# QSimPy: A Learning-centric Simulation Framework for Quantum Cloud Resource
Management

Hoa T. Nguyen1, Muhammad Usman2,3, and Rajkumar Buyya1
<EMAIL_ADDRESS>{muhammad.usman<EMAIL_ADDRESS>1Cloud Computing and Distributed Systems (CLOUDS) Laboratory,
School of Computing and Information Systems, The University of Melbourne,
Parkville, 3052, Victoria, Australia 2School of Physics, The University of
Melbourne, Parkville, 3052, Victoria, Australia 3Data61, CSIRO, Clayton, 3168,
Victoria, Australia

###### Abstract

Quantum cloud computing is an emerging computing paradigm that allows seamless
access to quantum hardware as cloud-based services. However, effective use of
quantum resources is challenging and necessitates robust simulation frameworks
for effective resource management design and evaluation. To address this need,
we proposed QSimPy, a novel discrete-event simulation framework designed with
the main focus to facilitate learning-centric approaches for quantum resource
management problems in cloud environments. Underpinned by extensibility,
compatibility, and reusability principles, QSimPy provides a lightweight
simulation environment based on SimPy, a well-known Python-based simulation
engine for modeling dynamics of quantum cloud resources and task operations.
We integrate the Gymnasium environment into our framework to support the
creation of simulated environments for developing and evaluating reinforcement
learning-based techniques for optimizing quantum cloud resource management.
The QSimPy framework encapsulates the operational intricacies of quantum cloud
environments, supporting research in dynamic task allocation and optimization
through DRL approaches. We also demonstrate the use of QSimPy in developing
reinforcement learning policies for quantum task placement problems,
demonstrating its potential as a useful framework for future quantum cloud
research.

###### Index Terms:

quantum cloud computing, quantum resource management, reinforcement learning,
discrete-event simulation

##  I Introduction

Quantum computing, characterized by its capacity to tackle impractically
complex computation tasks for classical systems, offers revolutionary
potential across multiple domains, such as drug discovery [1], machine
learning [2], and optimization [3]. However, this emerging paradigm faces
significant practical barriers, notably the operational challenges and
substantial costs associated with operating quantum hardware [4]. The advent
of quantum cloud computing [5, 6] offers a pathway to overcome these barriers,
providing access to quantum computing capabilities through cloud services.
Figure 1 illustrates the overview of the contemporary quantum cloud computing
environment and an example of how end-users interact with cloud-based quantum
applications or services.

Figure 1: A high-level view of quantum cloud computing environments and a
sample procedure from user invocation to quantum task execution.

Currently, classical systems are utilized to host these quantum applications
[7], which define the quantum execution logic using a quantum software
development kit (SDK), such as Qiskit [8]. Typically, end-users can manually
build and compile quantum circuits within a cloud-based integrated development
environment, such as Jupyter Notebook, and submit them for execution in a
quantum system. Another comprehensive approach involves deploying quantum
applications or functions as cloud-based services, offering API endpoints for
external invocation [9]. Through this service model, the user can invoke a
quantum service via a cloud gateway; then, a corresponding quantum circuit is
generated and compiled. These compiled circuits are then enqueued within a
broker, which is responsible for the task scheduling and the subsequent
dispatch to the selected quantum system for execution.

Despite the benefits, integrating quantum computing into cloud environments
introduces complex resource management challenges. These challenges arise from
the unique properties of quantum computations, such as the different execution
models compared to classical computation tasks [9] and the current limitations
of Noisy intermediate-scale quantum (NISQ) hardware [10]. Additionally, the
heterogeneous nature of quantum cloud resources [7], varying in architectural
design and capability, complicates the effective allocation and scheduling of
quantum tasks.

Current literature primarily focuses on quantum algorithms and physical
quantum operation simulations [11], with less emphasis on the holistic
simulation of quantum cloud environments that include both hardware and
service management aspects. Most existing quantum simulators, such as Qiskit
Aer [8], or cloud simulators, such as CloudSim [12], do not support the
modeling of quantum cloud computing environments. In this domain, our prior
work, iQuantum [13], stands as a pioneering Java-based framework utilizing
CloudSim for modeling quantum cloud environments. iQuantum has laid a critical
foundation; however, its Java-centric nature presents challenges when
interfacing with the predominantly Python-based quantum software and machine
learning libraries, despite potential Java-Python interoperability solutions
such as Py4J [14]. Given the predominance of Python in the quantum software
landscape [11], as seen in SDKs like Qiskit and Cirq and machine learning
libraries like TensorFlow, PyTorch, and Ray, the demand for a Python-native
simulation tool is evident. Furthermore, the specific demands of machine
learning integration for dynamic resource management in quantum cloud
computing pose a steep learning curve for those who are unfamiliar with
CloudSim-base methodologies in the cloud computing domain.

TABLE I: Representative works related to our study Work |  | Simulation
---
Approach
Focus |  | Programming
---
Language
| ML
---
Integration
Dataset Support
qgym [15] | - |  | Quantum compiler
---
benchmarking
Python | Gymnasium | -
DRAS-CQSim [16] | Discrete-event |  | RL-based HPC
---
cluster scheduling
Python |  | Tensorflow,
---
Keras
-
Jawaddi et al. [17] | Discrete-event |  | RL-based Classical
---
Cloud Environments
Java |  | Open AI Gym,
---
Keras
-
iQuantum [13] | Discrete-event |  | Quantum Cloud-Edge
---
Environments
Java | - | CSV
QSimPy (This work) | Discrete-event |  | RL-based Quantum
---
Resource Management
Python |  | Gymnasium,
---
Ray
QASM, CSV

To address these considerations and close the gap, we propose QSimPy, a
Python-based simulation framework specifically tailored for creating learning-
centric environments for modeling and evaluating resource management
strategies in quantum cloud computing. QSimPy is not only the Python-based
counterpart of iQuantum but also represents a starting point for the paradigm
shift toward a more expansive, integrated simulated environment compatible
with well-known frameworks in both the quantum computing and machine learning
ecosystems. Our framework aims to facilitate the development and evaluation of
resource management policies using reinforcement learning (RL) techniques. It
models not only quantum computation but also resource orchestration in a
cloud-based quantum computing environment, enabling the simulation of various
scenarios and policies to optimize resource utilization effectively. By
offering a learning-centric environment, QSimPy allows researchers and
practitioners to design and evaluate RL-based strategies for resource
management, addressing both the performance variability of quantum devices and
the fluctuating demands of quantum computing tasks. This simulation tool is
developed to advance quantum cloud resource management by offering a platform
for prototyping, modeling and conducting experimentation, ultimately enabling
the development of more efficient and robust resource management strategies
for quantum cloud computing.

###  I-A Main Contributions

This paper proposes QSimPy, a Python-based simulation framework that leverages
the SimPy [18] discrete-event simulation approach to focus on reinforcement
learning techniques for resource management in quantum cloud environments. The
main contributions and novel aspects of our work are summarized as follows:

  * •

We present the system model specifically designed for the quantum cloud
resource management problem, tailored for the reinforcement learning
approaches. This model serves as a foundational hypothetical basis for
simulating and analyzing the dynamics of quantum task execution in cloud
environments.

  * •

We propose QSimPy, a Python-based discrete-event simulation framework,
leveraging the extensive ecosystem of Python libraries tailored for machine
learning and quantum computing. QSimPy offers a flexible and powerful
environment for developing complex resource management algorithms with machine
learning techniques. Its design focuses on ease of integration with external
libraries, making it a potential starting point for collaborations between
researchers and practitioners in quantum and cloud computing to enhance
quantum cloud resource management.

  * •

We demonstrate QSimPy’s capabilities through its application to a quantum task
placement problem. This proof of concept implementation showcases how QSimPy
integrates seamlessly with industry-grade RL environments and algorithm
libraries, such as Gymnasium and Ray RLlib. Our results confirm the
framework’s functionality and potential as a benchmark tool for future quantum
cloud computing resource management research.

These contributions highlight QSimPy’s innovative aspects and its potential to
significantly impact the field of quantum cloud computing. By integrating
state-of-the-art RL methodologies with a flexible simulation environment,
QSimPy offers a novel tool that enhances the study and practical management of
cloud-based quantum computing resources.

###  I-B Paper Organization

The rest of the paper is organized as follows: Section II reviews the relevant
literature, identifying the research gaps that our study aims to bridge.
Section III outlines the system model and problem formulation for resource
management within a quantum cloud computing environment tailored to
reinforcement learning approaches. Section IV details the design, primary
components, and implementation of the QSimPy framework. Section V demonstrates
an exemplary application of our framework, implementing RL-based task
placement policies in a heterogeneous quantum cloud environment. Section VI
concludes the paper by summarizing our key findings and outlining future
research directions.

##  II Related Work

This section presents a brief review of related work, illustrating the unique
gap our research addresses in the domain of quantum cloud computing
simulation. We summarise several representative works and precedent studies in
Table I.

In the high-performance computing (HPC) cloud computing domains, DRAS-QCsim
[16] employs reinforcement learning for efficient cluster scheduling. However,
it does not specifically address quantum cloud environments. Other endeavors,
such as the work of Jawaddi et al. [17], explore resource management within
classical cloud environments using reinforcement learning yet do not cross the
bridge into supporting the quantum computing domain.

In the domain of quantum computing simulations, a variety of frameworks exist,
each with its distinct focus and methodology. For example, qgym [15] focuses
on quantum compiler benchmarking without delving into resource management of
the quantum cloud environment. Considering the modeling and simulation of
quantum cloud environments, our prior work, iQuantum [13], can be considered
as one of the first-of-its-kind frameworks. iQuantum is a Java-based modeling
and simulation framework that initiates the quantum cloud simulation, based on
the core simulation engine of CloudSim [12] but with a specific focus on
quantum cloud-edge computing environment modeling and simulation without
considering the integration of machine learning or reinforcement learning
techniques for optimization.

