# Using Game Engines and Machine Learning
to Create Synthetic Satellite Imagery
for a Tabletop Verification Exercise

Johannes Hoster
Berliner Hochschule für Technik
Sara Al-Sayed
Program on Science and Global Security, Princeton University
Felix Biessmann
Berliner Hochschule für Technik
Einstein Center Digital Future, Berlin
Alexander Glaser
Program on Science and Global Security, Princeton University
Einstein Center Digital Future, Berlin
Kristian Hildebrand
Berliner Hochschule für Technik
Igor Moric
Program on Science and Global Security, Princeton University
Tuong Vy Nguyen
Berliner Hochschule für Technik

###### Abstract

Satellite imagery is regarded as a great opportunity for citizen-based
monitoring of activities of interest. Relevant imagery may however not be
available at sufficiently high resolution, quality, or cadence—let alone be
uniformly accessible to open-source analysts. This limits an assessment of the
true long-term potential of citizen-based monitoring of nuclear activities
using publicly available satellite imagery. In this article, we demonstrate
how modern game engines combined with advanced machine-learning techniques can
be used to generate synthetic imagery of sites of interest with the ability to
choose relevant parameters upon request; these include time of day, cloud
cover, season, or level of activity onsite. At the same time, resolution and
off-nadir angle can be adjusted to simulate different characteristics of the
satellite. While there are several possible use-cases for synthetic imagery,
here we focus on its usefulness to support tabletop exercises in which simple
monitoring scenarios can be examined to better understand verification
capabilities enabled by new satellite constellations and very short revisit
times.

##  1 Introduction

Satellites have been used since the 1950s for Earth observation, first by
governments with reconnaissance missions shrouded in secrecy, but increasingly
now also by commercial providers and by a growing community of open-source
analysts. There are currently about 700 imaging satellites in orbit, and some
constellations realize revisit times on the order of 20 minutes. Maxar
Technologies, one of the main commercial satellite operators, currently
acquires an average of about 80 terabytes of imagery per day. Over the next
few years, these and other collection efforts will grow to an immense archive
of digital data, and there is a widely shared expectation—or hope—that broad
and open access to open-source information will enable the early or timely
detection of non-compliance with relevant international agreements. Here, we
are particularly interested in monitoring compliance with nuclear
nonproliferation and arms-control agreements, but similar opportunities are
also emerging in the context of environmental and carbon-emission monitoring,
emergency response and human-rights monitoring, and archaeological-site
monitoring. This article is part of an ongoing project that seeks to
systematically assess the long-term potential of satellite imagery for
monitoring and verification purposes.

Working with real satellite imagery, however, has strong limitations. For one,
access to high-resolution imagery can be extremely expensive when required at
scale, i.e., on a daily basis for multiple sites or larger geographical
regions. More importantly perhaps, the number of relevant existing facilities
of interest is relatively small, making it more difficult to draw broader
conclusions based on rather limited datasets. To overcome both these
constraints, this project seeks to generate and leverage synthetic satellite
imagery for notional sites showing various industrial sites or other human
activities. This enables us to simulate the capabilities of arbitrary
satellite constellations, while controlling relevant parameters of the imagery
such as seasons, revisit times, or resolution and allowing for other gradual
and abrupt changes. An overview of our method is illustrated in Figure 1. We
can then use this imagery to develop and examine concrete scenarios, in which
parties are either compliant or non-compliant with an agreement.

(a) Game engine render

(b) Input modality for structural guidance

(c) Synthesized image without details

(d) Synthesized image with reinserted details

Figure 1: Overview of our method from game engine render, via control input
modality for structural guidance, to reinsertion of details into the
synthesized image

##  2 Related Work

#### Synthetic datasets supporting nuclear agreement monitoring:

The first synthetic imagery datasets were mostly of the non-overhead kind and
efforts for their generation were driven by autonomous driving R&D,
representative datasets being KITTI [KITTI] and SYNTHIA [SYNTHIA]. While there
are many dedicated synthetic overhead (aerial or satellite) dataset generation
efforts for different object classes (e.g., buildings, infrastructure such as
roads, bridges, and dams, vehicles) or contexts of interest (e.g., urban
scenes, agriculture, wildlife) and different computer vision applications
(e.g., segmentation, classification, object detection and tracking, autonomous
navigation, scene understanding, surveying, change detection, time-series
analysis) – representative datasets being Synthinel-1 [Synthinel-1] and
RarePlanes [RarePlanes] (satellite) and UnityShip [UnityShip] and BIRDSAI
[BIRDSAI] (aerial), to the best of our knowledge, there are none in the
nuclear context, except the efforts by Gastelum et al. [SafeguardsDataset] –
if not of the overhead kind – and Nguyen et al. [Nguyen2023]. The former
represents work at the Sandia National Laboratories using a game engine to
generate synthetic imagery of uranium hexafluoride containers – part of the
nuclear fuel cycle – against various backgrounds typical of nuclear scenarios
of interest, for computer vision applications. The work by Nguyen et al., on
the other hand, focuses on text-conditioned synthetic satellite imagery
generation of nuclear power plants. Apart from the latter work, there are no
datasets serving the purpose of a comprehensive assessment of agreement
monitoring using the current capabilities of commercial satellite
constellations. That being said, synthetic data in the form of text, imagery,
or video are leveraged or foreseen to be leveraged in the nuclear safeguards
and nonproliferation context for specific applications, as suggested by the
meeting agenda of the Joint Technical Exchange on Synthetic Data for
Safeguards and Nonproliferation: ESARDA Verification Technologies &
Methodologies and INMM Open Source and Geospatial Information Working Groups,
2022, which was attended by the second author of this paper.

#### Text-conditioned image generation:

Text-to-image generation provides an intuitive and comprehensible way of
conditional image synthesis, with large models like DALL-E 2
[ramesh2022hierarchical] already demonstrating how powerful they can be.
However, the issue with text-conditioned image generation alone is that
there’s little control over the exact outcome. While there has been
substantial progress in terms of image quality with text-conditioning for some
aspects of images, controlling the output of image generation models precisely
remains challenging. In this paper, we propose to first create a render of the
model of interest, in this case a nuclear power plant, using the game engine
then use feature maps like canny edge and depth map of the render as an
additional guidance mechanism, which provides a way to further control the
image generation process.

#### Applications for dataset generation:

Since computer vision models require a large amount of labeled data to be
trained, generating synthetic data using machine learning techniques or game
engines or a combination thereof is a simple and straightforward approach to
create datasets. Unity Perception [borkman2021unity], for example, provides a
highly customizable toolset for this task. Other applications are BlenderProc
[denninger2019blenderproc], NVIDIA Isaac Gym [makoviychuk2021isaac], and
NVISII [morrical2021nvisii]. All of these tools provide semantic or instance
segmentation, bounding-box generation, and also, in some cases, support for
other vision tasks such as depth maps, keypoints, normals, or optical flow.
However, our work doesn’t focus on traditional machine vision tasks, such as
distinguishing individual objects from their cluttered background, nor on
understanding indoor scenes, but on a very specific task for which a dedicated
kernel modelling algorithm is better suited.

#### Game engines for system design, testing, and validation:

While game engines in addition to being used to generate synthetic imagery are
also being used to build simulators of aerial systems for surveillance [OVVV]
and autonomous navigation [AirSim], our tabletop exercise may inform the
design, using game engines or other CAD tools, of the ideal satellite
constellation for high-confidence global and comprehensive monitoring of
nuclear activities that patches present technical gaps. Having imagery from
the ideal satellite constellation at hand, one can properly assess the
shortcomings of nuclear activity monitoring based on the current capabilities
of contemporary commercial satellite constellations.

##  3 Method

We combine the strengths of game engines and machine-learning models to
generate synthetic satellite images with configurable content and realistic
appearance. Specifically, we use the Unity3D [Unity] game engine to create a
procedurally generated model of a nuclear power plant with randomizable
parameters like camera distance and off-nadir angle, plant structure, time-of-
day, cloud cover, and level of activity onsite. This provides the possibility
to design the content of the created image as desired.

###  3.1 Imagery Generation Using Game Engine

Since procedural level generation is widely used in the game industry, using a
game engine such as Unity allows to create algorithms that can generate
different layouts of pre-defined structures that typically make up a nuclear
power plant, with customizable constraints. Since we employ the CoAdapter
model later to make enhancements to the game engine render of the nuclear
power plant model towards more realistic imagery, as explained in Section 3.2,
we focused more on the plant structure instead of the graphical appearance of
the game engine render. That is, we created untextured models of the reactor,
cooling towers, stacks, and tetromino-shaped buildings, and we chose a basic
grid system to place the models in different layouts. Immediately after the
plant model is constructed (typically in less than a second), details
reflecting the level of activity onsite, such as the number of cars in the
parking lot (2(b)) or the amount of steam emanating from the cooling towers
(2(c)), can be added automatically. The density of these details can be preset
or randomized to control the level of activity onsite.

(a) Basic structure

(b) Cars added

(c) Steam added

Figure 2: Visualization of different components of the model from basic
structure to high level of activity onsite

Since the encoded configurable model of the plant is 3-dimensional, the camera
can be moved freely around the model. Thus, images could be created of several
instantiations of the model with arbitrary off-nadir angles (Figure 3).

(a) Off-nadir 10°

(b) Off-nadir 20°

(c) Off-nadir 30°

(d) Off-nadir 40°

(e) Off-nadir 50°

Figure 3: Visualization of different off-nadir angles

Game engines also enable the rendering of models under different conditions
such as the time-of-day (Figure 4) or with different cloud parameters (Figure
5). Overall, our approach enables the creation of large high-variability
datasets in a short time.

(a) Morning

(b) Day

(c) Evening

(d) Night

Figure 4: Visualization of different times-of-day

(a) Low cloud coverage

(b) Medium cloud coverage

(c) High cloud coverage

(d) Extreme cloud coverage

Figure 5: Visualization of different cloud coverage

###  3.2 Multimodally-guided Satellite Imagery Synthesis

Utilizing the game engine with virtual models of the relevant elements enables
the generation of many images in a short time with different features.
However, despite the extensive graphic achievements in recent years, the
output of the engine alone still lacks realism to a large degree, impeding the
successful execution of downstream tasks.

We solve this problem by feeding the images of the game engine into a T2I
[mou2023t2iadapter] Composable Adapter (CoAdapter) model [T2I], which is based
on machine learning. The CoAdapter allows for multiple input modalities to
define the appearance as well as content of the image. The most promising
input modalities to capture the layout of the input image are the canny edge,
depth map, and sketch modalities, which can be seen along with the
corresponding results in Figure 6. It is also possible to use multiple input
modalities at once (Figure 7).

(a) Canny edge

(b) Depth map

(c) Sketch

Figure 6: Visualization of different input modalities (top) and corresponding
results (bottom). In addition to the visual input modality, the text prompt
“Satellite image of a nuclear power plant” was used.

(a) Canny + Depth

(b) Canny + Sketch

(c) Depth + Sketch

(d) Canny + Depth + Sketch

Figure 7: Results from multiple different input modalities with text prompt,
“Satellite image of a nuclear power plant”

While most features like plant structure, time-of-day, cloud coverage, and
level of activity onsite can be adjusted within the game engine, we decided to
capture seasonality through text prompts and different guidance scales of the
input modalities, rather than implementing a comparatively costly additional
step in the game engine that would probably only lead to mediocre results. The
output is shown in Figure 8. While most of the results are convincing, in the
case of using a high guidance scale of the text prompt for “fall” and
“winter”, the model generates a background from a low-elevation viewpoint even
though the text prompt was “Satellite image of a nuclear power plant in
fall/winter”. This shows that the model is biased towards typical viewpoints,
as satellite imagery is underrepresented in the training data. So the guidance
scale of the text prompt shouldn’t be set too high in order to prevent the
model from generating incoherent results.

