# NLP-based Cross-Layer 5G Vulnerabilities Detection via Fuzzing Generated
Run-Time Profiling

Zhuzhu Wang  School of Systems and Enterprises
Stevens institute of technology
Hoboken, USA
<EMAIL_ADDRESS>Ying Wang  School of Systems and Enterprises
Stevens institute of technology
Hoboken, USA
<EMAIL_ADDRESS>
###### Abstract

The effectiveness and efficiency of 5G software stack vulnerability and
unintended behavior detection are essential for 5G assurance, especially for
its applications in critical infrastructures. Scalability and automation are
the main challenges in testing approaches and cybersecurity research. In this
paper, we propose an innovative approach for automatically detecting
vulnerabilities, unintended emergent behaviors, and performance degradation in
5G stacks via run-time profiling documents corresponding to fuzz testing in
code repositories. Piloting on srsRAN, we map the run-time profiling via
Logging Information (LogInfo) generated by fuzzing test to a high dimensional
metric space first and then construct feature spaces based on their timestamp
information. Lastly, we further leverage machine learning-based classification
algorithms, including Logistic Regression, K-Nearest Neighbors, and Random
Forest to categorize the impacts on performance and security attributes. The
performance of the proposed approach has high accuracy, ranging from $93.4\%$
to $95.9\%$, in detecting the fuzzing impacts. In addition, the proof of
concept could identify and prioritize real-time vulnerabilities on 5G
infrastructures and critical applications in various verticals.

###### Index Terms:

5G systems, LogInfo, fuzzing test, natural language processing, machine
learning

##  I Introduction

The fifth generation (5G) cellular network holds promise for achieving the
vision of universal connectivity. It enables various application scenarios and
verticals, including connected vehicles, remote robotic surgery, etc., which
has a profound impact on society[1]. Meanwhile, there is a paradigm shift in
the core network architecture of telecommunication systems in the 5G networks,
which have transitioned to software-defined infrastructure and thereby reduced
reliance on hardware-based network functions. This change has dramatically
improved network efficiency, performance, and robustness, however, it has also
made the network more vulnerable, as software systems are often more
susceptible to disruption than hardware systems. Therefore, rigorous testing
against vulnerabilities becomes critical in 5G and future G systems [2].
Security concerns, connection failures, and performance latency have often
been a topic of interest to academia and industry.

Fuzzing technology is an effective statistic-based testing technique in
detecting unknown vulnerabilities of a product or protocol by automating the
generation and execution of a large number of random test cases. Many
researchers used fuzzy testing techniques to check the security of
communication protocols, including Packet Forwarding Control
Protocol[PFCP][3], 5G Non-Access-Stratum[NAS] protocol[4], etc. However,
existing research on fuzz testing suffers from challenges in utilizing fuzz
testing in network protocols and software implementations: scalability of
search space.

The exponential increase of fuzzing testing case corresponding to the search
space leads to the challenge of providing a complete picture of the entire
security threat or vulnerability with mere fuzzing and raise insurmountable
difficulties to the computation complexity when search space scales. To
address the scalability issue in fuzz testing search space, formal
verification uses mathematical models to precisely describe a system.
Recently, Megha et al.[5] used an automatic verification tool called ProVerif
to verify the 5G Extensible Authentication Protocol and Authentication and Key
Agreement[5G-EAP-AKA] protocol. Jingjing et al.[6] based on the Scyther model
checker to study the 5G Extensible Authentication Protocol and Transport Layer
Security[5G-EAP-TLS] protocol. However, these methods rely on an in-depth
understanding of the 5G specifications and require high time and labor costs
to build the mathematical model.

To reduce challenges in computation complexity and reliance on human expertise
in specific projects for vulnerability detection, our objective in this study
is to semantically predict vulnerabilities based on the profiling traces of
various 5G platforms. We propose an innovative approach for detecting
vulnerabilities and unintended emergent behaviors in general 5G stacks
automatically based on NLP models with fuzz testing profiling via LogInfo. The
method uses semantic information for vulnerability detection, which not only
predicts potential detection before the occurrence of failures and
vulnerability but also breaks the application scenario barrels and allows for
an implementation-independent and cross-platform solution. The proposed
solution can be applied to different protocols as well as software stacks.

Event logging records detailed run-time information for a system, which
supports engineers in understanding system behavior and tracking potential
problems. The log file is the text data generated by the printout code
embedded in the program by the program developer to assist in debugging, which
is used to record the variable information and program execution status when
the program is running, etc. And therefore log messages are a valuable
resource used for many tasks, such as performance monitoring [7] and anomaly
detection [8]. It can locate the specific log and event information and also
locate the exception request instance. Based on the NLP technology,
calculating the center distance according to the time interval, we use log
files to successfully detect if there is a vulnerability under the fuzzing
test.

The contributions of this paper are summarized as follows:

  * •

An NLP-based LogInfo vulnerability detection solution for 5G fuzz testing is
designed and implemented. The BERT-based model is trained for sentence-to-
vector, and the weighed periodic distances are calculated to determine
vulnerabilities and unintended emergent behaviors.

  * •

The design solution discovers insights for potential vulnerabilities and
unintended emergent behaviors via the pattern comparison between fuzzed
testing and standardized testing. With the featured scalability and
automation, it enables zero-day attack resistance and enhances real-time
sustainability.

  * •

Machine-learning-based classification models are developed for classifying the
type of vulnerabilities. We also demonstrate its practical value of light-
weighted and high accuracy.

The remainder of this paper is organized as follows. Section II summarizes the
related work. Section III describes the necessary preliminary knowledge of the
5G protocol stack. In section IV, we present the design and implementation of
the proposed approach. Section V discusses and analyzes the experimental
results. Section VI concludes the paper and discusses the future work plan,
respectively.

##  II Related work

Previous work in cellular network security includes both formal verification
[9] [10] and fuzzy testing [11][12]. A formal model is constructed and
security properties are extracted to formally validate the model. Formal
verification can be effective in detecting vulnerabilities in protocol
designs. Manual methods of analysis and verification Manual methods of
vulnerability analysis and verification have produced some attacks, but it
requires extensive experience from security researchers and lacks
universality. Fuzzing has also been applied to security test analysis of
protocols, but it requires manual sample construction and is a semi-automatic
approach. In [13], the author utilizes NLP as a guide to discover
vulnerabilities in LTE specifications. In addition, many studies have used
text analysis techniques to automatically discover various errors in different
domains, such as SSL/TLS implementations[14], API Misuse Detection [15].

