# Adversarial Robustness of Deep Learning:
Theory, Algorithms, and Applications

Wenjie Ruan  University of ExeterExeterUK78229<EMAIL_ADDRESS>,  Xinping
Yi  University of LiverpoolLiverpoolUK<EMAIL_ADDRESS>and  Xiaowei
Huang  University of LiverpoolLiverpoolUK<EMAIL_ADDRESS>
(2021)

###### Abstract.

This tutorial aims to introduce the fundamentals of adversarial robustness of
deep learning, presenting a well-structured review of up-to-date techniques to
assess the vulnerability of various types of deep learning models to
adversarial examples. This tutorial will particularly highlight state-of-the-
art techniques in adversarial attacks and robustness verification of deep
neural networks (DNNs). We will also introduce some effective countermeasures
to improve robustness of deep learning models, with a particular focus on
adversarial training. We aim to provide a comprehensive overall picture about
this emerging direction and enable the community to be aware of the urgency
and importance of designing robust deep learning models in safety-critical
data analytical applications, ultimately enabling the end-users to trust deep
learning classifiers. We will also summarize potential research directions
concerning the adversarial robustness of deep learning, and its potential
benefits to enable accountable and trustworthy deep learning-based data
analytical systems and applications.

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
30th ACM International Conference on Information and Knowledge Management;
November 1–5, 2021; Virtual Event, QLD, Australia††booktitle: Proceedings of
the 30th ACM International Conference on Information and Knowledge Management
(CIKM ’21), November 1–5, 2021, Virtual Event, QLD, Australia††price:
15.00††doi: 10.1145/3459637.3482029††isbn: 978-1-4503-8446-9/21/11††ccs:
Computing methodologies Artificial intelligence

##  1\. Rationale

In recent years, we witness significant progress has been made in deep
learning, which can achieve human- or superhuman-level performance on various
data analytical tasks, such as image data recognition (Russakovsky et al.,
2015), natural language processing (Collobert et al., 2011), and medical data
analysis (Esteva et al., 2019). Given the prospect of a broad deployment of
DNNs in a wide range of applications, concerns regarding the safety and
trustworthiness of deep learning have been recently raised (Goodfellow et al.,
2018; Xu et al., 2020). There is significant research that aims to address
these concerns, with many publications appearing since the year of 2014 (Huang
et al., 2020). As we seek to deploy deep learning systems not only on virtual
domains, but also in real systems, it becomes critical that a deep learning
model can obtain satisfactory performance, but which are truly robust and
reliable. Although many notions of robustness and reliability exist in
different communities, one particular topic in machine learning community that
has attract enormous attention in recent years is the adversarial robustness
of deep learning: a deep learning model is fragile or extremely non-robust to
an input that is adversarially perturbed, and such perturbations usually are
invisible or insensible by humans (Goodfellow et al., 2018). This is although
a very specific notion of robustness in general, but one that concerns the
safety and trustworthiness of modern deep learning systems.

This tutorial seeks to provide a broad, hands-on introduction to the topic
concerning adversarial robustness: the widespread vulnerability of state-of-
the-art deep learning models to adversarial misclassification (i.e.,
adversarial examples). The goal is to combine both formal mathematical
treatments and practical tools and applications that highlight some of the key
methods and challenges concerning the adversarial robustness. This tutorial
will specifically concentrate on three major research progress on this
direction - adversarial attacks, defences and verification. As detailed in
Section 5, some tutorials regarding this emerging direction already appeared
in flagship conferences in machine learning, AI and computer vision
communities, including IJCAI 2021111Towards Robust Deep Learning Models:
Verification, Falsification, and Rectification in IJCAI 2021
(https://tutorial-ijcai.trustai.uk/), ECML/PKDD 2021222https://tutorial-
ecml.trustai.uk/, ICDM 2021333https://tutorial.trustdeeplearning.com/, CVPR
2020, KDD 2019444Recent Progress in Zeroth Order Optimization and Its
Applications to Adversarial Robustness in Data Mining and Machine Learning, in
KDD 2019, CVPR 2020, etc. Our tutorial is fundamentally different to those
existing ones, which is i) more comprehensive: we not only cover adversarial
attacks but particularly concentrate on verification-based approaches which is
able to establish formal robustness guarantees; ii) more application-oriented:
in the second part of our tutorial, we emphasize one particular defence
technique that can significantly improve robustness of DNNs - adversarial
training, which will shed a light on the development of robust deep learning
models for real-world data analytical applications. The specific differences
to each current similar tutorials are detailed in Section 5.

This tutorial can alarm the community to be aware of the safety
vulnerabilities of deep learning on real-world data analytical solutions
despite its appealing performance. We also envision, through this tutorial, AI
and data mining researchers and engineers get a sense on how to evaluate the
robustness of deep learning models (e.g., via adversarial
attacks/perturbations and verification-based approaches) and how to
design/train robust deep learning models (e.g., via defence, especially
adversarial training).

##  2\. Content details

  * •

Introduction to adversarial robustness: this part will introduce the concept
of adversarial robustness by showing some examples from computer vision (Zhang
et al., 2021b), natural language processing (Jin et al., 2020a), medical
systems (Wu et al., 2021), and autonomous systems (Wu and Ruan, 2021).
Specifically, we will demonstrate the vulnerabilities of various types of deep
learning models to different adversarial examples. We will also highlight the
dissimilarities of research focuses on adversarial robustness from different
communities, i.e., attack, defense and verification.

  * •

Adversarial attacks: this part will detail some famous adversarial attack
methods with an aim to provide some insights of why adversarial examples exit
and how to generate adversarial perturbation effectively and efficiently.
Specifically, we will present six typical adversarial attacks, including
L-BFGS (Szegedy et al., 2013), FGSM (Goodfellow et al., [n.d.]), C&W (Carlini
and Wagner, 2017), ZeroAttack (Tu et al., 2019), spatial-transformed attacks
(Xiao et al., 2018), universal attacks (Zhang et al., 2020). In the end of
this part, we will also briefly introduce some adversarial attacks on other
domains, including attacks on sentiment analysis systems (Jin et al., 2020a),
attacks on 3D point cloud models (Hamdi et al., 2020), attacks on audio
recognition systems (Abdullah et al., 2019).

  * •

Adversarial defense: this part will present an overview of state-of-the-art
robust optimisation techniques for adversarial training (Madry et al., 2018),
with emphasis on distributional robustness and the interplay between
robustness and generalisation. In particular, adversarial training with Fast
Gradient Method (FGM) (Goodfellow et al., [n.d.]), Projected Gradient Method
(PGM) (Kurakin et al., 2017) will be introduced briefly, followed by the
advanced methods promoting distributional robustness (Sinha et al., 2018) from
the viewpoints of robustness versus accuracy (Zhang et al., 2019), supervised
versus semi-supervised learning (Miyato et al., 2018b), and the exploitation
of local and global data information (Zhang and Wang, 2019; Qian et al.,
2021). In addition, the interplay between robustness and generalisation will
be discussed with respect to generalisable robustness and robust
generalisation. A variety of regularisation techniques such as spectral
normalisation (Miyato et al., 2018a), Lipschitz regularisation (Virmaux and
Scaman, 2018), and weight correlation regularisation (Jin et al., 2020b) to
promote generalisable robustness will be discussed, together with some recent
advances to improve robust generalisation (Schmidt et al., 2018; Yin et al.,
2019; Zhang et al., 2021a).

  * •

Verification and validation: this part will review the state-of-the-art on the
verification techniques for checking whether a deep learning model is
dependable. First, we will discuss verification techniques for checking
whether a convolutional neural network is robust against an input, including
constraint solving based techniques (MILP, Reluplex (Katz et al., 2017)),
approximation techniques (MaxSens (Xiang et al., 2018), AI2 (Gehr et al.,
2018), DeepSymbol (Li et al., 2019)), and global optimisation based techniques
(DLV (Huang et al., 2017), DeepGO (Ruan et al., 2018; Ruan et al., 2019),
DeepGame (Wicker et al., 2018; Wu et al., 2020)). Second, we will discuss some
software testing based methods which generate a large set of test cases
according to coverage metrics, including e.g., DeepXplore (Pei et al., 2017),
and DeepConcolic (Sun et al., 2018, 2019; Berthier et al., 2021), and their
extension to recurrent neural networks (Huang et al., 2021). The dependability
of a learning model can then be estimated through the test cases. Third, we
will discuss how to extend these above techniques to work with a reliability
notion which considers all possible inputs in an operational scenario (Zhao et
al., 2020). This requires the consideration of robustness and generalisation
in a holistic way (Jin et al., 2020b; Zhao et al., 2021). Finally, we will
summarize and outlook current state of this research field and future
perspectives.

