\phi\approx 6.02$ for
$\xi=52$, $d=0.45$ and $\beta=-100$.

[width=0.5]New_plots/Pot_param_1.eps [width=0.5]New_plots/Pot_param_2.eps

Figure 2: Uplifted inflaton potentials with $dS$ minima for different values
of $\beta$, $d$ with fixed value of $\xi$ (upper figure) and different values
of $\beta$, $\xi$ with fixed value of $d$ (lower figure). The $dS$ vacua are
uplifted as well as shifted in right direction keeping the slow-roll plateau
same.

[width=0.6]New_plots/3D.eps

Figure 3: Three dimensional plot of the 2-field inflaton potential of Eq. (54)
against the $\phi$ and $u$. The auxiliary field $u$ remains at its minimum and
its component corresponding to the potential has almost no effect, during
inflation.

At the end, we would like to mention that with the slow-roll potential of Eq.
(59) and following the formalism in Ref. [10] we have obtained the values of
some cosmological parameters such as, scalar power spectrum ($\Delta_{s}$):
$3.38\times 10^{-4}$ \- $3.60\times 10^{-4}$, tensor power spectrum
($\Delta_{t}$): $2.1015\times 10^{-7}$ \- $2.1018\times 10^{-7}$, number of
e-folds ($N$): $55.0$ \- $56.7$, scalar spectral index ($n_{s}$): $0.9652$ \-
$0.9662$, tensor spectral index ($n_{t}$): $(-7.28\times 10^{-5})$ \-
$(-7.76\times 10^{-5})$ and tensor-to-scalar ratio ($r$): $5.8\times 10^{-4}$
\- $6.2\times 10^{-4}$ at $k=0.001$ \- $0.009$ Mpc-1 for $\xi=52$, $d=0.45$
and $\beta=-100$. We plan to report the details of these calculations in a
future publication.

In conclusion, we have derived, effectively, a single-field slow-roll inflaton
potential from Kähler moduli stabilizations in type IIB/F-theory.

###### Acknowledgements.

The authors acknowledge the University Grants Commission for the CAS-II
program. AL acknowledges CSIR, the Government of India for NET fellowship. AS
and CS acknowledge Government of West Bengal for granting them Swami
Vivekananda fellowship.

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