# User Preference-aware Fake News Detection

Yingtong Dou1, Kai Shu2, Congying Xia1, Philip S. Yu1, Lichao Sun3 1Department
of Computer Science, University of Illinois at Chicago, Chicago, IL, USA
2Department of Computer Science, Illinois Institute of Technology, Chicago,
IL, USA  3Department of Computer Science and Engineering, Lehigh University,
Bethlehem, PA, USA  ydou5, cxia8<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(2021)

###### Abstract.

Disinformation and fake news have posed detrimental effects on individuals and
society in recent years, attracting broad attention to fake news detection.
The majority of existing fake news detection algorithms focus on mining news
content and/or the surrounding exogenous context for discovering deceptive
signals; while the endogenous preference of a user when he/she decides to
spread a piece of fake news or not is ignored. The confirmation bias theory
has indicated that a user is more likely to spread a piece of fake news when
it confirms his/her existing beliefs/preferences. Users’ historical, social
engagements such as posts provide rich information about users’ preferences
toward news and have great potentials to advance fake news detection. However,
the work on exploring user preference for fake news detection is somewhat
limited. Therefore, in this paper, we study the novel problem of exploiting
user preference for fake news detection. We propose a new framework,
$\mathsf{UPFD}$, which simultaneously captures various signals from user
preferences by joint content and graph modeling. Experimental results on real-
world datasets demonstrate the effectiveness of the proposed framework. We
release our code and data as a benchmark for GNN-based fake news detection:
https://github.com/safe-graph/GNN-FakeNews.

Data Mining; Social Media Analysis; Fake News Detection

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
44th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 11–15, 2021; Virtual Event, Canada.††booktitle:
Proceedings of the 44th International ACM SIGIR Conference on Research and
Development in Information Retrieval (SIGIR ’21), July 11–15, 2021, Virtual
Event, Canada††price: 15.00††isbn: 978-1-4503-8037-9/21/07††doi:
10.1145/XXXXXX.XXXXXX††ccs: Computing methodologies Machine learning††ccs:
Information systems Social networks

##  1\. Introduction

In recent years, social media has enabled the wide dissemination of
disinformation and fake news– false or misleading information disguised in
news articles to mislead consumers (Zhou and Zafarani, 2020; Shu et al.,
2017). Disinformation has resulted in deleterious effects and raised serious
concerns, demanding novel approaches for fake news detection.

Among various fake news detection techniques, fact-checking is the most
straightforward approach; however, it is usually labor-intensive to acquire
evidence from domain experts (Hassan et al., 2017). In addition, computational
approaches using feature engineering or deep learning have shown many
promising results (Wang et al., 2018; Karimi et al., 2018; Ruchansky et al.,
2017; Chandra et al., 2020). For example, SAFE (Zhou et al., 2020) and
FakeBERT (Kaliyar et al., [n.d.]) used the TextCNN (Zhang and Wallace, 2015)
and BERT (Devlin et al., 2018; Sun et al., 2020b; Sun et al., 2020a) to encode
the news textual information, respectively; GCNFN (Monti et al., 2019) and
GNN-CL (Han et al., 2020) leveraged the GCN (Kipf and Welling, 2017) to encode
the news propagation patterns on social media (e.g., news sharing cascading
among social media accounts). However, these methods focus on modeling news
content and its user exogenous context and ignore the user endogenous
preferences.

Sociological and psychological studies on journalism have theorized the
correlation between user preferences and their online news consumption
behaviors (Shu et al., 2019). For example, Naíve Realism (Ross and Ward, 1995)
indicates that consumers tend to believe that their perceptions of reality are
the only accurate views, while others who disagree are regarded as uninformed,
irrational, or biased; and Confirmation Bias theory (Nickerson, 1998) reveals
that consumers prefer to receive information that confirms their existing
views. For instance, a user believes the election fraud would probably share
similar news with a supportive stance, and the news asserting election is
stolen would attract users with similar beliefs (Abilov et al., 2021). To
model user endogenous preferences, existing works have attempted to utilize
historical posts as a proxy and have shown promising performance to detect
sarcasm (Khattri et al., 2015), hate speech (Qian et al., 2018), and fake news
spreaders (Rangel et al., 2020) on social media.

Figure 1. The proposed $\mathsf{UPFD}$ framework for user preference-aware
fake news detection. Given the news piece and its engaged users on social
media, we extract the exogenous context as a news propagation graph and encode
the endogenous information based on user historical posts and news texts. The
endogenous and exogenous information are fused using a GNN encoder. The final
news embedding, composed of user engagement embedding and news textual
embedding, is fed into the neural classifier to predict the news’ credibility.

In this paper, we consider the historical posts of social media users as their
endogenous preference in news consumption. We propose an end-to-end fake news
detection framework named User Preference-aware Fake Detection
($\mathsf{UPFD}$) to model endogenous preference and exogenous context jointly
(as shown in Figure 1). Specifically, $\mathsf{UPFD}$ consists of the
following major components: (1) To model the user endogenous preference, we
encode news content and user historical posts using various text
representation learning approaches. (2) To obtain the user exogenous context,
we build a tree-structured propagation graph for each news based on its
sharing cascading on social media. The news post is regarded as the root node,
and other nodes represent the users who shared the same news posts. (3) To
integrate the endogenous and exogenous information, we take the vector
representations of news and users as their node features and employ Graph
Neural Networks (GNNs) (Hamilton et al., 2017; Monti et al., 2019) to learn a
joint user engagement embedding. The user engagement embedding and news
textual embedding are used to train a neural classifier to detect fake news.
Our major contributions can be summarized as follows:

  * •

We study a novel problem of user preference-aware fake news detection on
social media;

  * •

We propose a principled way to exploit both endogenous preference and
exogenous context jointly to detect fake news; and

  * •

We conduct extensive experiments on real-world datasets to demonstrate the
effectiveness of $\mathsf{UPFD}$ for detecting fake news.

##  2\. Our Approach

In this section, we present the details of the proposed framework for fake
news detection named $\mathsf{UPFD}$ (User Preference-aware Fake News
Detection). As shown in Figure 1, our framework has three major components.
First, given a news piece, we crawl the historical posts of the users engaged
in the news to learn user endogenous preference. We implicitly extract the
preferences of engaged users by encoding historical posts using text
representation learning techniques (e.g., word2vec (Mikolov et al., 2013),
BERT (Devlin et al., 2018)). The news textual data is encoded using the same
approach. Second, to leverage user exogenous context, we build the news
propagation graph according to its engagement information on social media
platforms (e.g., retweets on Twitter). Third, we devise a hierarchical
information fusion process to fuse the user endogenous preference and
exogenous context. Specifically, we obtain the user engagement embedding using
GNN as the graph encoder, where the news and user embeddings encoded by the
text encoder are used as their corresponding node features in the news
propagation graph. The final news embeddings are composed by the concatenation
of user engagement embedding and news textual embedding.

Next, we will introduce how we encode endogenous preference, extract the
exogenous context, and fuse both information.

###  2.1. Endogenous Preference Encoding

It is non-trivial to explicitly model the endogenous preference of a user only
using his/her social network information. Similar to (Ahmad and Siddique,
2017; Khattri et al., 2015; Qian et al., 2018) which model the users’
personality, sentiment and stance using their historical posts, we leverage
the historical posts of a user to encode his/her preference implicitly.
However, none of the previous fake news datasets contain such information. In
this paper, we select the FakeNewsNet dataset (Shu et al., 2020b) which
contains news content and its social engagement information on Twitter. Then
we use the Twitter Developer API (Developer, 2021) to crawl historical tweets
of all accounts that retweeted the news in FakeNewsNet.

To obtain rich historical information for user preference modeling, we crawl
the recent two hundred tweets for each account, so as to near 20 million
tweets being crawled in total. For inaccessible users whose accounts are
suspended or deleted, we use randomly sampled tweets from accessible users
engaging the same news as its corresponding historical posts. Because deleting
the inaccessible users will break the intact news propagation cascading and
results in a less effective exogenous context encoder. We also remove the
special characters, e.g., “@” characters and URLs, before applying text
representation learning methods.