(a) Spring

(b) Summer

(c) Fall

(d) Winter

Figure 8: Visualization of different seasonality through prompt engineering
with default 10 (top) and high 15 (bottom) guidance scale. In addition to the
visual input modality, the text prompt “Satellite image of a nuclear power
plant in [season]” was used.

The same way seasonality can be changed, the environment can be specified by
the text prompt, as shown in Figure 9.

(a) "in a forest"

(b) "in the desert"

(c) "by a coastline"

(d) "in the mountains"

Figure 9: Visualization of different environments through prompt engineering
with default 10 (top) and high 15 (bottom) guidance scale. In addition to the
visual input modality, the text prompt “Satellite image of a nuclear power
plant in/by/on [environment]” was used.

The problem of disappearing details, such as the steam from the cooling
towers, clouds, or cars in the parking lot, can easily be solved by extracting
those details from the game engine render and adding them to the image
synthesized by the CoAdapter model, as illustrated in Figure 1. This process
can be automated, as each object can be separately rendered in the game
engine. More general image features such as the lighting mood at different
times-of-day, as illustrated in Figure 4, can be added to the synthesized
image by creating a blend between both images.

By using the stylistic features of a real satellite image along with the text
prompt and content-related features of a synthetic rendering, it is possible
to generate a realistic image that includes the desired elements presented in
a more realistic style, as illustrated in Figure 10.

(a) Satellite image of a real nuclear power plant

(b) Synthesized result from color map of reference image

(c) Final image with details from game engine render

Figure 10: Synthesized results of canny edge with visual input modality of
real satellite image and reinserted details

##  4 Discussion

###  4.1 Method Limitations and Desiderata

Despite the advantages of combining game engines and machine learning, there
are still limitations to our method. While we showed that arbitrary views of
procedurally generated layouts of pre-modelled structures can result in a
large number of synthetic satellite images, the quality of which can be
greatly improved through the multimodally-guided CoAdapter model, and
illustrated how details can be reinserted into the final images, on closer
inspection the images are still not entirely realistic. This is due to the
fact that the reinserted details are limited by progress on game engine
graphics, which would need to be manually optimized for each image generated
by the CoAdapter model, for example by adjusting the details to the specific
lighting. And although the CoAdapter model can narrow the domain gap between
game engine renderings and real images, it lacks the ability to produce photo-
realistic results. Streets are often simplified, structures asymmetrical, and
shadows sometimes inconsistent – all indications that the image is synthetic.

In summary, what has been achieved so far is a procedure for generating
synthetic satellite images of a notional nuclear power plant. Starting with a
3D game engine model of the plant composed of its basic elements, the
rendering can be modified to account for variable conditions pertaining to
both the environment being imaged (time-of-day, cloud coverage, level of
activity onsite such as steam plumes and number of cars) as well as satellite
image acquisition (camera distance, off-nadir angle). The images are then
further enhanced in terms of appearance and content via the machine learning-
based CoAdapter model. Thanks to its multiple input modalities, the text
prompt given to the CoAdapter model can further be used to modify the images
to account for variable seasonality and surrounding environment. The realism
of the resulting imagery can also be enhanced by providing the CoAdapter model
with a real satellite image. Since the 3D model layout and the image
acquisition and environmental conditions are configurable, a large set of
images exhibiting high variability can be generated with minimal manual labor.
Determining the “best” combination of input modalities and guidance scale and
manually re-inserting details take several minutes per image – steps that
constitute the largest overhead. For future work, the detail re-insertion
process can be automated by rendering the details only and adding them
automatically to all variations of input modalities so that the “best” result
of the $2^{4}=16$ possible input modality combinations (canny, depth, sketch,
and color of reference photo) can be chosen from multiple proposals that can
be generated in under three minutes.

For the purposes of developing materials for a tabletop exercise a number of
additional improvements are needed to capture the full range of variability in
image acquisition and environmental conditions. First, we need to be able to
simulate imaging spatial resolution to reflect inherent finiteness, but also
to capture degradation effects with the variation of camera distance, off-
nadir angle, and environmental conditions such as time-of-day and weather and
atmospheric phenomena. Second, we want to be able to generate imagery in
different spectral bands, including the infrared band for the detection of
thermal indicators of nuclear activity and the microwave band, which enables
visibility through clouds and during night-time. Synthetic aperture radar
imagery is also potent and would be interesting to synthesize [Moric2022].
Finally, for the assessment of the power of satellite imagery in enabling
change detection and activity tracking, we need to be able to generate an
image sequence over a given temporal window at specific intervals conforming
to actual Earth-observation satellite operation, i.e., capturing satellite
orbital trajectory over the temporal window as well as the image acquisition
frequency and the corresponding variation in environmental conditions (which
can in turn influence image quality) over the acquisition times. The scope of
the required imagery to assess applicability to change detection and activity
tracking is expected to increase in this case to encompass  (a) a wider swath
of environing road networks as well as other types of infrastructure relevant
to nuclear material production, e.g., mines and ports;  (b) longer time
intervals corresponding to the duration of the nuclear activities of interest;
(c) different bands and spatial and temporal resolutions as required by the
nuclear activity indicators of interest.

###  4.2 Ethical and Political Considerations

#### Large-scale data collection and analysis:

The explosion in data, compute power, and algorithms in recent years has
prompted interest in agreement monitoring in the nuclear context relying on
open sources and machine learning-based analysis. Work by [Gastelum2018]
[Gastelum2018] and [Feldman2018] [Feldman2018] and other work featured in the
same Journal of Nuclear Materials Management 2018 issue exemplify the trend in
safeguards research, where the state-level concept is often the pretext
justifying large-scale data collection and analysis. However, there is dire
need to resolve trust issues between states based on a solid common
understanding, to say nothing of the trust issues between human and machine.
The tabletop exercise attempts to flesh out some of the underlying issues of
concern.

#### Bias perpetuation:

In projects involving the generation of synthetic imagery of a given object of
interest, pre-existing notions of the object of interest tend to influence the
procedure and in turn, the generated dataset. So far, the procedure is guided
by prenotions about the appearance of a nuclear power plant in three obvious
ways. First, the construction of the 3D model of a notional plant is inspired
by domain knowledge of the appearance of real plants. Second, a real image of
an existing nuclear power plant is used as reference for the CoAdapter model
to enhance the realism of the generated synthetic imagery. Third, when the
CoAdapter is guided by a text prompt that refers to a ‘nuclear power plant’,
the underlying machine learning procedure powering the text-to-image
generation produces an output that is ultimately influenced by training data
in the form of existing imagery representing the concept ‘nuclear power
plant’. Recalling to mind the lack of sufficient, representative real imagery
of nuclear power plants, this training dataset can safely be assumed to be
small, limiting the variability of the generated synthetic imagery. Together,
these three factors result in the output imagery manifesting a degree of bias
towards pre-existing notions that ideally one would like to reduce, if not
eliminate, in the context of a tabletop exercise that seeks to assess the
potential and limitations of the use of satellite imagery in a controlled
setup that guards against the participants’ being influenced by their
prenotions.

##  5 Conclusion

Our demonstration showcases how the discrepancy between 3D game engine renders
and real satellite imagery can be reduced by means of multimodally-guided
image synthesis. Additionally, our approach enables the creation of sizable
high-variability datasets in a brief span of time, making it particularly
useful in cases where comprehensive satellite imagery is rare, which is the
case for nuclear power plants. With this work, we have laid the groundwork for
the assessment of the long-term potential of satellite imagery for nuclear
activity monitoring and verification purposes.

With advances in machine learning and with the growth in high-quality real
imagery datasets for training, it’s to be expected that synthetic Earth-
observation imagery generation will improve in quality and that synthetic
imagery will proliferate. And while synthetic imagery has countless good uses,
it’s to be expected that malicious actors would use doctored images to stage
hoaxes or discredit reality, in both cases to mislead adversaries, entities
tasked with monitoring and verification, or publics. And while ‘deepfake
geography’ is a milennia-old practice, novel artificial intelligence
(AI)-generated deepfake satellite imagery introduces new challenges
[Zhao2021]. For example, satellite imagery’s characteristically low resolution
compared to other forms of imagery such as photography works to the favor of
deepfakes, since the latter are then easier to produce and deemed authentic by
the observer. The convincing power of deepfake satellite imagery can also be
attributed to the level of complexity and expense observers rightfully ascribe
to the long, intricate, and costly process of satellite imagery generation – a
process embedded in a sociotechnical complex mediated by science, economics,
politics, …[Verge-DeepfakeSatImagery]. As a form of disinformation, undetected
deepfake satellite imagery could have serious consequences for international
peace and security, and current governance structures are concernedly lagging
in keeping pace with AI developments.

## Acknowledgments

The authors thank the German Foundation for Peace Research (DSF) for their
support of this research effort as part of our ongoing project “Citizen-based
Monitoring for Peace & Security in the Era of Synthetic Media and Deepfakes.”


# Mutation of $n$-cotorsion pairs in triangulated categories

Huimin Chang  Department of Applied Mathematics, The Open University of China,
100039 Beijing, P. R. China<EMAIL_ADDRESS>and  Panyue Zhou  School of
Mathematics and Statistics, Changsha University of Science and Technology,
410114 Changsha, Hunan, P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this article, we define the notion of $n$-cotorsion pairs in triangulated
categories, which is a generalization of the classical cotorsion pairs. We
prove that any mutation of an $n$-cotorsion pair is again an $n$-cotorsion
pair. When $n=1$, this result generalizes the work of Zhou and Zhu for
classical cotorsion pairs. As applications, we give a geometric
characterization of $n$-cotorsion pairs in $n$-cluster categories of type $A$
and give a geometric realization of mutation of $n$-cotorsion pairs via
rotation of certain configurations of $n$-diagonals.

###### Key words and phrases:

mutation; $n$-cotorsion pair; triangulated category

###### 2020 Mathematics Subject Classification:

18E40; 05E10; 18G80

Huimin Chang is supported by the National Natural Science Foundation of China
(Grant No. 12301047). Panyue Zhou is supported by the National Natural Science
Foundation of China (Grant No. 12371034) and by the Hunan Provincial Natural
Science Foundation of China (Grant No. 2023JJ30008).

##  1\. Introduction

The notion of torsion pairs in abelian categories was first introduced by
Dickson [9], and the triangulated version was studied by Iyama and Yoshino
[14]. Later, cotorsion pairs in a triangulated category were introduced by
Nakaoka [19] to unify the abelian structures arising from $t$-structures [1]
and from cluster tilting subcategories [16, 17, 14]. Torsion pairs and
cotorsion pairs in a triangulated category can be transformed into each other
by shifting the torsion-free parts. Hence, it is equivalent to consider
torsion pairs and cotorsion pairs in triangulated categories.

Let $n$ be a positive integer. Motivated by some properties satisfied by
Gorenstein projective and Gorenstein injective modules over an Iwanaga-
Gorenstein ring, Huerta, Mendoza and Pérez [13, Definition 2.2] introduced the
notion of $n$-cotorsion (resp. left $n$-cotorsion, right $n$-cotorsion) pairs
in an abelian categories. It is worth to note that $1$-cotorsion pairs
coincide with the concept of complete cotorsion pairs. Recently, He and Zhou
[11] introduced $n$-cotorsion (resp. left $n$-cotorsion, right $n$-cotorsion)
pairs in extriangulated categories, which introduced by Nakaoka and Palu in
their seminal work [20]. Extriangulated categories simultaneously generalizes
exact categories and triangulated categories. Abelian categories and
extension-closed subcategories of a triangulated category are considered as
specific instances of extriangulated categories. Especially, when $n=1$, an
$n$-cotorsion pair is just a cotorsion pair in the sense of Nakaoka and Palu
[20, Defintion 4.1].

