11institutetext: Department of Cognitive Robotics, Delft University of
Technology, Delft, Netherlands<EMAIL_ADDRESS>22institutetext:
Institute of Robotics and Mechatronics, German Aerospace Center (DLR),
Oberpfaffenhofen, Germany. 33institutetext: Department of Computer Science,
University of Luxembourg, Luxembourg 44institutetext: Telefónica I+D, Spain
55institutetext: AEGIS IT Research GmbH, Germany 66institutetext: Université
catholique de Louvain, Belgium 77institutetext: Speech Technology and Machine
Learning Group, ETSI de Telecomunicación, Universidad Politécnica de Madrid,
Madrid, Spain 88institutetext: Alien Venture Studio, Alien Technology
Transfer, Rome, Italy 99institutetext: Department of Neurology, University
Medical Center Hamburg-Eppendorf (UKE), Hamburg, Germany 1010institutetext:
Institut Jean Nicod, École Normale Supérieure, Paris, France
1111institutetext: Research, Technology and Development Department, Digital
Industry Software, Helmond, Netherlands 1212institutetext: Department of
Information Technology, Uppsala University, Uppsala, Sweden 1313institutetext:
Division of Decision and Control Systems, School of Electrical Engineering and
Computer Science, KTH Royal Institute of Technology, Stockholm, Sweden
1414institutetext: Max Planck Institute for Software Systems, Kaiserslautern,
Germany 1515institutetext: Department of Electrical Engineering, Control
Systems Group, Eindhoven University of Technology, Eindhoven, Netherlands
1616institutetext: Netherlands Aerospace Centre (NLR), Amsterdam, Netherlands
1717institutetext: Donders Centre for Neuroscience, Radboud University,
Nijmegen, Netherlands 1818institutetext: Computer Science Department,
University of Pisa, Pisa, Italy 1919institutetext: Faculty of Philosophy,
Philosophy of Science and the Study of Religion, Ludwig Maximilian University
of Munich, Munich, Germany 2020institutetext: School of Engineering
Mathematics and Technology, University of Bristol, Bristol, United Kingdom
2121institutetext: Autonomous Systems Laboratory, Universidad Politécnica de
Madrid, Spain 2222institutetext: Cajal International Neuroscience Center
(CINC), Spanish National Research Council (CSIC), Madrid, Spain
2323institutetext: Department of Information Engineering and Computer Science,
University of Trento, Trento, Italy 2424institutetext: Department of
Information and Communications Engineering, Aalto University, Espoo, Finland
2525institutetext: Department of Electronic Engineering, Universitat
Politècnica de Catalunya, Barcelona, Spain 2626institutetext: Studio Stelluti,
Brussels, Belgium 2727institutetext: Artificial Intelligence Research
Institute (IIIA), Spanish National Research Council (CSIC), Barcelona, Spain

# Awareness in robotics: An early perspective from the viewpoint of the EIC
Pathfinder Challenge “Awareness Inside”

Cosimo Della Santina 1122 Carlos Hernandez Corbato 11 Burak Sisman 11 Luis A.
Leiva 33 Ioannis Arapakis 44 Michalis Vakalellis 55 Jean Vanderdonckt 66 Luis
Fernando D’Haro 77 Guido Manzi 88 Cristina Becchio 99 Aïda Elamrani 1010
Mohsen Alirezaei 1111 Ginevra Castellano 1212 Dimos V. Dimarogonas 1313
Arabinda Ghosh 1414 Sofie Haesaert 1515 Sadegh Soudjani 1414 Sybert Stroeve
1616 Paul Verschure 1717 Davide Bacciu 1818 Ophelia Deroy 1919 Bahador Bahrami
1919 Claudio Gallicchio 1818 Sabine Hauert 2020 Ricardo Sanz 2121 Pablo
Lanillos 2222 17 17 Giovanni Iacca 2323 Stephan Sigg 2424 Manel Gasulla 2525
Luc Steels 2626 Carles Sierra 2727

###### Abstract

Consciousness has been historically a heavily debated topic in engineering,
science, and philosophy. On the contrary, awareness had less success in
raising the interest of scholars in the past. However, things are changing as
more and more researchers are getting interested in answering questions
concerning what awareness is and how it can be artificially generated. The
landscape is rapidly evolving, with multiple voices and interpretations of the
concept being conceived and techniques being developed. The goal of this paper
is to summarize and discuss the ones among these voices that are connected
with projects funded by the EIC Pathfinder Challenge called “Awareness
Inside”, a nonrecurring call for proposals within Horizon Europe that was
designed specifically for fostering research on natural and synthetic
awareness. In this perspective, we dedicate special attention to challenges
and promises of applying synthetic awareness in robotics, as the development
of mature techniques in this new field is expected to have a special impact on
generating more capable and trustworthy embodied systems.