To encode the news textual information and user preferences, we employ two
types of text representation learning approaches based on language
pretraining. Instead of training on the local corpus, the word embeddings
pretrained on a large corpus are supposed to encode more semantic similarities
between different words and sentences. For pretrained word2vec vectors, we
choose the 680k 300-dimensional vectors pretrained by spaCy (Honnibal and
Montani, 2017). We also employ pretrained BERT embeddings to encode the
historical tweets and news content as a sequence (Devlin et al., 2018) using
bert-as-a-service (Xiao, 2018).

Next, we elaborate the details of applying the above text representation
learning models. spaCy includes pretrained vectors for 680k words, and we
average the vectors of existing words in combined recent 200 tweets to get
user preference representation. The news textual embedding is obtained
similarly. For the BERT model, we use the cased BERT-Large model to encode the
news and user information. The news content is encoded using BERT with maximum
input sequence length (i.e., 512 tokens). Due to BERT’s input sequence length
limitation, we could not use BERT to encode 200 tweets as one sequence, so we
resort to encode each tweet separately and average them afterward to obtain a
user’s preference representation. Generally, the tweet text is way shorter
than the news text, we empirically set the max input sequence length of BERT
as 16 tokens to accelerate the tweets encoding time.

###  2.2. Exogenous Context Extraction

Given a news piece on social media, the user exogenous context is composed of
all users that engaged with the news. We utilize the retweet information of
news pieces to build a news propagation graph. As the toy example of the news
propagation graph shown in Figure 1, it is a tree-structured graph where the
root node represents the news piece, and other nodes represent users who share
the root news. In this paper, we investigate the fake news propagation on
Twitter as a proof-of-concept use case. To build propagation networks in
Twitter, we follow a similar strategy used in (Shu et al., 2020a; Monti et
al., 2019; Han et al., 2020). Specifically, we define a new piece as $v_{1}$,
and $\\{v_{2},\dotsc,v_{n}\\}$ as a list of users that retweeted $v_{1}$
ordered by time. We define two following rules to determine the news
propagation path:

  * •

For any account $v_{i}$, if $v_{i}$ retweets the same news later than at least
one following accounts in $\\{v_{1},\dotsc,v_{n}\\}$, we estimate the news
spreads from the account with the latest timestamp to account $v_{i}$. Since
the latest tweets are first presented in the timeline of the Twitter app, and
thus have higher probabilities to be retweeted.

  * •

If account $v_{i}$ does not follow any accounts in the retweet sequences
including the source account, we conservatively estimate the news spreads from
the accounts with the most number of followers. Because tweets from accounts
with more followers have a higher chance to be viewed/retweeted by other users
according to the Twitter content distributing rules.

Based on the above rules, we can build the news propagation graphs on Twitter.
Note that this approach can be applied to other social media platforms like
Facebook as well.

###  2.3. Information Fusion

Previous works (Lu and Li, 2020; Monti et al., 2019; Han et al., 2020) have
demonstrated that fusing the user features with a news propagation graph could
boost the fake news detection performance. Since the GNN can encode both node
feature and graph structure in an end-to-end manner, it is a good fit for our
task. Specifically, we propose a hierarchical information fusion approach. We
first fuse the endogenous and exogenous information using the GNN. With a GNN,
the news textual embedding and user preference embedding can be taken as node
features. Given the news propagation graph, most GNNs aggregate the features
of its adjacent nodes to learn the embedding of a node. Like previous GNN-
based graph classification models (Xu et al., 2018; Ying et al., 2018), we
apply a readout function over all node embeddings to obtain the embedding of a
news propagation graph. The readout function makes the mean pooling operation
over all node embeddings to get the graph embedding (i.e., user engagement
embedding). Second, since the news content usually contains more explicit
signals regarding the news’ credibility (Chandra et al., 2020). We fuse the
news textual embedding and user engagement embedding by concatenation as the
ultimate news embedding to enrich the news embedding information.

The fused news embedding is finally fed into a two-layer Multi-layer
Perceptron (MLP) with two output neurons representing the predicted
probabilities for fake and real news. The model is trained using a binary
cross-entropy loss function and is updated with SGD.

Table 1. Dataset and graph statistics.

Dataset |  | #Graphs
---
(#Fake)
| #Total
---
Nodes
| #Total
---
Edges
| #Avg. Nodes
---
per Graph
Politifact (POL) |  | 314
---
(157)
41,054 | 40,740 | 131
Gossipcop (GOS) |  | 5464
---
(2732)
314,262 | 308,798 | 58

##  3\. Experiments

In the experiments, we want to address two following questions: RQ1: How are
the performances of the proposed $\mathsf{UPFD}$ framework compared to
previous works? RQ2: What are the contributions of endogenous/exogenous
information and other variants of the proposed framework?

###  3.1. Experimental Setup

####  3.1.1. Dataset.

Previous works have proposed a couple of fake news datasets with news pieces
from different websites and their fact-checking information (Wang, 2017;
Potthast et al., 2018). To investigate both the user preference and
propagation pattern of fake news, we choose the FakeNewsNet dataset (Shu et
al., 2020b). It contains fake and real news information from two fact-checking
websites and the related social engagement from Twitter. The dataset
statistics are shown in Table 1.

####  3.1.2. Baselines

We compare the $\mathsf{UPFD}$ with fake news detection models that utilize
different information. Many baseline methods leverage extra information like
image information which is not included in the FakeNewsNet (Shu et al.,
2020b). To ensure a fair comparison, we implement the baselines only with the
parts for encoding the news content, user comments, and news propagation
graph. CSI (Ruchansky et al., 2017) employs an LSTM to encode the news content
information to detect fake news. SAFE (Zhou et al., 2020) uses the TextCNN
(Zhang and Wallace, 2015) to encode the news textual information. GCNFN (Monti
et al., 2019) is the first fake news detection framework to encode the news
propagation graph using GCN (Kipf and Welling, 2017). It takes the profile
information and comment textual embeddings as the user feature. GNN-CL (Han et
al., 2020) encodes the news propagation graph using DiffPool (Ying et al.,
2018), a GNN designed for graph classification. The node features are
extracted from user profile attributes on Twitter. The list of ten profile
feature names can be found in (Han et al., 2020; Lu and Li, 2020) We also add
two baselines that apply MLP directly on news textual embeddings encoded by
word2vec and BERT.

####  3.1.3. Experimental Settings

We implement all models using PyTorch, and all GNN models are implemented with
PyTorch-Geometric package (Fey and Lenssen, 2019). We use unified graph
embedding size (128), batch size (128), optimizer (Adam), and L2
regularization weight (0.001), train-val-test split (20%-10%-70%) for all
models. The experimental results are averaged over five different runnings.
Other hyper-parameters for each model are reported with the code.

###  3.2. RQ1: Performance Evaluation

Table 2. The fake news detection performance of baselines and our model. Stars
denote statistically significant under the t-test ($\ast\;p\leq
0.05,\ast\ast\;p\leq 0.01$, $\ast\ast\ast\;p\leq 0.001$).

| Model | POL | GOS
---|---|---|---
ACC | F1 | ACC | F1
News Only |  SAFE (Zhou et al., 2020) | 73.30 | 72.87 | 77.37 | 77.19
CSI (Ruchansky et al., 2017) | 76.02 | 75.99 | 75.20 | 75.01
BERT+MLP | 71.04 | 71.03 | 85.76 | 85.75
word2vec+MLP | 76.47 | 76.36 | 84.61 | 84.59
News + User |  GNN-CL (Han et al., 2020) | 62.90 | 62.25 | 95.11 | 95.09
GCNFN (Monti et al., 2019) | 83.16 | 83.56 | 96.38 | 96.36
$\mathsf{UPFD}$ (ours) | $\mathbf{84.62^{\ast}}$ | $\mathbf{84.65^{\ast}}$ | $\mathbf{97.23^{\ast\ast}}$ | $\mathbf{97.22^{\ast\ast\ast}}$

Table 2 shows the fake news detection performance of $\mathsf{UPFD}$ and six
baselines. First, we can observe that $\mathsf{UPFD}$ has the best performance
comparing to all baselines. $\mathsf{UPFD}$ outperforms the best baseline
GCNFN around 1% on both datasets with statistical significance. The
experimental results of $\mathsf{UPFD}$ and GCNFN demonstrate that the user
comments (used by GCNFN) are also beneficial to fake news detection; and the
user endogenous preference could impose additional information when user
comment information is limited. Second, since all baselines either encode the
news content or user comments without considering the historical posts, we can
tell that leveraging the historical posts as user endogenous preferences could
improve the fake news detection performance. Note that the $\mathsf{UPFD}$
with the best performance on the both datasets uses BERT as the text encoder
and GraphSAGE as the graph encoder.