##  3\. Target audience and prerequisites

This tutorial motivates and explains a topic of emerging importance for AI,
and it is particularly devoted to anyone who is concerning the safety and
robustness of deep learning models. The target audience would be data mining
and AI researchers and engineers who wish to learn how the techniques of
adversarial attacks and verification as well as adversarial training can be
effectively used in evaluating and improving the robustness of deep learning
models. No knowledge of the tutorial topics is assumed. A basic knowledge of
deep learning and statistical pattern classification is requested.

##  4\. Benefits

Deep learning techniques now is not only pervasive in the community of
computer vision and machine learning, but also widely applied on data
analytical systems. For researchers and industrial practitioners who are
developing safety-critical systems such as health data analytics, malware
detection, and automatic disease diagnosis, the robustness and reliability of
deep learning models are profoundly important. CIKM, as a flagship conference
in data mining and knowledge management, has attracted huge amount of data
scientists and engineers and many of them are using deep learning techniques.
We envision, by presenting a tutorial concerning the robustness of deep
learning at CIKM’21, the community can i) be aware the vulnerability of deep
learning models, ii) understand why such vulnerability exits in deep learning
and how to evaluate its adversarial robustness, and iii) know how to train a
robust deep learning model. We believe, those mentioned goals are appealing to
the audience in CIKM’21. In the meantime, a tutorial555Recent Progress in
Zeroth Order Optimization and Its Applications to Adversarial Robustness in
Data Mining and Machine Learning, in CVPR2020, KDD 2019. conerning similar
topic already appeared in SIGKDD’19 and ICDM 2020, two flagship conferences in
data mining. As such, we believe it is necessary and urgent to propose a more
comprehensive tutorial concentrating this topic in CIKM’21 as well.

##  5\. Difference to similar tutorials

  * •

Rigorous Verification and Explanation of ML Models, in AAAI 2020. Link:
https://alexeyignatiev.github.io/aaai20-tutorial/

    * –

The overlapping will be on the verification part, where the above tutorial
only considers from the logic/binary perspective, while we will cover
comprehensively constraint-solving based methods, approximation methods, and
global optimisation methods. The other two topics of this tutorial, i.e.,
adversarial attack (Part-I) and defense (Part-III), are not covered in the
above tutorial.

  * •

Adversarial Machine Learning, in AAAI 2019, AAAI 2018. Link:
https://aaai19adversarial.github.io/index.html#org

    * –

The above tutorial focuses on adversarial attacks of classifier evasion and
poisoning as well as the corresponding defense techniques, while this tutorial
places the emphasis on the fundamentals of adversarial examples (Part I) and
generalisable robust optimisation techniques for defense (Part III), which are
not covered by the above tutorial. In addition, the topic of verification of
this tutorial (Part II) is not covered at all in the above one.

  * •

Adversarial Machine Learning, in IJCAI 2018, ECCV 2018, ICCV 2017. Link:
https://www.pluribus-one.it/research/sec-ml/wild-patterns

    * –

The above tutorial concentrates on demonstrating vulnerability of various
machine learning models and the design of learning-based pattern classifiers
in adversarial environments. Our tutorial, however, is primarily about
adversarial robustness of deep neural networks, especially the safety
verification (Part-II) and adversarial defense (Part-III) on DNNs are not
covered by the above tutorial. The only overlapping will be the part of
adversarial attacks, but ours is more comprehensive and deep learning-focused.

  * •

Adversarial Robustness: Theory and Practice, in NIPS 2018
Link: https://adversarial-ml-tutorial.org/

    * –

The above tutorial was given two years ago, concentrating on the verification-
based approaches to establishing formal guarantees for adversarial robustness.
Then it presents adversarial training and regularization-based methods for
improving robustness. Our tutorial will be more up-to-date. The major
differences are: 1) in adversarial attacks, we present more recent and
advanced adversarial examples, such as universal and spatial-transformed one;
2) in verification, we are more comprehensive, except for the constraint-
solving based methods, we also cover the approximation and global optimisation
methods; and (3) in adversarial defense, our focus is on generalisable
adversarial training via advanced spectral regularisation.

  * •

Recent Progress in Zeroth Order Optimization and Its Applications to
Adversarial Robustness in Data Mining and Machine Learning, in CVPR 2020, KDD
2019.
Link: https://sites.google.com/view/adv-robustness-zoopt

    * –

The above tutorial concentrates on Zero-order optimization methods with a
particular focus on black-box adversarial attacks to DNNs. Our tutorial is
more comprehensive, which not only covers a wider range of adversarial
attacks, but also presents verification approaches that can establish formal
guarantees on adversarial robustness, as well as review state-of-the-art
adversarial training methods that can defence adversarial attacks and improve
DNN’s robustness.

##  6\. Relevant Experience on the Topic

###  6.1. Relevant Tutorial Experience

  * •

W. Ruan, X. Yi, X. Huang, Tutorial “Adversarial Robustness of Deep Learning
Theory, Algorithms, and Applications”, The 20th IEEE International Conference
on Data Mining (ICDM 2020), 17-20 Nov 2020, Sorrento, Italy
Link: https://tutorial.trustdeeplearning.com/

  * •

W. Ruan, E. Botoeva, X. Yi, X. Huang, Tutorial ”Towards Robust Deep Learning
Models: Verification, Falsification, and Rectification”, The 30th
International Joint Conference on Artificial Intelligence (IJCAI 2021), 21-26
Aug 2021, Canada
Link: http://tutorial-ijcai.trustai.uk/

  * •

W. Ruan, X. Yi, X. Huang, Tutorial “Adversarial Robustness of Deep Learning
Theory, Algorithms, and Applications”, The 2021 European Conference on Machine
Learning and Principles and Practice of Knowledge Discovery in Databases
(ECML/PKDD 2021), 13-17 Sep 2021, Virtual
Link: http://tutorial-ecml.trustai.uk/

###  6.2. Citations of Relevant Works

The following are the selected papers published by the presenters which are
related to this tutorial666The citation numbers are from Google Scholar on 22
August 2021.

  * (1)

Safety verification of deep neural networks, CAV 2017, Citation = 623

  * (2)

Structural Test Coverage Criteria for Deep Neural Networks, in ACM
Transactions on Embedded Computing Systems 2018, Citation = 196

  * (3)

Concolic Testing for Deep Neural Networks, in ASE 2018, Citation = 183

  * (4)

Feature-guided black-box safety testing of deep neural networks, in TACAS
2018, Citation = 162

  * (5)

Reachability Analysis of Deep Neural Networks with Provable Guarantees, in
IJCAI 2018, Citation = 163

  * (6)

A survey of safety and trustworthiness of deep neural networks: Verification,
testing, adversarial attack and defence, and interpretability, in Computer
Science Review, 2020. Citation = 111

  * (7)

A game-based approximate verification of deep neural networks with provable
guarantees, in Theoretical Computer Science, 2020. Citation = 61

  * (8)

Global Robustness Evaluation of Deep Neural Networks with Provable Guarantees
for the Hamming Distance, in IJCAI 2019, Citation = 53

The overall citations of the above papers that are closely related to this
tutorial are over 1,200 since 2017. The details are as below:

  * •

A seminal paper (1), on the safety verification of deep learning, has
attracted 600+ Google Scholar citations, which is one of the first papers on
the verification of deep learning.

  * •

A few papers including (5) (7) (8), on the verification of neural networks
through global optimisation algorithms, have attracted 250+ citations.

  * •

A few papers including (2) (3) (4) (5), on the adversarial attacks of the
robustness of neural networks, have attracted 500+ citations.

  * •

A recent survey paper (6), closely aligned with the topic of this tutorial,
has attracted 100+ Google scholar citations since its publication in 2020.

##  7\. Brief Resumes of Presenters

####  7.0.1. Dr Wenjie Ruan

Dr Wenjie Ruan is a Senior Lecturer of Data Science at University of Exeter,
UK. Previously, he has worked at Lancaster University as a lecturer, and
University of Oxford as a postdoctoral researcher. Dr Ruan got his PhD from
University of Adelaide, Australia. His series of research works on Device-free
Human Localization and Activity Recognition for Supporting the Independent
Living of the Elderly have received Doctoral Thesis Excellence from The
University of Adelaide. He was also the recipient of the prestigious DECRA
fellowship from ARC (Australian Research Council). Dr Ruan has published 30+
top-tier papers in top venues such as AAAI, IJCAI, ICDM, UbiComp, CIKM, ASE,
etc. His recent work on reachability analysis on deep learning is one of the
most citable papers in IJCAI’18 (150+ citations since 2018), and his work on
testing-based falsification on deep learning is also one of the most citable
papers in ASE’18 (150+ citations since 2018). Dr. Ruan has served as Senior
PC, or PC member for over 10 conferences including IJCAI, AAAI, ICML, NeurlPS,
CVPR, ICCV, AAMAS, ECML-PKDD, etc. His homepage is: http://wenjieruan.com/.