###### keywords:

Awareness, Robotics, Autonomous Systems, Living Machines, Artificial
Consciousness

## Acknowledgements

This work is supported by the EU EIC projects: SYMBIOTIK, grant number 637107;
ASTOUND, grant number 101071191; SymAware, grant number 101070802; CAVAA,
grant number 10039052; EMERGE, grant number 101070918; METATOOL, grant number
101070940; SUST(AI)N, grant number 101071179; and VALAWAI, grant number
101070930.

##  1 Introduction

According to the Cambridge English Dictionary [2], awareness is the “knowledge
that something exists, or understanding of a situation or subject at the
present time based on information or experience”. It may seem that, according
to this broad definition, awareness is already widespread in robotics. Think,
for example, of a drone creating an internal representation of its environment
using a SLAM algorithm [20], a manipulator detecting an interaction with its
environment through an observer [26], or a social robot acquiring knowledge on
the logical structure of its environment [14, 7] or its body [17]. If we
accept this broad definition, it is evident that a common framework that
enables the discussion of levels of awareness across domains is missing. Even
more, it is intuitively clear that the concept may be loaded with more complex
significance than this interpretation suggests. For example, awareness is
sometimes semantically associated with the term consciousness, which is a much
more debated concept. Researchers have proposed several definitions of the
latter, including philosophical [25], psychological [16], architectural [10],
neural correlates [9], and computer science theory [5].

As Artificial Intelligence (AI) progresses in reproducing a growing number of
human capabilities [23, 21, 13], it is thus natural to ask if and how
synthetic awareness can be produced and embodied into physical agents. This
challenge will necessarily pass through broadening awareness to include
different species beyond humans and even recognizing awareness of inanimate
objects. Ultimately, questions that need answering include: What is awareness
exactly? Can this concept be formalized philosophically or technically? What
distinguishes an aware agent from a non-aware one? What is the role of
embodiment? Does awareness improve the performance of robots? Is it even
ethical to endow artificial agents with awareness?

In this vein, in June 2021, the European Innovation Council (EIC) opened a
Pathfinder Challenge call for proposals with the explicit aim of pushing the
boundaries of AI [11]. The call challenged the scientific community to define
and achieve true awareness and understand awareness beyond the perception of
surroundings or self-awareness. The call also pointed towards the role that
awareness could play in both clinical settings and technology, also given the
importance that trustworthy AI has in this field for the EU roadmap. The call
suggested that robots or decision support systems that appear ‘aware’ could
play a role in gaining human trust.

Eight projects were selected among the applicants, covering a large portion of
the European research landscape: SYMBIOTIK, ASTOUND, SymAware, CAVAA, EMERGE,
METATOOL, SUST(AI)N, VALAWAI. The aim of this paper is to introduce this
multi-faceted view to awareness research and to provide some preliminary
discussion on how these activities will impact the robotics field in the
future. In the rest of the manuscript, we will review how each project looks
at the challenge. In doing that, we observe that the views on awareness vary
from project to project. At the same time, we show that—although not all of
them are intended to generate direct applications for robotics—they all have
the potential to generate disruptive innovation in this field.