An important motivation for mutation comes from cluster algebras. Cluster
algebras were introduced by Fomin and Zelevinsky [10] in order to give an
algebraic and combinatorial framework for the positive and canonical basis of
quantumn groups. The mutation of clusters was defined in cluster algebras. As
a categorization of cluster algebra, Buan et al [6] introduced cluster
categories. Cluster categories are triangulated categories by Keller[15]. The
rigid indecomposable objects in cluster categories correspond to cluster
variables and cluster tilting subcategories correspond to clusters. Just like
mutation of cluster, mutation of cluster tilting subcategory was studied
through replacing one indecomposable object by a unique other indecomposable
object such that one gets a cluster tilting subcategory again. See [6] for
details. As a generalization, mutation of cluster tilting subcategories in
arbitrary Krull-Schmidt $K$-linear triangulated categories was studied in [2,
14, 21]. Iyama and Yoshino [14] introduced a more general concept mutation in
triangulated categories that makes mutation of cluster tilting subcategory a
special case. Zhou and Zhu [23] proved that any mutation of a torsion pair is
again a torsion pair.

In this paper, we define $n$-cotorsion pairs in triangulated categories, which
is a generalization of the classical cotorsion pairs. It should be noted that
the notion of $n$-cotorsion pairs in triangulated categories we defined in
this paper is different from that defined by He and Zhou in extriangulated
categories and deduced to triangulated categories. We prove that any mutation
of an $n$-cotorsion pair is again an $n$-cotorsion pair. When $n=1$, this
result generalizes a work of Zhou and Zhu for classical cotorsion pairs. As
applications, we give a geometric characterization of $n$-cotorsion pairs in
$n$-cluster categories of type $A$ and give a geometric realization of
mutation of $n$-cotorsion pairs via rotation of certain configurations of
$n$-diagonals.

This paper is organized as follows. In Section 2 we give an overview of
cotorsion pairs in triangulated categories, and mutation of subcategories of
triangulated categories. In Section 3 we define $n$-cotorsion pairs in
triangulated categories, recall from [14] the construction of subfactor
triangulated categories and study its compatibility with $n$-cotorsion pairs,
and prove the main result. In Section 4 we give a geometric characterization
of $n$-cotorsion pairs in $n$-cluster categories of type $A$ via certain
configurations of $n$-diagonals, and introduce the rotation of such
combinatorial models, which can give a geometric realization of mutation of
$n$-cotorsion pairs in $n$-cluster categories of type $A$.

### Conventions

In this paper, $K$ stands for an algebraically closed field. All additive
categories considered are assumed to be Krull-Schmidt, i.e. any object is
isomorphic to a finite direct sum of objects whose endomorphism rings are
local. Let $\mathcal{C}$ be a triangulated category, we denote by $\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(X,Y)$ the set of morphisms from $X$ to $Y$ in
$\mathcal{C}$. We denote the composition of $f\in\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(X,Y)$ and $g\in\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(Y,Z)$ by $g\circ f\in\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(X,Z)$. When we say that $\mathscr{X}$ is a
subcategory of $\mathcal{C}$, we always mean that $\mathcal{C}$ is a full
subcategory which is closed under isomorphisms, direct sums and direct
summands. We denote by $\mathscr{X}^{\perp}$ (resp. ${}^{\perp}\mathscr{X}$)
the subcategory whose objects are $M\in\mathcal{C}$ satisfying $\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(\mathscr{X},M)=0$ (resp. $\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(M,\mathscr{X})=0$). We use $\mathop{\rm
Ext}\nolimits_{\mathcal{C}}^{i}(X,Y)$ to denote $\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(X,Y[i])$, $i\in\mathbb{Z}$, where $[1]$ is the
shift functor of $\mathcal{C}$. For an object $X\in\mathcal{C}$, $\mathop{\rm
add}\nolimits X$ means the additive closure of $X$. Let $\mathscr{X}$ and
$\mathscr{Y}$ be subcategories of $\mathcal{C}$. We denote by
$\mathscr{X}\ast\mathscr{Y}$ the collection of objects in $\mathcal{C}$
consisting of all such $C\in\mathcal{C}$ with triangles

$X\to C\to Y\to X[1]$

where $X\in\mathscr{X}$ and $Y\in\mathscr{Y}$.

##  2\. Preliminaries

We recall the definition of cotorsion pairs in triangulated categories, and
mutation of subcategories of triangulated categories.

###  2.1. Cotorsion pairs in triangulated categories

We briefly review the definition of the classical cotorsion pairs and some
related results in triangulated categories from [14, 7, 23].

######  Definition 2.1.

Let $\mathscr{X}$ and $\mathscr{Y}$ be subcategories of a triangulated
category $\mathcal{C}$.

  * (1)

The pair $(\mathscr{X},\mathscr{Y})$ is called a torsion pair [14] if

$\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(\mathscr{X},\mathscr{Y})=0\;\text{and}\;\mathcal{C}=\mathscr{X}\ast\mathscr{Y}.$

The subcategory $\mathcal{I}(\mathscr{X})=\mathscr{X}\cap\mathscr{Y}[-1]$ is
called the core of the torsion pair.

  * (2)

The pair $(\mathscr{X},\mathscr{Y})$ is called a cotorsion pair [19] if

$\mathop{\rm
Ext}\nolimits^{1}_{\mathcal{C}}(\mathscr{X},\mathscr{Y})=0\;\text{and}\;\mathcal{C}=\mathscr{X}\ast\mathscr{Y}[1].$

The subcategory $\mathcal{I}(\mathscr{X})=\mathscr{X}\cap\mathscr{Y}$ is
called the core of the cotorsion pair.

######  Remark 2.2.

By definition, we know that $(\mathscr{X},\mathscr{Y})$ is a cotorsion pair if
and only if $(\mathscr{X},\mathscr{Y}[1])$ is a torsion pair.

######  Definition 2.3.

Let $\mathscr{X}$ be a subcategory of a triangulated category $\mathcal{C}$
and $n\geq 1$ be an integer.

  * (1)

We call $\mathscr{X}$ an $(n+1)$-rigid subcategory if $\mathop{\rm
Ext}\nolimits^{i}_{\mathcal{C}}(\mathscr{X},\mathscr{X})=0$ for $1\leq i\leq
n$.

  * (2)

For an object $C$ in $\mathcal{C}$, a right $\mathscr{X}$-approximation of $C$
in $\mathcal{C}$ is a map $X\rightarrow C$, with $X$ in $\mathscr{X}$, such
that for all objects $Y$ in $\mathscr{X}$, the sequence

$\mathop{\rm Hom}\nolimits_{\mathcal{C}}(Y,X)\rightarrow\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(Y,C)\rightarrow 0$

is exact. Dually, we have the concept of left $\mathscr{X}$-approximation of
$C$.

  * (3)

We call $\mathscr{X}$ contravariantly finite if any object $C\in\mathcal{C}$
admits a right $\mathscr{X}$-approximation, $\mathscr{X}$ covariantly finite
if any object $C\in\mathcal{C}$ admits a left $\mathscr{X}$-approximation, and
$\mathscr{X}$ functorially finite if both $\mathscr{X}$ contravariantly finite
and covariantly finite.

  * (4)

We call $\mathscr{X}$ an $(n+1)$-cluster tilting subcategory if $\mathscr{X}$
is functorially finite and satisfying

$\mathscr{X}=\bigcap\limits_{i=1}^{n}\mathscr{X}[-i]^{\perp}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathscr{X}[i].$

Note that when $n=1$, 2-rigid subcategory is simply called rigid and 2-cluster
tilting subcategory is called cluster tilting.

######  Lemma 2.4.

Let $\mathscr{X}$ and $\mathscr{Y}$ be subcategories of a triangulated
category $\mathcal{C}$.

  * (1)

The pair $(\mathscr{X},\mathscr{Y})$ is a cotorsion pair if and only if the
following hold.

    * (a)

$\mathscr{X}{{}^{\bot}}[-1]=\mathscr{Y}$;

    * (b)

$\mathscr{X}={{}^{\bot}}\mathscr{Y}[1]$;

    * (c)

$\mathscr{X}$ is contravariantly finite or $\mathscr{Y}$ is covariantly
finite.

  * (2)

$(\mathscr{X},\mathscr{X})$ is a cotorsion pair if and only if $\mathscr{X}$
is cluster tilting.

  * (3)

The core $\mathcal{I}(\mathscr{X})=\mathscr{X}\cap\mathscr{Y}$ of the
cotorsion pair $(\mathscr{X},\mathscr{Y})$ is rigid.

###  2.2. Mutation of subcategories of triangulated categories

In this subsection, we review from [14] the notion of mutation of
subcategories of triangulated categories.

The following result is well known and straightforward to check.

######  Lemma 2.5.

[5, Lemma 1.1] Let $\mathscr{X}$ be a subcategory of $\mathcal{C}$ and $C$ be
an object of $\mathcal{C}$.

  * (1)

If there is a right $\mathscr{X}$-approximation of $C$, then there is a
minimal right $\mathscr{X}$-approximation of $C$, unique up to isomorphism.

  * (2)

If $f:X\rightarrow C$ is a minimal right $\mathscr{X}$-approximation of $C$,
then each right $\mathscr{X}$-approximation is, up to isomorphism, of the form
$f\oplus 0:X\oplus X^{{}^{\prime}}\rightarrow C$.

######  Definition 2.6.

[14, Definition 2.5] Fix a functorially finite rigid subcategory $\mathcal{D}$
of $\mathcal{C}$. For a subcategory $\mathscr{X}$ of $\mathcal{C}$, put

$\mu^{-1}_{\mathcal{D}}(\mathscr{X}):=(\mathcal{D}\ast\mathscr{X}[1])\cap{{}^{\bot}}\mathcal{D}[1].$

That is, $\mu^{-1}_{\mathcal{D}}(\mathscr{X})$ consists of all
$M\in\mathcal{C}$ such that there exists a triangle

$X\stackrel{{\scriptstyle f}}{{\rightarrow}}D\rightarrow M\rightarrow X[1]$

with $X\in\mathscr{X}$ and a left $\mathcal{D}$-approximation $f$. Dually, for
a subcategory $\mathscr{Y}$ of $\mathcal{C}$, put

$\mu_{\mathcal{D}}(\mathscr{Y}):=(\mathscr{Y}[-1]\ast\mathcal{D})\cap\mathcal{D}[-1]^{\bot}.$

That is, $\mu_{\mathcal{D}}(\mathscr{Y})$ consists of all $M\in\mathcal{C}$
such that there exists a triangle

$M\rightarrow D\stackrel{{\scriptstyle g}}{{\rightarrow}}Y\rightarrow M$

with $Y\in\mathscr{Y}$ and a right $\mathcal{D}$-approximation $g$.

In this case, $\mu^{-1}_{\mathcal{D}}(\mathscr{X})$ is called the forward
$\mathcal{D}$-mutation of $\mathscr{X}$ and $\mu_{\mathcal{D}}(\mathscr{Y})$
is called the backward $\mathcal{D}$-mutation of $\mathscr{Y}$.

######  Remark 2.7.

It is clear that
$\mu_{\mathcal{D}}(\mathcal{D})=\mathcal{D}=\mu^{-1}_{\mathcal{D}}(\mathcal{D})$.
When $\mathcal{D}=0$, we have
$\mu^{-1}_{\mathcal{D}}(\mathcal{M})=\mathcal{M}[1]$ and
$\mu_{\mathcal{D}}(\mathcal{M})=\mathcal{M}[-1]$.