Different from the above works, our research provides a systematic framework
to discover vulnerabilities from real log files, which is an expression of the
specific implementation of the protocol. As detailed guidance on the
implementation of protocols is still lacking, code flaws can introduce new
security risks at the implementation level of 5G protocols. Especially for
lack of testing for unexpected data during the implementation of 5G protocol
stacks, which can lead to potential security risks not being detected
promptly, thus affecting the robustness of the protocol implementation. By
analyzing the related log files, we can identify vulnerabilities and propose
solutions promptly.

##  III BACKGROUND

###  III-A 5G protocol stack

A protocol stack is a prescribed hierarchy of software layers, and the 5G
protocol stack between the UE and eNB shown in Fig. 1 can be divided into
three layers. Starting from the bottom layer, these layers are divided into
the physical layer, the data link layer, and the network layer.

  * •

Physical layer: The physical layer transmits information through the air
interface between the UE and the gNB.

  * •

The data link layer is responsible for connecting the physical layer to the
network layer and is divided into three parts into three subparts. Starting
from the bottom layer, these subparts are the Media Access Control (MAC)
protocol, the Radio Link Control (RLC) protocol, and the Packet Data
Convergence Protocol (PDCP).

  * •

Network layer: The network layer, similar to the data link layer, is divided
into three subparts. The subcomponents of the network layer are the non-access
layer (NAS), the radio resource control (RRC), and the Internet protocol (IP).
Among them, NAS is divided into Mobility Management (NAS-MM) and NAS Session
Management (NASSM), which interact with Access and Mobility Function (AMF) and
Session Management Function (SMF), respectively.

The log files used for this experiment are based on the work in [16], and
piloting RRC protocols in the srsRAN[17] platform. RRC protocols are designed
as one of the critical cellular protocols for radio resource management, which
not only provides the upper layers with wireless resource parameters from the
network system but also control the main parameters and behavior of the lower
layers.

Figure 1: 5G Stack Architecture.

##  IV The proposed approach

###  IV-A System description

Fig. 1 illustrates a typical description of 5G architecture. There are three
main entities (roles) and we explain them below.

Figure 2: 5G architecture.

  * •

User Equipment (UE): represents a subscriber’s mobile device, such as a cell
phone, tablet, or modem.

  * •

Radio access network (RAN): represents the telecommunications network that
connects UE to the core network via a radio connection. There are two main
forms: Next Generation e-NodeB (ng-eNB) and Next Generation NodeB (gNB). The
gNB allows 5G UE to connect with the 5G NG(New Generation) core using the 5G
NR(New Radio)air interface.

  * •

Core network (CN): represents the network that provides services to mobile
subscribers through the RAN. It also serves as a gateway to other networks,
such as to the public switched telephone network or the public cloud.

###  IV-B Approach overview

Fig. 3 shows the designed 5G stacks’ unintended behaviors and vulnerability
detection via fuzzing-generated run-time profiling based on NLP. Here, we used
command-level fuzzing to detect vulnerabilities. Specifically, exchanged
legitimate messages will be observed and collected to the fuzzing message
candidate pool, and at the command level, commands are replaced by other
commands in the same physical channel to test whether any communication error
state occurs. The detailed running profile will be recorded in the log files.

The proposed method is shown in Fig. 3 and mainly consists of 4 parts:
generating and collecting log files, natural language processing, dimensional
reduction, and classification. Below is a detailed explanation of each step:

Figure 3: System design and specific experimental protocols.

####  IV-B1 Data collection

We fuzz the RRC protocols and tunneled Non-Access Stratum(NAS) protocols
through message recorder, injection, and modification. The srsRAN system is
monitored using packet capture software and logs are generated using packet
capture software. When srsRAN executes, the gNB is initiated, and its
respective outputs are sent to a log file. According to whether the
communication was successfully achieved, we divide the logs into two parts:
successfully connected and failed connected.

####  IV-B2 Natural language processing

Log files are semi-structured language, we could not use them directly. And
therefore, we choose the sentence-BERT model to obtain the sentence vector.
Specifically, the log files are pre-processed and input into the model for
training to obtain a semantic representation of the text: the sentence vector.

####  IV-B3 Dimensional reduction

To avoid dimensional disasters and speed up the training process, we use the
t-SNE model to perform down-scaling operations.

####  IV-B4 Classification

Without losing generality, we chose common classification algorithms as our
model, such as Logistic Regression, KNeighbors, Random Forest, etc.

###  IV-C Experimental setup

####  IV-C1 Profiling Acquiring via Fuzz Testing

Fuzzy testing is the generation and injection of unexpected inputs called
”fuzzy” inputs, by slightly changing a fundamental change and introducing each
change into the system to observe its effect. Here, the srsUE executable has
been modified to replace the field values in the message of interest with the
fuzzy values of the current test case. Then, the new fuzzy message is sent
from the UE to the gNodeB(gNB). During the fuzzing, the recorded states
include the following information: ‘message time’, ‘original bytes’, ‘RRC
channel’, ‘message type’, and ‘physical channel’. When the monitor in gNB
detects the rrcConnectionSetupComplete message, the testing case will be
terminated and labeled as a successful connection. When the monitor in gNB
cannot detect the rrcConnectionSetupComplete message within a predefined
timeout limit (600 seconds in the proof of concept experiments), it is
considered a failed connection. The inclusion of this keyword indicates that
the RRC connection is completed and communication can proceed. Profiling files
that do not include this specific keyword indicate an effective fuzzing that
leads to an unsuccessful connection. By annotating the profiling files with a
search for the keyword rrcConnectionSetupComplete, all files are labeled as
either success ($=1$) or fail ($=0$).

However, relying on keyword searching results is a fragile approach that can
be misleading in case of changes to the keyword expression or format changes
in the profiling. Thus, we use keyword searching as the ground truth and
propose a semantic-based robust detection mechanism.

The binary annotation provides critical information on the effectiveness or
impact of fuzzing cases. However, annotating a case as successful ($=1$) is
not necessarily an indicator that the fuzzing did not have an impact on
communication. The latency generated by the fuzzing could lead to severe
Denial of Service in scaled cases with a large number of users and
significantly reduce the system’s capacity. To address this, we also annotate
the profiling with the duration of the RRC connection.