Our proposed QSimPy aims to fill the noticeable gap in the current landscape
of simulation software. It stands out by adopting a discrete-event approach
tailored to the intricate problem of quantum cloud resource management through
reinforcement learning approaches. With QSimPy, we have developed a Python-
based framework that is not only compatible with the massive ecosystem of
software development kits, libraries, and other frameworks in quantum
computing and machine learning but also simplifies the usage, customization,
and further extension for more complex quantum cloud environment modeling and
simulation. Unlike its predecessors, such as CloudSim [12] and iQuantum [13],
QSimPy is designed to be compatible with industry-graded machine learning
environments and libraries, such as Gymnasium and Ray RLlib, facilitating the
implementation and validation of advanced resource management strategies. Our
framework is also built to support a variety of datasets, including OpenQASM
[19] quantum circuits and CSV-based quantum circuit features, ensuring its
versatility and applicability across various synthetic quantum computing tasks
dataset generation. Our framework not only addresses the limitations of
existing simulators but also pioneers a pathway for future studies and
developments in quantum cloud resource management algorithms. By bridging the
gap between classical and quantum resource orchestration, QSimPy serves as a
potential starting point for event-based simulation and optimization of
quantum tasks within cloud environments, a step towards further investigation
to harness the full potential of quantum cloud computing.

Figure 2: An overview of the QSimPy Framework.

##  III RL-based Problem Formulation and Model

###  III-A Problem Formulation

The quantum cloud resource management (QCRM) problem can be defined as the
process of determining the optimal policy to place incoming quantum tasks in
the best-suited quantum computation resource so that the optimization
objective can be achieved successfully. This objective can vary from different
criteria, including 1) time-awareness (such as minimizing the total task
completion time), 2) precision-awareness (such as minimizing the error rates
in the task execution results), and 3) utilization-awareness (such as
maximizing the overall resource utilization). A complex problem can also take
multiple objectives into account at the same time.

###  III-B Reinforcement Learning-based QSTAR Model

Let QCRM problem $\mathcal{M}$ in the reinforcement learning setting be a set
of components denoted as a QSTAR model as follows:

$\mathcal{M}=\\{\mathbb{Q},\mathbb{S},\mathbb{T},\mathbb{A},\mathbb{R}\\}$ (1)

where:

  1. 1.

Quantum Resources: $\mathbb{Q}=\\{Q_{1},Q_{2},...,Q_{n}\\}$ denoted the list
of $n$ quantum cloud computation resources, or quantum nodes (QNodes). Each
QNode $Q$ can also be modeled as $Q=\\{q_{1},q_{2},...,q_{k}\\}$ where
$q_{1},...,q_{k}$ are $k$ different computation capacity metrics such as the
number of qubits, quantum volume [20], circuit layer operation per seconds
(CLOPS) [21], native quantum gates, qubit connectivity, and error rates.

  2. 2.

Quantum Tasks: $\mathbb{T}=\\{T_{1},T_{2},...,T_{m}\\}$ denotes the list of
$m$ quantum tasks (QTask) that need to be executed on one of the available
quantum nodes. Each QTask $T$ can be presented as
$T=\\{t_{1},t_{2},...,t_{l}\\}$ where $t_{1},...,t_{l}$ are $l$ different
attributes of QTask such as the number of qubits, circuit layers, required
quantum gates, qubit connectivity, and other quality of service (QoS) metrics.

  3. 3.

State space: $\mathbb{S}$ is all possible state of the agent’s observation
from the quantum cloud environment, which includes 1) information of all
available quantum resources (QNodes) and 2) information of current QTasks to
be scheduled. The feature vector of $n$ quantum nodes $Q_{i}\in\mathbb{Q}$,
each quantum node has $k$ features, at time step $\tau\in\mathcal{T}$ can be
presented as:

$\mathcal{F}^{\mathbb{Q}}_{\tau}=\\{f^{q_{z}}_{Q_{i}}|\forall
Q_{i}\in\mathbb{Q},1\leq i\leq n,1\leq z\leq k\\}$ (2)

where $i$ is the index of the quantum node, and $z$ is the index of the
quantum node’s metric.

The feature vector of current quantum task (QTask) $T_{j}\in\mathbb{T}$ need
to be scheduled, which has $l$ features at time step $\tau\in\mathcal{T}$ can
be presented as:

$\mathcal{F}^{T_{j}}_{\tau}=\\{f^{t_{p}}_{T_{j}}|T_{j}\in\mathbb{T},1\leq
j\leq m,1\leq p\leq l\\}$ (3)

where $j$ is the index of the current quantum task, and $p$ is the index of
the current quantum task’s metric.

Therefore, the state space $\mathbb{S}$ of the overall system can be
represented as follows:

$\mathbb{S}=\\{s_{\tau}|s_{\tau}=(\mathcal{F}^{\mathbb{Q}}_{\tau},\mathcal{F}^{T_{j}}_{\tau}),\forall\tau\in\mathcal{T},T_{j}\in\mathbb{T}\\}$
(4)

  4. 4.

Action space: $\mathbb{A}$ is all possible actions that can be taken without
the quantum cloud environment. An action can be defined as a mapping of a
QTask to an available QNode. The action $a_{\tau}$ at time step $\tau$ is an
placement of QTask $T_{j}$ to quantum node $Q_{i}$ can be defined as

$a_{\tau}=\\{Q_{i},T_{j}\\},Q_{i}\in\mathbb{Q},T_{j}\in\mathbb{T},$ (5)

Thus, the Action space is equivalent to the set of all available quantum nodes
at the data center:

$\mathbb{A}=\mathbb{Q}$ (6)

  5. 5.

Reward function $\mathbb{R}$ determine the reward or penalty that assigned to
each action $a_{\tau}$. A sample reward function can be defined as follows:

$r_{\tau}=\begin{cases}\delta^{+}&\text{if }\text{success}=1\\\
\delta^{-}&\text{if }\text{success}=0\end{cases}$ (7)

where $\delta^{+}$ is the reward (typically a positive number) applied when
the QTask can be successfully executed, and $\delta^{-}$ and is the penalty
(typically a negative number) applied if the QTask cannot be executed if the
action violates the predefined constraints of the problem. This is to guide
the agent to avoid taking similar action in the future.

##  IV The QSimPy Framework

###  IV-A Design Overview

The design of our QSimPy simulation framework is driven by the following
principles:

  * •

Extensibility: QSimPy is designed to seamlessly accommodate future
enhancements and extensions. By adopting a modular design approach, each
framework component is designed to be easily extendable, allowing researchers
and practitioners to incorporate new features or functionalities without
compromising the integrity of the existing framework. This extensibility
ensures that QSimPy remains adaptable to evolving research needs and
technological advancements in the field of quantum hardware and quantum cloud
computing.

  * •

Compatibility: This is a cornerstone of the QSimPy framework, ensuring
interoperability with existing tools, libraries, and standards commonly
utilized in the quantum computing and machine learning communities. QSimPy is
designed to integrate easily with industry-standard machine learning
environments and libraries, such as Gymnasium and Ray RLlib, facilitating the
adoption and validation of advanced resource management strategies.
Additionally, QSimPy supports a variety of datasets, including OpenQASM and
CSV, enhancing its compatibility with diverse quantum computing tasks and
research methodologies.

  * •

Reusability: QSimPy emphasizes reusability by providing a flexible and
versatile framework for quantum cloud simulation and resource management. The
framework’s modular design encourages the reuse of components across different
simulation scenarios and research projects, promoting efficiency and reducing
development overhead.

###  IV-B Architecture Design and Functionality

####  IV-B1 Core Simulation Engine

The core simulation engine of QSimPy is built on top of the robust foundation
provided by SimPy [18], a well-established and versatile discrete-event
simulation Python library. A discrete-event simulation is an approach that
models the operation of a system as a sequence of events in time. Each event
occurs at an instant in time and marks a change of state in the system [22].
By leveraging SimPy, QSimPy inherits a proven framework for representing
complex, stochastic systems where interdependent events drive the evolution of
system states. This approach is particularly advantageous for modeling quantum
cloud environments, as it allows for the proper representation of asynchronous
and concurrent operations within quantum computation resources.

####  IV-B2 Dataset Generation and Modeling

As the quantum cloud workload dataset is not yet available, generating a
reliable synthetic dataset is essential for designing and evaluating quantum
cloud resource management policies. Figure 3 illustrates the modeling and
dataset generation process. We develop a QTask dataset generation module for
QSimPy by extracting circuit features from the MQTBench [23] quantum circuit
dataset and mapping with other QTask features, such as arrival time and shots
(task execution repetition) based on the defined distribution. We also use
quantum system calibration data from quantum cloud providers (such as IBM
Quantum) for modeling quantum node instances.

Figure 3: Feature extraction and modeling for (a) quantum computation tasks
(QTasks) and (b) quantum computation nodes (QNodes) in QSimPy

####  IV-B3 Gymnasium-based Environment

We have devised an environment wrapper utilizing Gymnasium [24], a leading API
standard in the reinforcement learning domain. This integration enables the
simulation framework to establish an interface for reinforcement learning
algorithms, allowing them to interact with quantum cloud computing
environments. Gymnasium standardized API facilitates the consistent
development and benchmarking of RL agents within our simulation platform.

####  IV-B4 RL Framework Connector

The Gymnasium-compliant QSimPy environment paves the way for interoperability
with a suite of state-of-the-art reinforcement learning frameworks. Prominent
among these are Ray RLlib [25], Stable Baselines 3 [26], and Tianshou [27],
which serve as popular tools for implementing and training RL agents. We
incorporate the Ray ecosystem to further strengthen QSimPy’s extensibility and
scalability, specifically leveraging the Ray RLlib for RL algorithm
implementation and Ray Tune [28] for systematic hyperparameter optimization.
This approach ensures that our framework facilitates cutting-edge RL-based
quantum resource management strategies.

###  IV-C Simulation Workflow

The simulation workflow of QSimPy, as outlined in Algorithm 1, orchestrates
the interaction between quantum task execution and resource management through
a discrete-event simulation process.