##  3\. Mutation of $n$-cotorsion pairs

In this section, we fix an integer $n\geq 1$.

###  3.1. $n$-cotorsion pairs in triangulated categories

We define $n$-cotorsion pairs in a triangulated category, which is a
generalization of the classical cotorsion pairs.

######  Definition 3.1.

Let $\mathcal{C}$ be a triangulated category and $\mathscr{X},\mathscr{Y}$ be
two subcategories of $\mathcal{C}$. The pair $(\mathscr{X},\mathscr{Y})$ is
called an $n$-cotorsion pair if the following conditions hold.

  1. (1)

$\mathscr{X}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathscr{Y}[i]$ for all $1\leq
i\leq n$;

  2. (2)

$\mathscr{Y}=\bigcap\limits_{i=1}^{n}\mathscr{X}[-i]^{\perp}$ for all $1\leq
i\leq n$;

  3. (3)

$\mathscr{X}$ is contravariantly finite and $\mathscr{Y}$ is covariantly
finite.

######  Remark 3.2.

It is easy to check that the pair $(\mathscr{X},\mathscr{X})$ is an
$n$-cotorsion pair in $\mathcal{C}$ if and only if $\mathscr{X}$ is
$(n+1)$-cluster tilting. When $n=1$, the concept of $1$-cotorsion pair is
compatible with the classical definition of a cotorsion pair in the sense of
Nakaoka [19].

######  Remark 3.3.

In [11], He and Zhou defined the notion of $n$-cotorsion pairs in
extriangulated categories. Note that exact categories and triangulated
categories are extriangulated categories. However, when their concept
degenerates into triangulated categories, it does not align with our defined
$n$-cotorsion pairs. In particular, in the case of finite triangulated
categories, i.e. there are only finitely many indecomposable objects up to
isomorphisms, their $n$-cotorsion pairs satisfy our defined $n$-cotorsion
pairs. Therefore, we also adopt the term $n$-cotorsion pair.

Now we give some examples of $n$-cotorsion pairs.

######  Example 3.4.

  1. (1)

Let $\mathcal{C}_{A_{3}}^{2}$ be the 2-cluster category of type $A_{3}$. We
give an example of a $2$-cotorsion pair in $\mathcal{C}_{A_{3}}^{2}$, see
Example 4.13 for details.

  2. (2)

Let $\mathcal{C}_{D_{4}}^{3}$ be the 3-cluster category of type $D_{4}$. The
AR-quiver of $\mathcal{C}_{D_{4}}^{3}$ is shown in Figure 1. Let
$\mathscr{X}=\\{(1,5),(1,8),(4,8),(3,7),(2,6)\\}$ and
$\mathscr{Y}=\\{(1,8),(1,11)_{r},(1,11)_{g},(10,20)_{r},(10,20)_{g},(9,19)_{r},(9,19)_{g},(8,18)_{r},(8,18)_{g}\\}$
be two subcategories of $\mathcal{C}_{D_{4}}^{3}$. One can check directly from
the AR-quiver of $\mathcal{C}_{D_{4}}^{3}$ that
$\mathscr{X}=\bigcap\limits_{i=1}^{3}{{}^{\bot}}\mathscr{Y}[i]$ and
$\mathscr{Y}=\bigcap\limits_{i=1}^{3}\mathscr{X}[-i]^{\perp}$. Thus,
$(\mathscr{X},\mathscr{Y})$ is a $3$-cotorsion pair.

(1,5)(1,8)$(1,11)_{r}$$(1,11)_{g}$$(4,14)_{r}$$(7,17)_{g}$$(10,20)_{r}$$(3,13)_{g}$$(6,16)_{r}$$(9,19)_{g}$$(2,12)_{r}$$(5,15)_{g}$$(8,18)_{r}$$(1,11)_{g}$(4,8)(4,11)$(4,14)_{g}$(7,11)(7,14)$(7,17)_{r}$(10,14)(10,17)$(10,20)_{g}$(3,7)(3,10)$(3,13)_{r}$(6,10)(6,13)$(6,16)_{g}$(9,13)(9,16)$(9,19)_{r}$(2,6)(2,9)$(2,12)_{g}$(5,9)(5,12)$(5,15)_{r}$(8,12)(8,15)$(8,18)_{g}$(1,5)(1,8)$(1,11)_{r}$
Figure 1. The AR-quiver of $\mathcal{C}_{D_{4}}^{3}$

######  Definition 3.5.

Let $(\mathscr{X},\mathscr{Y})$ be an $n$-cotorsion pair in $\mathcal{C}$. We
call $\mathcal{I}(\mathscr{X})=\mathscr{X}\cap\mathscr{Y}$ the core of
$(\mathscr{X},\mathscr{Y})$.

######  Remark 3.6.

By the definition of $n$-cotorsion pair, it is easy to see the core
$\mathcal{I}(\mathscr{X})$ is an $(n+1)$-rigid subcategory.

###  3.2. Compatibility with subfactor triangulated categories

Let $\mathcal{C}$ be a Hom-finite Krull-Schmidt triangulated category over $K$
with a Serre functor $\mathbb{S}$ from now on. Recall from [3] that a functor
$\mathbb{S}\colon\mathcal{C}\rightarrow\mathcal{C}$ is called a Serre functor
if there is a bifunctorial isomorphism

$D\mathop{\rm Hom}\nolimits_{\mathcal{C}}(X,Y)\cong\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(Y,\mathbb{S}X),$

for any objects $X,Y\in\mathcal{C}$. We put

$\mathbb{S}_{n}=\mathbb{S}\circ[-n]:\mathcal{C}\rightarrow\mathcal{C}.$

Recall that $\mathcal{C}$ is $n$-Calabi-Yau if and only if $\mathbb{S}_{n}$ is
an identity functor.

In this subsection, we fix a functorially finite $(n+1)$-rigid subcategory
$\mathcal{D}$ of $\mathcal{C}$ which satisfying the condition
$\mathbb{S}\mathcal{D}=\mathcal{D}[n+1]$. Then it is easy to check that
$\bigcap\limits_{i=1}^{n}\mathcal{D}[-i]^{\perp}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathcal{D}[i]$,
which is denoted by $\mathcal{Z}$. The quotient category
$\mathfrak{U}:=\mathcal{Z}/\mathcal{D}$ is called a subfactor triangulated
category which is defined by the following data.

  * (1)

The objects in $\mathfrak{U}$ are the same as $\mathcal{Z}$.

  * (2)

The morphism space $\mathop{\rm Hom}\nolimits_{\mathfrak{U}}(X,Y)$ is defined
as

$\mathop{\rm Hom}\nolimits_{\mathfrak{U}}(X,Y):=\mathop{\rm
Hom}\nolimits_{\mathcal{Z}}(X,Y)/\mathcal{D}(X,Y)$

for each $X,Y\in\mathcal{Z}$, where $\mathcal{D}(X,Y)$ is the subspace of
$\mathop{\rm Hom}\nolimits_{\mathcal{Z}}(X,Y)$ consisting of morphisms
factoring through objects in $\mathcal{D}$.

It is proved in [14] that $\mathfrak{U}$ carries a natural triangulated
structure inherited from the triangulated structure of $\mathcal{C}$ as
follows.

  * $\bullet$

For any object $X\in\mathfrak{U}$, choose a left $\mathcal{D}$-approximation
$f:D\rightarrow X$ and extend it to a triangle

$X\stackrel{{\scriptstyle f}}{{\rightarrow}}D\rightarrow Z\rightarrow X[1],$

then the shift of $X$ in $\mathfrak{U}$ is defined to be $Z$, denoted by
$X\langle 1\rangle$.

  * $\bullet$

For any triangle $X\stackrel{{\scriptstyle
a}}{{\rightarrow}}Y\stackrel{{\scriptstyle
b}}{{\rightarrow}}Z\stackrel{{\scriptstyle c}}{{\rightarrow}}X[1]$ in
$\mathcal{C}$ with $X,Y,Z\in\mathcal{Z}$, since $\mathop{\rm
Hom}\nolimits_{\mathcal{C}}(Z,D[1])=0$, there is a commutative diagram:

|  |  |  |  |
---|---|---|---|---|---
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{c}$$\textstyle{X[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\langle
1\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{X[1]}$

Now we consider the complex

$X\stackrel{{\scriptstyle\bar{a}}}{{\rightarrow}}Y\stackrel{{\scriptstyle\bar{b}}}{{\rightarrow}}Z\stackrel{{\scriptstyle\bar{d}}}{{\rightarrow}}X\langle
1\rangle$

in $\mathfrak{U}$. We define triangles in $\mathfrak{U}$ as the complexes
obtained in this way.

The following results are useful.

######  Lemma 3.7.

Let $\mathcal{D}$ be a functorially finite $(n+1)$-rigid subcategory of
$\mathcal{C}$ satisfying $\mathbb{S}\mathcal{D}=\mathcal{D}[n+1]$ and
$\mathcal{Z}=\bigcap\limits_{i=1}^{n}\mathcal{D}[-i]^{\perp}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathcal{D}[i]$.
We have the following results.

  * (1)

[14, Theorem 4.7] The subfactor category
$\mathfrak{U}:=\mathcal{Z}/\mathcal{D}$ forms a triangulated category with a
Serre functor $\mathbb{S}_{n}\circ\langle n+1\rangle$. In particular, if
$\mathcal{C}$ is $(n+1)$-Calabi–Yau, then so is $\mathfrak{U}$.

  * (2)

[14, Lemma 4.8] For any $X,Y\in\mathcal{Z}$, there exists an isomorphism

$\mathfrak{U}(X,Y\langle i\rangle)\cong\mathcal{C}(X,Y[i])$

for any $1\leq i\leq n$.

  * (3)

[14, Theorem 4.9] There exists a one-one correspondence between
$(n+1)$-cluster tilting subcategories of $\mathcal{C}$ containing
$\mathcal{D}$ and $(n+1)$-cluster tilting subcategories of $\mathfrak{U}$.

The following lemma is useful.

######  Lemma 3.8.

Let $(\mathscr{X},\mathscr{Y})$ be an $n$-cotorsion pair in $\mathcal{C}$ with
core $\mathcal{I}(\mathscr{X})$. Then
$\mathcal{D}\subset\mathcal{I}(\mathscr{X})$ if and only if
$\mathcal{D}\subset\mathscr{X}\subset\mathcal{Z}$ and
$\mathcal{D}\subset\mathscr{Y}\subset\mathcal{Z}$.

###### Proof.

The ‘if’ part is trivial since
$\mathcal{I}(\mathscr{X})=\mathscr{X}\cap\mathscr{Y}$. To prove the ‘only if’
part, let $X$ be an arbitrary object in $\mathscr{X}$. Since
$\mathcal{D}\subset\mathcal{I}(\mathscr{X})\subset\mathscr{Y}$, we have
$\mathop{\rm Hom}\nolimits(X,\mathcal{D}[i])=0$ for $1\leq i\leq n$. It
follows that $X\in\mathcal{Z}$ and hence $\mathscr{X}\subset\mathcal{Z}$.
Similarly, we have $\mathscr{Y}\subset\mathcal{Z}$. ∎

By the lemma above, one can consider the relationship between $n$-cotorsion
pairs in $\mathcal{C}$ whose cores contain $\mathcal{D}$ and $n$-cotorsion
pairs in the subfactor triangulated category $\mathfrak{U}$. In the following,
for a subcategory $\mathscr{W}$ of $\mathcal{C}$ satisfying
$\mathcal{D}\subset\mathscr{W}\subset\mathcal{Z}$, we denote
$\overline{\mathscr{W}}$ the subcategory of $\mathcal{Z}$. It is clear that
any subcategory of $\mathcal{Z}$ has this form.