Log messages in the profiling files take the format like this: Timestamp
[Layer ] Level Content (e.g., 17:52:25.246 [RLC ] Info DRB1 Tx SDU), among
them, PHY-layer log messages have some additional details: Timestamp [Layer]
Level [Subframe] Channel: Content (e.g., 17:52:26.094 [PHY1] Info [05788]
PDSCH: l_crb=1, harq=0, snr= 22.1 dB, CW0: tbs=55, mcs=22, rv=0, crc=OK, it=1,
dec_time=12 us).

The srsENB profiling files could be configured by log levels to display each
layer of the network stack correspondingly. Contents of each profiling file
may include the time stamp of each step, and log information for semantic
information abstraction. It details the steps in the srsRAN platform enabling
hex message output. We adapt our models to run on various log levels,
including debug, info, warning, error, and none. A visual example of the
srsENB profiling file is shown in Fig.4.

Figure 4: Example of srsENB events.

####  IV-C2 Data pre-processing

As shown in Fig.4, each of the log files contains a varying number of lines
and each line includes a different amount of words, including non-alphanumeric
characters. Firstly, we use regular expressions to remove these non-
alphanumeric characters. It should be emphasized that the sentences input to
the subsequent model here are those LogInfo belonging to the same timestamp.
In addition, for the timestamp information, we extract it and save it
separately but not as a corpus for training.

####  IV-C3 Definition and calculation of distance

As raw logs are often semi-structured textual messages, they need to be
converted into digital features firstly. And therefore, we choose natural
language processing tools to map every log event to matrices. The aim is to
learn semantic information about the logs [18]. Here, we adopt the
Bidirectional Encoder Representations from the Transformers (BERT) model. BERT
is a useful tool for calculating text similarity, which is published by
researchers at Google AI Language [19]. The BERT model leverages the attention
mechanism in the transformer, training methods of the mask language model, and
word order information encoded in the embedding to implement the simultaneous
use of contextual and word order information. The BERT model can be trained by
combining the contexts in all layers of the neural network to obtain
contextually relevant bi-directional feature representations. The trained
model is thus able to understand the semantics in conjunction with the
context.

In the actual training process, we mainly refer to[20] and use[21] as a pre-
training model for our corpus training.

Algorithm 1 Sentenc2vec based on sentence-BERT & t-SNE

0: $N_{th}$ log file’s sentences $\chi_{i}$ =
$\\{s_{1}^{N},s_{2}^{N},...,s_{n}^{N}\\}$ after pre-processing

0: Coordinate mapping for each sentence: $\textbf{Y}^{N}$

1: For $N_{th}$ log file, given a set of input sentences $\chi_{i}$ =
$\\{s_{1}^{N},s_{2}^{N},...,s_{n}^{N}\\}$

2: Training these sentences to obtain sentence vectors:
$[\textbf{x}_{1}^{N},\textbf{x}_{2}^{N},...,\textbf{x}_{n}^{N}]$ (via
sentence-BERT)

3: for each $i,j\in[1,n]$ do

4: if $i\neq j$ then

5:
$p_{j|i}=\frac{exp(-||\textbf{x}_{i}^{n}-\textbf{x}_{j}^{n}||^{2}/2\delta_{i}^{2})}{\sum_{k\neq
i}exp(-||\textbf{x}_{k}^{n}-\textbf{x}_{i}^{n}||^{2}/2\delta_{i}^{2})}$;

6: else

7: set $p_{i|i}=0$

8: end if

9: $p_{ij}=\frac{p_{j|i}+p_{i|j}}{2N}$;

10: end for

11: Initialize $2$-dimensional vectors [$\textbf{y}_{1}^{N}$, …,
$\textbf{y}_{n}^{N}$] based on $\mathcal{N}(0,10^{-4}\textit{I})$

12: for each $i,j\in[1,n]$ do

13: if $i\neq j$ then

14: $q_{j|i}=\frac{(1+||\textbf{y}_{i}-\textbf{y}_{j}||^{2})^{-1}}{\sum_{k\neq
l}(1+||\textbf{y}_{k}-\textbf{y}_{l}||^{2})^{-1}}$;

15: else

16: set $q_{i|i}=0$;

17: end if

18: $q_{ij}=\frac{q_{j|i}+q_{i|j}}{2N}$;

19: end for

20: Given learning rate $\eta$ and momentum $\alpha(t)$

21: Gradient optimization formula: $\frac{\delta C}{\delta
y_{i}}=4\sum_{j}(p_{ij}-q_{ij})(y_{i}-y_{j})(1+||{y}_{i}-{y}_{j}||^{2})^{-1}$;

22: repeat

23:
$\textbf{y}_{i+1}^{N}=\textbf{y}_{i}^{N}+\eta\frac{dC}{dY}+\alpha(t)(\textbf{y}_{i-1}^{N}-\textbf{y}_{i-2}^{N})$;

24: until Minimize:
$KL(P||Q)=\sum\limits_{i}\sum\limits_{j}p_{ij}\log\frac{p_{ij}}{q_{ij}}$

After the training of the BERT model, we obtain a sentence vector of dimension
512. To facilitate the computation and reduce memory, the extraction of valid
information, and the rejection of useless information, we reduce the
dimensionality of the sentence vector. The T-Distribution Stochastic Neighbour
Embedding(t-SNE) was chosen for the dimensionality reduction algorithm. The
t-SNE is a machine learning algorithm for dimension reduction, which
visualizes high-dimensional data by giving the position of each data point on
two maps, proposed by Laurens van der Maaten and Geoffrey Hinton in 2008 [22].
As a nonlinear dimensionality reduction algorithm, t-SNE is very suitable for
visualizing high-dimensional data down to 2 or 3 dimensions, by using a small
distance to generate a large gradient for dissimilar points to repel them. But
it is not infinitely large to avoid dissimilar issues being too far away.
Algorithm 1 shows the specific vector training and dimensionality reduction
process.

After inputting the log files into algorithm 1, we get a 2-dimensional
coordinates vector for each sentence from LogInfo. For each log event, there
is a unique timestamp corresponding to it. And therefore, we calculate the
time duration from starting time to each log event that occurred. The accuracy
of the calculation is measured in seconds. Under this condition, each second
interval will contain multiple log events. For every running time interval, we
calculate the corresponding center point. Here, it is important to note that
because the first 4 seconds are the time that trying to build a connection,
and therefore, the values of $T$ are ${0,4,5,6,...,40}$. Then, we calculate
the distance of every point. The specific calculation formula is shown in
Algorithm 2.