1 Initialize QSimPy Gymnasium environment;

2 Instantiate QSimPy computation resources (QNodes);

3 Generate quantum computing tasks (QTasks);

4 Instantiate QBroker for managing resources and tasks;

5 Connect RL Actor to the environment for decision-making ;

6 for  _each QTask in the system_ do

7 Sends QTask to QBroker;

8 QBroker collects observations from the environment;

9 QBroker sends the observation to the RL Actor;

10 RL Actor performs training and returns an action;

11 QBroker dispatched QTask to a QNode based on the action;

12 if  _QNode is available_ then

13 Generate the execution event;

14 Update the state of QNodes;

15

16 else

17 Queue QTask in QNode for later execution;

18

19 end if

20 QBroker collects observations from the environment;

21 QBroker sends the observation to the RL Actor;

22 RL Actor calculates reward and improve the policy;

23

24 end for

Algorithm 1 QSimPy Simulation Logic

The simulation initiates by establishing the Gymnasium environment, which acts
as a standardized interface for reinforcement learning agents. Then,
computational resources, QNodes, are instantiated to simulate the quantum
computing resources. Besides, quantum computing tasks (QTasks), are generated
and fed into the system, emulating the workload of quantum computations. A
QBroker component is responsible for the allocation and management of these
task placements. It serves as an intermediary between the QTasks and the
QNodes, observing the state of the environment and delegating tasks
accordingly. The RL Actor, a reinforcement learning agent, is connected to the
simulation for dynamic decision-making. It receives the environment’s state
from the QBroker and computes an action that dictates the subsequent
allocation of the QTasks. These actions are based on the objective of
optimizing resource utilization or other resource management objectives within
the quantum cloud environment.

After the action determination from the RL Actor, the QBroker dispatches
QTasks to QNodes, delegation on their availability. If a QNode is ready for
task execution, the corresponding event is generated, and the QNode state is
updated. Otherwise, the QTask is queued, awaiting future execution. The
simulation proceeds iteratively, with the QBroker continuously collecting
observations and interacting with the RL Actor, which refines its policy based
on the received state information and the rewards calculated from the actions
taken. This iterative process underpins the RL agent’s ability to learn and
adapt its task placement strategy over time, striving for an optimized
management of quantum computational resources.

The use of reinforcement learning within our simulation framework introduces a
layer of learning-centric adaptability, enabling the system to improve its
resource allocation decisions through continuous interaction with the quantum
cloud environment. This approach encapsulates the dynamism and
unpredictability inherent in quantum cloud resource management, providing a
simulated environment for researchers to explore and develop scheduling and
orchestrating algorithms.

###  IV-D Implementation

QSimPy is implemented as a Python package to facilitate the environmental
setup of reinforcement learning-based quantum cloud resource management
problems. The core simulation engine of QSimPy is developed based on SimPy, a
well-known Python library for discrete-event simulation. All environmental
implementations use the Gymnasium package and comply with the most popular
Reinforcement learning frameworks, such as stable-baselines3, Tianshou, and
Ray RLlib. We mainly test and suggest using Ray RLlib for the reinforcement
learning algorithm implementation. The implementation of QSimPy can be
accessed on GitHub at https://github.com/Cloudslab/qsimpy.

##  V Example of using QSimPy

This section discusses an example to demonstrate how to set and use the QSimPy
environment step-by-step, followed by an exemplary result to show the
potential use of the reinforcement learning approach for quantum task
placement over heuristic approaches.

###  V-A Example scenario: DRL-based Quantum Task Scheduling

We consider a quantum cloud data center comprising of five QNodes, with qubit
numbers, varied from 7 to 127, and the qubit topologies follow five IBM
Quantum systems [29], including ibm_washington, ibm_kolkata, ibm_hanoi,
ibm_perth, and ibm_lagos. The detailed specifications of all QNodes are
depicted in Table II. The model of QNode in our study adopts four benchmarking
metrics for a quantum system as proposed by Wack et al. [21]. These metrics
include the number of qubits, the quantum volume (QV), the rate of circuit
layer operations per second (CLOPS), and the depth-1 circuit operations per
second (D1CPS). For the purposes of our simulations, we adopt the D1CPS values
for each QNode from the empirical evaluations presented in [21]. These values
are used to estimate the execution time for each incoming QTask, with the
calculation being contingent on the number of depth-1 circuit layers that the
QTask encompasses.

TABLE II: QNode specifications based on IBM Quantum systems QNode Model | Qubits | QV | CLOPS | D1CPS
---|---|---|---|---
washington | 127 | 64 | 850 | 16967.5
kolkata | 27 | 128 | 2000 | 39900
hanoi | 27 | 64 | 2300 | 45935
perth | 7 | 32 | 2900 | 57905
lagos | 7 | 32 | 2700 | 53865

For workload, we generated the synthetic dataset of 49000 QTasks based on
quantum circuits extracted from 12 quantum algorithms/applications of the
MQTBench dataset [23] with the qubit number varied from 2 qubits to 27 qubits
as follows:

  1. 1.

Amplitude Estimation

  2. 2.

Deutsch-Jozsa

  3. 3.

Greenberger-Horne-Zeilinger (GHZ) state

  4. 4.

Quantum Fourier Transformation (QFT)

  5. 5.

Entangle QFT

  6. 6.

Quantum Neural Network

  7. 7.

Quantum Phase Estimation exact

  8. 8.

Quantum Phase Estimation inexact

  9. 9.

Random Circuit

  10. 10.

Real Amplitudes ansatz with Random Parameters

  11. 11.

Efficient SU2 ansatz with Random Parameters

  12. 12.

Two Local ansatz with random parameters

All QTasks are divided into 1900 subsets. Each subset contains random circuits
of all the applications above to simulate the stochastic nature of quantum
cloud computing environments. To simplify the demonstration purpose, we assume
that an average of 25 QTasks arrive every minute, with the arrival time
following the uniform distribution.

Problem statement: Given all quantum computation resources (QNodes) and
continuous incoming QTasks as aforementioned, design a reinforcement learning
policy to optimize the quantum task scheduling by minimizing the total
completion time of incoming tasks, in which task completion time is the sum of
waiting time and execution time.

###  V-B Setting up quantum cloud environment using QSimPy

The structure of the source code for creating the Gymnasium environment using
QSimPy is as depicted in Code 1.

⬇

import gymnasium as gym

from qsimpy import QTask, QNode, Broker, Dataset

...

class QSimPyEnv(gym.Env):

def __init__(self, config, dataset):

...[environment initialization]

def reset(self, *,seed, option):

...[reset operation]

def step(self, action):

...[step definition based on action]

...[reward calculation]

...[terminate/done condition]

def close(self):

...[other close operation]

Code 1: Sample structure for the Gymnasium-based QSimPy environment

To model the quantum computation resources (QNodes), users can use our
predefined QNode model based on IBM Quantum system information as Code 2. User
can also define their own customized QNode configuration using the QNode
object in the QSimPy package.

⬇

def setup_quantum_resources(self):

qnode_ids = range(self.n_qnodes)

qnode_names = [ "washington", "kolkata", "hanoi", "perth", "lagos",]

self.qnodes = [IBMQNode.create_ibmq_node(self.qsp_env, qid, qname) for qid,
qname in zip(qnode_ids, qnode_names)]

self.broker = Broker(self.qsp_env, self.qnodes)

... [truncated]

Code 2: Sample code snippet for model quantum nodes and broker

Then, we can generate incoming QTasks based on the workload dataset (as
depicted in Code 3) and define the QTask submission function (as depicted in
Code 4) to connect with QSimPy engine to simulate all the execution events.

⬇

def generate_qtasks(self):

qtasks = self.qdataset.get_subset(self.round)

self.qtasks = [QTask(id, arrival_time, qtask_data) for id, arrival_time,
qtask_data in zip(qtask_ids, qtask_arrival, qtask_values)]

... [truncated]

Code 3: Sample code snippet for generating QTasks

⬇

def submit_task_to_qnode(self, qtask, qnode_id):

qtask_execution = self.broker.submit_qtask_to_qnode(qtask,
self.qnodes[qnode_id])

self.qsp_env.process(qtask_execution)

...[truncated]

Code 4: Sample code snippet for submitting a QTask to a QNode

To achieve the task completion time minimization objective, we can define a
simple reward function as follows:

$r_{\tau}=\begin{cases}\dfrac{1}{time(T_{i})}&\text{if }\text{success}=1\\\
-10&\text{if }\text{success}=0\end{cases}$ (8)

where $time(T_{i})$ is the total completion time of QTask $T_{i}$ and $-10$ is
penalty value. If QTask execution is successful, we apply the reward as the
inverse value of its completion time, as the reinforcement learning algorithm
will try to maximize the reward. Otherwise, if the execution fails, we apply
the penalty with a large negative value (for example, -10) to urge the agent
to avoid taking similar action later on. This reward function is defined in
the step function of the environment based on the result of the
submit_task_to_qnode function. Besides, users can define a customized
environment wrapper to normalize and rescale the observation and reward if
needed, leveraging the features of Gymnasium. Figure 4 illustrates the main
steps involved in the simulation process of submitting each incoming QTask to
QSimPy during the RL training.

Figure 4: The sequence of main steps in the action decision-making process for
each incoming QTask within the QSimPy simulation environment.

First, a QTask object is generated with attributes from the quantum circuit
dataset. A QTask submission event is then scheduled according to its
designated arrival time, and the QTask object is dispatched to the QBroker for
processing. The QBroker subsequently acquires the current state of all QNodes
and, integrating this with the incoming QTask’s data, compiles an observation
to relay to the RL actor. Utilizing Ray RLlib, the RL actor evaluates the
observation and determines the appropriate action, which it communicates back
to the QBroker. Following the action directive, the QBroker orchestrates task
placement by forwarding the QTask to the specified QNode. Post-placement, the
QBroker retrieves a new observation for the RL actor, which then assesses the
reward and iteratively refines its policy, thereby enhancing its decision-
making for future tasks.