######  Lemma 3.9.

Let $\mathscr{X}$ be a subcategory of $\mathcal{C}$ satisfying
$\mathcal{D}\subset\mathscr{X}\subset\mathcal{Z}$. Then

  * (1)

$\mathscr{X}$ is a contravariantly finite subcategory of $\mathcal{C}$ if and
only if $\overline{\mathscr{X}}$ is a contravariantly finite subcategory of
$\mathfrak{U}$.

  * (2)

$\mathscr{X}$ is a covariantly finite subcategory of $\mathcal{C}$ if and only
if $\overline{\mathscr{X}}$ is a covariantly finite subcategory of
$\mathfrak{U}$.

  * (3)

$\mathscr{X}$ is a functorially finite subcategory of $\mathcal{C}$ if and
only if $\overline{\mathscr{X}}$ is a functorially finite subcategory of
$\mathfrak{U}$.

###### Proof.

We only prove the first assertion, the others can be proved similarly.

Suppose $\mathscr{X}$ is a contravariantly finite subcategory of
$\mathcal{C}$. Let $Z$ be an object of $\mathcal{Z}$. Since $\mathscr{X}$ is
contravariantly finite, there exists an object $X\in\mathscr{X}$ such that
$X\stackrel{{\scriptstyle f}}{{\rightarrow}}Z$ is a right
$\mathscr{X}$-approximation of $Z$ in $\mathcal{C}$. It is easy to check that
$X\stackrel{{\scriptstyle\overline{f}}}{{\rightarrow}}Z$ is a right
$\overline{\mathscr{X}}$-approximation of $Z$ in $\mathfrak{U}$.

Now suppose $\overline{\mathscr{X}}$ is a contravariantly finite subcategory
of $\mathfrak{U}$. Since $\mathcal{D}$ is a functorially finite subcategory of
$\mathcal{C}$, so is $\mathcal{Z}$ by Proposition 2.4 in [14] and its dual.
Let $C$ be an object of $\mathcal{C}$. Since $\mathcal{Z}$ is a functorially
finite subcategory of $\mathcal{C}$, there exists an object $Z\in\mathcal{Z}$
such that $Z\stackrel{{\scriptstyle f}}{{\rightarrow}}C$ is a right
$\mathcal{Z}$-approximation of $C$ in $\mathcal{C}$. By assumption, there
exists an object $X\in\mathscr{X}$ such that
$X\stackrel{{\scriptstyle\overline{g}}}{{\rightarrow}}Z$ is a right
$\overline{\mathscr{X}}$-approximation of $Z$ in $\mathfrak{U}$. For any
morphism $X^{\prime}\stackrel{{\scriptstyle h}}{{\rightarrow}}C$ with
$X^{\prime}\in\mathscr{X}\subset\mathcal{Z}$, since $Z\stackrel{{\scriptstyle
f}}{{\rightarrow}}C$ is a right $\mathcal{Z}$-approximation of $C$ in
$\mathcal{C}$, there exists a morphism
$X^{\prime}\stackrel{{\scriptstyle\ell}}{{\rightarrow}}Z$ such that
$h=f\circ\ell$. Since $X\stackrel{{\scriptstyle\overline{g}}}{{\rightarrow}}Z$
is a right $\overline{\mathscr{X}}$-approximation of $Z$ in $\mathfrak{U}$,
there exists a morphism
$X^{\prime}\stackrel{{\scriptstyle\overline{k}}}{{\rightarrow}}X$ such that
$\overline{l}=\overline{g}\circ\overline{k}$, i.e.
$\overline{l}-\overline{g}\circ\overline{k}=\overline{0}$. So there exists an
object $D\in\mathcal{D}$ and two morphisms $X^{\prime}\stackrel{{\scriptstyle
a_{1}}}{{\rightarrow}}D$ and $D\stackrel{{\scriptstyle
a_{2}}}{{\rightarrow}}Z$ such that $\ell=g\circ k+a_{2}\circ a_{1}$. We will
show that $X\oplus D\xrightarrow{(f\circ g,f\circ a_{2})}C$ is a right
$\mathscr{X}$-approximation of $C$ in $\mathcal{C}$ and thus complete the
proof. For any morphism $X^{\prime}\stackrel{{\scriptstyle
h}}{{\rightarrow}}C$ with $X^{\prime}\in\mathscr{X}$, we have found a morphism
$X^{\prime}\xrightarrow{\binom{k}{a_{1}}}X\oplus D$ such that
$h=f\circ\ell=f\circ(g\circ k+a_{2}\circ a_{1})=(f\circ g,f\circ
a_{2})\circ\binom{k}{a_{1}}$, so $X\oplus D\xrightarrow{(f\circ g,f\circ
a_{2})}C$ is a right $\mathscr{X}$-approximation of $C$ in $\mathcal{C}$. ∎

######  Theorem 3.10.

Let $(\mathscr{X},\mathscr{Y})$ be an $n$-cotorsion pair in $\mathcal{C}$ with
$\mathcal{D}\subset\mathcal{I}(\mathscr{X})$. Then
$(\overline{\mathscr{X}},\overline{\mathscr{Y}})$ is an $n$-cotorsion pair in
$\mathfrak{U}$ with
$\mathcal{I}(\overline{\mathscr{X}})=\overline{\mathcal{I}(\mathscr{X})}$.
Moreover, the map
$(\mathscr{X},\mathscr{Y})\mapsto(\overline{\mathscr{X}},\overline{\mathscr{Y}})$
is a bijection from the set of $n$-cotorsion pair in $\mathcal{C}$ whose cores
contain $\mathcal{D}$ to the set of $n$-cotorsion pair in $\mathfrak{U}$.

###### Proof.

Since $(\mathscr{X},\mathscr{Y})$ is an $n$-cotorsion pair in $\mathcal{C}$
with $\mathcal{D}\subset\mathcal{I}(\mathscr{X})$, we have
$\mathcal{D}\subset\mathscr{X}\subset\mathcal{Z}$ and
$\mathcal{D}\subset\mathscr{Y}\subset\mathcal{Z}$ by Lemma 3.8. So
$\overline{\mathscr{X}}$ and $\overline{\mathscr{Y}}$ are subcategories of
$\mathfrak{U}$. By Lemma 3.7, there is an isomorphism
$\mathfrak{U}(\mathscr{X},\mathscr{Y}\langle
i\rangle)\cong\mathcal{C}(\mathscr{X},\mathscr{Y}[i])$ for any $1\leq i\leq
n$, so $\mathscr{X}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathscr{Y}[i]$ for all
$1\leq i\leq n$ if and only if
$\overline{\mathscr{X}}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\overline{\mathscr{Y}}\langle
i\rangle$ for all $1\leq i\leq n$. Moreover, By Lemma 3.9, we get
$(\overline{\mathscr{X}},\overline{\mathscr{Y}})$ is an $n$-cotorsion pair in
$\mathfrak{U}$. In this case, we have
$\mathcal{I}(\overline{\mathscr{X}})=\overline{\mathscr{X}}\cap\overline{\mathscr{Y}}=\overline{\mathscr{X}\cap\mathscr{Y}}=\overline{\mathcal{I}(\mathscr{X})}$.

The bijection from the set of $n$-cotorsion pair in $\mathcal{C}$ whose cores
contain $\mathcal{D}$ to the set of $n$-cotorsion pair in $\mathcal{Z}$ is
obvious by Lemma 3.8, Lemma 3.7 and Lemma 3.9. ∎

###  3.3. Mutation of $n$-cotorsion pairs

Let $\mathcal{C}$ be a Hom-finite Krull-Schmidt triangulated category over $K$
with a Serre functor $\mathbb{S}$. In this subsection, we study mutation of
$n$-cotorsion pairs.

######  Definition 3.11.

Fix a functorially finite $(n+1)$-rigid subcategory $\mathcal{D}$ of
$\mathcal{C}$. For a subcategory $\mathscr{X}$ of $\mathcal{C}$, put

$\mu^{-1}_{\mathcal{D}}(\mathscr{X}):=(\mathcal{D}\ast\mathscr{X}[1])\cap\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathcal{D}[i].$

That is, $\mu^{-1}_{\mathcal{D}}(\mathscr{X})$ consists of all
$M\in\mathcal{C}$ such that there exists a triangle

$X\stackrel{{\scriptstyle f}}{{\rightarrow}}D\rightarrow M\rightarrow X[1]$

with $X\in\mathscr{X}$ and a left $\mathcal{D}$-approximation $f$. Dually, for
a subcategory $\mathscr{Y}$ of $\mathcal{C}$, put

$\mu_{\mathcal{D}}(\mathscr{Y}):=(\mathscr{Y}[-1]\ast\mathcal{D})\cap\bigcap\limits_{i=1}^{n}\mathcal{D}[-i]^{\perp}.$

That is, $\mu_{\mathcal{D}}(\mathscr{Y})$ consists of all $M\in\mathcal{C}$
such that there exists a triangle

$M\rightarrow D\stackrel{{\scriptstyle g}}{{\rightarrow}}Y\rightarrow M$

with $Y\in\mathscr{Y}$ and a right $\mathcal{D}$-approximation $g$.

In this case, $\mu^{-1}_{\mathcal{D}}(\mathscr{X})$ is called the forward
$\mathcal{D}$-mutation of $\mathscr{X}$ and $\mu_{\mathcal{D}}(\mathscr{Y})$
is called the backward $\mathcal{D}$-mutation of $\mathscr{Y}$.

######  Remark 3.12.

We also have that
$\mu_{\mathcal{D}}(\mathcal{D})=\mathcal{D}=\mu^{-1}_{\mathcal{D}}(\mathcal{D})$.
When $\mathcal{D}=0$, we have
$\mu^{-1}_{\mathcal{D}}(\mathcal{M})=\mathcal{M}[1]$ and
$\mu_{\mathcal{D}}(\mathcal{M})=\mathcal{M}[-1]$. When $n=1$, our definition
of mutation is just Iyama-Yoshino’s definition (see Definition 2.6). It’s
important to note that the concept of mutation mentioned later specifically
refers to our definition, which is Definition 3.11.

The following result was proved in [14, Proposition 2.7] for the case $n=1$.
But their proof can be applied for the general case without any change.

######  Proposition 3.13.

Let $\mathcal{D}$ be a functorially finite $(n+1)$-rigid subcategory of
$\mathcal{C}$ satisfying $\mathbb{S}\mathcal{D}=\mathcal{D}[n+1]$ and
$\mathcal{Z}=\bigcap\limits_{i=1}^{n}\mathcal{D}[-i]^{\perp}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathcal{D}[i]$.
Then the maps $\mu^{-1}_{\mathcal{D}}(-)$ and $\mu_{\mathcal{D}}(-)$ are
mutually inverse on the set of subcategories $\mathcal{M}$ of $\mathcal{C}$
satisfying $\mathcal{D}\subset\mathcal{M}\subset\mathcal{Z}$.

Now we give the main result of this paper.