Algorithm 2 Features’ value calculation method based on center distance

0: $N_{th}$ log file’s 2-dimensional vectors $\textbf{Y}^{N}$ =
$\\{y_{t}^{1},y_{t}^{2}\\}$ based on the algorithm 1

0: Every second’s ($t$) distance of central point ${Distance}_{t}$

1: Initialize time interval $T=\\{0,4,...,40\\}$

2: for each $t\in T$ do

3: $\bar{y}_{t}^{1}=\frac{\sum_{i=1}^{n}y_{i}^{1}}{n}$;

4: $\bar{y}_{t}^{2}=\frac{\sum_{i=1}^{n}y_{i}^{2}}{n}$

5: end for

6: for each $t\in T$ do

7: Calculate distance:
${Distance}_{t}=\sqrt{\bar{y}_{t}^{1}+\bar{y}^{2}_{t}}$;

8: end for

To validate our approach, we select mainstream binary classification
algorithms as our training model. These classification algorithms are provided
by scikit-learn and are very easy to select and train, including Support
Vector Machine, k-Nearest Neighbors, Logistic Regression, etc.

Figure 5: The visualization of the partial log files

##  V Experimental results and analysis

In this section, we test the validity of our model with real log files. As
mentioned earlier, these log files were generated by fuzzing tests applied to
the 5G radio resource control (RRC) protocol piloting on srsRAN. Fig. 5
presents the result of the partial log files after distance calculation. Every
point represents the distance from the center of all event logs generated
during this second to the origin when the system runs to this second.

Fig. 6 presents the Receiver Operating Characteristics(ROC) and Area Under the
ROC(AUC) of different classifiers. Here, we choose five common binary
classification algorithms from sci-kit-learn as our classification models.

Figure 6: ROC and AUC results corresponding to different algorithms.

Through further analysis, we found that the main difference between failed log
files and passed log files occurs between the 10th and 17th seconds as shown
in Fig. 7, which means we do not need to use the whole log file to input the
prediction model. As shown in Fig. 8, we generate the accuracy distributions
using different time intervals and find that only takes until the tenth second
for the accuracy rate to reach a high level, which means we could determine in
advance if the connection is successful.

(a) Time interval from 0s to 25s

(b) Time interval from 26s to 40s

Figure 7: The distribution of data features

Figure 8: Accuracy distribution changes at different time intervals

Figure 9: Distribution of time to establish the connection between UE and ENB

##  VI Conclusions and future work

This paper studies the vulnerability detection method from an NLP perspective,
aiming to provide a new solution idea. We have demonstrated the design and
implementation of our method for real log files generated by fuzzing tests
based on the 5G RRC protocol and achieved good practical performance.
Detecting vulnerabilities automatically in 5G systems is difficult and complex
work and our approach provides a new possible solution idea.

As shown in Fig. 9, although these cases show the final result is connecting
successfully, there are obvious and different delay situation that happens
compared to the log files without any fuzzing injection. In real situations,
some severe delays are equivalent to connection failures. And therefore, what
we do next is to further analyze the relationship between fuzzing types and
delay.

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#  Deformed Kac-Moody Algebra and Noncommutative Fermi Theory in Two-
Dimensions

<EMAIL_ADDRESS>and M.R.B.
<EMAIL_ADDRESS>
Cybersecurity and Systems Unit, Universiti Sains Islam Malaysia,
Bandar Baru Nilai, 71800 Nilai, Negeri Sembilan, Malaysia

##  1 Introduction

Noncommutative field theories has been studied extensively in the previous two
decades [1, 2, 3, 4, 5, 6]. The noncommutativity appears naturally in quantum
mechanics when we study the motion of a charged particle moving in two-
dimensions under the presence of a constant, perpendicular magnetic field[2,
7] and more broadly in the context of quantum Hall effect[8, 9, 10]. Other
studies suggest the appearance of non-commutativity in the study of open
string field theory in the presence of $B$-field[14, 15, 16, 17]. Very
recently, a possible application in the emerging field of magnetic skyrmions
has been reported [11, 12, 13]

It is a well-established fact that the quantum corrections to the classical
action breakdown the gauge symmetry which is referred to as ” anomaly”. The
non-commutative counterpart of Chiral anomaly and its ramifications has been
studied in many papers ( for example, see [18, 19, 20, 21, 22, 23]).

Noncommutative field theories in two-dimensions have attracted a lot of
attention during the last couple of years; the general structure of gauge
theories can be read from [24], the formulation of two-dimensional
noncommutative gravity was given in [25]. The noncommutative counterpart of
other quantum regimes in two-dimensions like Sine-Gordon model and Thirring
model were presented in [26]and [27]respectively.

The deformation of the conformal symmetry in two-dimensions and the deformed
Kac-Moody algebra has been studied before [28, 29, 30]. In [30], Balachandran
and his collaborators investigated the deformed Kac-Moody algebra from a
general perspective, and they ensure that there are two ways for deforming the
Kac-Moody algebra: the first deformation is obtained by deforming the
oscillators while the second is obtained directly by deforming the generators
of Kac-Moody algebra . These approaches lead to different deformations of the
Kac-Moody algebra. It is also shown [28] that by twisting the commutation
relations between the creation and annihilation operators, the conformal
invariance can still be maintained in the two-dimensional Moyal plane.

In this paper, we continue along these lines and show how to obtain a
deformed333”deformed”throughout this paper means noncommutatively deformed.
Kac-Moody algebra starting from the massless noncommutative Fermi theory in
two-dimensions. We also calculate the higher-order corrections of the deformed
Kac-Moody algebra in powers of the antisymmetric tensor $\theta_{\mu\nu}$.

This article is organised as follows: in section 2, we give a general
derivation of the deformed Kac-Moody algebra starting from a massless
noncommutative Fermi theory in two-dimensions. In section 3, we calculate the
higher-order corrections to the deformed Kac-Moody algebra. The paper ends
with a conclusion and an appendix about the properties of the Moyal
$\star$-product.

##  2 The noncommutative deformation of Kac-Moody algebra

Affine Lie algebra is an infinite-dimensional Lie algebra constructed out of a
finite-dimensional simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ whose
generators obey the commutation relations [31], [32]

$[J^{A},J^{B}]=f^{AB}_{C}\;J^{C}$ (1)

where $f^{AB}_{C}$ are the antisymmetric structure constants. The affine Kac-
Moody commutation relations are defined by

$[J^{A}_{m},J^{B}_{n}]=f^{AB}_{C}\;J^{C}_{m+n}+m\;c\;\delta_{m+n,0}\delta^{AB}$
(2)

Where $\\{J^{A}_{m}:m\;\in\;\mathbb{Z}\\}$ are the corresponding Laurent modes
and $c$ is the central charge.