Table 3. Fake news detection performance on two datasets with different node
feature types and models. The bold (underlined) text indicates the best
(second best) performances on each dataset.

Feature | POL | GOS
---|---|---
GraphSAGE | GCNFN | GraphSAGE | GCNFN
ACC | F1 | ACC | F1 | ACC | F1 | ACC | F1
Profile | 77.38 | 77.12 | 76.94 | 76.72 | 92.19 | 92.16 | 89.00 | 88.96
word2vec | 80.54 | 80.41 | 80.54 | 80.41 | 96.81 | 96.80 | 94.97 | 94.95
BERT | 84.62 | 84.53 | 83.26 | 83.14 | 97.23 | 97.22 | 96.18 | 96.17

###  3.3. RQ2: Ablation Study

####  3.3.1. Encoder Variants

As we mentioned in Section 2.3, we employ different text encoders and GNNs to
encode the endogenous and exogenous information. In Table 3, we show the fake
news detection performance of two GNN variants using three different node
features. Note that “word2vec” and “BERT” represent features encoding the user
endogenous preferences while the “Profile” feature is regarded as a baseline.
GraphSAGE (Hamilton et al., 2017) is a GNN to learn node embeddings via
aggregating neighbor nodes information and GCNFN (Monti et al., 2019) is a
GNN-based fake news detection model which leverages two GCN layers to encode
the news propagation graph.

Table 3 shows that the endogenous features (word2vec and BERT) are
consistently better than the profile feature, which only encodes the user
profile information. We also observe that GraphSAGE and BERT have the average
best performance among other model and feature variants. It suggests that BERT
is better than word2vec for encoding textual features which has been verified
on other NLP tasks (Devlin et al., 2018). Note that the BERT performance could
be further improved via fine-tuning, and we leave it as future work.

Figure 2. The fake news detection performance of different variants of
$\mathsf{UPFD}$ framework. -END/-EXO represents the $\mathsf{UPFD}$ variant
without endogenous/exogenous information.

####  3.3.2. Framework Variants

To verify the effectiveness of endogenous preference and exogenous context, we
fix the text and graph encoder and design three $\mathsf{UPFD}$ variants that
remove the endogenous information, exogenous information or both of them.

Specifically, we employ the GCNFN (word2vec) as the graph (text) encoder for
both datasets, and remove news concatenation to ensure a fair comparison. The
$\mathsf{UPFD}$ variant without exogenous information (-EXO) is implemented by
removing all edges in the news propagation graph. Thus, -EXO encodes the news
embedding solely based on node features without exchanging information between
nodes. The $\mathsf{UPFD}$ variant without endogenous information (-END) takes
the user profile as node features and does not contain user endogenous
preference information. The $\mathsf{UPFD}$ variant without both endogenous
and exogenous information (-END & -EXO) replaces the node features of the -EXO
with user profile features.

Figure 2 shows the fake news detection performance for different
$\mathsf{UPFD}$ variants on two datasets. We can find that removing either
component from the $\mathsf{UPFD}$ will reduce its performance. Moreover,
jointly encoding the endogenous and exogenous information attains the best
performance. The accuracy of $\mathsf{UPFD}$/-EXO is 85.61%/81.63%, and the F1
score of $\mathsf{UPFD}$/-EXO is 85.97%/81.15% on Politifact. The accuracy of
$\mathsf{UPFD}$/-EXO is 95.47%/93.92%, and the F1 score of
$\mathsf{UPFD}$/-EXO is 95.46%/93.81% on Politifact. All the experimental
results are statistically significant under the t-test ($p\leq 0.01$). This
indicates that exogenous information (i.e., news propagation graph) is more
informative on Politifact since removing it results in a larger performance
drop. It is obvious that endogenous information contributes more to
performance gain than exogenous information. This observation further verifies
the necessity of modeling user endogenous preferences.

##  4\. Conclusion

In this paper, we argues that user endogenous news consumption preference
plays a vital role in the fake news detection problem. To verify this
argument, we collect the user historical posts to implicitly model the user
endogenous preference and leverage the news propagation graph on social media
as the exogenous social context of users. An end-to-end fake news detection
framework named $\mathsf{UPFD}$ is proposed to fuse the endogenous and
exogenous information and predict the news’ credibility on social media.
Experimental results demonstrate the advantage of modeling the user endogenous
preference.

###### Acknowledgements.

This work is supported in part by NSF under grants III-1763325, III-1909323,
and SaTC-1930941. Kai Shu is supported by the John S. and James L. Knight
Foundation through a grant to the Institute for Data, Democracy & Politics at
The George Washington University.

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# Improvement of response bandwidth and sensitivity of Rydberg Receiver using
multi-channel excitations

Jinlian Hu  Yuechun Jiao  Yunhui He  Hao Zhang  Linjie Zhang  Jianming Zhao
Suotang Jia  State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006,
P. R. China  Collaborative Innovation Center of Extreme Optics, Shanxi
University, Taiyuan 030006, China

###### Abstract

We investigate the response bandwidth of a superheterodyne Rydberg receiver at
a room-temperature vapor cell, and present an architecture of multi-channel
lasers excitation to increase the response bandwidth and keep sensitivity,
simultaneously. Two microwave fields, denoted as a local oscillator (LO)
$E_{LO}$ and a signal field $E_{Sig}$, couple two Rydberg states transition of
$|52D_{5/2}\rangle\to|53P_{3/2}\rangle$. In the presence of the LO field, the
frequency difference between two fields can be read out as an intermediate
frequency (IF) signal using Rydberg electromagnetically induced transparency
(EIT) spectroscopy. The bandwidth of the Rydberg receiver is obtained by
measuring the output power of IF signal versus the frequency difference
between two fields. The bandwidth dependence on the Rabi frequency of
excitation lasers is presented, which shows the bandwidth decrease with the
probe Rabi frequency, while it is quadratic dependence on the coupling Rabi
frequency. Meanwhile, we investigate the effect of probe laser waist on the
bandwidth, showing that the bandwidth is inversely proportional to the laser
waist. We achieve a maximum response bandwidth of the receiver about 6.8 MHz.
Finally, we design an architecture of multi-channel lasers excitation for
increasing the response and keeping the sensitivity, simultaneously. Our work
has the potential to extend the applications of Rydberg atoms in
communications.

Rydberg receiver,

response bandwidth,

multi-channel lasers excitation,

microwave sensor,

###### keywords:

Research

##  1 Introduction

Rydberg atom-based electrometry has made a significant progress [1] in the
last decade due to its property of large polarizabilities and microwave-
transition dipole moments [2] and their advantages in the measurement of weak
signals with high sensitivity, calibration-free, and intrinsic accuracy [3,
4]. An optical Rydberg electromagnetically induced transparency (Rydberg-EIT)
spectroscopy [5] and Autler–Townes (AT) splitting [6] has been used to detect
the properties of electric field over a wide frequency range from DC to over 1
THz [7, 8, 9, 10, 11, 12, 13], as well as terahertz imaging [14] and sub-
wavelength imaging of MW electric-field distributions [15]. Recently, the use
of Rydberg-atom superheterodyne technique [16, 13] by adding an auxiliary
local oscillator (LO) field greatly improve the sensitivity of the Rydberg
receiver to 55 nVcm-1Hz-1/2, and later to 30 nVcm-1Hz-1/2 by adding an optical
pumping technique [17]. The superheterodyne technique also provides a method
to determine the phase, frequency and angle-of-arrival of the signal field
[18].