####  7.0.2. Dr Xinping Yi

Dr Xinping Yi is a Lecturer (Assistant Professor) of Electrical Engineering at
the University of Liverpool, UK. He received his Ph.D. degree from Telecom
ParisTech, Paris, France. Prior to Liverpool, he worked at Technische
Universitat Berlin, Germany, EURECOM, France, UC Irvine, US, and Huawei
Technologies, China. Dr Yi’s recent research lies in deep learning theory with
emphasis on generalisation and adversarial robustness. Dr Yi has published 40+
papers in IEEE transactions such as IEEE Transactions on Information Theory
(TIT), and machine learning conferences such as ICML, NeurIPS. He has served
as programme committee members and reviewers at a number of international
conferences and journals, such as ICML, ICLR, IJCAI, CVPR, ICCV, ISIT, ICC,
TIT, Proceedings of IEEE, JSAC, TWC, Machine Learning. His homepage is:
https://sites.google.com/site/xinpingyi00/

####  7.0.3. Dr Xiaowei Huang

Dr Xiaowei Huang is Reader of Computer Science at the University of Liverpool,
UK. Prior to Liverpool, he worked at Oxford and UNSW Sydney. Dr Huang’s
research concerns the safety and trustworthiness of autonomous intelligent
systems. He is now leading a research group working on the verification,
validation, and interpretability of deep neural networks. Dr Huang is the
principle investigator of several Dstl projects concerning the safety and
assurance of artificial intelligence, and the Liverpool lead on a H2020
project on the foundation of trustworthy autonomy and an EPSRC project on the
security of neural symbolic architectures. He is actively engaged with both
formal verification and artificial intelligence communities and has given a
number of invited talks on topics related to the safety of artificial
intelligence. Dr Huang has published 50+ papers in international conferences
such as AAAI, IJCAI, CAV, TACAS, ASE, etc., and has served in program
committees of 20+ international conferences. His homepage is:
https://cgi.csc.liv.ac.uk/~xiaowei/

##### Acknowledgement

Wenjie Ruan is supported by Offshore Robotics for Certification of Assets
(ORCA) Partnership Resource Fund (PRF) on Towards the Accountable and
Explainable Learning-enabled Autonomous Robotic Systems (AELARS)
[EP/R026173/1].  XH has received funding from the European Union’s Horizon
2020 research and innovation programme under grant agreement No 956123, and is
also supported by the UK EPSRC (through the Offshore Robotics for
Certification of Assets [EP/R026173/1] and End-to-End Conceptual Guarding of
Neural Architectures [EP/T026995/1]).

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aainstitutetext: Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russiabbinstitutetext: Physics Department, Ariel University,
Ariel 40700, Israelccinstitutetext: Moscow Institute of Physics and
Technology, 141700 Dolgoprudny, Russiaddinstitutetext: Kharkevich Institute
for Information Transmission Problems, 127994 Moscow, Russia

# Periods of the multiple Berglund–Hübsch–Krawitz mirrors

Alexander Belavin b Vladimir Belavin d Gleb Koshevoy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in
Berglund-Hübsch-Krawitz (BHK) mirror symmetry. We show that for any pair of
Calabi–Yau orbifolds that are BHK mirrors of a loop–chain type pair of
Calabi–Yau manifolds in the same weighted projective space the periods of the
holomorphic nonvanishing form coincide.

###### Keywords:

Mirror Symmetry, Calabi-Yau manifolds, Compactification

††dedication: In memory of Boris Dubrovin.

##  1 Introduction

Calabi-Yau manifolds (CY) arise in the context of spacetime supersymmetric
compactifications of string theories. An important property of CY manifolds is
mirror symmetry Gepner ; Greene ; Hori , which reflects geometric relation
between pairs of CY manifolds. Namely, for a pair of n-dimensional Calabi-Yau
manifolds $X$ and $Y$, cohomologies are isomorphic,
$H^{p,q}(X,\mathbb{C})=H^{n-p,q}(Y,\mathbb{C})$. Calabi-Yau manifolds possess
the structure of complex and Kähler manifold which admit deformations (see
Candelas:1990pi ), giving rise to the moduli spaces $M_{C}(X)$ and $M_{K}(X)$.
Mirror symmetry can be considered as matching of the Special geometries
Strominger  on the moduli spaces

$M_{C}(X)\simeq M_{K}(Y)\;,\quad M_{K}(X)\simeq M_{C}(Y)\;.$ (1)

Here we focus on the class of CY manifolds, defined as hypersurfaces or
orbifolds in weighted projective spaces
$\mathbb{P}^{4}_{(k_{1},k_{2},k_{3},k_{4},k_{5})}$ cut out by non-degenerate
quasihomogeneous polynomials consisting of five monomials. The nondegeneracy
requires that these polynomials must be a sum of the polynomials of three
basic types, classified by M. Kreuzer and H. Skarke Kreuzer . For this class
of CY manifolds it was suggested by Berglund and Hübsch in Berglund  that
their mirrors are obtained as particular orbifolds of the projective spaces
defined by the “transposed” polynomials. The Berglund-Hübsch mirror
construction was generalized by Krawitz in Krawitz . Chiodo and Ruan Chiodo
proved that the Berglund-Hübsch-Krawitz (BHK) mirrors form a mirror pair on
the level of Chen-Ruan orbifold cohomology.

The multiple mirror phenomenon occurs when a given CY threefold possesses more
than one mirror in different weighted projective spaces. More specifically the
phenomenon is the following. Some weighted projective spaces allow the
existence of a few CY manifolds defined by the polynomials that belong to
different types of Kreuzer-Skarke class. In such cases the Berglund-Hübsch-
Krawitz (BHK) mirrors of these two CY manifolds generally appear in two
different weighted projective spaces. The examples of such phenomenon are
given in the Table 1, which shows all the cases when the weighted projective
space allow the simultaneous occurrence of CY-manifolds of loop and chain
types of Kreuzer-Skarke class CY manifolds. The problem of birationality of
multiple BHK-mirrors has been investigated early in the case of one-parameter
CY family Kelly ; Shoemaker ; Clarke ; Clader , where it was shown that the
different BHK mirrors are birational. The derived equivalence of such multiple
BHK-mirrors was obtained in Favero . In ABMB , the periods of the multiple
mirrors were calculated in one case of multi-parameter CY family, and it was
found that they match. In this paper we prove that the periods of the
nonvanishing holomorphic form $\Omega$ coincide for multiple BHK mirrors of
the loop and chain types. We assume that, using a similar technique, one can
also prove coincidence of the periods for other types of multiple BHK mirrors.

In G  it was shown using the technique of Griffiths-Dwork diagrams, that if
two invertible pencils ($\sum_{j}\prod_{i}x_{ij}^{M_{ij}}+t\prod_{i}x_{i}$)
with reversible $M$ have the same dual weights, then their Picard-Fuchs
equations are the same. It is equivalent to the statement that one-parameter
deformations of several BHK mirrors have the same Picard-Fuchs equations. This
statement is consistent with ours as a special case. The paper G  used another
tools to show the coincidence of Picard-Fucks equations for one parameter
deformation of invertible polynomials with multiple mirrors. Note that the
coincidence of the periods does not follow automatically from this even in
this particular case.

In Section 2 we briefly recall the BHK mirror construction. In Section 3, we
consider the expressions for the periods of the holomorphic form $\Omega$ for
two different BHK mirrors and show their coincidence.

##  2 Berglund–Hübsch–Krawitz mirror construction

As was mentioned above among all possible CY manifolds there is a class
Kreuzer , which is defined in the weighted projective space

$\mathbb{P}^{4}_{(k_{1},k_{2},k_{3},k_{4},k_{5})}=\Big{\\{}(x_{1},\dots,x_{5})\in\mathbb{C}^{5}\setminus\\{0\\}\
\Big{|}\ x_{i}\sim\lambda^{k_{i}}x_{i}\Big{\\}}$ (2)

by a quasi-homogeneous polynomial

$W^{0}_{M}(x)=\sum_{i=1}^{5}\prod_{j=1}^{5}x_{j}^{M_{ij}}\;.$ (3)

Such polynomials obey the following properties:

  * 1)

Matrix $M$ is integer, and invertible;

  * 2)

The polynomial $W^{0}_{M}$ is quasi-homogeneous, i.e., there exist positive
integers $k_{i}$ (weights of the projective space), and $d$ (degree), such
that

$\sum_{j=1}^{5}M_{ij}k_{j}=d,\ \ \ \forall i,$ (4)

where $d=\sum_{j=1}^{5}k_{j}$, which is CY condition for the orbifold in the
wighted projective space.