One way to visualize the algebra is to define the generators on
$\mathbb{S}^{1}$=$\\{z\;\in\;\mathbb{C}:|z|=1\\}$ parametrised by
$\theta\;\in\;[0,2\pi)$, therefore we can expand the generators as

$J^{A}(\theta)=\sum_{m\;\in\;\mathbb{Z}}\;e^{im\theta}\;J^{A}_{m}$ (3)

We plug this expansion in the general Kac-Moody commutator (2) . After some
straightforward calculations we find the following relations between the
currents

$[J^{A}(\theta_{1}),J^{B}(\theta_{2})]=2\pi\;f^{AB}_{C}\;J^{C}(\theta_{1})\;\delta(\theta_{1}-\theta_{2})-2\pi
i\;c\;\delta^{AB}\;\delta^{{}^{\prime}}(\theta_{1}-\theta_{2})$ (4)

where we have used the identities
$2\pi\;\delta(\theta)=\sum_{m\;\in\;\mathbb{Z}}\;e^{im\theta}$ and $-2\pi
i\delta(\theta)=\sum_{m\;\in\;\mathbb{Z}}\;m\;e^{im\theta}$.
The term $2\pi i\;c\;\delta^{AB}\;\delta^{{}^{\prime}}(\theta_{1}-\theta_{2})$
is called a ” Schwinger term”.

From physical perspective, the Kac-Moody algebra appears for example in the
context of non-Abelian bosonization in two dimensions [33] and in the study of
$SU(2)$ chiral spin currents algebra in the Luttinger-Tomonaga liquid [34].

Our main purpose in this paper is to construct a similar algebra starting from
the noncommutative Fermi theory in two dimensions.

We adopt the following conventions throughout this paper
:$\gamma^{0}=\sigma^{1}\;,\gamma^{1}=i\sigma^{2}$ and the chirality operator
in two-dimensions is
$\gamma_{3}=\gamma^{1}\gamma^{0}=i\sigma^{2}\sigma^{1}=\sigma^{3}$ , where
$\sigma^{\alpha}$ stands for the Pauli matrices.

The Euclidean version of the theory is obtained readily after the Wick
rotation ($x^{0}\rightarrow-ix^{2}$), that also changes the signature of the
metric from $(+,-)$ to $(-,-)$ and $\gamma^{o}\rightarrow-i\gamma^{2}$. The
Wick rotation of the gauge potential is given by $A_{0}(x)\rightarrow
iA_{2}(x)$.

The massless Fermi action in two-dimensional noncommutative Euclidean space-
time is given by

$S=\int d^{2}x\;\overline{\psi}(x)\star
i\gamma^{\mu}(\partial_{\mu}-i\;\hat{A}_{\mu}^{a}\;T^{a})\star\psi(x).$ (5)

$\psi$ is the Fermionic field, $\gamma_{\mu}$’s are the gamma matrices in two-
dimensions, $\hat{A}_{\mu}$ is a one-form noncommutative gauge field. The
covariant derivative is defined as
$D_{\mu}=\partial_{\mu}-i\hat{A^{a}_{\mu}}(x)T^{a}$ and $\star$ is the star-
product ( see appendix A for more details on the $\star$-product) which we
define it as

$f(x)\;\star\;g(x)=e^{i\frac{\theta_{\mu\nu}}{2}\;\frac{\partial}{\partial{\zeta_{\mu}}}\;\otimes\frac{\partial}{\partial{\eta_{\nu}}}}\;f(x+\zeta)\;g(x+\eta)\mid_{\zeta=\eta=0}.$
(6)

Then the corresponding path integral is

$\mathcal{Z}=\int\mathcal{D}\overline{\psi}\;\mathcal{D}\psi\;e^{S}$ (7)

where $S$ is the action functional.

The noncommutative Chiral-vector and vector currents are defined in two-
dimensions respectively as

$\displaystyle\hat{j^{a\;\mu}_{3}}(x)=\overline{\psi}(x)\;\gamma^{\mu}\gamma_{3}T^{a}\;\star\psi(x),$
(8)
$\displaystyle\hat{j^{a\;\mu}}(x)=\overline{\psi}(x)\;\gamma^{\mu}T^{a}\star\psi(x).$
(9)

Under chiral transformation with an infinitesimal $\beta^{a}(x)$ parameter ,
we can write the chirally rotated spinors as

$\displaystyle\psi^{{}^{\prime}}(x)=\mathrm{exp}[i\beta^{a}(x)T^{a}\gamma_{3}]\star\;\psi(x),$
(10)
$\displaystyle\overline{\psi^{{}^{\prime}}}(x)=\overline{\psi}(x)\star\;\mathrm{exp}[i\beta^{a}(x)T^{a}\gamma_{3}],$
(11)

where $\gamma_{3}$ is the chirality operator in two-dimensions.
After this chiral transformation, the path integral becomes

$\mathcal{Z}^{{}^{\prime}}=\int\mathcal{D}\overline{\psi}\;\mathcal{D}\psi\;\mathbb{J}\;\star\mathrm{exp}[\int
d^{2}x\;\big{(}\overline{\psi}\star
i\gamma^{\mu}(\partial_{\mu}-i\hat{A}^{a}_{\mu}T^{a})\star\psi+\beta^{a}(x)\star
D_{\mu}\hat{j}_{3}^{a\;\mu}(x)\big{)}]$ (12)

$\mathbb{J}$ is the Jacobian of the transformation. We know from [27] that the
noncommutative anomaly in two-dimensions is
$\epsilon^{\mu\nu}\hat{F}_{\mu\nu}(x)$ up to a constant , where
$\hat{F}_{\mu\nu}$ is the noncommutative two-forms field strength444 The
noncommutative field strength is the ordinary two-forms field strength plus
some terms in order of $\theta$ which we call it the antisymmetric constant
tensor[14, 35, 36].The first order corrections to the one-form noncommutative
vector field and two-forms noncommutative field strength are respectively
$A^{1}_{\gamma}=-\frac{1}{4}\theta^{\kappa\lambda}\big{\\{}A_{\kappa},\partial_{\lambda}A_{\gamma}+F_{\lambda\gamma}\big{\\}}$
(13)
$F^{1}_{\gamma\rho}=-\frac{1}{4}\theta^{\kappa\lambda}\big{(}\big{\\{}A_{\kappa},\partial_{\lambda}F_{\gamma\rho}+D_{\lambda}F_{\gamma\rho}\big{\\}}-2\big{\\{}F_{\gamma\kappa},F_{\rho\lambda}\big{\\}}\big{)}$
(14) ; thus the divergence of Chiral current is