In addition, the Rydberg atom-based receiver has an application in wireless
communication, which has been used to receive baseband signals in a formation
of amplitude modulation (AM) [19, 20, 21], frequency modulation (FM) [22], and
phase modulation (PM) [23, 24, 25]. However, temporal dynamics of Rydberg
sensors are predominantly limited at sub-microsecond timescales [26, 27],
which limits its applications for detecting more complex waveforms and
transmitting data with high rates. Some investigations aim to improve the data
transfer capacity by employing multiple bands and multiple channels [28, 29],
multiple species [30] and spatially distributed atomic receiver array [31].
Further, the time-dependent response of Rydberg receiver to pulse-modulated
electric fields is investigated under different laser conditions, which shows
the origins of Rydberg-Atom electrometer transient response [32, 33] by
investigation of the rise and fall times for the receiver response to a
transient signal, separately. The direct investigation of the response of a
Rydberg atom-based superheterodyne receiver is not reported yet. All the
aforementioned works use the EIT steady-state response so that the changes in
the probe beam can only be measured after atoms reach steady-state. Recently,
a new receiver concept based on spatiotemporal shaping of the probe beam
within a Rydberg vapor cell is proposed to increase the response speed of
Rydberg sensors to greater than 100 MHz [34].

In this work, we investigate the response bandwidth of a Rydberg atom-based
superheterodyne receiver. The dependence of response bandwidth on the
excitation laser conditions is investigated, including the probe laser Rabi
frequency, the probe beam waist and the coupling laser Rabi frequency. The
results show the bandwidth decreases with the probe Rabi frequency, while
shows quadratic increasing with the coupling Rabi frequency. In addition, we
show the bandwidth is inversely proportional to the laser waist. A maximum
response bandwidth of the receiver about 6.8 MHz is achieved by optimizing the
laser parameters. The small probe laser Rabi frequency and small probe beam
waist can increase the response bandwidth, while it causes a reduction of
Rydberg atoms number, which lowers the sensitivity of receiver. Finally, we
design a structure of two-channel lasers excitation, which realizes the
increasing bandwidth, meanwhile guarantee the sensitivity. Our results show
the improved response bandwidth is consistent with the reduction of beam
waist, e.g. the reduction of transit time.

##  2 Experimental setup

Figure 1: (a) Energy level diagram for the Rydberg EIT-AT system. A probe
laser ($\Omega_{p}$) drives atoms from $|1\rangle$ to $|2\rangle$ and a
coupling laser ($\Omega_{c}$) drives atoms from $|2\rangle$ to $|3\rangle$
transition. A local MW field ($\Omega_{L}$) resonantly couples the transition
of $|3\rangle$ to $|4\rangle$, which causes the AT splitting of EIT spectrum,
while the weak signal field ($\Omega_{S}$) has a detuning of $\delta_{f}$. In
the presence of the LO field, the Rydberg atoms work as an atomic mixer to
down-convert the signal field to an intermediate frequency with a frequency of
$\delta_{f}$, which causes an oscillation in the transmission of the probe
laser. (b) Sketch of the experimental setup for single channel Rydberg
receiver. A coupling laser and a probe laser counter-propagate through a
cylindrical room-temperature cesium cell. The transmission of the probe laser
is detected by an avalanche photodiode (APD). Two RF fields, noted as the LO
field and the signal field ($E_{LO}$ and $E_{SIG}$), are applied transversely
to the laser beams propagating through the vapor cell by two identical horn
antennas. (c) Diagram of two-channel Rydberg receiver for increasing the
response bandwidth. Both the probe and coupling lasers are split into two
beams using a 50/50 beam splitter (BS) and counter-propagate through the cell.
One channel is right above the other. The beam waists of each probe and
coupling laser are set to $\omega_{p}/\sqrt{2}$ and $\omega_{c}/\sqrt{2}$.
After propagating through the cell, two separated probe lasers pass through a
dichroic mirror (DM) and are merged into a detector.

We perform the experiment in a cylindrical cesium room-temperature vapor cell
with $\phi$ 25 mm $\times$ 20 mm. A relevant four-level Rydberg EIT-AT scheme
with 133Cs atoms and a diagram of setup for single-channel Rydberg receiver
are illustrated in Figs. 1 (a) and (b). A green Rydberg coupling laser
($\lambda_{c}$ = 510 nm) and a near-infrared probe laser ($\lambda_{p}$ = 852
nm) counter-propagate through the cell, where the probe laser drives the
transition of $|1\rangle\to|2\rangle$
($|6S_{1/2},F=4\rangle\to|6P_{3/2},F^{\prime}=5\rangle$) with Rabi frequency
$\Omega_{p}$ and the coupling laser further excites the atoms to the Rydberg
state $|3\rangle$ ($|52D_{5/2}\rangle$) with Rabi frequency $\Omega_{c}$, thus
establishing the EIT spectroscopy to enhance the probe transmission. The EIT
signal is detected by measuring the transmission of the probe laser with an
avalanche photodiode (APD). The probe and coupling lasers keep co-linear
polarization along the y-axis and their $1/e^{2}$ beam waist $\omega_{p}$ and
$\omega_{c}$ are varied. Two MW fields with co-linear polarization along the
y-axis, denoted as a strong local oscillator (LO) field $E_{LO}$ and a weak
signal field $E_{Sig}$, are emitted from two identical horn antennas (A-info
LB-20180-SF) and incident to the Rydberg system simultaneously. In our
experiment, we fix the $E_{LO}$ = 5.5 mV/cm ($\Omega_{L}$ = 2$\pi\times$ 12.5
MHz) and $E_{Sig}$ = 1.2 mV/cm ($\Omega_{S}$ = 2$\pi\times$ 2.27 MHz) for the
experiments. The choose of LO MW filed strength is used to obtain the maximum
sensitivity, and its value could also have an effect on the response time
[26]. The LO field is resonant with the transition of two adjacent Rydberg
states $|3\rangle\to|4\rangle$ ($|52D_{5/2}\rangle\to|53P_{3/2}\rangle$) with
frequency of 5.038 GHz, leading to AT splitting of EIT spectrum, which is
proportional to the Rabi frequency of LO field, $\Omega_{L}$. While the weak
signal field couples two Rydberg states with a detuning of $\delta_{f}$ and
Rabi frequency of $\Omega_{S}$. In the presence of the LO field, the Rydberg
atoms work as an atomic mixer to down-convert the signal field to an
intermediate frequency with frequency of $\delta_{f}$, which causes an
oscillation in the transmission of the probe laser, shown in Fig. 1 (a), where
the oscillation signal is interrogated by a spectrum analyzer (ROHDE $\&$
SCHWARZ FSVA3013). Fig. 1 (c) shows the diagram of a 2-channel version of a
Rydberg receiver for increasing the response bandwidth. One channel is right
above the other, which make sure the oscillation signals in-phase detected by
the PD. The distance between two channels is 10 mm to exclude the exchange of
Rydberg atoms between two channels. Both the probe and coupling lasers are
split into two beams using 50/50 beam splitter (BS) and counter-propagate
through the cell, and the beam waists of probe and coupling laser are set to
$\omega_{p}/\sqrt{2}$ and $\omega_{c}/\sqrt{2}$, such keeping the probe
(coupling) Rabi frequency of each channel equal to the probe (coupling) Rabi
frequency in single channel, while the only difference is that the excitation
volume of each channel is equal to the half of single channel. After
propagating through the cell, two separated probe lasers pass through a
dichroic mirror (DM) and are merged into a detector.

##  3 Results and discussions

Figure 2:  (a) Measured response bandwidths versus the probe laser Rabi
frequencies ($\Omega_{p}$) for five different sets of probe beam waists. The
coupling laser beam waists were adjusted during the experiments to maintain
their slightly larger than the probe, and its Rabi frequency is fixed to
$\Omega_{c}$ = 2$\pi\times$ 4.55 MHz. The response bandwidth decreases with
the probe laser Rabi frequency for each probe beam waist. (b) The dependence
of response bandwidths on the probe beam waists with fixing probe laser Rabi
frequency and coupling laser Rabi frequency $\Omega_{p}$ = 2$\pi\times$ 5.22
MHz, $\Omega_{c}$ = 2$\pi\times$ 4.55 MHz. The atomic response bandwidth
decreases with the probe beam waist. The solid line shows the fitting result.
Inset: The output power as a function of $\delta_{f}$ obtained with a spectrum
analyzer at the probe beam waist $\omega_{p}$ = 65 $\mu m$. The 3 dB bandwidth
is about 1.5 MHz. The error bars are standard deviations of 10 measurements
for all data sets.