  * 3)

The polynomial $W^{0}_{M}$ is non-degenerate outside the origin.

From these conditions it follows, that the $W^{0}_{M}$ is given by a sum of
polynomials of one of three types Kreuzer :

$\displaystyle x_{1}^{A_{1}}+x_{2}^{A_{2}}+...+x_{n}^{A_{n}}-\text{Fermat}\;,$
(5) $\displaystyle
x_{1}^{A_{1}}x_{2}+x_{2}^{A_{2}}x_{3}+...+x_{n}^{A_{n}}-\text{Chain}\;,$
$\displaystyle
x_{1}^{A_{1}}x_{2}+x_{2}^{A_{2}}x_{3}+...+x_{n}^{A_{n}}x_{1}-\text{Loop}\;.$

Then it follows that the equation $W^{0}_{M}=0$ defines a Calabi-Yau orbifold
$X^{0}_{M}$. We use below the following properties of the matrix $M$. There
exist positive integer numbers $\bar{k}_{j}$ such that

$\sum_{i=1}^{5}\bar{k}_{i}M_{ij}=\bar{d},\ \ \ \forall j,$ (6)

where $\bar{d}=\sum_{i=1}^{5}\bar{k}_{i}$. The set $\bar{k}_{j}$ is nothing
but the set of weights of another weighted projective space
$\mathbb{P}^{4}_{(\bar{k}_{1},\bar{k}_{2},\bar{k}_{3},\bar{k}_{4},\bar{k}_{5})}$
where $X_{M^{T}}$, i.e. the mirror of the CY orbifold $X^{0}_{M}$, which will
be defined below. Let $B_{ij}$ denote the inverse matrix of $M$, i.e.
$\sum_{k}B_{ik}M_{kj}=\delta_{ij}$. Then using (4),(6) we get the relations
(see e.g. Chiodo ):

$\sum_{j=1}^{5}B_{ij}=\frac{k_{i}}{d}\;,\quad\sum_{i=1}^{5}B_{ij}=\frac{\bar{k}_{j}}{\bar{d}}\;.$
(7)

The CY orbifold $X^{0}_{M}$ admits a deformation of the complex structure,
which is realized by a polynomial $W_{M}$, which is a deformation of the
$W^{0}_{M}$. The full family $X_{M}$ is given by zero locus of

$W_{M}=\sum_{i=1}^{5}\prod_{j=1}^{5}x_{j}^{M_{ij}}+\sum_{l=1}^{h}\varphi_{l}e_{l}(x),$
(8)

where $e_{l}$ are some quasi-homogeneous monomials (see below). The
$\varphi_{l}$ are moduli of the complex structure of the family $X_{M}$, and
$h:=h_{21}$ is the Hodge number of the family $X_{M}$, which is equal to the
dimension of the complex structures moduli space of $X_{M}$.

Berglund and Hübsch proposed Berglund  that the mirror partner $X_{M^{T}}^{0}$
of the orbifold $X_{M}^{0}$ is defined as a quotient of the zero-locus of the
polynomial

$W^{0}_{M^{T}}(z)=\sum_{i=1}^{5}\prod_{j=1}^{5}x_{j}^{(M^{T})_{ij}}$ (9)

in the weighted projective space
$\mathbb{P}^{4}_{(\bar{k}_{1},\bar{k}_{2},\bar{k}_{3},\bar{k}_{4},\bar{k}_{5})}$.
This quotient is realized by some subgroup of the phase symmetry group of
$W^{0}_{M^{T}}$.

This approach has been generalized by Krawitz Krawitz  as follows. Let
$\text{Aut}(M)$ be the group of the phase symmetries of $W^{0}_{M}=0$, and let
$\operatorname{SL}(M)$ be its maximal subgroup, which preserves the
nonvanishing holomorphic form $\Omega$. We will call $\operatorname{SL}(M)$
the maximal admissible group. We denote by
$J_{M}\subseteq\operatorname{SL}(M)$ the subgroup that consists of the phase
symmetries induced by $\mathbb{C}^{*}$ action on
$\mathbb{P}^{4}_{(k_{1},k_{2},k_{3},k_{4},k_{5})}$. Choose a group $G_{0}$ to
be some subgroup of $\operatorname{SL}(M)$ containing $J_{M}$ and let
$G=G_{0}/J_{M}$. It can be shown that the quotient space $Z(M,G):=X_{M}/G$ is
a CY orbifold. The same can be applied to the transposed polynomial
$W_{M^{T}}^{0}$ to get CY orbifold $Z(M^{T},G^{T}):=X_{M^{T}}/G^{T}$. Krawitz
has shown how to choose the dual group $G^{T}$ such that $Z(M,G)$ and
$Z(M^{T},G^{T})$ form a mirror pair. Below we remind this mirror constuction
Krawitz  in more details.

Let CY hypersurface $X_{M}^{0}$ be defined, as above, by the zero locus of
$W_{M}^{0}(x)$ and let $\omega=e^{2\pi i/d}$. The polynomial $W_{M}^{0}(x)$,
as well as the form $\Omega$, is invariant under the action of the group
$J_{M}$ defined as

$x_{i}\mapsto\omega^{k_{i}}x_{i},\ \ \ \omega^{d}=1.$ (10)

Let $q_{i}(M)$ be generators of ${Aut}(M)$, which act on the coordinates as

$q_{i}(M):x_{j}\mapsto e^{2\pi iB_{ji}}x_{j}.$ (11)

In these terms, the generator of the group $J_{M}$

$\hat{J}=\prod_{i=1}^{5}q_{i}(M).$ (12)

Let $G_{0}$ be some admissible group such that $J_{M}\subseteq G_{0}\subseteq
SL(M)$ and the quotient group

$G:=G_{0}/J_{M}.$ (13)

Then the orbifold defined as $Z(M,G):=X_{M}/G$ is a Calabi-Yau manifold. Note
that $X_{M}$ itself is a special case of $Z(M,G)$ when $G_{0}=J_{M}$.

The BHK construction begins with the polynomial $W_{M}$ and some group
$G_{0}$. Let monomials

$e_{l}:=\prod_{j=1}^{5}x_{j}^{S_{lj}},$ (14)

where $\sum_{j}S_{lj}k_{j}=d$, be elements of the $G_{0}$-invariant subring of
the Milnor ring $\mathbb{C}[x_{1},\dots,x_{5}]/\langle\frac{\partial
W_{M}}{\partial x_{j}}\rangle$ Lerche . Then the quotient by the group $G$ is
defined by the zero locus of the polynomial

$W_{M}=\sum_{i=1}^{5}\prod_{j=1}^{5}x_{j}^{M_{ij}}+\sum_{l=1}^{h}\varphi_{l}\prod_{j=1}^{5}x_{j}^{S_{lj}},$
(15)

where $\varphi_{l}$ are moduli of complex structure deformations, and
$h:=h_{21}(X)$ is Hodge number.111With a slight abuse of notation, we use here
$h$ for the Hodge number of the orbifold, which we used before for the
original manifold $X_{M}$. We denote by $e_{h}$ the monomial with $S_{h,i}=1$,
which plays a distinguished role,

$e_{h}=\prod_{i=1}^{5}x_{i}.$ (16)

Let $\rho_{s}$ be the generators of the group $G_{0}$, then by definition

$\rho_{s}\cdot\prod_{j=1}^{5}x_{j}^{S_{lj}}=\prod_{j=1}^{5}x_{j}^{S_{lj}},\ \
\ l=1,\cdots,h.$ (17)

We can also define the similar data for the polynomial $W_{M}^{T}$ and for
some of its admissable group $G^{T}_{0}$. Taking the quotient
$G^{T}:=G_{0}^{T}/J_{M^{T}}$, we get the Calabi-Yau orbifold $Z(M^{T},G^{T})$.