######  Theorem 3.14.

Let $(\mathscr{X},\mathscr{Y})$ be an $n$-cotorsion pair in $\mathcal{C}$ and
$\mathcal{D}\subset\mathcal{I}(\mathscr{X})$ be a functorially finite
subcategory of $\mathcal{C}$ satisfying
$\mathbb{S}\mathcal{D}=\mathcal{D}[n+1]$. Then the pairs

$(\mu^{-1}_{\mathcal{D}}(\mathscr{X}),\mu^{-1}_{\mathcal{D}}(\mathscr{Y}))\hskip
5.69054pt\mbox{and}\hskip
8.53581pt(\mu_{\mathcal{D}}(\mathscr{X}),\mu_{\mathcal{D}}(\mathscr{Y}))$

are $n$-cotorsion pairs in $\mathcal{C}$ and we have

$\mathcal{I}(\mu^{-1}_{\mathcal{D}}(\mathscr{X}))=\mu^{-1}_{\mathcal{D}}(\mathcal{I}(\mathscr{X}))\hskip
5.69054pt\mbox{and}\hskip
8.53581pt\mathcal{I}(\mu_{\mathcal{D}}(\mathscr{X}))=\mu_{\mathcal{D}}(\mathcal{I}(\mathscr{X})).$

###### Proof.

By Remark 3.6, we have that $\mathcal{I}(\mathscr{X})$ is $(n+1)$-rigid, then
so is $\mathcal{D}$. So both $\mu^{-1}_{\mathcal{D}}(\mathscr{X})$ and
$\mu^{-1}_{\mathcal{D}}(\mathscr{Y})$ are well-defined. We denote
$\mu^{-1}_{\mathcal{D}}(\mathscr{X})$, $\mu^{-1}_{\mathcal{D}}(\mathscr{Y})$
and $\mu^{-1}_{\mathcal{D}}(\mathcal{I}(\mathscr{X}))$ by
$\mathscr{X}^{\prime}$, $\mathscr{Y}^{\prime}$ and $\mathcal{I}^{\prime}$
respectively. As in the previous subsection, we denote by
$\mathcal{Z}=\bigcap\limits_{i=1}^{n}\mathcal{D}[-i]^{\perp}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathcal{D}[i]$
and $\mathfrak{U}=\mathcal{Z}/\mathcal{D}$ the subfactor triangulated
category. By Theorem 3.10, we have that
$(\overline{\mathscr{X}},\overline{\mathscr{Y}})$ is an $n$-cotorsion pair in
$\mathfrak{U}$ and
$\mathcal{I}(\overline{\mathscr{X}})=\overline{\mathcal{I}(\mathscr{X})}$.
Note that its forward 0-mutation $(\overline{\mathscr{X}}\langle
1\rangle,\overline{\mathscr{Y}}\langle 1\rangle)$ is also an $n$-cotorsion
pair in $\mathfrak{U}$. By [14, Proposition 4.4], we have that
$\overline{\mathscr{X}}\langle 1\rangle=\overline{\mathscr{X}^{\prime}}$ and
$\overline{\mathscr{Y}}\langle 1\rangle=\overline{\mathscr{Y}^{\prime}}$. Then
by Theorem 3.10, we have that the pair
$(\mathscr{X}^{\prime},\mathscr{Y}^{\prime})$ is an $n$-cotorsion pair in
$\mathcal{C}$ and
$\mathcal{I}(\overline{\mathscr{X}^{\prime}})=\overline{\mathcal{I}(\mathscr{X}^{\prime})}$.
This means that
$\mathcal{I}^{\prime}=\overline{\mathcal{I}(\mathscr{X})}\langle
1\rangle=\mathcal{I}(\overline{\mathscr{X}})\langle
1\rangle=\mathcal{I}(\overline{\mathscr{X}})\langle
1\rangle=\mathcal{I}(\overline{\mathscr{X}^{\prime}})=\overline{\mathcal{I}(\mathscr{X}^{\prime})}$.
Therefore,
$\mathcal{I}(\mu^{-1}_{\mathcal{D}}(\mathscr{X}))=\mu^{-1}_{\mathcal{D}}(\mathcal{I}(\mathscr{X}))$.
The assertion for
$\mathcal{I}(\mu_{\mathcal{D}}(\mathscr{X}))=\mu_{\mathcal{D}}(\mathcal{I}(\mathscr{X}))$
can be proved similarly. ∎

##  4\. A geometric realization of mutation of $n$-cotorsion pairs in
$n$-cluster categories of type $A_{m}$

In this section, we recall from [4, 18, 8] a geometric description of
$n$-cluster categories of type $A_{m}$, denoted by $\mathcal{C}_{A_{m}}^{n}$.
Then we give a geometric characterization of $n$-cotorsion pairs in
$\mathcal{C}_{A_{m}}^{n}$ via certain configurations of $n$-diagonals, and
introduce the rotation of such combinatorial models, which can give a
geometric realization of mutation of $n$-cotorsion pairs in
$\mathcal{C}_{A_{m}}^{n}$.

###  4.1. $n$-cluster categories of type $A_{m}$ and its geometric realization

Let $H$ be a finite dimensional hereditary algebra over a field $K$, and
$\mathcal{H}=\mathcal{D}^{b}(H)$ be the bounded derived category of $H$. The
Auslander-Reiten translate of $\mathcal{H}$ is denoted by $\tau$ and the shift
functor of $\mathcal{H}$ is denoted by $[1]$. Fix an integer
$m\in\mathbb{Z}_{\geq 1}$ throughout this subsection.

######  Definition 4.1 ([15]).

The orbit category $\mathcal{D}^{b}(H)/\tau^{-1}[n]$ is called the $n$-cluster
category of $\mathcal{H}$, and is denoted by $\mathcal{C}^{n}(\mathcal{H})$.

Specially, when $H=KQ$ is the path algebra over a Dynkin quiver of type
$A_{m}$, the $n$-cluster category $\mathcal{C}^{n}(\mathcal{H})$, denoted by
$\mathcal{C}_{A_{m}}^{n}$, is called the $n$-cluster category of type $A_{m}$,
which we consider in this section.

Now we summarize some known facts about $\mathcal{C}_{A_{m}}^{n}$ from [6,
15].

######  Proposition 4.2.

  1. (1)

The category $\mathcal{C}_{A_{m}}^{n}$ has Auslander-Reiten triangles and
Serre functor $\mathbb{S}=\tau[1]$, where $\tau$ is the AR-translate and $[1]$
is the shift functor in $\mathcal{C}_{A_{m}}^{n}$.

  2. (2)

The category $\mathcal{C}_{A_{m}}^{n}$ is a Krull-Schmidt and $(n+1)$-Calabi-
Yau triangulated category.

Since $\mathcal{C}_{A_{m}}^{n}$ is $(n+1)$-Calabi-Yau, $[n+1]$ is a Serre
functor. Hence any subcategory $\mathcal{D}$ of $\mathcal{C}_{A_{m}}^{n}$
satisfies the condition $\mathbb{S}\mathcal{D}=\mathcal{D}[n+1]$. We have the
following useful result.

######  Lemma 4.3.

[22, Lemma 3] Suppose $X,Y\in\mathcal{C}_{A_{m}}^{n}$. Then

$\mathrm{dim}_{K}\mathop{\rm
Ext}\nolimits^{i}_{\mathcal{C}_{A_{m}}^{n}}(X,Y)=\mathrm{dim}_{K}\mathop{\rm
Ext}\nolimits^{n+1-i}_{\mathcal{C}_{A_{m}}^{n}}(Y,X)$

for all $i=1,\ldots,n$.

Let $\Pi$ be a regular $(n(m+1)+2)$-gon, $m,n\in\mathbb{Z}_{\geq 1}$, with
vertices numbered clockwise from 1 to $n(m+1)+2$. We regard all operations on
vertices of $\Pi$ modulo $n(m+1)+2$ in this section. A diagonal of $\Pi$ is a
straight line between two non-adjacent vertices $i$ and $j$ of $\Pi$, denoted
by $(i,j)$. Thus $(i,j)=(j,i)$.

######  Definition 4.4.

A diagonal $\alpha$ is called an $n$-diagonal if $\alpha$ divides $\Pi$ into
an $(nj+2)$-gon and $(nk+2)$-gon for some integers $j,k$.

######  Remark 4.5.

For a diagonal $\alpha=(i,j)$ of $\Pi$ with $i<j$, $\alpha$ is an $n$-diagonal
if and only if $j-i\equiv 1\mathop{\rm mod}\nolimits n$.

There is a translation $\tau_{n}$ on the set of $n$-diagonals, which sends the
$n$-diagonal $(i,j)$ to $(i-n,j-n)$, i.e. $\tau_{n}(i,j)=(i-n,j-n)$.

######  Definition 4.6.

([18, 22]) Suppose $u=(c_{1},c_{2}),v=(d_{1},d_{2})$ are two $n$-diagonals in
$\Pi$ with $c_{1}<c_{2},d_{1}<d_{2}$. The $n$-diagonals $u$ and $v$ are called
crossing if their endpoints are all distinct and come in the order
$c_{1},d_{1},c_{2},d_{2}$ when moving around the polygon in one direction or
the other, i.e. $c_{1}<d_{1}<c_{2}<d_{2}$ or $d_{1}<c_{1}<d_{2}<c_{2}$.

By Definition above, the $n$-diagonals $u$ and $\tau_{n}(u)$ always cross for
all $u$ in $\Pi$, see Lemma 3.1 in [18].

######  Proposition 4.7.

[4, Proposition 5.4] There exists a bijection between $n$-diagonals in $\Pi$
and indecomposable objects in $\mathrm{ind}\;\mathcal{C}_{A_{m}}^{n}$.

The bijection above induces a bijection between the subcategories of
$\mathcal{C}_{A_{m}}^{n}$ and the sets of $n$-diagonals in $\Pi$. For a
subcategory $\mathscr{X}$ of $\mathcal{C}_{A_{m}}^{n}$, we denote the
corresponding set of $n$-diagonals in $\Pi$ by $\widetilde{\mathscr{X}}$. For
a set of $n$-diagonals $\mathfrak{X}$ in $\Pi$, we denote the corresponding
subcategory of $\mathcal{C}_{A_{m}}^{n}$ by $E(\mathfrak{X})$. We sometimes
use an $n$-diagonal $(i,j)$ to represent an indecomposable object $M$ in
$\mathcal{C}_{A_{m}}^{n}$ without confusion, denote by $M=(i,j)$.

######  Example 4.8.

Suppose $n=2,m=3$. Then the AR-quiver of $\mathcal{C}_{A_{3}}^{2}$ is shown as
Figure 2.

|  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  |  |
$\textstyle{(1,8)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(3,10)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(2,5)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(4,7)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(6,9)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,8)}$$\textstyle{(1,6)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(3,8)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(5,10)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(2,7)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(4,9)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,6)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(3,6)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(5,8)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(7,10)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(2,9)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(1,4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 2. The AR-quiver of $\mathcal{C}_{A_{3}}^{2}$

######  Lemma 4.9.

[22, Proposition 2] Suppose $u,v$ are two $n$-diagonals in $\Pi$, and
$M_{u},M_{v}$ are the corresponding indecomposable objects in
$\mathcal{C}_{A_{m}}^{n}$. Then

  * (1)

$u$ does not cross $v$ if and only if $\mathop{\rm
Ext}\nolimits^{i}_{\mathcal{C}_{A_{m}}^{n}}(M_{u},M_{v})=0$ for all $1\leq
i\leq n$.

  * (2)

$M_{u}[n]=M_{\tau_{n}u}$

###  4.2. $n$-cotorsion pairs in $\mathcal{C}_{A_{m}}^{n}$

In this subsection, we give a geometric characterization of $n$-cotorsion
pairs in $\mathcal{C}_{A_{m}}^{n}$.

Let $\mathfrak{X}$ be a set of $n$-diagonals in $\Pi$. We define

$\mathop{\rm
nc}\nolimits\mathfrak{X}=\\{u\in\Pi\text{\;is\;an\;}n\text{-diagonal\;}|\;u\text{\;does\;not\;cross\;any\;}n\text{-diagonals\;
in\;}\mathfrak{X}\\}.$

By Lemma 4.9, $\mathop{\rm nc}\nolimits\mathfrak{X}$ corresponds to the
subcategory
$\bigcap\limits_{i=1}^{n}\mathscr{X}[-i]^{\perp}=\bigcap\limits_{i=1}^{n}{{}^{\bot}}\mathscr{X}[i]$.