For extraction of the response bandwidth, we collect a series of sinusoidal
oscillation signals such as in Fig. 1 (a) and measure their output power using
the spectrum analyzer as a function of $\delta_{f}$ by changing the signal
field frequency with fixing LO field frequency. The response bandwidth of the
Rydberg receiver is defined as the frequency corresponding to the output of
the spectrum analyzer attenuating 3dB of its power. The inset in Fig. 2 (b)
shows an example that the output power attenuates with increasing the
$\delta_{f}$, corresponding to the 3 dB bandwidth about 1.5 MHz.

The response bandwidths of Rydberg system consist of the time of reaching
steady state and the transit time [32], which are related to Rabi frequency of
excitation lasers and the excitation volume. When we study the steady-state
part, the transit time is fixed, while when we study the transit time, the
time of build steady-state is fixed. In Fig. 2 (a), we demonstrate the effects
of probe Rabi frequency ($\Omega_{p}$) on the response bandwidths of the
superheterodyne receiver for five different sets of probe beam waists. Each
data set in Fig. 2 (a) is taken with fixing the coupling laser Rabi frequency
$\Omega_{c}$ = 2$\pi\times$ 4.55 MHz and the coupling laser waist slightly
larger than the probe laser waist. For giving more details of experimental
parameters, as an example for $\omega_{p}$ = 65 $\mu m$, the probe Rabi
frequency is varied from 2$\pi\times$ 5.22 MHz to 2$\pi\times$ 12.49 MHz,
corresponding to probe power varied from 0.22 $\mu$W to 1.26 $\mu$W, while the
coupling laser beam waist is $\omega_{c}$ = 80 $\mu m$ and its power is 10 mW.
All results show that the response bandwidths decrease with $\Omega_{p}$ at
certain beam waist, which is consistent with the previous work that the probe
Rabi frequency lows to 2$\pi\times$ 0.65MHz [32]. Therefore, it is
advantageous to use lower $\Omega_{p}$ to get a fast response for fast-timing
applications. This is because the steady state is rapidly built at lower
$\Omega_{p}$, in response to effective alteration of the EIT by the MW field.
The results also show the bandwidths are disparate for different probe beam
waists with the same $\Omega_{p}$.

Then, with fixing the probe Rabi frequency and coupling Rabi frequency to
$\Omega_{p}$ = 2$\pi\times$ 5.22 MHz (the probe laser power is varied from
0.22 $\mu$W to 4.70 $\mu$W to keep its Rabi frequency constant) and
$\Omega_{c}$ = 2$\pi\times$ 4.55 MHz (the coupling laser power is varied from
10.0 mW to 165.0 mW to keep its Rabi frequency constant), we investigate the
dependence of response bandwidths on the probe beam waists by varying the
waist of probe beam from 65 $\mu m$ to 300 $\mu m$. The results in Fig. 2(b)
show the response bandwidth depends strongly on the probe beam waist. The
varied beam waist changes the transit time, which is used to describe the time
that the Rydberg atoms move in and out the interaction volume [33]. The
transit time can be expressed as:

$\tau_{t}=\frac{2\cdot\omega}{v},$ (1)

where $\omega$ is the beam waist, $v$ is the average velocity of Cs atoms,
which is 218 m/s at room temperature. Using above equation, we get the transit
time is 0.59 $\mu$s and 2.75 $\mu$s for probe beam waist of 65 $\mu m$ and 300
$\mu m$, respectively. We simply fit the experimental data using
$\frac{A}{\tau_{t}}$, yielding A = 0.46 that account for other factors
contributing to the bandwidth, e.g. the time of reaching steady-state, which
is related to Rabi frequency of probe and coupling lasers. The solid line
shows the fitting result, which agrees well with the data, except for the 65
$\mu m$ width. This is because the strong divergence of the probe laser makes
the average beam waist through the cell be about 114 $\mu m$, which is larger
than 65 $\mu m$, so the effective Rabi frequency is lower than estimated. As
the above study showed, the lower Rabi frequency gets the higher response, so
the actual measured bandwidth is larger than the fitting point. Our
investigation shows that the response bandwidth is inversely proportional to
the beam waist $\omega$.

Figure 3: The response bandwidths of the Rydberg system versus the coupling
Rabi frequencies ($\Omega_{c}$) with fixed $\omega_{p}$ = 65 $\mu m$ and
$\Omega_{p}$ = 2$\pi\times$ 5.22 MHz, which shows increasing with the coupling
laser Rabi frequency. The solid line shows the fitting result. Inset: The
output power as a function of $\delta_{f}$ obtained with a spectrum analyzer
at coupling Rabi frequency $\Omega_{c}$ = 2$\pi\times$ 14.55 MHz. The 3 dB
bandwidth is about 6.8 MHz. The error bars are standard deviations of 10
measurements for all data sets.

Next, we measure the response bandwidths of the Rydberg system versus the
coupling Rabi frequencies ($\Omega_{c}$) with fixed $\omega_{p}$ = 65 $\mu m$,
$\omega_{c}$ = 80 $\mu m$ and $\Omega_{p}$ = 2$\pi\times$ 5.22 MHz. The
results are presented in Fig. 3, which shows the response bandwidth increases
with the $\Omega_{c}$ varied from 2$\pi\times$ 4.55 MHz to 2$\pi\times$ 14.55
MHz, corresponding coupling laser power varied from 10.0 mW to 102.2 mW. The
higher coupling Rabi frequencies can accelerate reach the steady state, such
getting a faster response. The bandwidth could be continued to scale if we
have stronger coupling laser power before it is saturated. The inserted
subplot shows an example for the output power as a function of $\delta_{f}$
obtained with a spectrum analyzer at coupling Rabi frequency $\Omega_{c}$ =
2$\pi\times$ 14.55 MHz. The 3 dB bandwidth is 6.8 MHz. We fit the data using
$A\cdot x^{a}+B$, yielding a = 2.2, which shows a roughly quadratic
relationship between response bandwidths and coupling Rabi frequencies. The
solid line shows the fitting result.

As the LO field breaks the EIT and the signal electric field much weaker than
the coupling laser, the response speed of Rydberg atoms is mainly determined
by the timescale of establishment EIT. So we consider a three-level system to
understand the dependence of response bandwidths on the coupling Rabi
frequencies, we theoretically analyze the relationship between steady-state
time $\tau_{s}$ and coupling laser Rabi frequency $\Omega_{c}$. The dynamics
of the system are governed by Lindblad master equation ($\hbar=1$)

$\dot{\rho}(t)=-i[H,\rho]+\mathcal{L}(\rho),$ (2)

where $\rho$ is the density matrix, $H$ is the Hamiltonian of three-level
atomic system, and $\mathcal{L}(\rho)$ is the Lindblad operator describing the
dissipation. We assume the initial condition of the system is in the ground
state $|1\rangle$, i.e. $\rho_{11}(0)=1,\rho_{22}(0)=0,\rho_{33}(0)=0$. Here
we make the approximation that the contribution of the upper transition dipole
amplitude $\rho_{23}$ is neglected. Since the energy levels 2 and 3 are barely
populated by atoms and the coherence is neglected, so $\rho_{23}$ = 0. We also
assume $\dot{\rho}_{13}=0$, which is mainly due to the fast-responding spin
wave between levels 1 and 3, whose contribution is negligible when the EIT
reaches the steady-state. By solving Eq. (2), we can obtain the time evolution
of the coherence between levels 1 and 2,