###  V-C DRL Training Experiments and Results

We implement the task scheduling policy using the Deep Q Network (DQN) [30] in
QSimPy environment. We use Ray RLlib for implementing the policy with the
hyperparameters set as follows: learning rate is 0.01; replay buffer capacity
is 60000; the number of atoms is 10; the number of steps is 5; prioritized
replay alpha, beta, and epsilon are set to 0.5, 0.5, and 3e-6, respectively.
We evaluate the policy’s performance over different numbers of training
iterations based on episode reward values. We also observe the episode lengths
of over 100 iterations, as these metrics reflect the number of task placements
taken in each episode.

Figure 5: Episode rewards of different iterations showed that the DRL-based
task scheduling policy has been trained in QSimPy environment Figure 6:
Average of episode length has been reduced over time and reached the optimal
value around 25 after 15 iterations

Figure 5 shows the episode reward values for the $10^{5}$ time steps taken
over 100 iterations. Initially, the agent takes random actions, resulting in a
lower negative reward as the incoming task cannot be executed successfully
(for example, the scheduled QNode does not have enough qubits to execute
QTask). After the agent learns and improves from previous experience, it is
clear that the average episode reward increases over time and converges after
around 15 training iterations. Besides, the average number of episode lengths
is reduced over time and reaches the optimal value of around 25. This result
indicates that the number of violated actions (or rescheduling actions)
decreases. Therefore, we have gained confidence that the agent has learned a
policy that minimizes the total completion time, satisfying the initial
quantum task scheduling objective.

##  VI Conclusions and Future Work

In this work, we proposed QSimPy, a novel discrete-event simulator and
learning-centric environment architected for the rigorous demands of quantum
cloud computing resource management problems. Its design harmoniously aligns
with the needs of high-level, abstract modeling and practical, application-
oriented research. By leveraging a Gymnasium-based environment and connecting
with the Ray ecosystem for reinforcement learning integration, QSimPy allows
for the creation and refinement of resource management policies that are
adaptive, robust, and efficient. Our study confirms the framework’s capability
to support the design, development, and evaluation of learning-based quantum
cloud resource management techniques.

Several promising directions for further research can be considered for
extending QSimPy’s capabilities. The framework’s adaptability to evolving
quantum hardware and algorithms lays the groundwork for extensive, real-world
deployment scenarios. New enhancements can include support for distributed
quantum computing models and incorporating quantum error correction
simulations. Additionally, diversifying the range of reinforcement learning
algorithms tested within the QSimPy environment can deepen our understanding
of their efficacy in quantum task scheduling.

Software availability
The QSimPy framework with the source code of proof-of-concept implementation
are released on Github at (github.com/Cloudslab/qsimpy) as an open-source tool
under the GPL-3.0 license.

## Acknowledgment

Hoa Nguyen acknowledges the support from the Science and Technology
Scholarship Program for Overseas Study for Master’s and Doctoral Degrees,
Vingroup, Vietnam.

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[1][#1] [1][#1]
[1][#1]

#  Inflationary universe from anomaly-free $F(R)$-gravity

Aroonkumar<EMAIL_ADDRESS>and Kazuharu
<EMAIL_ADDRESS>1Department of Mathematical Sciences,
University of Zululand, P Bag X1001, Kwa-Dlangezwa 3886, South Africa
2Faculty of Natural Sciences, Mangosuthu University of Technology, Jacobs
4026, South Africa
3Division of Human Support System, Faculty of Symbiotic Systems Science,
Fukushima University, Fukushima 960-1296, Japan

###### Abstract

By adding a three dimensional manifold to an eleven dimensional manifold in
supergravity, we obtain the action of $F(R)$-gravity and find that it is
anomaly free. We calculate the scale factor of the inflationaty universe in
this model, and observe that it is related to the slow-roll parameters. The
scalar-tensor ratio R_(scalar-tensor) is in good agreement with experimental
data.

PACS numbers: 98.80.-k, 04.50.Gh, 11.25.Yb, 98.80.Qc
Keywords: $F(R)$-gravity, Manifold, Inflation, Universe

##  I Introduction

One of the best models for considering an inflationary universe is suggested
by modified theories of gravity like $f(R)$ gravity, whose Lagrangian is a
function $f(R)$ of the scalar curvature $R$ in a 4-dimensional universe p1 ;
p2 ; p3 ; p4 ; p5 . The special property of these theories is the existence of
the effective cosmological constant in such a way that early-time inflation
and late-time cosmic acceleration are naturally unified within a single theory
p1 . Some of these $f(R)$ gravity theories, especially a power-law mechanism
and Yang-Mills $f(R)$ gravity, give the best fit values compatible with
cosmological parameters like the tensor-to-scalar ratio within the permitted
ranges derived by the Planck and BICEP2 observations p2 . In some other
versions of these theories, a dynamical scalar field ($\varphi$) is coupled to
gravity ($f(R,\varphi)$) whose dynamics allow for exit from inflation and
which gives rise to the correct amount of inflation in agreement with
observational data p3 . Also, in some modified gravity theories, $F$ is a
generic function of the curvature scalar and the Gauss-Bonnet topological
parameter. Cosmological dynamical results which are obtained by two effective
masses (lengths) correspond to both of these parameters and work,
respectively, at early and very early epochs of the evolution of the universe
p4 . Furthermore, modified teleparallel gravity leads to second order
equations and solves the particle horizon problem in a spatially flat cosmic
space by providing an initial exponential expansion without resorting to an
inflaton particle p5 . For more reviews on the connections between inflation,
the dark energy problem and modified gravity theories see, for instance, p6 .

Now, the question that arises is whether $F(R)$ gravity could be anomaly free
or not? Also, it is required to consider the role of anomaly cancellation in
the evolution of the universe during the inflationary epoch. Recently, the
exact forms of two-dimensional and four-dimensional anomalies from higher
derivative gravity and $F(R)$ gravity have been calculated p7 . On the other
hand, the conformal-anomaly driven inflation in $F(R)$ gravity without using
the scalar-tensor representation has been studied. It has been shown that in
$F(R)$ gravity, the curvature perturbations with their high amplitude which
are consistent with observations, are created during inflation p8 . Motiviated
by these works, we propose a theory which lives on an (11+3)-dimensional
manifold with two 11-dimensional manifolds and one 3-dimensional manifold. We
will show that our universe is a part of an 11-dimensional manifold which is
connected with the other 11-dimensional manifold by an extra 3-dimensional
manifold. Two 11-dimensional manifolds interact with each other via exchanging
fields which move along the 3-dimensional manifold. These fields are the main
cause for the appearance of $F(R)$-gravity in the four dimensional universe.

This paper consists of two main parts. In section II, we show that by adding
3-dimensional manifold to an 11-dimensional one, $F(R)$-gravity emerges. In
section III, we obtain the cosmological parameters like the tensor to scalar
ratio, and compare with experimental data.

##  II  Emergence of $F(R)$-gravity on an (11+3)-dimensional manifold

In this section, we will assume that our universe has been constructed on a
D3-brane. Thus, the action of gravity and matter in a 4-dimensional universe
should be equal to the action of a D3-brane (See Figure 1). This means that
strings which live on a D3-brane produces gravity and different types of
matter. Thus, each string ($X^{i}$) can be expanded in terms of curvatures
($R$), gauge fields ($F$) and scalars ($\phi$) (See Figure 2). According to
the Horava-Witten mechanism b1 ; b2 , this D3-brane can be a part of a bigger
11-dimensional manifold on which the action of fields should be anomaly free
(See Figure 3). However, using the relation between strings and fields, we
notice that the gauge variation of the action on this manifold is not zero,
and some extra anomalies emerge. To remove these anomalies, we have to add an
extra 11-dimensional manifold which is connected with the first 11-dimensional
manifold by a 3-dimensional manifold. The extra fields which are produced by
the interaction of the two manifolds produce $F(R)$-gravity (See Figure 4).

---
Figure 1:  4-dimensional universe on D3-brane.
---
Figure 2:  Different shapes of strings produce different types of fields.
---
Figure 3:  D3-brane is a part of 11-dimensional manifold.
---
Figure 4:  Two 11-dimensional manifolds interact via 3-dimensional manifold in
14-dimensional manifold.

First, let us introduce the action of the D3-brane which is given by D3 :

$\displaystyle S_{D3}$ $\displaystyle=$ $\displaystyle-T_{D3}\int
d^{4}y\sqrt{-det(\bar{\gamma}_{ab}+2\pi l_{s}^{2}F_{ab})},$
$\displaystyle\bar{\gamma}_{ab}$ $\displaystyle=$ $\displaystyle
g_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu},$ $\displaystyle F_{ab}$
$\displaystyle=$ $\displaystyle\partial_{a}A_{b}-\partial_{b}A_{a}$ (1)

where $A_{b}$ is the gauge field, $F_{ab}$ is the field strength, $X^{\mu}$ is
the string, $g_{\mu\nu}$ is the metric, $T_{D3}$ is the tension and $l_{s}$ is
the string length. Substituting $X^{0}=t$ and doing some mathematical
calculations, the acion of the D3-brane in equation (1) is given by D3 :

$\displaystyle S_{D3}$ $\displaystyle=$ $\displaystyle-T_{D3}\int
d^{4}y\sqrt{1+g_{ij}\partial_{a}X^{i}\partial^{a}X^{j}-4\pi^{2}l_{s}^{4}F_{ab}F^{ab}},$
(2)

To construct our universe on a D3-brane, this action should be equal to the
action of fields and gravity in a 4-dimensional universe. The action of matter
and gravity is given by:

$\displaystyle S_{Gravity-Matter}$ $\displaystyle=$ $\displaystyle-\int
d^{4}y\sqrt{-g}\Big{(}R-g_{ab}\partial^{a}\phi\partial^{b}\phi-i\bar{\psi}\gamma^{a}\partial_{a}\psi+1\Big{)},$
(3)

where $\phi$ is the scalar field and $\psi$ is the fermionic field. Puting
equation (2) equal to equation (3), and doing some mathematical calculations,
we obtain the relation between strings and matter fields:

$\displaystyle S_{Gravity-Matter}$ $\displaystyle=$ $\displaystyle S_{D3}$
$\displaystyle\Longrightarrow X^{i}$ $\displaystyle=$ $\displaystyle\int
dy^{i}\sqrt{-1+4\pi^{2}l_{s}^{4}F_{ij}F^{ij}+\Big{(}\sqrt{-g}(g^{ij}R_{ij}-g_{ij}\partial^{i}\phi\partial^{j}\phi-i\bar{\psi}\gamma^{i}\partial_{i}\psi+1)\Big{)}^{2}},$
(4)

where we have assumed $T_{D3}\simeq 1$ and $4\pi^{2}l_{s}^{4}\simeq 1$. Using
the above relation between matter fields and strings, we can reconsider the
Horava-Witten mechanism. We will show that there are also other anomalies that
have been ignored in previous considerations. Our goal is to show that by
adding a 3-dimensional manifold to 11-dimensional spacetime in the Horava-
Witten mechanism, all anomalies can be removed and an action without anomaly
can be produced. This action is identical to the action of the modified
gravity theory presented in p1 ; p2 ; p3 ; p4 ; p5 ; p6 .