$\langle
D_{\mu}\;\hat{j^{a\;\mu}_{3}}(x)\rangle=\partial_{\mu}\langle\hat{j^{a\;\mu}_{3}}(x)\rangle+f^{abc}\;\hat{A^{b}_{\mu}}(x)\star\langle\hat{j^{c\mu}_{3}}(x)\rangle=-\frac{i}{4\pi}\epsilon^{\mu\nu}\;\hat{F^{a}_{\mu\nu}}(x)$
(15)

Here we assumed that the gauge group generators satisfy the normalization
condition ${\rm Tr}(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$ and the averaged
quantities are defined by

$\langle\mathcal{O}(x)\rangle=\int\mathcal{D}\overline{\psi}\mathcal{D}\psi\;\mathcal{O}(x)\star\mathrm{exp}[\int
d^{2}x\;\overline{\psi}\star
i\gamma^{\mu}(\partial_{\mu}-i\hat{A^{a}_{\mu}}T^{a})\star\psi].$ (16)

Under vector-like transformations

$\displaystyle\psi^{{}^{\prime}}=\mathrm{exp}[i\;\alpha^{a}(x)\;T^{a}]\star\psi(x),$
(17)
$\displaystyle\overline{\psi^{{}^{\prime}}}=\overline{\psi}(x)\star\mathrm{exp}[-i\;\alpha^{a}(x)\;T^{a}]$
(18)

we obtain a similar relation for the vector current :

$\langle
D_{\mu}\;\hat{j^{a\;\mu}}(x)\rangle=\partial_{\mu}\langle\hat{j^{a\;\mu}}(x)\rangle+f^{abc}\;\hat{A^{b}_{\mu}}(x)\star\langle\hat{j^{c\mu}}(x)\rangle=0.$
(19)

We normalize the antisymmetric tensor according to $\epsilon^{12}=1$, note
that the Dirac-delta function $\delta(x-y)$ and the $\epsilon^{\mu\nu}$
symbols are defined as the ordinary commutative case.

In the massless case, as a special property of living in two-dimensions, we
may write the following relation between the Chiral and vector currents:

$\hat{j^{a\mu}_{3}}(x)=\overline{\psi}(x)\;\star\gamma^{\mu}T^{a}\gamma_{3}\psi(x)=-\epsilon^{\mu\nu}\;\overline{\psi}(x)\;\star
T^{a}\gamma_{\nu}\psi(x)=-\epsilon^{\mu\nu}\;\hat{j^{a}_{\nu}}(x).$ (20)

We functionally differentiate (15) with respect to $\hat{A^{b}_{\nu}}(y)$ and
send $\hat{A^{b}_{\nu}}$ to zero:

$\displaystyle\frac{\delta}{\delta\hat{A^{b}_{\nu}}(y)}\langle
D_{\mu}\hat{j^{a\;\mu}_{3}}(x)\rangle=\partial_{\mu}\langle\mathcal{T}\hat{j^{a\;\mu}_{3}}(x)\star\hat{j^{b\;\nu}}(y)\rangle+f^{abc}\;\delta^{2}(x-y)\star\langle\hat{j^{c\;\nu}_{3}}(x)\rangle$
(21)
$\displaystyle=-\frac{i\epsilon^{\mu\nu}\;\delta_{ab}}{2\pi}\partial{\mu}\delta^{2}(x-y).$

In the same manner, we obtain the following relation for the noncommutative
vector current :

$\displaystyle\frac{\delta}{\delta\hat{A^{b}_{\nu}}(y)}\langle
D_{\mu}\hat{j^{a\;\mu}}(x)\rangle=\partial_{\mu}\langle\mathcal{T}\hat{j^{a\;\mu}}(x)\star\hat{j^{b\;\nu}}(y)\rangle+f^{abc}\;\delta^{2}(x-y)\star\langle\hat{j^{c\;\nu}}(x)\rangle=0$
(22)

We apply the Bjorken-Johonson-Low (BJL) prescription to convert the time-
ordering $\mathcal{T}$ in the Lorentz covariant calculations to the normal
time-ordering $T$ which is not necessarily Lorentz covariant.555Suppose
$\hat{X},\hat{Y}$ some noncommutative quantities then the BJL prescription can
be build as follows  $\lim_{k_{2}{\to}\infty}\int
d^{2}x\;e^{ikx}\langle\mathcal{T}\hat{X}(x)\star\hat{Y}(0)\rangle=0$
$\displaystyle\lim_{k_{2}{\to}\infty}\int d^{2}x\;e^{ikx}\langle
T\hat{X}(x)\star\hat{Y}(0)\rangle=\int
d^{2}xe^{ikx}\langle\mathcal{T}\hat{X}(x)\star\hat{Y}(o)\rangle-\lim_{k_{2}{\to}\infty}\int
d^{2}x\;e^{ikx}\langle\mathcal{T}\hat{X}(x)\star\hat{Y}(0)\rangle$ For more
details about the usage of BJL prescription in the noncommutative case,see
[37] where this prescription is applied in both higher-derivatives scalar
field and noncommutative scalar field theory.

The equations (21) and (22) can be written as

$\partial_{\mu}\langle
T\hat{j^{a\;\mu}_{3}}(x)\star\hat{j^{b2}}(y)\rangle+f^{abc}\langle\hat{j^{c\;2}}\rangle\star\delta^{2}(x-y)=-\frac{i\delta_{ab}}{2\pi}\partial_{1}\delta^{2}(x-y)$
(23)

$\partial_{\mu}\langle
T\hat{j^{a\;\mu}_{3}}(x)\star\hat{j^{b1}}(y)\rangle+f^{abc}\langle\hat{j^{c\;1}}\rangle\star\delta^{2}(x-y)=0$
(24)

$\partial_{\mu}\langle
T\hat{j^{a\;\mu}}(x)\star\hat{j^{2}}(y)\rangle+f^{abc}\langle\hat{j^{c\;2}}\rangle\star\delta^{2}(x-y)=0$
(25)