######  Theorem 4.10.

Let $\mathscr{X}$ be a subcategory of $\mathcal{C}_{A_{m}}^{n}$, and
$\widetilde{\mathscr{X}}$ be the corresponding set of $n$-diagonals in $\Pi$.
Then the following statements are equivalent.

  * (1)

$(\mathscr{X},\bigcap\limits_{i=1}^{n}\mathscr{X}[-i]^{\perp})$ is an
$n$-cotorsion pair.

  * (2)

$\widetilde{\mathscr{X}}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}$.

###### Proof.

Since $\mathcal{C}_{A_{m}}^{n}$ is a finite triangulated category, any
subcategory $\mathscr{X}$ of $\mathcal{C}_{A_{m}}^{n}$ is functorially finite.
The equivalence is obvious by Lemma 4.9. ∎

We recall from [8] the definition of Ptolemy diagram and give a necessary
condition for an $n$-cotorsion pair.

######  Definition 4.11.

Let $\mathfrak{X}$ be a set of $n$-diagonals in $\Pi$. Then $\mathfrak{X}$ is
called a Ptolemy diagram if for any two crossing $n$-diagonals $u=(i,j)$ and
$v=(k,\ell)$ in $\mathfrak{X}$, those of $(i,k),(i,\ell),(j,k),(j,\ell)$,
which are $n$-diagonals are in $\mathfrak{X}$.

######  Lemma 4.12.

Let $\mathscr{X}$ be a subcategory of $\mathcal{C}_{A_{m}}^{n}$, and
$\widetilde{\mathscr{X}}$ be the corresponding set of $n$-diagonals in $\Pi$.
If $\widetilde{\mathscr{X}}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}$, then $\widetilde{\mathscr{X}}$ is a
Ptolemy diagram.

###### Proof.

By Theorem 4.10, it is enough to show for any subcategory $\mathscr{Y}$ of
$\mathcal{C}_{A_{m}}^{n}$, $\mathop{\rm nc}\nolimits\widetilde{\mathscr{Y}}$
is a Ptolemy diagram.

Suppose $u=(i,j)$ and $v=(k,\ell)$ are two crossing $n$-diagonals in
$\mathop{\rm nc}\nolimits\widetilde{\mathscr{Y}}$, and suppose $(i,k)$ is an
$n$-diagonal, we need to show $(i,k)\in\mathop{\rm
nc}\nolimits\widetilde{\mathscr{Y}}$.

If $(i,k)\not\in\mathop{\rm nc}\nolimits\widetilde{\mathscr{Y}}$, then there
exists an $n$-diagonal $w\in\widetilde{\mathscr{Y}}$ such that $w$ crosses
$(i,k)$. Oberserve that every $n$-diagonal $w\in\widetilde{\mathscr{Y}}$
crossing $(i,k)$ crosses either $u$ or $v$, which is a contradiction with the
assumption $u,v\in\mathop{\rm nc}\nolimits\widetilde{\mathscr{Y}}$. ∎

Note that when $n=1$, the converse of Lemma 4.12 is true by [12, Proposition
2.7]. However, the converse of Lemma 4.12 is not true in general case. We see
an example below.

######  Example 4.13.

Suppose $n=2,m=3$. The AR-quiver of $\mathcal{C}_{A_{3}}^{2}$ has been shown
in Figure 2.

  * (1)

Let $\mathscr{X}=\\{(1,4),(1,6),(3,6)\\}$ be a subcategory of
$\mathcal{C}_{A_{3}}^{2}$. Then $\widetilde{\mathscr{X}}$ is a Ptolemy
diagram. However, $\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}=\\{(1,6),(1,8),(7,10),(6,9)\\}$ and
$\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}=\\{(1,4),(1,6),(3,6),(2,5)\\}$, so we have
$\widetilde{\mathscr{X}}\neq\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}$. Thus,
$(\mathscr{X},\mathscr{X}[-1]^{\perp}\cap\mathscr{X}[-2]^{\perp})$ can not be
a $2$-cotorsion pair.

  * (2)

Let $\mathscr{Y}=\\{(1,4),(1,6),(3,6),(2,5)\\}$ be a subcategory of
$\mathcal{C}_{A_{3}}^{2}$. We have $\mathop{\rm
nc}\nolimits\widetilde{\mathscr{Y}}=\\{(1,6),(1,8),(7,10),(6,9)\\}$ and
$\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{Y}}=\\{(1,4),(1,6),(3,6),(2,5)\\}$, so we have
$\widetilde{\mathscr{Y}}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{Y}}$. Thus,
$(\mathscr{Y},\mathscr{Y}[-1]^{\perp}\cap\mathscr{Y}[-2]^{\perp})$ is a
$2$-cotorsion pair.

###  4.3. Mutation of $n$-cotorsion pairs in $\mathcal{C}_{A_{m}}^{n}$

######  Definition 4.14.

Let $\mathscr{X}$ be a subcategory of $\mathcal{C}_{A_{m}}^{n}$ and
$\widetilde{\mathscr{X}}$ be the corresponding set of $n$-diagonals in $\Pi$
satisfying $\widetilde{\mathscr{X}}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}$. The set of $n$-diagonals in
$\widetilde{\mathscr{X}}$ that does not cross any diagonals in
$\widetilde{\mathscr{X}}$ is called the frame of $\widetilde{\mathscr{X}}$,
denoted by $F_{\widetilde{\mathscr{X}}}$.

By the assumption above, the subcategory $\mathscr{X}$ is a first half of an
$n$-cotorsion pair in $\mathcal{C}_{A_{m}}^{n}$ by Theorem 4.10, and the
subcategory $E(F_{\widetilde{\mathscr{X}}})$ which corresponds to
$F_{\widetilde{\mathscr{X}}}$ is the core of the $n$-cotorsion pair
$(\mathscr{X},\bigcap\limits_{i=1}^{n}\mathscr{X}[-i]^{\perp})$, i.e.
$\mathcal{I}(\mathscr{X})=E(F_{\widetilde{\mathscr{X}}})$.

######  Definition 4.15.

  * (1)

For any $n$-diagonal $(i,j)$ of $\Pi$, its rotation in $\Pi$, denoted by
$\rho_{\Pi}(i,j)$, is defined to be $(i-n,j-n)$, i.e.

$\rho_{\Pi}(i,j)=(i-n,j-n).$

  * (2)

Let $\mathfrak{D}$ be a set of non-crossing $n$-diagonals of $\Pi$.

    * (i)

The $n$-diagonals in $\mathfrak{D}$ divide $\Pi$ into polygons, called
$\mathfrak{D}$-cells. Thus, any $n$-diagonals $(i,j)$ which neither is in
$\mathfrak{D}$ nor crosses any diagonals in $\mathfrak{D}$ is an $n$-diagonal
of a $\mathfrak{D}$-cell. We call the rotation of $(i,j)$ in this
$\mathfrak{D}$-cell the $\mathfrak{D}$-rotation of $(i,j)$, and denote it by
$\rho_{\mathcal{D}}(i,j)$.

    * (ii)

Let $\mathfrak{X}$ be a set of diagonals of $\Pi$ satisfying
$\mathfrak{X}=\mathop{\rm nc}\nolimits\mathop{\rm nc}\nolimits\mathfrak{X}$.
The $\mathfrak{D}$-rotation of $\mathfrak{X}$ is defined as

$\rho_{\mathfrak{D}}(\mathfrak{X}):=\\{\rho_{\mathfrak{D}}(i,j)|(i,j)\in\mathfrak{X}\backslash\mathfrak{D}\\}\cup\mathfrak{D}.$

######  Example 4.16.

Suppose $n=2,m=3$. The AR-quiver of $\mathcal{C}_{A_{3}}^{2}$ has been shown
in Figure 2. Let

$\mathfrak{X}=\\{(1,4),(1,6),(1,8),(7,10),(6,9)\\}.$

It is easy to check the set $\mathfrak{X}$ satisfies $\mathfrak{X}=\mathop{\rm
nc}\nolimits\mathop{\rm nc}\nolimits\mathfrak{X}$ and
$F_{\mathfrak{X}}=\\{(1,4),(1,6)\\}$. Let $\mathfrak{D}=\\{(1,6)\\}\subset
F_{\mathfrak{X}}$. Then $\mathfrak{D}$ divides $\Pi$ into two polygons. The
$\mathfrak{D}$-rotation of $\mathfrak{X}$ is

$\rho_{\mathfrak{D}}(\mathfrak{X})=\\{(2,5),(1,6),(6,9),(1,8),(7,10)\\},$

where for example $\rho_{\mathfrak{D}}(1,4)=(2,5).$ Let
$\mathfrak{Y}=\rho_{\mathfrak{D}}(\mathfrak{X})$. It is clear that
$\mathfrak{Y}$ still satisfies $\mathfrak{Y}=\mathop{\rm
nc}\nolimits\mathop{\rm nc}\nolimits\mathfrak{Y}$.

See Figure 3 and Figure 4 for geometric explanations.

$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$
Figure 3. The set $\mathfrak{X}$ satisfies $\mathfrak{X}=\mathop{\rm
nc}\nolimits\mathop{\rm nc}\nolimits\mathfrak{X}$ with
$\mathfrak{D}=\\{(1,6)\\}$
$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$$1$$2$$10$$9$$8$$7$$6$$5$$4$$3$
Figure 4. The set $\mathfrak{Y}=\rho_{\mathfrak{D}}(\mathfrak{X})$ still
satisfies $\mathfrak{Y}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\mathfrak{Y}$

Now we prove that rotation of a set of $n$-diagonals $\mathfrak{X}$ of $\Pi$
satisfying $\mathfrak{X}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\mathfrak{X}$ gives a geometric model of mutation of $n$-cotorsion
pairs in $\mathcal{C}_{A_{m}}^{n}$.

######  Theorem 4.17.

Let $\mathscr{X}$ be a subcategory of $\mathcal{C}_{A_{m}}^{n}$,
$\widetilde{\mathscr{X}}$ be the corresponding set of $n$-diagonals in $\Pi$
satisfying $\widetilde{\mathscr{X}}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\widetilde{\mathscr{X}}$, and $F_{\widetilde{\mathscr{X}}}$ be the
frame of $\widetilde{\mathscr{X}}$. Then the following hold.

  * (1)

The frame $F_{\widetilde{\mathscr{X}}}$ of $\widetilde{\mathscr{X}}$
corresponds to the core of the $n$-cotorsion pair
$(\mathscr{X},\bigcap\limits_{i=1}^{n}\mathscr{X}[-i]^{\perp})$, i.e.
$\mathcal{I}(\mathscr{X})=E(F_{\widetilde{\mathscr{X}}})$, where
$E(F_{\widetilde{\mathscr{X}}})$ represents the subcategory of
$\mathcal{C}_{A_{n}}^{m}$ which corresponds to $F_{\widetilde{\mathscr{X}}}$.

  * (2)

Let $\mathfrak{D}\subset F_{\widetilde{\mathscr{X}}}$ be a set of non-crossing
$n$-diagonals and $E(\mathfrak{D})$ be the corresponding subcategory of
$\mathcal{C}_{A_{m}}^{n}$, we have that

$\mu_{E(\mathfrak{D})}(\mathscr{X})=E(\rho_{\mathfrak{D}}(\widetilde{\mathscr{X}})).$

Consequently, a rotation of a set of $n$-diagonals $\mathfrak{X}$ of $\Pi$
satisfying $\mathfrak{X}=\mathop{\rm nc}\nolimits\mathop{\rm
nc}\nolimits\mathfrak{X}$ is again a set of $n$-diagonals
$\mathfrak{Y}=\rho_{\mathfrak{D}}(\mathfrak{X})$ of $\Pi$ satisfying
$\mathfrak{Y}=\mathop{\rm nc}\nolimits\mathop{\rm nc}\nolimits\mathfrak{Y}$.