##  2 SYMBIOTIK: Context-aware adaptive visualizations for critical decision
making

Information Visualization (InfoVis) technologies provide a way to expand and
enhance our innate cognitive abilities and enable us to accomplish a range of
tasks, from simple (e.g., company revenue assessment) to intractable (e.g.,
air traffic control). However, depending on the timing and context, we may
have a greater or lesser ability to make rapid and effective decisions. Yet
our ability to do so based on fast-flowing data streams may be decisive. The
project SYMBIOTIK proposes a user-centric approach guided by awareness and
emotion-sensing capabilities [19], inspired by symbiosis principles where
humans and machines cooperate and co-evolve to support decision-making
processes. To fulfill this goal, we must define what “awareness” is and how to
implement it computationally. Yet researchers have shed little light on the
issue of the origins of our subjective experiences, also known as the “hard
problems” [8]. Much progress has been made with what is described as “easy
problems,” which relate to access-to-information issues, like understanding
our ability to categorize environmental stimuli and the deliberate control of
behavior. Solving the hard problems depends on developing a greater
understanding of human working memory. In this regard, according to Humphrey
[15], awareness has a strong social dimension that cannot be ignored. Humans
have existed in social groups for tens of thousands of years and thus had to
predict, understand, and manipulate the behavior of others. Also, according to
Baars [3], awareness is the main component of our cognitive processing system,
responsible for functions such as adaptation and learning, decision-making,
self-monitoring, and self-maintenance. Inspired by this prior body of
research, we propose that awareness is a latent property of a properly
organized system. In addition, we argue that it is necessary to consider
social and collaborative aspects of user-AI interaction with InfoVis systems,
orchestrating highly complex and collaborative ways of working while
augmenting our visual functions to empower our cognitive functions. Taken
together, these innovations will lead to better approaches for shaping the
future of adaptive interfaces [1]. In the context of robotics, these
innovations can potentially change how people interact with machines in a more
natural and user-friendly way. We further need to incorporate basic elements
of social functioning (or even priming) by mean of an altruistic, multi-agent
environment, as well as emotion recognition capabilities (via cross-modal
communication) for establishing the sensing and the affective, which play a
fundamental role in our social interactions within humans as well as within
machines.

##  3 ASTOUND: Improving social competences of virtual agents through
artificial consciousness based on the Attention Schema Theory

The ASTOUND project aims to provide an integrative approach to awareness
engineering to establish artificial consciousness in machines. The overarching
goal is to develop an AI architecture for Artificial Consciousness based on
the Attention Schema Theory (AST), which reconciles some of the current
cognitive neuroscience theories of consciousness, stating that subjective
awareness results from constructing an internal model of the “state of
attention”. Then, implementing the developed architecture into a contextually
aware virtual agent (chatbot) to verify the hypothesis that an artificial
consciousness based on AST will unambiguously improve performance in a task of
natural language understanding. In this path, we are also targeting to define
novel ways to measure the presence and level of consciousness in both humans
and machines.

In the context of ASTOUND, we use awareness and artificial consciousness
mostly as interchangeable terms, both referring to the quality of a
conversational system that allows a coherent and contextualized discussion,
self-regulation, and ensuring both short- and long-term interactions with
human participants. We relate artificial consciousness to the capability of
agents to predict where their own attention and the attention of others is
going to focus next. That capability elicits a representation of the activity
of attending to something, and that representation enables a form (e.g.,
verbal) of reporting about the attending activity itself. Here, the proposed
architecture will feature an appropriate language registry, adaptation to the
interlocutor, and long-term coherence. In addition, it provides larger
capabilities for explainability, personalization, multimodal interaction, and
understanding.

Currently, conversational systems (chatbots) are a trending area of research,
development, and innovation (including robotics as part of the embodiment
integration of such systems), as they allow natural interactions,
understanding, and reasoning capabilities, as well as provide explanations
over actions and environment. The output of the ASTOUND project will increase
such capabilities thanks to our research in multimodal processing (text,
audio, and image), contextualization, accessibility (especially for deaf
people), and automatic evaluation of such systems. We expect a relevant
impact, especially in the robotics domain of Cobots, due to the capability of
models enriched with an Attention Controller based on an Attention Schema to
generate and maintain a model of the attention of others as a proxy to their
goals and intentions. In other terms, collaborative robots will be much more
effective (and safe) when they are provided with the capability to monitor and
correctly predict the objectives of their human partners. In addition, we are
working on ethical aspects to make the chatbot compliant with the EU AI act to
guarantee a high-quality and safe interaction with users while also being
meaningful. Finally, the ASTOUND project targets a high impact as we are
building on top of state-of-the-art models and technologies (e.g.,
Transformer-based pre-trained models) and focused on existing problems such as
contextualization, controllability, evaluation, and explanation.