At this stage, we introduce the Horava-Witten mechanism in 11-dimensional
spacetime. In this theory, the bosonic part of the action in 11-dimensional
supergravity (SUGRA) is given by b1 ; b2 :

$\displaystyle S_{\rm Bosonic-SUGRA}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{\kappa}^{2}}\int
d^{11}x\sqrt{g}\Big{(}-\frac{1}{2}R-\frac{1}{48}G_{IJKL}G^{IJKL}\Big{)}+S_{\rm
CGG},$ $\displaystyle S_{\rm CGG}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{3456\bar{\kappa}^{2}}\int_{M^{11}}d^{11}x\varepsilon^{I_{1}I_{2}...I_{11}}C_{I_{1}I_{2}I_{3}}G_{I_{4}...I_{7}}G_{I_{8}...I_{11}},$
(5)

where $\varepsilon^{I_{1}I_{2}..I_{k}}$ is the rank-$k$ Levi-Civita
pseudotensor, and CGG is used to denote the product term of the three-form
field $C_{I_{1}I_{2}I_{3}}$ and four-form field $G_{IJKL}$, which are directly
related to the gauge field $A^{I}$, field strength $F^{IJ}$ and Ricci
curvature $R^{IJ}$ b2 :

$\displaystyle G_{IJKL}$ $\displaystyle=$
$\displaystyle-\frac{3}{\sqrt{2}}\frac{\kappa^{2}}{\lambda^{2}}\varepsilon(x^{11})(F_{[IJ}F_{KL]}-R_{[IJ}R_{KL]})+...,$
$\displaystyle\delta C_{ABC}$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}}{6\sqrt{2}\lambda^{2}}\delta(x^{11})\operatorname{tr}(\epsilon_{C}F_{AB}-\epsilon_{C}R_{AB}),$
$\displaystyle G_{11ABC}$ $\displaystyle=$
$\displaystyle(\partial_{11}C_{ABC}\pm\text{23 permutations of the indices
$11$ and
$ABC$})+\frac{\kappa^{2}}{\sqrt{2}\lambda^{2}}\delta(x^{11})\omega_{ABC},$
$\displaystyle\delta\omega_{ABC}$ $\displaystyle=$
$\displaystyle\partial_{A}\operatorname{tr}(\epsilon F_{BC})+\text{cyclic
permutations of}~{}ABC,$ $\displaystyle F^{IJ}$ $\displaystyle=$
$\displaystyle\partial^{I}A^{J}-\partial^{J}A^{I},$ $\displaystyle R_{IJ}$
$\displaystyle=$
$\displaystyle\partial_{I}\Gamma^{B}_{JB}-\partial_{J}\Gamma^{B}_{IB}+\Gamma^{A}_{JB}\Gamma^{B}_{IA}-\Gamma^{A}_{IB}\Gamma^{B}_{JA},$
$\displaystyle\Gamma_{IJK}$ $\displaystyle=$
$\displaystyle\partial_{I}g_{JK}+\partial_{K}g_{IJ}-\partial_{J}g_{IK},$
$\displaystyle G_{IJ}$ $\displaystyle=$ $\displaystyle
R_{IJ}-\frac{1}{2}Rg_{IJ},$ (6)

where $\epsilon$ and $\epsilon_{C}$ characterize infinitesimal gauge
transformations b2 . Here, $\varepsilon(x^{11})$ is 1 for $x^{11}>0$ and $-1$
for $x^{11}<0$ and also
$\delta(x^{11})=\frac{1}{2}\partial\varepsilon(x^{11})/\partial x^{11}$ is the
Dirac delta function. As usual b2 , $\operatorname{tr}$ is 1/30th of the trace
$\operatorname{Tr}$ in the adjoint representation for $E_{8}\times E_{8}$. The
ellipsis ($...$) denotes terms that are regular near $x^{11}=0$ and hence
vanish there b2 . It is clear from the above equation that G-fields can be
produced by joining F-fields or R-fields. This means that G-fields are
produced by joining two strings, each of which produces one gauge field or
curvature ($R$) (See Figure 5).

---
Figure 5:  G-fields are produced by joining two strings.

The gauge variation of the CGG term in the action yields the following
equation b2 :

$\displaystyle\delta S_{\rm CGG}|_{11}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{3456\bar{\kappa}^{2}}\int_{M^{11}}d^{11}x\varepsilon^{I_{1}I_{2}...I_{11}}\delta
C_{I_{1}I_{2}I_{3}}G_{I_{4}...I_{7}}G_{I_{8}...I_{11}}$ (7)
$\displaystyle\approx$
$\displaystyle-\frac{\bar{\kappa}^{4}}{128\lambda^{6}}\int_{M^{10}}\Sigma_{n=1}^{5}(\operatorname{tr}F^{n}-\operatorname{tr}R^{n}+\operatorname{tr}(F^{n}R^{5-n})),$

where
$\operatorname{tr}F^{n}=\operatorname{tr}(F_{[I_{1}I_{2}}..F_{I_{2n-1}I_{2n}]})=\varepsilon^{I_{1}I_{2}..I_{2n-1}I_{2n}}F_{I_{1}I_{2}}..F_{I_{2n-1}I_{2n}}$
or
$\operatorname{tr}R^{n}=\operatorname{tr}(R_{[I_{1}I_{2}}..R_{I_{2n-1}I_{2n}]})=\varepsilon^{I_{1}I_{2}..I_{2n-1}I_{2n}}R_{I_{1}I_{2}}..R_{I_{2n-1}I_{2n}}$
. The above terms cancel the anomaly of ($S_{\rm Bosonic-SUGRA}$) in
11-dimensional manifolds b2 :

$\displaystyle\delta S_{\rm CGG}|_{11}=-\delta S^{\rm anomaly}_{\rm Bosonic-
SUGRA}.$ (8)

Thus, $S_{\rm CGG}$ is necessary for anomaly cancellation. Our goal now is to
find a good rationale for its inclusion. We also answer the issue of the
relation between CGG terms in 11-dimensional supergravity and $F(R)$-gravity.
In fact, we suggest a theory in which $F(R)$-gravity terms appear in the
supergravity action without being added by hand. To this end, we choose a
unified form for all particles and fields by using Nambu-Poisson brackets and
properties of string fields ($X$). Solving equation (4) and assuming that
changes of fields with respect to the coordinates be ingnorable, we obtain b9
; A1 :

$\displaystyle X^{I}$ $\displaystyle\approx$ $\displaystyle
y^{I}+y_{J}F^{IJ}-y_{J}R^{IJ}+y_{J}\partial^{I}\phi\partial^{J}\phi-
iy_{J}\bar{\psi}\gamma^{J}\partial^{I}\psi+...$
$\displaystyle\Rightarrow\\{X^{I},X^{J}\\}$ $\displaystyle=$
$\displaystyle\Sigma_{I,J}\varepsilon^{I^{\prime}J^{\prime}}\frac{\partial
X^{I}}{\partial y^{I^{\prime}}}\frac{\partial X^{J}}{\partial y^{J^{\prime}}}$
(9) $\displaystyle=$ $\displaystyle
F^{IJ}-R^{IJ}+\partial^{I}\phi\partial^{J}\phi-i\bar{\psi}\gamma^{J}\partial^{I}\psi+....$

where we introduced the scalar field $\phi$. Also, $A^{I}$ is the gauge field
and $R$ is the curvature (R). In fact, the origin of all matter is the same
and they are different shapes of strings. In the static state, all strings can
be described by a unit vector ($X^{i}$). When these strings interact with each
other or move, the initial symmetry is broken and fields and particles emerge.
Using 4-dimensional instead of 2-dimensional brackets, we may derive the GG
term $G_{IJKL}G^{IJKL}$ in supergravity in terms of strings ($X$) b9 ; A1 :

$\displaystyle G^{IJKL}$ $\displaystyle=$
$\displaystyle\\{X^{I},X^{J},X^{K},X^{L}\\}=\varepsilon^{I^{\prime}J^{\prime}K^{\prime}L^{\prime}}\frac{\partial
X^{I}}{\partial y^{I^{\prime}}}\frac{\partial X^{J}}{\partial
y^{J^{\prime}}}\frac{\partial X^{K}}{\partial y^{K^{\prime}}}\frac{\partial
X^{L}}{\partial y^{L^{\prime}}}$ $\displaystyle\Downarrow$ $\displaystyle\int
d^{11}x\sqrt{g}\Big{(}G_{IJKL}G^{IJKL}\Big{)}$ $\displaystyle=$
$\displaystyle\int
d^{11}x\sqrt{g}\Big{(}\varepsilon_{I^{\prime}J^{\prime}K^{\prime}L^{\prime}}\frac{\partial
X_{I}}{\partial y_{I^{\prime}}}\frac{\partial X_{J}}{\partial
y_{J^{\prime}}}\frac{\partial X_{K}}{\partial y_{K^{\prime}}}\frac{\partial
X_{L}}{\partial
y_{L^{\prime}}}\varepsilon^{I^{\prime\prime}J^{\prime\prime}K^{\prime\prime}L^{\prime\prime}}\frac{\partial
X^{I}}{\partial y^{I^{\prime\prime}}}\frac{\partial X^{J}}{\partial
y^{J^{\prime\prime}}}\frac{\partial X^{K}}{\partial
y^{K^{\prime\prime}}}\frac{\partial X^{L}}{\partial
y^{L^{\prime\prime}}}\Big{)}.$ (10)