The way to choose a group $G^{T}$ such that the manifold $Z(M^{T},G^{T})$ will
be the mirror for $Z(M,G)$ is as follows Krawitz . The generators
$\rho_{l}^{T}$ of the group $G_{0}^{T}$ should be constructed using the
exponents $S_{li}$ of the invariant monomials (14) as follows

$\rho_{l}^{T}:=\prod_{i=1}^{5}q_{i}(M^{T})^{S_{li}},$ (18)

where $q_{i}(M^{T})$ are the generators of $\operatorname{Aut}(M^{T})$ acting
on each coordinate $z_{j}$ in
$\mathbb{P}^{4}_{(\bar{k}_{1},\bar{k}_{2},\bar{k}_{3},\bar{k}_{4},\bar{k}_{5})}$
as

$q_{i}(M^{T}):z_{j}\mapsto e^{2\pi iB_{ji}^{T}}z_{j}=e^{2\pi iB_{ij}}z_{j}.$
(19)

It follows that the generators $\rho_{l}^{T}$ of the group $G_{0}^{T}$ act on
the coordinates $z_{j}$ as

$\rho_{l}^{T}:z_{j}\mapsto e^{2\pi i\sum_{i}S_{li}B_{ij}}z_{j}.$ (20)

The mirror Calabi-Yau family $Z(M^{T},G^{T})$ is given by the zero locus of

$W_{M^{T}}:=\sum_{i=1}^{5}\prod_{j=1}^{5}z_{j}^{M^{T}_{ij}}+\sum_{r=1}^{\bar{h}}\psi_{r}\prod_{j=1}^{5}z_{j}^{T_{rj}}=\sum_{i=1}^{5}\prod_{j=1}^{5}z_{j}^{M_{ji}}+\sum_{r=1}^{\bar{h}}\psi_{r}\prod_{j=1}^{5}z_{j}^{T_{rj}},$
(21)

where $\psi_{r}$ are moduli of the complex structure of the family
$Z(M^{T},G^{T})$ and monomials $\bar{e}_{r}:=\prod_{j=1}^{5}z_{j}^{T_{rj}}$
are invariant under the $G_{0}^{T}$ action (20):

$\rho_{l}^{T}\cdot\bar{e}_{r}=e^{2\pi
i\sum_{ij}B_{ij}S_{li}T_{rj}}\prod_{j=1}^{5}z_{j}^{T_{rj}}=\prod_{j=1}^{5}z_{j}^{T_{rj}}.$
(22)

It is also necessary to take into account the quasi-homogeneity condition

$\sum_{i=1}^{5}T_{mi}\bar{k}_{i}=\bar{d}.$ (23)

It is convenient to consider the exponents $S_{li}$, and $T_{rj}$ as
nonnegative integer five-component vectors $(\vec{S}_{l})_{j}=S_{lj}$, and
$(\vec{T}_{r})_{i}=T_{ri}$. It was shown in ABBE  that the condition (22) can
be reformulated in terms of the pairing of these vectors, defined by the
matrix $B$, as follows

$(\vec{S}_{l},\vec{T}_{r}):=\sum_{i,j=1}^{5}B_{ij}S_{li}T_{rj}\in\mathbb{Z}.$
(24)

The relations (23) and (24) are very restrictive because the entries of the
matrices $S_{li}$ and $T_{rj}$ are integers, while the entries of the inverse
matrix $B_{ij}$ are rational. Therefore, if the vectors $\vec{S}_{l}$ are
defined as above, these relations have a finite number of solutions for the
vectors $\vec{T}_{r}$. The CY orbifold cut out by $W_{M^{T}}$ is the mirror
for $Z(M,G)$, as it is shown in Krawitz .

##  3 Periods of BHK mirrors and their coincidence

In this section we use the relation (24) to prove our basic statement that the
periods of multiple mirrors coincide. We will focus on the case when two
hypersurfaces in the weighted projective space simultaneously come from two
polynomials of the loop and chain types that “point” in the same direction. We
will assume also that the admissible group $G_{0}=J_{M}$ preserves both
polynomials.

Let $X_{M}$ be the CY family $X_{M}$ as defined in Section 2. We regard $X(M)$
as the family of CY manifolds, since the polynomial $W_{M}(x,\phi)$ is a
function of two sets of variables $x$ and parameters $\\{\phi_{l}\\}$. Then
the mirror family $Y(M^{T})$ is defined in
$\mathbb{P}_{\vec{\bar{k}}}=\mathbb{P}^{4}_{(\bar{k}_{1},\bar{k}_{2},\bar{k}_{3},\bar{k}_{4},\bar{k}_{5})}$
by zeros of the polynomial

$W_{M^{T}}(x)=\sum_{i=1}^{5}\prod_{j=1}^{5}x_{j}^{M^{T}_{ij}}+\sum_{r=1}^{\bar{h}}\psi_{r}\prod_{j=1}^{5}x_{j}^{T_{rj}}\;,$
(25)

where it is assumed that the relationships
$\sum_{j=1}^{5}T_{rj}\bar{k}_{j}=\bar{d}$ and
$\sum_{ij}B_{ij}S_{li}T_{rj}\in\mathbb{Z}$ are fulfilled.

Periods of the holomorphic form $\Omega$ for CY family $X(M)$ can be rewritten
as the contour integrals ABKA

$\sigma_{\vec{\mu}}(\phi)=\oint_{\Gamma_{\vec{\mu}}}e^{-W_{M}(x,\phi)}d^{5}x\;.$
(26)

The number of the contours $\Gamma_{\vec{\mu}}$ is equal to
$\text{dim}\,R(M)=2h+2$. Here $R(M)$ is so-called chiral ring which is subring
of polynomials in $x_{1},...,x_{5}$ modulo $\\{\frac{\partial
W^{0}_{M}}{\partial x_{i}}\\}$ of degree $0$,$d$, $2d$ or $3d$. The ring
$R(M)$ consists of monomials

$e_{\vec{\mu}}(x)=\prod_{i=1}^{5}x_{i}^{\mu_{i}}\;,$ (27)

where

$\sum_{i=1}^{5}k_{i}\mu_{i}=0,\,\,d,\,\,2d,\,\,3d$ (28)

and $\mu_{i}$ are non-negative integers satisfying some constraints:222Below
$A_{i}$ are defined according to eqs.(4) and (5).
In the Fermat case

$0\leq\mu_{i}\leq A_{i}-2\;.$ (29)

In the Loop case

$0\leq\mu_{i}\leq A_{i}-1\;.$ (30)

A convenient set of the chiral ring monomials in the Chain case was described
in Kreuzer3  (see also Kreuzer2 , Krawitz ).

From (28) it follows that the ring $R(M)$ is graded,

$R(M)=\mathbb{R}^{0}(M)\oplus R^{d}(M)\oplus R^{2d}(M)\oplus R^{3d}(M)$ (31)

and $\text{dim}\,R^{0}(M)=\text{dim}\,R^{3d}(M)=1$, while
$\text{dim}\,R^{d}(M)=\text{dim}\,R^{2d}(M)=h$. The monomials
$e_{l}(x)=\prod_{j=1}^{5}x_{j}^{S_{lj}},\,l=1,...,h$ are elements of the
subspace $R^{d}(M)$. Similar statements can be formulated for the mirror
orbifold $Y(M^{T})$.
Since

$W_{M}(x,\phi)=W_{M}^{0}(x)+\sum_{l=1}^{h}\phi_{l}e_{l}(x)\;,$ (32)

the periods have the form

$\displaystyle\sigma_{\vec{\mu}}(\phi_{1},...,\phi_{h})=\int_{\Gamma_{\vec{\mu}}}d^{5}xe^{-W_{M}^{0}(x)}e^{-\sum_{l=1}^{h}\phi_{l}e_{l}(x)}=\sum_{\\{m_{l}\\}}\prod_{l=1}^{h}\frac{\phi_{l}^{m_{l}}}{m_{l}!}\int_{\Gamma_{\vec{\mu}}}d^{5}xe^{-W_{M}^{0}(x)}\prod_{j=1}^{5}x_{j}^{\sum_{l=1}^{h}m_{l}S_{lj}}.$
(33)

Using the standard cohomology technique ABKA , we obtain the following
explicit expression

$\sigma_{\vec{\mu}}(\phi_{1},...,\phi_{h})\sim\sum_{\\{m_{l}\\}\in\Sigma_{\vec{\mu}}}C(\\{m_{l}\\})\prod_{l=1}^{h}\frac{\phi_{l}^{m_{l}}}{m_{l}!}\;,$
(34)

where

$C(\\{m_{l}\\})=\prod_{i=1}^{5}\Gamma\left(\sum_{j}(m_{l}S_{lj}+1)B_{ji}\right)\;.$
(35)

The canonical choice of $\Gamma_{\vec{\mu}}$ defines it as a cycle dual to the
element of the base of the chiral ring $e_{\vec{\mu}}$

$\int_{\Gamma_{\vec{\mu}}}e_{\vec{\nu}}(x)e^{-W_{0}(x)}d^{5}x=\delta_{\vec{\mu},\vec{\nu}}\;.$
(36)

It leads to the following definition of $\Sigma_{\vec{\mu}}$ ABKA . The set of
non-negative integers $m_{l}$, where $l=1,...,h$, belongs to
$\Sigma_{\vec{\mu}}$, if one can find such non-negative integers $n_{j}$,
$j=1,...,5$ that

$m_{l}S_{li}=({\vec{\mu}})_{i}+\sum_{j}n_{j}M_{ji}\;.$ (37)

From (35) we see that the periods are determined by the sets of $h$
5-component vectors $\sum_{j=1}S_{lj}B_{ji}$, where $l=1,..,h$, as well as by
the sets $\\{m_{l}\\}$, $l=1,...,h$.