##  4 SymAware: Symbolic logic framework for situational awareness in mixed
autonomy

A substantial increase in the deployment of both aerial and ground vehicles
with high autonomy is anticipated, necessitating human involvement only in
specific circumstances. These vehicles are expected to operate not only in
controlled environments but also in dynamic outdoor settings for
transportation, logistics, and mobility. In current unpredictable real-world
conditions, human situational awareness (SA), risk perception, coordination,
and decision-making ensure safety and resilience. However, as operations
become more autonomous, relying on the experience of human operators becomes
less feasible. In the existing literature, human-based SA models explain how
these situations may change over time, indicate who may be affected, and
explain how it may be controlled [6, 12]. Nevertheless, existing robotic
agents, including cars, drones, and care assistants, must make decisions based
on information beyond what can currently be incorporated into the human-based
models of SA.

SymAware addresses the fundamental need for a new conceptual framework for SA
in multi-agent systems (MASs) by providing structural and formal modeling of
awareness in its various dimensions, sustaining awareness by learning in
social contexts, quantifying risks based on limited knowledge, and formulating
risk-aware negotiation of task distributions. This framework is aligned with
the internal models and specifications of robot agents, facilitating the safe
concurrent operation of autonomous agents and humans working together.

Figure 1: The architecture of an awareness-enabled MAS proposed in [24] as
part of the project SymAware. The architecture provides insights into the
internal structure of an awareness-enabled agent $i$, which interacts with
another awareness-enabled agent $j$.

To date, we have proposed a novel architecture for MASs, providing a
comprehensive framework that incorporates knowledge, perception,
communication, situational awareness, planning, and control to enable
effective coordination and functioning of the MAS [24]. The first version of
the proposed architecture, shown in Fig. 1, consists of the knowledge
component, estimation and perception components, communication modules,
situational awareness block, planning and control block, and the physical
system. The SA block comprises four elements: (i) state, indicating the
current system state combined with environmental information like object
presence or obstacles; (ii) intent, deducing other artificial or human agents’
intentions by communication or by observing and interpreting their actions to
understand their goals; (iii) uncertainty, managing inherent uncertainty and
incomplete information in the environment and perception module; and (iv)
risk, evaluating potential risks by detecting obstacles, anomalies, or
environmental changes to identify threats.

The implementation and validation of the SymAware approach in awareness
engineering will involve two specific scenarios: (i) modeling, simulating, and
assessing risks in traffic management for unmanned aircraft systems used in
drone operations within urban environments, accounting for potential
disturbances and hazards and (ii) simulating and analyzing higher levels of
autonomous vehicles, ensuring both functional safety in the face of failures
and the Safety of the Intended Functionality.

##  5 CAVAA: Counterfactual assessment and valuation for awareness
architecture

The CAVAA project is built on the hypothesis that awareness serves survival in
a world governed by hidden states, to deal with the “invisible”, from
unexplored environments, projections of the future, to social interaction that
depends on the internal states of other agents, conventions, and norms. CAVAA
proposes that awareness is the acuity of experience of the conscious scene,
reflecting a constructed virtual world, a hybrid of perceptual evidence,
memory states, and inferred “unobservables” extended in space and time.
Consciousness, in turn, is seen as the coherent experience that results from
the large-scale integration of parallel subconscious processing of perception,
affect, memory, cognition and action along the neuroaxis supported by a
dedicated transient memory system that supports unification, virtualisation,
norm extraction and valuation. The CAVAA project builds on the Distributed
Adaptive Control theory of mind and brain and will realise this theory of
awareness instantiated as an integrated computational architecture and its
components to explain awareness in biological systems and engineer it in
technological ones. It will realise the underlying computational components of
perception, memory, virtualisation, simulation, and integration, embody the
architecture in robots and artificial agents, and validate it across a range
of use cases involving the interaction between multiple humans and artificial
agents, using accepted measures and behavioural correlates of awareness. Use
cases will address robot foraging, social robotics, computer game benchmarks
and human-generated decision trees in a health coach. These benchmarks will
focus on resolving trade-offs, e.g. between search efficiency and robustness,
and assess the acceptance of human users of aware technology. CAVAA’s
awareness engineering is accompanied by an ethics framework towards human
users and aware artefacts in the broader spectrum of trustworthy AI,
considering shared ontologies, intention complementarity, behavioural
matching, empathy, relevance of outcomes, reciprocity, counterfactuals and
projections towards new future scenarios, and to predict the impact of
choices. CAVAA will deliver a better user experience because of its
explainability, adaptability, and legibility. CAVAA’s integrated framework
redefines how we look at the relationship between humans, other species and
smart technologies because it makes the invisible visible.