$\dot{\rho}_{12}=-(i\Delta_{p}+\gamma_{12})\rho_{12}+i\frac{\Omega_{p}}{2}-\frac{\Omega_{c}^{2}}{4[i(\Delta_{p}+\Delta_{c})+\gamma_{13}]}\rho_{12},$
(3)

where $\gamma_{ij}=(\Gamma_{i}+\Gamma_{j})/2$, $\Gamma_{i(j)}$ is the
spontanous decay rate. $\Delta_{p}$ and $\Delta_{c}$ are the detunings of the
probe laser and the coupling laser respectively, $\Omega_{p}$ and $\Omega_{c}$
are the Rabi frequencies of the probe laser and the coupling laser. We derive
the analytical solution using Eq. (3)

$\rho_{12}(t)=\frac{2\Omega_{p}\gamma_{12}+C_{1}e^{-\frac{t\sigma_{1}}{4((\Delta_{p}+\Delta_{c})-i\gamma_{13})}}+2i\Omega_{p}(\Delta_{p}+\Delta_{c})}{\sigma_{1}},$
(4)

where
$\sigma_{1}=4\gamma_{12}(\Delta_{p}+\Delta_{c})+4\Delta_{p}\gamma_{13}-4i\gamma_{12}\gamma_{13}-i\Omega_{c}^{2}+4i\Delta_{p}(\Delta_{p}+\Delta_{c})$
and $C_{1}$ is the amplitude of the coherence of the probe field, can be any
value by changing the experimental conditions. As the Im$(\rho_{12})$ denotes
the absorption of the probe laser, we find only the exponential part evolves
with time,

$e^{-\beta
t}=e^{-t\frac{4((\Delta_{p}+\Delta_{c})^{2}+\gamma_{13}^{2})\gamma_{12}+\gamma_{13}\Omega_{c}^{2}}{4((\Delta_{p}+\Delta_{c})^{2}+\gamma_{13}^{2})}},$
(5)

Then steady-state time $\tau_{s}$ is

$\tau_{s}=\frac{1}{\beta}=\frac{4((\Delta_{p}+\Delta_{c})^{2}+\gamma_{13}^{2})}{4((\Delta_{p}+\Delta_{c})^{2}+\gamma_{13}^{2})\gamma_{12}+\gamma_{13}\Omega_{c}^{2}},$
(6)

The response bandwidth $B$ is the inverse of $\tau_{s}$, so we can get
$B\propto\frac{1}{\tau_{s}}\propto\Omega_{c}^{2}$ with $\Delta_{p}$ =
$\Delta_{c}$ = 0, which also shows the quadratic relationship. Note that our
theoretical analysis is used to obtain the dependence of the bandwidth on the
coupling Rabi frequency in the regime of $\Omega_{c}>\Omega_{p}$, may not suit
to $\Omega_{c}<\Omega_{p}$.

Figure 4:  (a) Comparison of sensitivity between two-channel scheme (red
hollow circles) and single channel scheme (black triangles). Each probe
(coupling) Rabi frequency of two-channel structure is equal to the single
channel, and the total interaction volume of the two-channel structure is same
with that of single channel. The results show the sensitivity about 477
nVcm-1Hz-1/2 for the single channel and the sensitivity about 468 nVcm-1Hz-1/2
for the two-channel, which are almost same for both scheme. (b) Comparison of
response bandwidth between two-channel scheme (red hollow circles) and single
channel scheme (black triangles). The results show the two-channel scheme (
red hollow circles, 872 kHz) has an improvement compared with the single
channel scheme (black triangles,578 kHz). The improved factor is 872/578
$\approx$ 1.5, which is consistent with the reduction of beam waist by
$\sqrt{2}$, e.g. the reduction of transit time. The error bars are standard
deviations of 10 measurements for all data sets.

The above investigations reveal we get fast response for smaller probe Rabi
frequency and smaller probe beam waist. However, this will cause a reduction
of Rydberg atoms number, further lowering the sensitivity of receiver. In
order to keep faster response and high sensitivity at the same time, we design
a structure of two-channel lasers excitation, see Fig. 1(c). In two-channel
structure, the probe (coupling) Rabi frequency of each channel is same and
equal to the single channel Rabi frequency. The beam waists of probe and
coupling lasers in each channel are set to $\omega_{p}/\sqrt{2}$ and
$\omega_{c}/\sqrt{2}$, such that the laser Rabi frequency and excitation
volume of two-channel design are same as the single channel. In our
experiment, we set $\Omega_{p}$ = 2$\pi\times$ 7.55 MHz and $\Omega_{c}$ =
2$\pi\times$ 4.55 MHz for both schemes, where $\omega_{p}$ = 92 $\mu m$,
$\omega_{c}$ = 131 $\mu m$, probe power 0.92 $\mu$W and coupling power 26.8 mW
for the single channel scheme, and $\omega_{p}$ = 65 $\mu m$, $\omega_{c}$ =
93 $\mu m$, probe power 0.46 $\mu$W and coupling power 13.4 mW for the two-
channel scheme. Fig. 4(a) and Fig. 4(b) show the comparison of the sensitivity
and response bandwidth of two-channel scheme with the single channel scheme,
respectively. In Fig. 4(a), we test the detectable signal as a function of the
MW power with $\delta_{f}$ = 10 kHz for two-channel scheme (red hollow
circles) and single-channel scheme (black triangles). After calibration [35],
we get the sensitivity about 477 nVcm-1Hz-1/2 for the single channel and the
sensitivity about 468 nVcm-1Hz-1/2 for the two-channel, which shows the
sensitivity for both scheme are almost same. In Fig. 4(b), the measurement of
bandwidth shows the single channel scheme (black triangles) is about 578 kHz,
while the two-channel scheme has been improved to 872 kHz. The improved factor
is 872/578 $\approx$ 1.5, which is consistent with the reduction of beam waist
by $\sqrt{2}$, e.g. the reduction of transit time.

##  4 Conclusions

We have investigated the response bandwidth of a Rydberg atom-based
superheterodyne receiver system on the Rabi frequency of excitation lasers and
the excitation volume. We show the response bandwidth decreases with the probe
Rabi frequency, while it shows a quadratic relationship with the coupling Rabi
frequency. The investigation of dependence of the response bandwidth on the
laser waist shows it is inversely proportional to the laser waist. We achieved
a response bandwidth of the receiver about 6.8 MHz by optimizing the laser
parameters. In addition, we give a theory to describe the experimental
results, which shows good agreement. Finally, we designed a two-channel lasers
excitation structure to get faster response and keep high sensitivity
simultaneously. The improved response bandwidth is consistent with the
reduction of beam waist, e.g. the reduction of transit time. Our work has the
potential to extend the applications of Rydberg atoms in communications.

## Acknowledgements

## Funding

This work is supported by the National Key Research and Development Program of
China (2022YFA1404003); National Natural Science Foundation of China (Grant
Nos. 61835007, 12120101004, 62175136, 12241408); the Scientific Cooperation
Exchanges Project of Shanxi province (Grant No. 202104041101015); Program for
Changjiang Scholars and Innovative Research Team in University (IRT 17R70);
1331 project of Shanxi province;

## Abbreviations

local oscillator (LO), intermediate frequency (IF), electromagnetically
induced transparency (EIT).

## Availability of data and materials

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Competing interests

The authors declare that they have no competing interests.

## Authors’ contributions

J.H. and Y.J. did the experiment. Y.H. calculated the theory. All authors
contributed to all the parts of the paper. All authors read and approved the
final manuscript.

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# Foliation of a space associated to vertex operator algebra on a Riemann
surface via characters

A. Zuevsky  Institute of Mathemtics
Czech Academy of Sciences
Zitna 25, Prague
Czech Republic<EMAIL_ADDRESS>
###### Abstract.

We construct the foliation of a character space associated to a vertex
operator algebra on Riemann surfaces. The foliation is defined by means of the
structure of characters for vertex operator algebras considered on Riemann
surfaces. We prove that the computation of a general genus $g$ correlation
functions determines a foliation on the space of characters a sewn Riemann
surface. Certain further applications of the definition are proposed.

###### Key words and phrases:

Vertex operator algebra; Riemann surfaces; Character functions; Foliation

##  1\. Introduction

The theory of vertex operator algebra (VOA) characters considered on Riemann
surfaces is a rapidly developing field of studies. Algebraic nature of methods
applied in this field helps to understand and compute the structure of VOA
character spaces. On the other hand, the geometric side of VOA characters sits
in associating their formal parameters with local coordinates on a manifold.
Depending on the geometry, one can obtain various consequences for a vertex
operator algebra and its space of characters. One is able to study the
geometry of a manifold using the algebraic structure of a VOA defined on it,
and, in particular, it is important to consider foliations of associated
spaces.