###### Proof.

It is obvious that (1) holds.

In order to prove (2), it is enough to show that for any $n$-diagonal
$(i,j)(i<j)$ in $\widetilde{\mathscr{X}}$ which is not in $\mathfrak{D}$, if
$\rho_{\mathfrak{D}}(i,j)=(k,\ell)$, then
$\mu_{E(\mathfrak{D})}(E(i,j))=E(k,\ell)$.

Let $\mathfrak{X}^{\prime}$ be the $\mathfrak{D}$-cell containing $(i,j)$.
Since $\rho_{\mathfrak{D}}(i,j)=(k,\ell)$, $(k,\ell)$ is the rotation of
$(i,j)$ in $\mathfrak{U}^{\prime}$. That means $(k,\ell)$ is an $n$-diagonal
of $\mathfrak{X}^{\prime}$, so $(k,\ell)$ does not cross any diagonal in
$\mathfrak{D}$, we have
$E(k,\ell)\in\bigcap\limits_{i=1}^{n}E(\mathfrak{D})[-i]^{\perp}$.

Since $(i,j)$ is an $n$-diagonal, $(k,\ell)$ crosses $(i,j)$. Without loss of
generality, we assume $k<i<\ell<j$. By the definition of rotation, we have
$(k,\ell)=(i-n,j-n)$. By the AR-quiver of $\mathcal{C}_{A_{m}}^{n}$, it is
easy to check that the crossing diagonals $(i,j)$ and $(k,\ell)$ induce a
triangle in $\mathcal{C}_{A_{m}}^{n}$

$E(k,\ell)\rightarrow E(i,\ell)\oplus E(k,j)\stackrel{{\scriptstyle
f}}{{\rightarrow}}E(i,j)\rightarrow E(k,\ell)[1]$

By the definition of rotation, $(i,\ell)$ and $(k,j)$ are edges of
$\mathfrak{X}^{\prime}$. Thus, $E(i,\ell)$ (resp. $E(k,j)$) is an zero object
or a non-zero object in $E(\mathfrak{D})$, i.e. $E(i,\ell)\oplus E(k,j)\in
E(\mathfrak{D})$. Applying $\mathop{\rm Hom}\nolimits(D,-)$ to the above
triangle for any $D\in E(\mathfrak{D})$, we have an exact sequence

$\mathop{\rm Hom}\nolimits(D,E(i,\ell)\oplus E(k,j))\rightarrow\mathop{\rm
Hom}\nolimits(D,E(i,j))\rightarrow\mathop{\rm Hom}\nolimits(D,E(k,\ell)[1]),$

where $\mathop{\rm Hom}\nolimits(D,E(k,\ell)[1])=0$ since
$E(k,\ell)\in\bigcap\limits_{i=1}^{n}E(\mathfrak{D})[-i]^{\perp}$. This
implies that $f$ is a right $E(\mathfrak{D})$-approximation and it is minimal
since $E(k,\ell)$ is indecomposable. So by definition,
$\mu_{E(\mathfrak{D})}(E(i,j))=E(k,\ell)$ and we complete the proof of
$\mu_{E(\mathfrak{D})}(\mathscr{X})=E(\rho_{\mathfrak{D}}(\widetilde{\mathscr{X}})).$
Then the last assertion follows from Theorem 3.14. ∎

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CMS CR-2016/411

Measurement of top quark polarisation in t-channel single top quark production

Andres Tiko on behalf of the CMS collaboration

National Institute of Chemical Physics and Biophysics
Tallinn, Estonia

> A first measurement of the top quark spin asymmetry, sensitive to the top
> quark polarisation, in $t$-channel single top quark production is presented.
> It is based on a sample of pp collisions at a centre-of-mass energy of 8 TeV
> corresponding to an integrated luminosity of 19.7 $\mathrm{fb^{-1}}$. A
> high-purity sample of $t$-channel single top quark events with an isolated
> muon is selected. Signal and background components are estimated using a fit
> to data. A differential cross section measurement, corrected for detector
> effects, of an angular observable sensitive to the top quark polarisation is
> performed. The differential distribution is used to extract a top quark spin
> asymmetry of $0.26\pm 0.03\textrm{(stat)}\pm 0.10\textrm{(syst)}$, which is
> compatible with a p-value of $4.6\%$ with the standard model prediction of
> 0.44.

> PRESENTED AT
>
>
>
>
> $9^{th}$ International Workshop on Top Quark Physics
> Olomouc, Czech Republic, September 19–23, 2016

##  1 Introduction

The top quark is the heaviest elementary particle discovered so far. Its
lifetime (${\approx 4\times 10^{-25}}$ s) is much shorter than the typical
timescales of quantum chromodynamics (QCD).

Therefore, top is the only quark that decays before hadronising. Furthermore,
the Standard Model (SM) predicts that only left-handed quarks are produced at
the Wtb vertex. Thus, top quark’s decay products retain memory of its spin in
their angular distributions, providing a probe to investigate the structure of
the Wtb vertex.

In electroweak $t$-channel single top quark production, shown in figure 1, SM
predicts that produced top quarks are highly polarised, as a consequence of
the V–A coupling structure, along the direction of the momentum of the
spectator quark ($\mathrm{q^{\prime}}$), which recoils against the top quark
[2, 3].

New physics models can lead to a depolarisation in production by altering the
coupling structure [4, 5, 6, 7]. In this way, measuring single top quark
polarisation is an important test of SM.

Figure 1: Single top $t$-channel production and decay.

The measurement is performed with the CMS detector [8] and published in [1].

##  2 Event selection and estimation of signal and background contributions

The decay products of the single top quark can be seen in Figure 1. We select
the events where the W-boson decays to a muon and a corresponding neutrino,
due to which we see a considerable amount of missing energy in the detector.
We also observe 2 jets, one of which is b-tagged, corresponding to the jet
from the top quark decay. The second jet (“spectator jet”) is light. There can
also be a second b-jet, produced in association with the top quark, but
usually it escapes detection.

The largest background processes mimicking the signal signature are top-
antitop pair production ($\mathrm{t\overline{t}}$), production of a W-boson in
association with jets (W+jets) and QCD multijet production. To discriminate
between the signal and background processes, we train boosted decision tree
(BDT) classifiers.

The first BDT, $\mathrm{BDT}_{\mathrm{multijet}}$, is trained to separate
signal and QCD multijet events. The most important variable is the transverse
mass of the reconstructed W boson. The distribution can be seen in Figure 2.

Another BDT, $\mathrm{BDT}_{\mathrm{W}/\mathrm{t\overline{t}}}$, is trained
for separating out W+jets and $\mathrm{t\overline{t}}$ backgrounds. Here, the
variables having the largest influence on the BDT are the pseudorapidity of
the light jet and reconstructed top quark mass.

The same same distribution is used to fit the contributions of signal and
background components to data.

Figure 2: Distributions of the $\mathrm{BDT}_{\mathrm{multijet}}$
discriminant, on the left, and the
$\mathrm{BDT}_{\mathrm{W}/\mathrm{t\overline{t}}}$ discriminant, on the right.
The predictions are normalised to the results of the fit described in Section
2. The bottom panels in both plots show the ratio between observed and
predicted event counts, with a shaded area to indicate the systematic
uncertainties affecting the background prediction and vertical bars indicating
statistical uncertainties.

##  3 Polarisation and spin asymmetry

In this analysis, the top quark spin asymmetry

$A_{X}\equiv\frac{1}{2}\cdot
P_{\mathrm{t}}\cdot\alpha_{X}=\frac{N(\uparrow)-N(\downarrow)}{N(\uparrow)+N(\downarrow)}\,$
(1)

is used to probe the coupling structure, where $P_{\mathrm{t}}$ represents the
top quark polarisation in production and $\alpha_{X}$ denotes the degree of
the angular correlations of one of its decay products, denoted $X$, with
respect to the spin of the top quark, the so-called spin-analysing power. The
variables ${N(\uparrow)}$ and ${N(\downarrow)}$ are defined, for each top
quark decay product from the decay chain ${\rm t}\rightarrow{\rm
bW}\rightarrow{\rm b}\mu\nu$, as the number of instances in which that decay
product is aligned or antialigned, respectively, relative to the direction of
the recoiling spectator quark momentum. We choose muon as the spin analyser
($X=\mu$) because $\alpha_{\mu}=1$ SM at leading order, as well as for its
high identification efficiency in CMS.

The angle between a top quark decay product $X$ and an arbitrary polarisation
axis $\vec{s}$ in the top quark rest frame, $\theta^{*}_{X}$, is distributed
according to the following differential cross section:

$\frac{1}{\sigma}\frac{d\sigma}{d\cos\theta^{*}_{X}}=\frac{1}{2}(1+P^{(\vec{s})}_{\mathrm{t}}\alpha_{X}\cos\theta^{*}_{X})=\left(\frac{1}{2}+A_{X}\cos\theta^{*}_{X}\right).$
(2)

In SM, the top quark spin tends to be aligned with the direction of the
spectator quark, so we expect a rising slope for the angle between the muon
and the spectator jet. Hence, an excess of antialigned would be a clear
indication of an anomalous coupling structure.

Figure 3 (left) shows the reconstructed distribution of
$\cos{\theta^{*}_{\mu}}$ (for
$\mathrm{BDT}_{\mathrm{W}/\mathrm{t\overline{t}}}>0.45$) The observed
distribution is expected to differ from the parton-level prediction because of
detector effects and the kinematic selection applied, with the most
significant effect being the relatively small number of selected events close
to $\cos{\theta^{*}_{\mu}}=1$. An overall trend in the ratio between data and
simulation is observed that suggests a slightly less asymmetric shape than
predicted by the SM.

Figure 3: On the left, the distribution of $\cos{\theta^{*}_{\mu}}$ in the
signal region (after background rejection and fit) The bottom panel shows the
ratio between observed and predicted event counts, with a shaded area to
indicate the systematic uncertainties affecting the background prediction, and
vertical bars indicating statistical uncertainties. On the right, the
normalised $\cos{\theta^{*}_{\mu}}$distribution compared to predictions. The
inner (outer) bars represent the statistical (total) uncertainties.

##  4 Unfolding

In this analysis, we perform background subtraction and unfolding on the
measured $\cos{\theta^{*}_{\mu}}$distribution.

Unfolding the distribution to parton level accounts for distortions from
detector acceptance, selection efficiencies, imperfect top quark
reconstruction, and the approximation of treating the untagged jet direction
as the spectator quark direction. We use a regularised matrix inversion
procedure for unfolding [9]. The unfolded distribution is shown on the right
in Figure 3.

The value of asymmetry is obtained from the unfolded distribution by a
$\chi^{2}$-fit according to Eq. (2), taking into account correlations. The
results are tested for bias by injecting anomalous Wtb-coupling events as
pseudo-data.

##  5 Results

For the asymmetry, we obtain a value of: $\mathrm{A_{\mu}}=0.26\pm
0.03\textrm{(stat)}\pm 0.10\textrm{(syst)}$. Compared to the SM prediction of
0.44 by POWHEG at NLO, the result is compatible with a p-value of p( data — SM
) = 4.6%, which corresponds to a difference of 2.0 standard deviations.
Separate results for top quarks and antiquarks are compatible with the
combined result.

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