Let’s now prove that the periods of CY orbifolds $Y(M^{T}(1))$ and
$Y(M^{T}(2))$, defined in two different weighted projective spaces
$\mathbb{P}_{\vec{k}(1)}$ and $\mathbb{P}_{\vec{k}(2)}$ are the same, because
they are the BHK mirrors of the CY manifolds $X(M(1))$ and $X(M(2))$ located
in the same projective space $\mathbb{P}_{\vec{k}}$.

In order to simplify notations we denote the initial loop matrix as $M(1)$,
and the initial chain matrix as $M(2)$. Recall that they satisfy the relations

$\sum_{j=1}^{5}(M(1))_{ij}k_{j}=\sum_{j=1}^{5}(M(2))_{ij}k_{j}=d=\sum_{j=1}^{5}k_{j}\;.$
(38)

We denote by $S_{lj}(1)\in\mathbb{Z}_{\geq 0}$ and
$S_{lj}(2)\in\mathbb{Z}_{\geq 0}$ the vectors of the exponents of the
monomials providing the deformation of the complex structure

$e_{l}(1)=\prod_{j=1}^{5}x_{j}^{S_{lj}(1)}\;,\quad
e_{l}(2)=\prod_{j=1}^{5}x_{j}^{S_{lj}(2)}$ (39)

and

$\sum_{j=1}^{5}S_{lj}(1)k_{j}=\sum_{j=1}^{5}S_{lj}(2)k_{j}=d\;.$ (40)

The sets of vectors $\vec{S}_{l}(1)$ and $\vec{S}_{l}(2)$ are slightly
different. It follows from the conditions for the sets of the chiral ring
monomials in the Loop and Chain cases and also from the fact that
$A_{i}(1)=A_{i}(2)$ for $i=1,2,3,4$ and $A_{5}(1)\neq A_{5}(2)$ in the
considered case.

However, from Table 1 in the appendix we come to the following important
observation: When a loop polynomial $W_{M{(1)}}$ and a chain polynomial
$W_{M{(2)}}$ appear in the same weighted projective space
$\mathbb{P}(k_{1},\ldots,k_{5})$ and are connected by Kreuzer-Skarke cleaves
Favero , in all these 111 cases listed in Table 1 the weight $k_{5}=1$.

From this fact we obtain that the sets of exponents of monomials of their
chiral rings shifted by the vector $(1,\ldots,1)$ not only belong to the same
4-dimensial lattice, but also both contain the integral basis of the lattice,
which consists of the four vectors $\vec{V_{a}},a=1,2,3,4$

$\begin{matrix}(-1,0,0,0,k_{1}),\cr(0,-1,0,0,k_{2}),\cr(0,0,-1,0,k_{3}),\cr(0,0,0,-1,k_{4}).\cr\end{matrix}$

Below we will use this observation to prove our main statement.

Recall that the mirrors $Y(1)$ and $Y(2)$ are CY orbifolds in the weighted
projective spaces $P_{\vec{k}(1)}$ and $P_{\vec{k}(2)}$ respectively. The
weights ${\vec{k}(1)}$ and ${\vec{k}(2)}$ satisfy the equations

$\sum_{j=1}^{5}(M^{T}(\alpha))_{ij}\bar{k}_{j}(\alpha)=\bar{d}(\alpha)=\sum_{j=1}^{5}\bar{k}_{j}(\alpha)\;,\quad\alpha=1,2\;.$
(41)

The two sets of the vectors $T_{mj}(1)$ and $T_{mj}(2)$, defining the complex
structure deformations, also satisfy the equations

$\sum_{j=1}^{5}(T(\alpha))_{rj}\bar{k}_{j}(\alpha)=\bar{d}(\alpha)\;,\quad\alpha=1,2\;.$
(42)

From the BHK mirror conditions (24) it follows

$(\vec{S}(\alpha)_{l},\vec{T}(\alpha)_{r})=\sum_{i,j=1}^{5}B(\alpha)_{ij}S(\alpha)_{li}T(\alpha)_{rj}\in\mathbb{Z}\;,\quad\alpha=1,2\;,$
(43)

where as before $B(\alpha)$ is the inverse matrix for $M(\alpha)$ and
$B^{T}(\alpha)$ is its transpose. From these equations and from the fact that
two sets of vectors $\vec{S}_{l}(1)$ and $\vec{S}_{l}(2)$ are linear
combinations of the same basis vectors $\vec{V_{a}}$ with integer coefficients
it follows that

$T_{ri}(1)B^{T}_{ij}(1)=T_{ri}(2)B^{T}_{ij}(2)\;.$ (44)

This equality implies also the equivalence of the pairings

$(\vec{S}(1)_{l},\vec{T}(1)_{r})=(\vec{S}(2)_{l},\vec{T}(2)_{r})\;,$ (45)

which in what follows for briefness is denoted $(\vec{S}_{l},\vec{T}_{r})$.

For the periods of the two BHK mirrors we have the following expressions
similar to (35)

$\sigma(1)_{\vec{\mu}(1)}(\phi_{1},...,\phi_{\bar{h}})\sim\sum_{\\{m_{r}\\}\in\Sigma(1)_{\vec{\mu}(1)}}C(1,\\{m_{r}\\})\prod_{r=1}^{\bar{h}}\frac{\psi_{r}^{m_{r}}}{m_{r}!}\;,$
(46)

where

$C(1,\\{m_{r}\\})=\prod_{i=1}^{5}\Gamma\left(\sum_{j}(m_{r}T(1)_{rj}+1)B^{T}(1)_{ji}\right).$
(47)

The condition $\\{m_{r}\\}\subset\Sigma_{\vec{\mu}(1)}$ means that there exist
non-negative integers $n_{j}$, $j=1,...,5$ such that

$m_{r}T(1)_{rj}=({\vec{\mu}(1)})_{j}+\sum_{k}n_{k}M^{T}(1)_{kj}.$ (48)

From (44) it follows that $\sum_{j}T(1)_{rj}B^{T}(1)_{ji}$, i.e. the factor
arising in the arguments of the gamma-functions in the formula (47) for the
period $\sigma(1)_{\vec{\mu}(1)}$, coincide with
$\sum_{j}T(2)_{rj}B^{T}(2)_{ji}$ in the corresponding expression for the
period $\sigma(2)_{\vec{\mu}(2)}$ of the second BHK mirror.
The sets of non-negative integers $m_{r}$ in the expressions for the periods
of both mirrors also coincide, which can be verified multiplying both parts of
the equation (48) by $B(1)_{ji}S_{li}$. Then we obtain the linear system

$m_{r}(\vec{T}(1)_{r},\vec{S}_{l})=(\mu(1),\vec{S}_{l})+S_{li}n_{i}.$ (49)

Similarly, for the second BHK mirror we get

$m_{r}(\vec{T}(2)_{r},\vec{S}_{l})=(\mu(2),\vec{S}_{l})+S_{li}n_{i}.$ (50)

Taking into account the equality (45), we find that the coefficients in both
systems of the linear equations (49) and (50) are the same. So that the sets
of integers $m_{r}$ in the expressions for the periods of both BHK mirrors
also coincide. Together with the equality (44) it proves the coincidence of
the periods of two BHK mirrors.

##  4 Conclusion

In this paper we showed that, for any pair of 3-dimensional Calabi–Yau
orbifolds that are multiple BHK mirrors of a loop–chain type pair of
Calabi–Yau manifolds in the same weighted projective space, the periods of the
holomorphic nonvanishing form coincide.

We assume that the key observation, leading to this statement, is true also
for the (n-2)-dimensional case. Namely, the following conjecture is true. Let
$W_{1}$ and $W_{2}$ be a loop and a chain polynomials of the form as in (2.4)
in the same weighted projective space $\mathbb{P}(k_{1},\ldots,k_{n})$. Then
$k_{n}=1$. From this conjecture we get that shifted exponents of the monomials
of the chiral rings for the loop $W_{1}$ and the chain $W_{2}$ both contain
the basis of the lattice

$\begin{matrix}(-1,0,0,\cdots,0,k_{1}),\cr(0,-1,0,\cdots,0,k_{2}),\cr\cdots\cr(0,\cdots,0,-1,k_{n-1}).\cr\end{matrix}$

It is not clear for us whether the birationality of the multiple BHK mirrors
is equivalent to the coincidence of their periods, we leave this question for
further investigation. Even if it is so, we find that the statement about the
coincidence of the multiple BHK mirrors is interesting, in particular taking
into account rather straightforward way it is obtained.

It will be interesting to consider possible generalizations of this statement
to a wider class of CY varieties.

## Acknowledgments

The work of A. Belavin has been supported by the Russian Science Foundation
under the grant 18-12-00439.