In this paper we introduce the formula for an $n$-point function for a vertex
operator algebra $V$ on a genus $g$ Riemann surface $\mathcal{S}^{(g)}$
obtained as a result of sewing of lower genus surfaces $\mathcal{S}^{(g_{1})}$
and $\mathcal{S}^{(g_{2})}$ of genera $g_{1}$ and $g_{2}$, $g=g_{1}+g_{2}$.
Using this formulation, we then introduce the construction of a foliation for
the character space for vertex operator algebra with formal parameters defined
on general sewn Riemann surfaces. Computations of a vertex operator algebra
characters allow us to define foliation of the space associated to a vertex
operator algebra with formal parameter associated to a local coordinate on a
genus $g$ Riemann surface sewn from lower genus Riemann surfaces. Such
foliations are important both for studies of character space for a vertex
operator algebras and possibly for studies of smooth manifolds in the frames
of Losik’s approach [5].

##  2\. Vertex operator algebras

A Vertex Operator Algebra (VOA) [3] $(V,Y,\mathbf{1},\omega)$ consists of a
$\mathbb{Z}$-graded complex vector space
$V=\bigoplus_{n\in\mathbb{Z}}\,V_{(n)}$ where $\dim V_{(n)}<\infty$ for each
$n\in\mathbb{Z}$, a linear map $Y:V\rightarrow{\rm End}(V)[[z,z^{-1}]]$ for a
formal parameter $z$ and pair of distinguished vectors: the vacuum
$\mathbf{1}\in V_{(0)}$ and the conformal vector $\omega\in V_{(2)}$. For each
$v\in V$, the image under the map $Y$ is the vertex operator

$\displaystyle Y(v,z)=\sum_{n\in\mathbb{Z}}v(n)z^{-n-1},$

with modes $v(n)\in{\rm End}(V)$, where $Y(v,z)\mathbf{1}=v+O(z)$. Vertex
operators satisfy locality, i.e., for all $u,v\in V$ there exists an integer
$k\geq 0$ such that

$\displaystyle(z_{1}-z_{2})^{k}\left[Y(u,z_{1}),Y(v,z_{2})\right]=0.$

The vertex operator of the conformal vector $\omega$ is

$\displaystyle Y(\omega,z)=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2},$

where the modes $L(n)$ satisfy the Virasoro algebra with central charge $c$

$\displaystyle[L(m),L(n)]=(m-n)L(m+n)+\frac{m^{3}-m}{12}\delta_{m,-n}c\,\mathrm{Id}_{V}.$

We define the homogeneous space of weight $k$ to be

$\displaystyle V_{(k)}=\\{v\in V|L(0)v=kv\\},$

and we write $(v)=k$ for $v\in V_{(k)}$.

##  3\. Construction of an $n$-point function at genus $g$ Riemann surface

Recall that a conformal field theory defined on a Riemann surface
$\mathcal{S}^{(g)}$ of genus $g$ [1] is determined by the set
$\left\\{Z^{(g)}_{V}\left(v_{1},z_{1};\ldots;v_{n},z_{n}\right)\right\\}$ of
all characters for all $n$, and all choices of points
$z_{i}\in\mathcal{S}^{(g)}$, and all choices of elements $v_{i}$ of
corresponding vertex operator algebra $V$.

Here we introduce the formula for an $n$-point character for a vertex operator
algebra $V$ on a genus $g$ Riemann surface $\mathcal{S}^{(g)}$ obtained as a
result of sewing of lower genus surfaces $\mathcal{S}^{(g_{1})}$ and
$\mathcal{S}^{(g_{2})}$ of genera $g_{1}$ and $g_{2}$, $g=g_{1}+g_{2}$. The
genus $g$ $n$-point character function for $a_{1},\ldots,a_{L}\in V$ and
$b_{1},\ldots,b_{R}\in V$, $L+R=n$ inserted at
$x_{1},\ldots,x_{L}\in{\mathcal{S}}^{(g_{1})}$ and
$y_{1},\ldots,y_{R}\in{\mathcal{S}}^{(g_{1})}$, respectively, can be then
defined by

$\displaystyle
Z^{(g)}_{V}\left(a_{1},x_{1};\ldots;a_{L},x_{L}|b_{1},y_{1};\ldots;b_{R},y_{R};\Omega_{1}^{(g_{1})},\Omega_{2}^{(g_{2})},\epsilon\right)$
$\displaystyle=\sum_{u\in V}Z^{(g_{1})}_{V}\left(Y(a_{1},x_{1})\ldots
Y(a_{L},x_{L})u;\Omega_{1}^{(g_{1})}\right)$ $\displaystyle\qquad\cdot
Z^{(g_{2})}_{V}\left(Y(b_{R},y_{R})\ldots
Y(b_{1},y_{1})\overline{u};\Omega_{2}^{(g_{2})}\right).$

Here $\epsilon$ is the sewing parameter [7] and $\Omega_{i}^{(g_{i})}$,
$i=1,2$ are period matrices for Riemann surfaces ${\mathcal{S}}^{(g_{i})}$. We
also assume that there exists non-degenerate bilinear form on $V$ [3]. On the
right hand side it is present in the definition of the dual state $\bar{u}$
with respect to such form.

There exists an algebraic procedure [8] recursively relating $n$-point
correlation functions to $(n-1)$-point functions for vertex operator algebras
on the torus. In [8] we find that the genus one $n$-point correlation
functions obey

$\displaystyle Z^{(1)}_{V}(v_{1},z_{1};\ldots;v_{n},z_{n};\tau)$
$\displaystyle=\mathrm{Tr}_{V}\left((v_{1})_{wt(v_{1})-1}Y(q_{z_{2}}^{L(0)}v_{2},q_{z_{2}})\ldots
Y(q_{z_{n}}^{L(0)}v_{n},q_{z_{n}})q^{L(0)-c/24}\right)$
$\displaystyle+\sum_{k=2}^{n}\sum_{j\geq
0}P_{1+j}(z_{1}-z_{k},\tau)Z^{(1)}_{V}(v_{2},z_{2};\ldots;v_{1}[j]v_{k},z_{k};\ldots;v_{n},z_{n};\tau),$

where $P_{l}(z,\tau)$ are Weierstrass functions [4], and square bracket on the
right hand side denotes the deformed mode for vertex operator algebra on the
torus [8].

This recursion procedure [8] allows us to reduce an $n$-point character to a
function of one-point character and can be derived for any genus Riemann
surface. There exist generalizations of this theorem for higher genera. We can
represent the genus $g$ recursion formula as:

$\displaystyle
Z^{(g)}_{V}\left(v_{1},z_{1};\ldots;v_{n},z_{n};\Omega^{(g)}\right)$ (3.1)
$\displaystyle\qquad=\sum\limits_{k=0}\mathbb{P}_{k,n}^{(g)}\left(A^{(g_{1})}_{1},A^{(g_{2})}_{2}\right)\cdot
f_{V,k,n}\left(I-A^{(g_{1})}_{1}A^{(g_{2})}_{2}\right)\cdot\mathbb{O}_{k,n}^{(g)},$

where $\mathbb{P}_{k,n}^{(g)}\left(A^{(g_{1})}_{1},A^{(g_{2})}_{2}\right)$ is
a polynomial of special infinite matrices $A^{(g_{1})}_{1}$, $A^{(g_{2})}$,
[7] associated to the period matrices for Riemann surfaces
${\mathcal{S}}^{(g_{1})}$ and ${\mathcal{S}}^{(g_{2})}$,
$\mathbb{P}^{(g)}_{k,n}$, $\mathbb{O}^{(g)}_{k,n}$ are some functions of
$(v_{1},z_{1};\ldots;v_{n},z_{n})$, $f_{V,k,n}$ is a function of
$\left(I-A^{(g_{1})}_{1}A^{(g_{2})}_{2}\right)$.