Equation (10) helps us to extract the CGG terms from the GG terms in
supergravity. To this aim, we will add a 3-dimensional manifold to the
11-dimensional manifold which connects it to other 11-dimensional manifold by
applying the properties of strings ($X$) in the Nambu-Poisson brackets A1 :

$\displaystyle X^{I}$ $\displaystyle\approx$ $\displaystyle
y^{I}+y_{J}F^{IJ}-y_{J}R^{IJ}+y_{J}\partial^{I}\phi\partial^{J}\phi-
iy_{J}\bar{\psi}\gamma^{J}\partial^{I}\psi+...$ $\displaystyle\Downarrow$
$\displaystyle\frac{\partial X^{I_{5}}}{\partial y^{I_{5}}}$
$\displaystyle\approx$ $\displaystyle\delta(y^{I_{5}})+...\quad\frac{\partial
X^{I_{6}}}{\partial y^{I_{6}}}\approx\delta(y^{I_{6}})+...\quad\frac{\partial
X^{I_{7}}}{\partial y^{I_{7}}}\approx\delta(y^{I_{7}})+...,$ (11)
$\displaystyle\int_{M^{N=3}}$ $\displaystyle\rightarrow$
$\displaystyle\int_{y^{I_{5}}+y^{I_{6}}+y^{I_{7}}}\varepsilon^{I^{\prime}_{5}I^{\prime}_{6}I^{\prime}_{7}}\frac{\partial
X^{I_{5}}}{\partial y^{I^{\prime}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{I^{\prime}_{6}}}\frac{\partial X^{I_{7}}}{\partial
y^{I^{\prime}_{7}}}=1+...,$ (12)

where ellipses (…) were used to represent higher-order derivatives. The
integration is over a 3-dimensional manifold with coordinates
($y^{I_{5}},y^{I_{6}},y^{I_{7}}$) and consequently, the integration can be
shown by $\int_{y^{I_{5}}+y^{I_{6}}+y^{I_{7}}}=\int dy^{I_{5}}\int
dy^{I_{6}}\int dy^{I_{7}}$). This result shows that ignoring fluctuations of
strings that leads to the production of fields, the result of integration over
each 3-dimensional manifold tends to one. When we add one manifold to another,
the integration will be the product of integration over each manifold.

Extending the manifold over additional dimensions extends the integration
volume element. By extending the 11-dimensional manifold in Eq. (10) with the
3-dimensional manifold of Eq. (12), we get A1 :

$\displaystyle\int_{M^{N=3}}\int_{M^{11}}\sqrt{g}\Big{(}G_{I_{1}I_{2}I_{3}I_{4}}G^{I_{1}I_{2}I_{3}I_{4}}\Big{)}=\int_{M^{11}+y^{I_{5}}+y^{I_{6}}+y^{I_{7}}}\sqrt{g}\epsilon^{I^{\prime}_{5}I^{\prime}_{6}I^{\prime}_{7}}G_{I_{1}I_{2}I_{3}I_{4}}G^{I_{1}I_{2}I_{3}I_{4}}\frac{\partial
X^{I_{5}}}{\partial y^{I^{\prime}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{I^{\prime}_{6}}}\frac{\partial X^{I_{7}}}{\partial y^{I^{\prime}_{7}}},$
(13)

where we notice that after making the identification A1 :

$\displaystyle C^{I_{5}I_{6}I_{7}}$ $\displaystyle=$
$\displaystyle\epsilon^{I^{\prime}_{5}I^{\prime}_{6}I^{\prime}_{7}}\frac{\partial
X^{I_{5}}}{\partial y^{I^{\prime}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{I^{\prime}_{6}}}\frac{\partial X^{I_{7}}}{\partial y^{I^{\prime}_{7}}},$
(14)

we recover the CGG action in $(11+3)$ dimensions A1 :

$S^{N=11+3}_{\rm
CGG}=\int_{M^{11}+y^{I_{5}}+y^{I_{6}}+y^{I_{7}}}\sqrt{g}G_{I_{1}I_{2}I_{3}I_{4}}G^{I_{1}I_{2}I_{3}I_{4}}C^{I_{5}I_{6}I_{7}}.$
(15)

This equation has three interesting results:  1. The CGG term that appears in
the supergravity action is a result of adding a 3-dimensional manifold to an
11-dimensional manifold. This manifold connects the first 11-dimensional
manifold to the second one.  2. Combining the 11-dimensional manifold with the
3-dimensional manifold yields 14-dimensional supergravity.  3. The shape of
the C-term is now clear in terms of string fields ($X^{I}$).  4. It is clear
that C-fields are in fact G-fields such that one end of the strings is placed
out of the 11-dimensional manifold and on the 3-dimensional manifold and for
this reason, C-fields seem to have three ends only (See Figure 6).

---
Figure 6:  C-fields are produced from G-fields when one end of strings is
placed on a 3-dimensional manifold and three ends are placed on an
11-dimensional manifold.

To verify that the theory is correct, we should be able to get back the gauge
variation of the CGG-action in Eq. (7) in terms of field strengths and
curvature. To this end, using Eqs. (9, 10, 12 and 14), we can calculate the
gauge variation of C A1 :

$\displaystyle X^{I}$ $\displaystyle\approx$ $\displaystyle
y^{I}+y_{J}F^{IJ}-y_{J}R^{IJ}+y_{J}\partial^{I}\phi\partial^{J}\phi-
iy_{J}\bar{\psi}\gamma^{J}\partial^{I}\psi+...$ $\displaystyle\Downarrow$
$\displaystyle\frac{\partial\delta_{A}X^{I}}{\partial y^{I}}$ $\displaystyle=$
$\displaystyle\delta(y^{I})$ $\displaystyle\Downarrow$
$\displaystyle\int_{M^{N=3}+M^{11}}\delta_{A}C^{I_{5}I_{6}I_{7}}$
$\displaystyle=$
$\displaystyle\int_{M^{N=3}+M^{11}}\Sigma_{I^{\prime}_{5}I^{\prime}_{6}I^{\prime}_{7}}\varepsilon^{I^{\prime}_{5}I^{\prime}_{6}I^{\prime}_{7}}\delta_{A}(\frac{\partial
X^{I_{5}}}{\partial y^{I^{\prime}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{I^{\prime}_{6}}}\frac{\partial X^{I_{7}}}{\partial y^{I^{\prime}_{7}}})$
(16) $\displaystyle=$
$\displaystyle\int_{M^{N=3}+M^{10}}\Sigma_{I^{\prime}_{5}I^{\prime}_{6}}\varepsilon^{I^{\prime}_{5}I^{\prime}_{6}}(\frac{\partial
X^{I_{5}}}{\partial y^{I^{\prime}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{I^{\prime}_{6}}})$ $\displaystyle=$
$\displaystyle\int_{M^{N=3}+M^{10}}(F^{IJ}-R^{IJ}+\partial^{I}\phi\partial^{J}\phi-i\bar{\psi}\gamma^{J}\partial^{I}\psi+....$
$\displaystyle=$
$\displaystyle\int_{M^{N=3}+M^{10}}(F^{IJ}-R^{IJ}+\partial^{I}\phi\partial^{J}\phi-i\bar{\psi}\gamma^{J}\partial^{I}\psi+....).$

where the ellipses (…) represent higher-order derivatives with respect to
fields. Using Eqs. (9, 10, 12, 14 and 16) we can calculate the gauge variation
of the CGG action in the equation of (15):