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## Appendix.

N $\\{k_{i}(2)\\}$ $\\{A_{i}\\}^{C}\&\\{A_{i}\\}^{C(mir)}$ $\\{k_{i}\\}$
$\\{A_{i}\\}^{L}\&\\{A_{i}\\}^{L(mir)}$ $\\{k_{i}(1)\\}$ 1 {126,42,70,13,1}
{2,3,3,14,239} {92,55,74,17,1} {2,3,3,14,147} {77,26,43,8,1} 2
{270,90,150,13,17} {2,3,3,30,31} {12,7,10,1,1} {2,3,3,30,19} {157,58,91,8,17}
3 {6,2,2,1,1} {2,3,5,10,11} {4,3,2,1,1} {2,3,5,10,7} {83,36,31,16,25} 4
{216,36,66,61,53} {2,6,6,6,7} {3,1,1,1,1} {2,6,6,6,4} {97,25,37,35,53} 5
{64,48,52,51,41} {4,4,4,4,5} {1,1,1,1,1} {4,4,4,4,4} {1,1,1,1,1} 6
{96,32,20,43,1} {2,3,8,4,149} {52,45,14,37,1} {2,3,8,4,97} {62,21,13,28,1} 7
{132,44,20,61,7} {2,3,11,4,29} {10,9,2,7,1} {2,3,11,4,19} {83,30,13,40,7} 8
{160,40,28,73,19} {2,4,10,4,13} {5,3,1,3,1} {2,4,10,4,8} {89,27,17,45,19} 9
{108,27,63,17,1} {2,4,3,9,199} {82,35,59,22,1} {2,4,3,9,117} {63,16,37,10,1}
10 {168,42,98,17,11} {2,4,3,14,29} {12,5,9,2,1} {2,4,3,14,17} {93,26,57,10,11}
11 {324,36,204,37,47} {2,9,3,12,13} {6,1,4,1,1} {2,9,3,12,7}
{151,22,109,20,47} 12 {120,24,54,31,11} {2,5,4,6,19} {8,3,4,3,1} {2,5,4,6,11}
{64,15,31,18,11} 13 {224,32,104,43,45} {2,7,4,8,9} {4,1,2,1,1} {2,7,4,8,5}
{34,7,19,8,15} 14 {180,15,115,49,1} {2,12,3,5,311} {146,19,83,62,1}
{2,12,3,5,165} {95,8,61,26,1} 15 {450,15,295,121,19} {2,30,3,5,41}
{20,1,11,8,1} {2,30,3,5,21} {221,8,151,62,19} 16 {126,18,78,29,1}
{2,7,3,6,223} {100,23,62,37,1} {2,7,3,6,123} {69,10,43,16,1} 17
{324,18,210,73,23} {2,18,3,6,25} {12,1,7,4,1} {2,18,3,6,13} {157,10,109,38,23}
18 {288,24,184,49,31} {2,12,3,8,17} {8,1,5,2,1} {2,12,3,8,9} {137,14,97,26,31}
19 {112,16,52,43,1} {2,7,4,4,181} {80,21,34,45,1} {2,7,4,4,101} {62,9,29,24,1}
20 {128,16,60,49,3} {2,8,4,4,69} {31,7,13,17,1} {2,8,4,4,38} {23,3,11,9,1} 21
{160,16,76,61,7} {2,10,4,4,37} {17,3,7,9,1} {2,10,4,4,20} {83,9,41,33,7} 22
{272,16,132,103,21} {2,17,4,4,21} {10,1,4,5,1} {2,17,4,4,11} {44,3,23,18,7} 23
{180,45,21,113,1} {2,4,15,3,247} {94,59,11,82,1} {2,4,15,3,153}
{111,28,13,70,1} 24 {300,75,21,193,11} {2,4,25,3,37} {14,9,1,12,1}
{2,4,25,3,23} {181,48,13,120,11} 25 {234,39,33,145,17} {2,6,13,3,19}
{8,3,1,6,1} {2,6,13,3,11} {127,24,19,84,17} 26 {100,20,36,41,3} {2,5,5,4,53}
{22,9,8,13,1} {2,5,5,4,31} {19,4,7,8,1} 27 {210,15,81,113,1} {2,14,5,3,307}
{144,19,41,102,1} {2,14,5,3,163} {111,8,43,60,1} 28 {480,15,189,257,19}
{2,32,5,3,37} {18,1,5,12,1} {2,32,5,3,19} {237,8,97,132,19} 29
{324,27,69,193,35} {2,12,9,3,13} {6,1,1,4,1} {2,12,9,3,7} {157,16,37,104,35}
30 {220,20,84,89,27} {2,11,5,4,13} {6,1,2,3,1} {2,11,5,4,7} {35,4,15,16,9} 31
{224,28,60,97,39} {2,8,7,4,9} {4,1,1,2,1} {2,8,7,4,5} {35,6,11,18,13} 32
{24,24,16,7,1} {3,2,3,8,65} {14,23,19,8,1} {3,2,3,8,51} {37,38,25,11,2} 33
{27,27,18,7,2} {3,2,3,9,37} {8,13,11,4,1} {3,2,3,9,29} {41,43,28,11,4} 34
{33,33,22,7,4} {3,2,3,11,23} {5,8,7,2,1} {3,2,3,11,18} {49,53,34,11,8} 35
{25,25,10,13,2} {3,2,5,5,31} {6,13,5,6,1} {3,2,5,5,25} {39,41,16,21,4} 36
{32,48,20,27,1} {4,2,4,4,101} {15,41,19,25,1} {4,2,4,4,86} {27,41,17,23,1} 37
{72,108,30,43,35} {4,2,6,6,7} {1,3,1,1,1} {4,2,6,6,6} {53,97,25,37,35} 38
{28,28,8,19,1} {3,2,7,4,65} {12,29,7,16,1} {3,2,7,4,53} {45,46,13,31,2} 39
{52,52,8,37,7} {3,2,13,4,17} {3,8,1,4,1} {3,2,13,4,14} {81,88,13,61,14} 40
{9,9,2,5,2} {3,2,9,5,11} {2,5,1,2,1} {3,2,9,5,9} {67,77,16,41,20} 41
{30,45,25,19,1} {4,2,3,5,101} {16,37,27,20,1} {4,2,3,5,85} {25,38,21,16,1} 42
{36,54,30,19,5} {4,2,3,6,25} {4,9,7,4,1} {4,2,3,6,21} {29,46,25,16,5} 43
{18,99,39,59,1} {12,2,3,3,157} {8,61,35,52,1} {12,2,3,3,149} {17,94,37,56,1}
44 {9,54,21,32,1} {13,2,3,3,85} {4,33,19,28,1} {13,2,3,3,81} {17,103,40,61,2}
45 {9,63,24,37,2} {15,2,3,3,49} {2,19,11,16,1} {15,2,3,3,47} {17,121,46,71,4}
46 {9,81,30,47,4} {19,2,3,3,31} {1,12,7,10,1} {19,2,3,3,30} {17,157,58,91,8}
47 {15,45,20,17,8} {7,2,3,5,11} {1,4,3,2,1} {7,2,3,5,10} {25,83,36,31,16} 48
{66,99,15,83,1} {4,2,11,3,181} {24,85,11,60,1} {4,2,11,3,157} {57,86,13,72,1}
49 {126,189,15,163,11} {4,2,21,3,31} {4,15,1,10,1} {4,2,21,3,27}
{107,166,13,142,11} 50 {21,42,9,32,1} {5,2,7,3,73} {8,33,7,24,1} {5,2,7,3,65}
{37,75,16,57,2} 51 {39,78,9,62,7} {5,2,13,3,19} {2,9,1,6,1} {5,2,13,3,17}
{67,141,16,111,14} 52 {27,81,12,59,10} {7,2,9,3,13} {1,6,1,4,1} {7,2,9,3,12}
{47,151,22,109,20} 53 {7,14,3,8,3} {5,2,7,4,9} {1,4,1,2,1} {5,2,7,4,8}
{15,34,7,19,8} 54 {30,75,21,53,1} {6,2,5,3,127} {12,55,17,42,1} {6,2,5,3,115}
{27,68,19,48,1} 55 {15,45,12,31,2} {7,2,5,3,37} {3,16,5,12,1} {7,2,5,3,34}
{27,83,22,57,4} 56 {72,48,84,11,1} {3,3,2,12,205} {56,37,94,17,1}
{3,3,2,12,149} {52,35,61,8,1} 57 {168,112,196,11,17} {3,3,2,28,29}
{8,5,14,1,1} {3,3,2,28,21} {116,83,141,8,17} 58 {60,24,78,17,1} {3,5,2,6,163}
{48,19,68,27,1} {3,5,2,6,115} {42,17,55,12,1} 59 {168,48,228,23,37}
{3,7,2,12,13} {4,1,6,1,1} {3,7,2,12,9} {104,35,157,16,37} 60 {32,24,52,19,1}
{4,4,2,4,109} {23,17,41,27,1} {4,4,2,4,86} {25,19,41,15,1}