Since the left hand side of the equation (24) is defined in terms of a time-
ordered product so we eliminate the right-hand side and set it to zero. We use
the relations
$\hat{j^{b\;1}}(y)=i\hat{j^{b\;2}_{3}}(y)$,$\hat{j^{c\;1}_{3}}(x)=i\hat{j^{c\;2}}(x)$
and $\partial_{\mu}\hat{j^{a\;\mu}}=\partial_{\mu}\hat{j^{a\;\mu}_{3}}=0$ to
write the equations (23), (24) and 25 as

$[\hat{j^{a\;2}_{3}}(x),\hat{j^{b\;2}}(y)]_{\star}\;\star\delta(x^{2}-y^{2})+f^{abc}\hat{j^{c\;2}}(x)\star\delta^{2}(x-y)=-\frac{i\delta_{ab}}{2\pi}\partial_{1}\delta^{2}(x-y)$
(26)

$[\hat{j^{a\;2}_{3}}(x),\hat{j^{b\;2}_{3}}(y)]_{\star}\;\star\delta(x^{2}-y^{2})+f^{abc}\hat{j^{c\;2}}(x)\star\delta^{2}(x-y)=0$
(27)

$[\hat{j^{a\;2}}(x),\hat{j^{b\;2}}(y)]_{\star}\;\star\delta(x^{2}-y^{2})+f^{abc}\hat{j^{c\;2}}(x)\star\delta^{2}(x-y)=0$
(28)

Note that $\partial_{\mu}\hat{j^{a\;\mu}_{3}}\neq 0$ in the massive Fermi
theory.

Let us define the following quantities

$\displaystyle\hat{j^{a\;2}_{L}}(x)=\frac{1}{2}[\hat{j^{a\;2}}(x)-\hat{j^{a\;2}_{3}}(x)],\;\;\;\;\;\hat{j^{a\;2}_{R}}(x)=\frac{1}{2}[\hat{j^{a\;2}}(x)+\hat{j^{a\;2}_{3}}(x)],$
(29)
$\displaystyle\hat{j^{b\;2}_{L}}(x)=\frac{1}{2}[\hat{j^{b\;2}}(x)-\hat{j^{b\;2}_{3}}(x)],\;\;\;\;\;\hat{j^{b\;2}_{R}}(x)=\frac{1}{2}[\hat{j^{b\;2}}(x)+\hat{j^{b\;2}_{3}}(x)],$
(30)
$\displaystyle\hat{j^{c\;2}_{L}}(x)=\frac{1}{2}[\hat{j^{c\;2}}(x)-\hat{j^{c\;2}_{3}}(x)],\;\;\;\;\;\hat{j^{c\;2}_{R}}(x)=\frac{1}{2}[\hat{j^{c\;2}}(x)+\hat{j^{c\;2}_{3}}(x)].$
(31)

and use them finally obtain the equal-time commutation relations

$[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]_{\star}=-f^{abc}\hat{j^{c\;2}_{L}}(x)\star\delta(x^{1}-y^{1})+\frac{i\delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1}),$
(32)

$[\hat{j^{a\;2}_{R}}(x),\hat{j^{b\;2}_{R}}(y)]_{\star}=-f^{abc}\hat{j^{c\;2}_{R}}(x)\star\delta(x^{1}-y^{1})-\frac{i\delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1}).$
(33)

Comparing 32and 33 with the deformed Kac-Moody algebra obtained from deforming
oscillators [30], we found in contrast the central charge ( Schwinger) term to
be unaffected by non-commutativity. This can be explained using the properties
of $\star$-product, namely the fact that Dirac delta function in the
noncommutative case is defined as $\int f(x)\star\delta^{2}(x-y)\;d^{2}x=f(y)$
for test function $f(x)$ which is identical to the commutative case plus the
fact that $\partial_{\mu}\star\delta^{2}(x-y)=\partial_{\mu}\delta^{2}(x-y)$.
Thus one can view the deformed Kac-Moody algebra introduced in [30] as the
most general deformation for all $SU(N)$ Kac-Moody algebras while in our work
is for specific $SU(2)$ Kac-Moody algebra with chiral and vector currents from
noncommutative Fermi theory in two-dimensions.

##  3 Deformed Kac-Moody algebra to all orders in $\theta$

In order to make a strong conclusion about the nature of deformed Kac-Moody
algebra 32 and 33, it is interesting to explore the higher-order corrections
in the antisymmetric noncommutative tensor $\theta^{\mu\nu}$.

The expansion of 32 gives

$\displaystyle[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]_{\star}=[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]-\sum_{n=1}^{\infty}\frac{f^{abc}}{n!}\hat{j^{c\;2}_{L}}(x)(\frac{1}{2}\theta^{\mu\nu}\partial^{\leftarrow}_{\mu}\partial^{\rightarrow}_{\nu})^{n}\delta(x^{1}-y^{1})$
(34)

where $[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]$ has the ordinary Kac-
Moody algebra structure and equals to

$[\hat{j^{a\;2}_{L}}(x),\hat{j^{b\;2}_{L}}(y)]=-f^{abc}\hat{j^{c\;2}_{L}}(x)\delta(x^{1}-y^{1})+\frac{i\delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1})$
(35)

Analogously we may write the higher-order expansion for 33 as

$\displaystyle[\hat{j^{a\;2}_{R}}(x),\hat{j^{b\;2}_{R}}(y)]_{\star}=[\hat{j^{a\;2}_{R}}(x),\hat{j^{b\;2}_{R}}(y)]-\sum_{n=1}^{\infty}\frac{f^{abc}}{n!}\hat{j^{c\;2}_{R}}(x)(\frac{1}{2}\theta^{\mu\nu}\partial^{\leftarrow}_{\mu}\partial^{\rightarrow}_{\nu})^{n}\delta(x^{1}-y^{1})$
(36)

and the corresponding ordinary Kac-Moody algebra as

$[\hat{j^{a\;2}_{R}}(x),\hat{j^{b\;2}_{R}}(y)]=-f^{abc}\hat{j^{c\;2}_{R}}(x)\delta(x^{1}-y^{1})+\frac{i\delta_{ab}}{4\pi}\partial_{1}\delta(x^{1}-y^{1})$
(37)

From the higher-order expansion we note that the deformed Kac-Moody algebra
can be written as ordinary Kac-Moody algebra plus infinitely many Lie algebra
structures of the form

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n!}\hat{j^{a\;2}_{j}}(x)(\frac{1}{2}\theta^{\mu\nu}\partial^{\leftarrow}_{\mu}\partial^{\rightarrow}_{\nu})^{n}\hat{j^{b\;2}_{j}}(y)-\sum_{n=1}^{\infty}\frac{1}{n!}\hat{j^{b\;2}_{j}}(y)(\frac{1}{2}\theta^{\mu\nu}\partial^{\leftarrow}_{\mu}\partial^{\rightarrow}_{\nu})^{n}\hat{j^{a\;2}_{j}}(x)$
(38)
$\displaystyle=-\sum_{n=1}^{\infty}\frac{f^{abc}}{n!}\hat{j^{c\;2}_{j}}(x)(\frac{1}{2}\theta^{\mu\nu}\partial^{\leftarrow}_{\mu}\partial^{\rightarrow}_{\nu})^{n}\delta(x^{1}-y^{1})$

where $j=L,R$.