$\displaystyle\delta S^{N=11+3}_{\rm CGG}$ $\displaystyle=$
$\displaystyle\delta\int_{M^{11}+M^{N=3}}\sqrt{g}\epsilon_{I_{1}I_{2}I_{3}I_{4}I^{\prime}_{1}I^{\prime}_{2}I^{\prime}_{3}I^{\prime}_{4}I_{5}I_{6}I_{7}}\epsilon^{\tilde{I}_{5}\tilde{I}_{6}\tilde{I}_{7}}(\frac{\partial
X^{I_{5}}}{\partial y^{\tilde{I}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{\tilde{I}_{6}}}\frac{\partial X^{I_{7}}}{\partial
y^{\tilde{I}_{7}}})G^{I_{1}I_{2}I_{3}I_{4}}G^{I^{\prime}_{1}I^{\prime}_{2}I^{\prime}_{3}I^{\prime}_{4}}$
(17) $\displaystyle=$
$\displaystyle\delta\int_{M^{11}+M^{N=3}}\sqrt{g}\epsilon_{I_{1}I_{2}I_{3}I_{4}I^{\prime}_{1}I^{\prime}_{2}I^{\prime}_{3}I^{\prime}_{4}I_{5}I_{6}I_{7}}\epsilon^{\tilde{I}_{5}\tilde{I}_{6}\tilde{I}_{7}}(\frac{\partial
X^{I_{5}}}{\partial y^{\tilde{I}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{\tilde{I}_{6}}}\frac{\partial X^{I_{7}}}{\partial y^{\tilde{I}_{7}}})$
$\displaystyle\hskip
36.135pt{}\times(\epsilon^{\tilde{I}_{1}\tilde{I}_{2}\tilde{I}_{3}\tilde{I}_{4}}\frac{\partial
X^{I_{1}}}{\partial y^{\tilde{I}_{1}}}\frac{\partial X^{I_{2}}}{\partial
y^{\tilde{I}_{2}}}\frac{\partial X^{I_{3}}}{\partial
y^{\tilde{I}_{3}}}\frac{\partial X^{I_{4}}}{\partial
y^{\tilde{I}_{4}}})(\epsilon^{\tilde{I}^{\prime}_{1}\tilde{I}^{\prime}_{2}\tilde{I}^{\prime}_{3}\tilde{I}^{\prime}_{4}}\frac{\partial
X^{I^{\prime}_{1}}}{\partial y^{\tilde{I}^{\prime}_{1}}}\frac{\partial
X^{I^{\prime}_{2}}}{\partial y^{\tilde{I}^{\prime}_{2}}}\frac{\partial
X^{I^{\prime}_{3}}}{\partial y^{\tilde{I}^{\prime}_{3}}}\frac{\partial
X^{I^{\prime}_{4}}}{\partial y^{\tilde{I}^{\prime}_{4}}})$ $\displaystyle=$
$\displaystyle\int_{M^{10}+M^{N=3}}\sqrt{g}\epsilon_{I_{1}I_{2}I_{3}I_{4}I^{\prime}_{1}I^{\prime}_{2}I^{\prime}_{3}I^{\prime}_{4}I_{5}I_{6}}\epsilon^{\tilde{I}_{5}\tilde{I}_{6}}(\frac{\partial
X^{I_{5}}}{\partial y^{\tilde{I}_{5}}}\frac{\partial X^{I_{6}}}{\partial
y^{\tilde{I}_{6}}})$ $\displaystyle\hskip
36.135pt{}\times(\epsilon^{\tilde{I}_{1}\tilde{I}_{2}\tilde{I}_{3}\tilde{I}_{4}}\frac{\partial
X^{I_{1}}}{\partial y^{\tilde{I}_{1}}}\frac{\partial X^{I_{2}}}{\partial
y^{\tilde{I}_{2}}}\frac{\partial X^{I_{3}}}{\partial
y^{\tilde{I}_{3}}}\frac{\partial X^{I_{4}}}{\partial
y^{\tilde{I}_{4}}})(\epsilon^{\tilde{I}^{\prime}_{1}\tilde{I}^{\prime}_{2}\tilde{I}^{\prime}_{3}\tilde{I}^{\prime}_{4}}\frac{\partial
X^{I^{\prime}_{1}}}{\partial y^{\tilde{I}^{\prime}_{1}}}\frac{\partial
X^{I^{\prime}_{2}}}{\partial y^{\tilde{I}^{\prime}_{2}}}\frac{\partial
X^{I^{\prime}_{3}}}{\partial y^{\tilde{I}^{\prime}_{3}}}\frac{\partial
X^{I^{\prime}_{4}}}{\partial y^{\tilde{I}^{\prime}_{4}}})$ $\displaystyle=$
$\displaystyle\int_{M^{10}+M^{N=3}}\sqrt{g}\epsilon_{I_{1}I_{2}I_{3}I_{4}I^{\prime}_{1}I^{\prime}_{2}I^{\prime}_{3}I^{\prime}_{4}I_{5}I_{6}}(\epsilon^{\tilde{I}_{4}\tilde{I}_{5}}\frac{\partial
X^{I_{4}}}{\partial y^{\tilde{I}_{4}}}\frac{\partial X^{I_{5}}}{\partial
y^{\tilde{I}_{5}}})(\epsilon^{\tilde{I}^{\prime}_{4}\tilde{I}_{6}}\frac{\partial
X^{I^{\prime}_{4}}}{\partial y^{\tilde{I}^{\prime}_{4}}}\frac{\partial
X^{I_{6}}}{\partial y^{\tilde{I}_{6}}})$ $\displaystyle\hskip
36.135pt{}\times(\epsilon^{\tilde{I}_{1}\tilde{I}_{2}}\frac{\partial
X^{I_{1}}}{\partial y^{\tilde{I}_{1}}}\frac{\partial X^{I_{2}}}{\partial
y^{\tilde{I}_{2}}})(\epsilon^{\tilde{I}^{\prime}_{1}\tilde{I}^{\prime}_{2}}\frac{\partial
X^{I^{\prime}_{1}}}{\partial y^{\tilde{I}^{\prime}_{1}}}\frac{\partial
X^{I^{\prime}_{2}}}{\partial
y^{\tilde{I}^{\prime}_{2}}})(\epsilon^{\tilde{I}_{3}\tilde{I}^{\prime}_{3}}\frac{\partial
X^{I_{3}}}{\partial\tilde{I}^{I_{3}}}\frac{\partial
X^{I^{\prime}_{3}}}{\partial y^{\tilde{I}^{\prime}_{3}}})$ $\displaystyle=$
$\displaystyle\int_{M^{10}+M^{N=3}}\sqrt{g}\Sigma_{n=1}^{5}\Big{(}\operatorname{tr}F^{n}-\operatorname{tr}R^{n}+\operatorname{tr}F^{n}R^{5-n}\Big{)}$
$\displaystyle+$
$\displaystyle\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{5}\Sigma_{m=0}^{5-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}R_{J_{1}J_{2}}..R_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{5}\Sigma_{m=0}^{5-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}R_{J_{1}J_{2}}..R_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{5}\Sigma_{m=0}^{n-5}\Sigma_{k=0}^{5-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}R_{J_{1}J_{2}}..R_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi\Big{)}$
$\displaystyle+$
$\displaystyle\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{5}\Sigma_{m=0}^{5-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{5}\Sigma_{m=0}^{5-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{5}\Sigma_{m=0}^{5-n}\Sigma_{k=0}^{5-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}.$

The first term in Eq. (17) cancels the anomaly in Eq. (7). However, the other
terms yield the action of modified gravity and matter. In fact, to remove the
anomalies, we have to add two extra actions to equation (5), one is related to
$F(R)$-gravity, and the second is corresponds to matter. We can write:

$\displaystyle S_{\rm SUGRA}$ $\displaystyle=$ $\displaystyle S_{\rm
GG}+S_{\rm CGG}+S_{\rm F(R)}+S_{\rm Matter},$ $\displaystyle S_{\rm GG}$
$\displaystyle=$ $\displaystyle\frac{1}{\bar{\kappa}^{2}}\int
d^{14}x\sqrt{g}\Big{(}-\frac{1}{2}R-\frac{1}{48}G_{IJKL}G^{IJKL}\Big{)},$
$\displaystyle S_{\rm CGG}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{3456\bar{\kappa}^{2}}\int_{M^{11}}d^{14}x\varepsilon^{I_{1}I_{2}...I_{11}}C_{I_{1}I_{2}I_{3}}G_{I_{4}...I_{7}}G_{I_{8}...I_{11}},$
$\displaystyle S_{\rm F(R)}$ $\displaystyle=$
$\displaystyle\int_{M^{11}+M^{N=3}}\sqrt{g}F(R,\phi,\psi)$ $\displaystyle
S_{\rm Matter}$ $\displaystyle=$
$\displaystyle\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{n-6}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
(18) $\displaystyle-$ $\displaystyle
i^{m}\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\int_{M^{11}+M^{N=3}}\sqrt{g}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\Sigma_{k=0}^{6-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}$

where $F(R,\phi,\psi)$ can be given by :

$\displaystyle F(R,\phi,\psi)$ $\displaystyle=$
$\displaystyle\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{n-6}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}R_{J_{1}J_{2}}..R_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
(19) $\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}R_{J_{1}J_{2}}..R_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\Sigma_{k=0}^{6-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}R_{J_{1}J_{2}}..R_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}$

This $F(R,\phi,\psi)$-gravity is now free of any anomaly on the (11 +
3)-dimensional manifold. In fact, this type of gravity is produced by
exchanging gravitons, scalars and fermions between two 11-dimensional
manifolds via one 3-dimensional manifold. The order of curvatures depends on
the number of dimensions of the manifolds and also on the number of scalars or
fermions which are attached to the curvatures. This result is in good
agreement with previous predictions in p7 ; p8 .

##  III Inflationary universe in $F(R,\phi,\psi)$-gravity

In this section, we construct a 4-dimensional universe on a (11+3)-dimensional
manifold. We will show that interaction between two 11-dimensional manifolds
in this system has a direct effect on the evolution of the universe during the
inflationary epoch. We will obtain the scale factor of the universe in terms
of the parameters of the system. Using the action in equation of (18) and the
relation between $G$, $C$, $F$ and $R$-terms in equation (6), we can obtain
the following field equations:

$\displaystyle R_{ij}-\frac{1}{2}Rg_{ij}$ (20) $\displaystyle+$
$\displaystyle(\frac{1}{2}g_{ij}F(R,\phi,\psi)-R_{ij}F^{\prime}(R,\phi,\psi)+\partial_{i}\partial_{j}F^{\prime}(R,\phi,\psi)-g_{ij}\partial_{i}\partial^{i}F^{\prime}(R,\phi,\psi))$
$\displaystyle+$
$\displaystyle(\frac{1}{2}g_{ij}(G_{IJKL}G^{IJKL})-R_{ij}(G_{IJKL}G^{IJKL})^{\prime}+\partial_{i}\partial_{j}(G_{IJKL}G^{IJKL})^{\prime}-g_{ij}\partial_{i}\partial^{i}(G_{IJKL}G^{IJKL})^{\prime})$
$\displaystyle+$
$\displaystyle\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}}(\frac{1}{2}g_{ij}(G_{IJKL}G^{IJKL})-R_{ij}(G_{IJKL}G^{IJKL})^{\prime}+\partial_{i}\partial_{j}(G_{IJKL}G^{IJKL})^{\prime}-g_{ij}\partial_{i}\partial^{i}(G_{IJKL}G^{IJKL})^{\prime})$
$\displaystyle+$
$\displaystyle(G_{IJKL}G^{IJKL})(\frac{1}{2}g_{ij}(\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}})-R_{ij}(\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}})^{\prime}+\partial_{i}\partial_{j}(\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}})^{\prime}-g_{ij}\partial_{i}\partial^{i}(\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}})^{\prime})$
$\displaystyle+$ $\displaystyle
g_{ij}[\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{n-6}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\Sigma_{k=0}^{6-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}]=0,$

where ′ denotes the derivative with respect to the curvature $R$ and

$\displaystyle\Big{(}\frac{\partial}{\partial
x_{I}}[\frac{\partial}{\partial(\partial_{I}\phi)}]-\frac{\partial}{\partial\phi}\Big{)}$
(21) $\displaystyle\times$
$\displaystyle[\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{n-6}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\Sigma_{k=0}^{6-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}]$
$\displaystyle\Big{(}\frac{\partial}{\partial
x_{I}}[\frac{\partial}{\partial(\partial_{I}\phi)}]-\frac{\partial}{\partial\phi}\Big{)}$
$\displaystyle\times$
$\displaystyle[G_{IJKL}G^{IJKL}+\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}}G_{IJKL}G^{IJKL}+F(R,\phi,\psi)]=0,$