N $\\{k_{i}(2)\\}$ $\\{A_{i}\\}^{C}\&\\{A_{i}\\}^{C(mir)}$ $\\{k_{i}\\}$
$\\{A_{i}\\}^{L}\&\\{A_{i}\\}^{L(mir)}$ $\\{k_{i}(1)\\}$ 61 {80,48,136,23,33}
{4,5,2,8,9} {2,1,4,1,1} {4,5,2,8,7} {18,13,35,6,11} 62 {88,16,124,35,1}
{3,11,2,4,229} {72,13,86,57,1} {3,11,2,4,157} {60,11,85,24,1} 63
{184,16,268,71,13} {3,23,2,4,37} {12,1,14,9,1} {3,23,2,4,25}
{120,11,181,48,13} 64 {132,24,186,35,19} {3,11,2,6,19} {6,1,8,3,1}
{3,11,2,6,13} {84,17,127,24,19} 65 {64,24,116,35,17} {4,8,2,4,13} {3,1,5,3,1}
{4,8,2,4,10} {45,19,89,27,17} 66 {48,60,114,29,37} {6,4,2,6,7} {1,1,3,1,1}
{6,4,2,6,6} {35,53,97,25,37} 67 {24,32,44,19,1} {5,3,2,4,101} {16,21,38,25,1}
{5,3,2,4,85} {20,27,37,16,1} 68 {140,56,26,197,1} {3,5,14,2,223}
{60,43,8,111,1} {3,5,14,2,163} {102,41,19,144,1} 69 {75,30,13,106,1}
{3,5,15,2,119} {32,23,4,59,1} {3,5,15,2,87} {109,44,19,155,2} 70
{85,34,13,121,2} {3,5,17,2,67} {18,13,2,33,1} {3,5,17,2,49} {123,50,19,177,4}
71 {105,42,13,151,4} {3,5,21,2,41} {11,8,1,20,1} {3,5,21,2,30}
{151,62,19,221,8} 72 {54,18,16,73,1} {3,6,9,2,89} {25,14,5,44,1} {3,6,9,2,64}
{77,26,23,105,2} 73 {78,26,16,109,5} {3,6,13,2,25} {7,4,1,12,1} {3,6,13,2,18}
{109,38,23,157,10} 74 {49,14,19,64,1} {3,7,7,2,83} {24,11,6,41,1} {3,7,7,2,59}
{69,20,27,91,2} 75 {72,18,22,97,7} {3,8,9,2,17} {5,2,1,8,1} {3,8,9,2,12}
{97,26,31,137,14} 76 {56,42,26,99,1} {4,4,7,2,125} {24,29,9,62,1}
{4,4,7,2,101} {45,34,21,80,1} 77 {64,48,26,115,3} {4,4,8,2,47} {9,11,3,23,1}
{4,4,8,2,38} {17,13,7,31,1} 78 {88,66,26,163,9} {4,4,11,2,21} {4,5,1,10,1}
{4,4,11,2,17} {23,18,7,44,3} 79 {50,30,34,83,3} {4,5,5,2,39} {8,7,4,19,1}
{4,5,5,2,31} {13,8,9,22,1} 80 {120,16,86,137,1} {3,15,4,2,223}
{70,13,28,111,1} {3,15,4,2,153} {82,11,59,94,1} 81 {216,16,158,245,13}
{3,27,4,2,31} {10,1,4,15,1} {3,27,4,2,21} {142,11,107,166,13} 82
{50,10,28,61,1} {3,10,5,2,89} {27,8,9,44,1} {3,10,5,2,62} {69,14,39,85,2} 83
{55,10,31,67,2} {3,11,5,2,49} {15,4,5,24,1} {3,11,5,2,34} {75,14,43,93,4} 84
{65,10,37,79,4} {3,13,5,2,29} {9,2,3,14,1} {3,13,5,2,20} {87,14,51,109,8} 85
{85,10,49,103,8} {3,17,5,2,19} {6,1,2,9,1} {3,17,5,2,13} {111,14,67,141,16} 86
{84,14,34,109,11} {3,12,7,2,13} {4,1,1,6,1} {3,12,7,2,9} {109,20,47,151,22} 87
{80,30,58,131,21} {4,8,5,2,9} {2,1,1,4,1} {4,8,5,2,7} {19,8,15,34,7} 88
{72,96,22,169,1} {5,3,12,2,191} {26,61,8,95,1} {5,3,12,2,165} {62,83,19,146,1}
89 {39,52,11,92,1} {5,3,13,2,103} {14,33,4,51,1} {5,3,13,2,89}
{67,90,19,159,2} 90 {45,60,11,107,2} {5,3,15,2,59} {8,19,2,29,1} {5,3,15,2,51}
{77,104,19,185,4} 91 {57,76,11,137,4} {5,3,19,2,37} {5,12,1,18,1}
{5,3,19,2,32} {97,132,19,237,8} 92 {42,70,26,113,1} {6,3,7,2,139}
{16,43,10,69,1} {6,3,7,2,123} {37,62,23,100,1} 93 {54,90,26,149,5}
{6,3,9,2,35} {4,11,2,17,1} {6,3,9,2,31} {47,80,23,132,5} 94 {14,14,8,31,3}
{5,4,7,2,13} {2,3,1,6,1} {5,4,7,2,11} {15,16,9,35,4} 95 {18,78,58,97,1}
{14,3,3,2,155} {8,43,26,77,1} {14,3,3,2,147} {17,74,55,92,1} 96 {9,42,31,52,1}
{15,3,3,2,83} {4,23,14,41,1} {15,3,3,2,79} {17,80,59,99,2} 97 {9,48,35,59,2}
{17,3,3,2,47} {2,13,8,23,1} {17,3,3,2,45} {17,92,67,113,4} 98 {9,60,43,73,4}
{21,3,3,2,29} {1,8,5,14,1} {21,3,3,2,28} {17,116,83,141,8} 99 {24,64,38,89,1}
{9,3,4,2,127} {10,37,16,63,1} {9,3,4,2,117} {22,59,35,82,1} 100
{24,80,46,109,5} {11,3,4,2,31} {2,9,4,15,1} {11,3,4,2,29} {22,75,43,102,5} 101
{36,84,34,127,7} {8,3,6,2,23} {2,7,2,11,1} {8,3,6,2,21} {32,77,31,116,7} 102
{21,56,19,85,8} {9,3,7,2,13} {1,4,1,6,1} {9,3,7,2,12} {37,104,35,157,16} 103
{32,40,38,77,5} {6,4,4,2,23} {3,5,3,11,1} {6,4,4,2,20} {27,35,33,67,5} 104
{20,30,22,59,9} {7,4,5,2,9} {1,2,1,4,1} {7,4,5,2,8} {11,18,13,35,6} 105
{48,18,58,67,1} {4,8,3,2,125} {28,13,21,62,1} {4,8,3,2,97} {37,14,45,52,1} 106
{84,18,106,115,13} {4,14,3,2,17} {4,1,3,8,1} {4,14,3,2,13} {61,14,81,88,13}
107 {9,6,13,16,1} {5,6,3,2,29} {5,4,5,14,1} {5,6,3,2,24} {29,20,43,53,4} 108
{21,12,31,37,4} {5,7,3,2,17} {3,2,3,8,1} {5,7,3,2,14} {33,20,51,61,8} 109
{27,12,41,47,8} {5,9,3,2,11} {2,1,2,5,1} {5,9,3,2,9} {41,20,67,77,16} 110
{48,40,62,113,25} {6,6,4,2,7} {1,1,1,3,1} {6,6,4,2,6} {37,35,53,97,25} 111
{12,18,22,31,1} {7,4,3,2,53} {6,11,9,26,1} {7,4,3,2,47} {21,32,39,55,2}

Table 1:  Here $\\{k_{i}\\}$ are weights arising simultaneously in loop and
chain types, with the corresponding exponents $\\{A_{i}\\}^{C}$ and
$\\{A_{i}\\}^{L}$ in (5). The $\\{k_{i}(2)\\}$ and $\\{A_{i}\\}^{C(mir)}$ are
the weights and the exponents of the Mirror CY for the chain manifold
$\\{A_{i}\\}^{C}$. The sets $\\{k_{i}(1)\\}$ and $\\{A_{i}\\}^{L(mir)}$ are
the same for the loop manifold $\\{A_{i}\\}^{L}$. The notation
$\\{A_{i}\\}^{C}\&\\{A_{i}\\}^{C(mir)}$ means that two sets coincide.