##  4 Conclusion

We have explicitly obtained the deformed Kac-Moody algebra starting from the
two-dimensional noncommutative Fermi theory. These deformations are different
from those given in paper [30] where the central charge term is modified by
the noncommutativity for deformations that are obtained directly from
deforming the oscillators. In our case the Schwinger term ( i.e. central
charge term ) is not affected by the non-commutativity. Finally, we conclude
that the deformed Kac-Moody algebra in the two-dimensional noncommutative
Fermi theory can be written as ordinary Kac-Moody algebra plus infinitely many
embedded Lie algebraic structures. Possible applications of our results can be
found in the study of $SU(2)$ chiral currents in the noncommutative Luttinger-
Tomonaga liquid and non-Abelian Bosonization of massless non-commutative Fermi
theory in two dimensions.

## Acknowledgment

One of the authors (M.W.A) is grateful to G. Thompson and K. S. Narain for
discussions on quantum anomalies. We thank our referee for the insightful
comments and suggestions that improved this paper. We are grateful to USIM for
support.

##  Appendix A Moyal $\star$-Product

Let $C^{\infty}(\mathbb{R}^{2n})$ be the space of real smooth functions,
$f:\mathbb{R}^{2n}\rightarrow\mathbb{C}^{n}$.Then given $f,g$ $\epsilon$
$C^{\infty}(\mathbb{R}^{2n})$, the Moyal $\star$-product is defined as

$f(x)\;\star\;g(x)=e^{i\frac{\theta_{\mu\nu}}{2}\;\frac{\partial}{\partial{\zeta_{\mu}}}\;\otimes\frac{\partial}{\partial{\eta_{\nu}}}}\;f(x+\zeta)\;g(x+\eta)\mid_{\zeta=\eta=0}$
(39)

where $\theta_{\mu\nu}=\theta\epsilon_{\mu\nu}$ is a real anti-symmetric
constant, and leads to $C^{\star}$-algebra. By expanding the previous formula
up to first order in $\theta$, we find that

$f\;\star
g=f\;q+\frac{i}{2}\;\theta^{\mu\nu}\;\partial_{\mu}f\;\partial_{\nu}g+\mathcal{O}(\theta^{2}).$
(40)

The star product has many properties like

$f\;\star g\mid_{\theta}=g\;\star f\mid_{-\theta}.$ (41)

Suppose $h$ is another smooth function. Then we have the following cyclic
property :

$\int_{-\infty}^{\infty}\;d^{D}x\;(f\star g\star
h)=\int_{-\infty}^{\infty}\;d^{D}x\;(h\star f\star
g)=\int_{-\infty}^{\infty}\;d^{D}x\;(g\star h\star f).$ (42)

The Leibniz rule holds in the $\star$-product case:

$\partial_{\mu}(f\star g)=(\partial_{\mu}f)\star g+f\star(\partial_{\mu}g).$
(43)

The ordinary commutators are generalized in the noncommutative case to the
Moyal $\star$-brackets

$[f,g]_{\star}=f\star\;g-g\star\;f.$ (44)

Then the deformed Jacobi identity would be written as

$[f,[g,h]_{\star}]_{\star}+[h,[f,g]_{\star}]_{\star}+[g,[h,f]_{\star}]_{\star}=0.$
(45)

It is important to notice the fact that any arbitrary commutator with Moyal
$\star$-product involves both ordinary commutators and anti-commutators [38]:

$f\star\;g-g\star\;f=[f,g]_{(\star,even)}+\\{f,g\\}_{(\star,odd)}$ (46)
$[f,g]_{(\star,even)}=[f,g]+(\frac{i}{2})^{2}\;\theta^{\mu\nu}\theta^{\rho\sigma}[\partial_{\mu}\partial_{\rho}f,\partial_{\nu}\partial_{\sigma}g]+\mathcal{O}(\theta^{4})$
(47)
$\\{f,g\\}_{(\star,odd)}=\frac{i}{2}\theta^{\mu\nu}\\{\partial_{\mu}f,\partial_{\nu}g\\}+(\frac{i}{2})^{3}\theta^{\mu\nu}\theta^{\rho\sigma}\theta^{\kappa\lambda}\\{\partial_{\mu}\partial_{\rho}\partial_{\kappa}f,\partial_{\nu}\partial_{\sigma}\partial_{\lambda}g\\}+\mathcal{O}(\theta^{5})$
(48)

We define the symmetric and anti-symmetric parts of a $\star$-product as

$\displaystyle(f\star g)_{s}=\frac{1}{2}(f\star g+g\star
f)=f\;\mathrm{cos}(\frac{1}{2}\overleftarrow{\partial_{\mu}}\theta^{\mu\nu}\overrightarrow{\partial_{\nu}})\;g$
(49) $\displaystyle\simeq
fg+(\frac{i}{2})^{2}\;\theta^{\mu\nu}\theta^{\rho\sigma}\;\partial_{\mu}\partial_{\rho}f\;\partial_{\nu}\partial_{\sigma}g+\mathcal{O}(\theta^{4}),$

$\displaystyle(f\star g)_{a}=\frac{1}{2}(f\star g-g\star
f)=if\;\mathrm{sin}(\frac{1}{2}\overleftarrow{\partial_{\mu}}\theta^{\mu\nu}\overrightarrow{\partial_{\nu}})\;g$
(50)
$\displaystyle\simeq(\frac{i}{2})\theta^{\mu\nu}\;\partial_{\mu}f\partial_{\nu}g+(\frac{i}{2})^{3}\theta^{\mu\nu}\theta^{\rho\sigma}\theta^{\kappa\lambda}\;\partial_{\mu}\partial_{\rho}\partial_{\kappa}f\;\partial_{\nu}\partial_{\sigma}\partial_{\lambda}g+\mathcal{O}(\theta^{5}).$

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