$\displaystyle\Big{(}\frac{\partial}{\partial
x_{I}}[\frac{\partial}{\partial(\partial_{I}\psi)}]-\frac{\partial}{\partial\psi}\Big{)}$
(22) $\displaystyle\times$
$\displaystyle[\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{n-6}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\Sigma_{k=0}^{6-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}]$
$\displaystyle\Big{(}\frac{\partial}{\partial
x_{I}}[\frac{\partial}{\partial(\partial_{I}\psi)}]-\frac{\partial}{\partial\psi}\Big{)}$
$\displaystyle\times$
$\displaystyle[G_{IJKL}G^{IJKL}+\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}}G_{IJKL}G^{IJKL}+F(R,\phi,\psi)]=0,$

$\displaystyle\Big{(}\frac{\partial}{\partial
x_{I}}[\frac{\partial}{\partial(\partial_{I}A_{J})}]-\frac{\partial}{\partial
A_{I}}\Big{)}$ (23) $\displaystyle\times$
$\displaystyle[\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{n-6}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2m-1}}\phi\partial^{I_{2m}}\phi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi\Big{)}$
$\displaystyle-$ $\displaystyle
i^{m}\Big{(}\Sigma_{n=1}^{6}\Sigma_{m=0}^{6-n}\Sigma_{k=0}^{6-n-m}\varepsilon^{J_{1}J_{2}..J_{2n-1}J_{2n}}F_{J_{1}J_{2}}..F_{J_{2n-1}J_{2n}}\varepsilon^{I_{1}I_{2}..I_{2m-1}I_{2m}}\bar{\psi}\gamma^{I_{1}}\partial^{I_{2}}\psi..\bar{\psi}\gamma^{2I_{m}-1}\partial^{I_{2m}}\psi$
$\displaystyle\times$
$\displaystyle\varepsilon^{I_{1}I_{2}..I_{2k-1}I_{2k}}\partial^{I_{1}}\phi\partial^{I_{2}}\phi..\partial^{I_{2k-1}}\phi\partial^{I_{2k}}\phi+...\Big{)}]$
$\displaystyle\Big{(}\frac{\partial}{\partial
x_{I}}[\frac{\partial}{\partial(\partial_{I}A_{J})}]-\frac{\partial}{\partial
A_{I}}\Big{)}$ $\displaystyle\times$
$\displaystyle[G_{IJKL}G^{IJKL}+\varepsilon^{I^{\prime}J^{\prime}K^{\prime}}C_{I^{\prime}J^{\prime}K^{\prime}}G_{IJKL}G^{IJKL}+F(R,\phi,\psi)]=0,$

For a 4-dimensional universe, we have $R=12H^{2}+6\dot{H}$, where
$H=\frac{\dot{a}}{a}$ is the Hubble parameter and $a$ is the scale factor of
the universe. Assuming that all fields depend only on time, and using the
relation between scalars, gauge fields, curvatures and strings in equation
(6), we can solve equations (20, 21, 22 and 23) simultanously and obtain:

$\displaystyle,a(t)$ $\displaystyle\approx$ $\displaystyle e^{-\int dtZ(t)},$
$\displaystyle Z(t)$ $\displaystyle\approx$
$\displaystyle\Sigma_{n=1}^{6}\beta_{n}t_{s}^{2n}(1-\frac{t}{t_{s}})^{2n}\Big{(}1+\Sigma_{m=0}^{6-n}\alpha_{m}ln^{2m}(t_{s}(1-\frac{t}{t_{s}}))[1+\Sigma_{k=0}^{12-2n-2m}\lambda_{k}e^{2\gamma_{k}kt_{s}(1-\frac{t}{t_{s}})}]\Big{)}$
$\displaystyle\phi(t)$ $\displaystyle\approx$
$\displaystyle(\psi)^{\frac{1}{2}}\approx\Sigma_{n=1}^{6}\beta_{n}t_{s}^{2n}(1-\frac{t}{t_{s}})^{2n}\Sigma_{m=0}^{6-n}\alpha_{m}ln^{-2m}(\frac{t}{t_{s}})\times
Z^{\frac{1}{4}}(t)$ $\displaystyle A^{I}(t)$ $\displaystyle\approx$
$\displaystyle\varepsilon^{I}\Sigma_{m=1}^{6}\alpha_{m}(t_{s}(1-\frac{t}{t_{s}})^{2m}\Sigma_{k=0}^{6-m}\lambda^{\prime}_{k}e^{-2\gamma^{\prime}_{k}k(\frac{t}{t_{s}})}\times
ln(Z(t))$ (24)

where $t_{s}$ is time of collidision between the two 11-dimensional manifolds.
This time is much larger than the present age of the universe ($t_{s}\lll
13.5Gyr$). The above result shows the coincidence with the birth of the
universe ($t=0$) on the 11-dimensional manifold. The scale factor grows with
time, while scalars, fermions and gauge fields decrease, and tend to zero at
($t=t_{s}$). This is because with the passage of time, the two 11-dimensional
manifolds come close to each other and all fields which live on the
3-dimensional manifold between them dissolve into the 11-dimensional
manifolds. By the disappearance of these fields, gravitons and $F(R)$-gravity
emerge that lead to the expansion and inflation of the universe (See Figure
7).

---
Figure 7:  Emergence of 4-dimensional universe on 11-dimensional manifold in
11+3-dimensional space-time.

We can test our model by calculating the magnitude of the slow-roll parameters
and the tensor-to-scalar ratio ($R_{tensor-scalar}$) defined in da  and
compare with previous predictions:

$\displaystyle H=\frac{\dot{a}}{a}=-\dot{Z}\Rightarrow$
$\displaystyle\varepsilon=-\frac{\dot{H}}{H^{2}}=\frac{\ddot{Z}}{(\dot{Z})^{2}}$
$\displaystyle\eta=-\frac{\ddot{H}}{2H\dot{H}}=\frac{\dddot{Z}}{2\dot{Z}\ddot{Z}}$
$\displaystyle R_{tensor-
scalar}=16\varepsilon=16\frac{\ddot{Z}}{(\dot{Z})^{2}}$ (25)

Obviously, during the inflationary epoch, the age of the universe (t) is very
smaller with respect to the time of collision between manifolds ($t_{s}$) and,
as a result, ($(t_{s}-t)^{-n}\ll 1$) where $n$ is an integer. Using equation
(25) and expanding the functions ($e^{O}$ and $ln$) by applying the Taylor
series, we can derive the following results:

$\displaystyle 0\ll t\ll
t_{s}\Rightarrow\eta\approx\Sigma\beta_{n}(t_{s}-t)^{-2n}(ln^{-2n}(t_{s}(1-\frac{t}{t_{s}}))+O(\text{smaller
terms}))\ll 1$
$\displaystyle\varepsilon\approx\Sigma\alpha_{m}(t_{s}-t)^{-2m}(e^{-2\gamma_{m}mt_{s}(1-\frac{t}{t_{s}})}+O(\text{smaller
terms}))\ll 1$ $\displaystyle\Rightarrow R_{tensor-scalar}\ll 1$ (26)

This equation indicates that the values of the slow-roll parameters and the
tensor-to-scalar ratio are very much smaller than the ones which are
consistent with the predictions of the experiments in ref pl . Thus, this type
of $F(R,\phi,\psi)$-gravity which is produced as due to the interaction of two
11-dimensional manifolds, creates the correct values for the experimental
parameters and can describe cosmological events.

##  IV Summary and conclusion

Motivated by the successes of modified theories of gravity such as $F(R)$
gravity, in this paper, we have investigated how we could make $F(R)$ gravity
anomaly-free. To achieve this, we considered the origin of the action of
$F(R)$-gravity on an (11+3)-dimensional space-time. However, we found that
this was not anomaly-free. So, we proposed a theory which lives on an
(11+3)-dimensional manifold with two 11-dimensional manifolds and one
3-dimensional manifold. We showed that our universe is a part of one
11-dimensional manifold which is connected with the other 11-dimensional
manifold by an extra 3-dimensional manifold. The two 11-dimensional manifolds
interact with each other via exchanging fields which move along the
3-dimensional manifold. These fields are the main cause for the appearance of
F(R)-gravity in the four dimensional spacetime. This is because with the
passage of time, the two 11-dimensional manifolds come close to each other and
all ﬁelds which live on the 3-dimensional manifold between them dissolve into
the 11-dimensional manifolds. As these fields disappear, gravitons and
F(R)-gravity emerge that lead to the expansion and inﬂation of the universe.
Hence, the interaction between the two 11-dimensional manifolds has a direct
efect on the evolution of the universe during the inflationary era. We have
calculated the scale factor of the inflationary universe, and examined our
model against WMAP and Planck experiments. We have obtained the slow-roll
parameters and shown that the tensor-scalar ratio is much smaller than unity
($R_{tensor-scalar}\lll 1$), which is consistent with experimental
observations.

## Acknowledgments

Aroonkumar Beesham acknowledges that this work is based on the research
supported wholly / in part by the National Research Foundation of South Africa
(Grant Numbers: 118511). The work of Kazuharu Bamba has been supported in part
by the JSPS KAKENHI Grant Number JP 25800136 and Competitive Research Funds
for Fukushima University Faculty (19RI017).

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