##  4\. Construction of a character space foliation over Riemann surfaces for
vertex operator algebra

The construction of a vertex operator algebra (VOA) $n$-point function of
arbitrary genus gives us a hint how to define a special-type foliation of a
space related to a VOA with the formal parameter associated to coordinates on
a Riemann surface (formed from two lower genus surfaces) of genus $g$ by means
of VOA characters.

A $p$-dimensional foliation $\mathcal{F}$ of an $n$-dimensional manifold
$\mathcal{M}$ is a covering by a system of domains $\\{U_{i}\\}$ of
$\mathcal{M}$ together with maps

$\phi_{i}:U_{i}\to\mathbb{C}^{n},$

such that for overlapping pairs $U_{i}$, $U_{j}$ the transition functions

$\varphi_{ij}:{\mathbb{C}}^{n}\rightarrow{\mathbb{C}}^{n},$

defined by

$\varphi_{ij}=\phi_{j}\phi_{i}^{-1},$

take the form

$\varphi_{ij}(x,y)=(\varphi_{ij}^{1}(x),\varphi_{ij}^{2}(x,y)),$

where $x$ denotes the first $n-p$ coordinates, and $y$ denotes the last $p$
co-ordinates. That is,

$\displaystyle\varphi_{ij}^{1}:$
$\displaystyle\mathbb{C}^{n-p}\to\mathbb{C}^{n-p},$
$\displaystyle\varphi_{ij}^{2}:$
$\displaystyle\mathbb{C}^{p}\to\mathbb{C}^{p}.$

In this paper, we would like to foliate the space

$\mathcal{M}_{n}=V^{\otimes^{n}}\times\mathcal{S}^{(g)},$

where $V$ is a vertex operator algebra, and $\mathcal{S}^{(g)}$ is a Riemann
surface of genus $g$.

######  Proposition 1.

The characters (3.1) determine a foliation on the space
$\mathcal{M}=\bigoplus_{n\geq 0}\mathcal{M}_{n}$.

###### Proof.

We consider a system of charts $\\{U_{m}\\}$, $m\in\mathbb{N}$, covering the
Riemann surface part of the space $\mathcal{M}_{n}$. Suppose certain points
$x_{i}$, $1\leq i\leq p\leq n$ are situated on $\mathcal{S}^{(g_{1})}$-part
and other $x_{i}$, $p+1\leq i\leq n$ are on $\mathcal{S}^{(g_{2})}$-part of
the Riemann surface $\mathcal{S}^{(g)}$. On the Riemann surface
$\mathcal{S}^{(g_{1})}$ of genus $g_{1}$, let us choose a particular point
$x_{i}$, $1\leq i\leq p\leq n$ with associated local coordinate $z_{i}$ around
$x_{i}$ which belongs to the domain $U_{i}$ of the system $\\{U_{m}\\}$. For
another point $x_{j}$, $1\leq j\neq i\leq p$ on $\mathcal{S}^{(g_{1})}$-part
we associate a domain $U_{j}$ intersecting with $U_{i}$. It can be always done
since we can move points $x_{i}$, $1\leq i\leq p$ around
$\mathcal{S}^{(g_{1})}$-part of the Riemann surface $\mathcal{S}^{(g)}$.

Now let us compute a $p\leq n$-point $V$-character
$Z^{(g_{1})}(v_{1},z_{1};\ldots;v_{p},z_{p})$ by means of the recursion
procedure described above reducing $p$-point character to a one-point
character $Z^{(g_{1})}(v_{i},z_{i})$ associated to our chosen point $x_{i}$ in
the domain $U_{i}$. Due to properties of vertex operators, the vertex algebra
elements $v_{1},\ldots,v_{p}$ can be chosen so that the expansion of the
character $Z^{(g_{1})}(v_{1},z_{1};\ldots;v_{p},z_{p})$ has a dimension $p$
polynomial nature, [1]. The procedure of computation of
$Z^{(g_{1})}(v_{1},z_{1};\ldots;v_{p},z_{p})$ by the reduction to one-point
character $Z^{(g_{1})}(v_{i},z_{i})$ defines a map

$\phi_{i}:V^{\otimes^{p}}\times\mathcal{S}^{(g_{1})}\rightarrow\mathbb{C}^{p}.$

The result of computation of $Z^{(g_{1})}(v_{1},z_{1};\ldots;v_{p},z_{p})$ can
re-written to reduce to another one-point function $Z^{(g_{1})}(v_{j},z_{j})$
associated to another coordinate $z_{j}$ around a point $x_{j}$ in the domain
$U_{j}$. Thus, we can define the inverse map

$\phi_{j}^{-1}:\mathbb{C}^{p}\rightarrow
V^{\otimes^{p}}\times\mathcal{S}^{(g_{1})}.$

For any set $x$ of pairs of non-coinciding points among $x_{1},\ldots,x_{p}$
on $\mathcal{S}^{(g_{1})}$ we then define the function for the intersecting
domains $U_{i}$ and $U_{j}$ on the Riemann surface $\mathcal{S}^{(g_{1})}$

$\varphi^{(g_{1})}_{ij}(x)=\phi_{i}\phi_{j}^{-1}(x).$

Exactly similar procedure is then performed for any set of $y$ non-coinciding
points among $x_{p+1},\ldots,x_{n}$ on the Riemann surface
$\mathcal{S}^{(g_{2})}$ of genus $g_{2}$ to define
$\varphi^{(g_{2})}_{ij}(y)$. Then the transition function is given by the map

$\varphi_{ij}:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n},$
$\varphi_{ij}(x,y)=\left(\varphi_{ij}^{(g_{1})}(x),\varphi_{ij}^{(g_{2})}(y)\right).$

As a result, we have defined a foliation of the space $\mathcal{M}_{n}$ by
means of characters for corresponding vertex operator algebras with the formal
parameter associated to a coordinate on a Riemann surface. The total space
$\mathcal{M}=\bigoplus_{n\geq 0}\mathcal{M}_{n}$ is foliated similar to
$\mathcal{M}_{n}$. ∎

The general situation is more complicated. Namely, we have to vary $n$, $p$,
the set of vertex operator elements $\left\\{v_{i}\right\\}$, $i=1,\ldots,n$,
and $g_{1}$, $g_{2}$ with $g=g_{1}+g_{2}$ (note also, that another parameter
can be provided by the type of grading of the vertex operator algebra).

##  5\. Further directions

The recursion procedure [8] plays a fundamental role in the theory of
characters for vertex operator algebras. It provides relations between $n$\-
and $n-1$\- point character functions. In our foliation picture, the recursion
procedure brings about relations among leaves of foliation. We work in the [2]
approach to definition and computation of cohomologies of vertex operator
algebras. Taking into account the above definitions and construction, we would
like to develop a theory of characteristic classes for vertex operator
algebras, and, in particular, for the space
$\mathcal{M}=\bigoplus_{n=0}^{\infty}V^{\otimes^{n}}\times\mathcal{S}^{(g)}$.
This can have a relation with Losik’s work [5] proposing a new framework for
singular spaces and new kind of characteristic classes. Losik defines a smooth
structure on the leaf space $M/\mathcal{F}$ of a foliation $\mathfrak{F}$ of
codimension $n$ on a smooth manifold $M$ that allows to apply to
$M/\mathcal{F}$ the same techniques as to smooth manifolds. Losik defined the
characteristic classes for a foliation as elements of the cohomology of
certain bundles over the leaf space $M/\mathcal{F}$. Similar to Losik’s
theory, we use certain bundles (of correlation functions) over a foliated
space.

On the other hand, we would like to apply methodology of vertex algebras in
order to complete Losik’s theory of characteristic classes. One can mention a
possibility to derive differential equations [2] for characters on separate
leaves of foliation. Such equations are derived for various genera and can be
used in frames of Vinogradov theory [6]. The structure of foliation (in our
sense) can be also studied from the automorphic function theory point of view.
Since on separate leaves one proves automorphic properties of characters, on
can think about ”global” automorphic properties for the whole foliation.

## Acknowledgements

The author would like to thank H. V. Lê, P. Somberg, and P. Zusmanovich for
related discussions. Research of the author was supported by the GACR project
18-00496S and RVO: 67985840.

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