##  6 EMERGE: Emergent awareness from minimal collectives

Figure 2: The project EMERGE deals with natural and artificial collaborative
awareness, emerging from the interaction of bodies and brains of multiple
heterogeneous agents. The figure showcases this idea in the context of swarms
of soft robots, which are one of the targeted application domains for the
activities of this project.

The EMERGE project [4] aims to develop a novel concept of collaborative
awareness for collectives of minimal artificial agents. In particular, the
exploration delves into how collectives of simple agents can develop a
representation of their mutual existence, environment, and cooperative
behavior toward realizing shared tasks and goals. EMERGE is built on a
scenario of artificial beings with no shared language and constrained
individual capabilities, which nevertheless leads to high-complexity behaviors
at the collective level. EMERGE delivers a philosophical, mathematical, and
technological framework that enables us to know how and where to allocate
awareness to achieve a goal through the collective optimally. To this end,
EMERGE uses nonlinear dynamical system theory to establish, analyze,
implement, and test a new AI framework (the Archetype Computing System) that
will make collaborative awareness emerge from the interplay of atomic units of
local awareness (see Fig. 2).

In this project, we identify and characterize a spectrum of global awareness
that we call collaborative awareness, which is capable of harmonizing multiple
local states belonging to the single agents of the collective in a single
coherent global awareness for the ensemble. We intend collaborative awareness
as an emergent process supporting complex, distributed, and loosely coupled
systems capable of high degrees of collaboration, self-regulation, and
interoperability without pre-defined protocols. EMERGE definition of
collaborative awareness follows a dimensional model, which allows for the
coexistence of multiple awareness aspects (temporal, spatial, bodily,
metacognitive, …) existing at different degrees in either fully dependent or
more dependent manners.

EMERGE targets the realization of the next generation of self-regulating
cyber-physical systems, leveraging collaborative awareness to deliver services
over loosely coupled collectives of virtual/physical entities. As such, the
project is expected to directly impact IoT applications, smart cities and
transportation, microservice-based ICT systems, biomedical nanodevices, and
robotics. The latter, in particular, is the primary target application for
EMERGE, which explicitly aims to develop three robotic use cases: (i) modular
soft robots, as an example of a physically distributed collective where the
body needs to self-organize to account for the dynamic addition of components;
(ii) robotic swarms, as an example of large scale minimal collective where
agents need coordination to achieve a collaborative goal; and (iii) cobots, as
a closer-to-market use case where interoperability is currently a significant
barrier. We also expect to generate a second stream of impacts by building on
the Archetype Computing System to develop neuromorphic and morphological
computing systems for green AI and edge AI applications.

##  7 METATOOL: Robots inventing tools

The METATOOL project investigates how robots can develop cognitive abilities
to invent tools as ancient humans did around three million years ago.
Archaeology, neuroscience, and robotics join forces to shed light on the
technological leaps that our ancestors achieved and to develop novel
technology inspired by human metacognition and awareness. Hence, in the same
way, tool invention was an outstanding technological milestone in human
history, robots inventing tools will be a major milestone in science and
technology.

Figure 3: The artificial awareness generated by METATOOL will make robots able
to use and invent tools, have self-awareness, and attain operationalizable
abstraction.

Research in METATOOL aims to endow current artificial intelligence systems and
robots with a deeper understanding of tools and the capability to monitor and
control their actions when interacting with the environment by means of tool-
extended bodies (see Fig. 3). This implies a mission-oriented understanding of
sensory flows from outside and from inside the robot in relation to the
performance of a physical task. We identify awareness as the capacity of
_real-time understanding of the sensory flows of the agent_ [22], and,
inspired by metacognitive theories of cognition, we in particular, focus on
the monitoring of the _uncertainty_ associated with the cognitive process
driving the performance of a task. Thus, analogous to humans, robots would be
able to make informed decisions/actions to select, discover, and innovate
tools, taking into account both $i)$ the task performance and $ii)$ its self-
confidence. We explore the use of a Bayesian computational model originated in
neuroscience, _i.e._ , active inference, to deploy this uncertainty monitoring
capacity into robots [18].

By understanding the processes of tool use and invention, we can create robots
that are more effective, resilient, and adaptive in open, changing
environments. Robots having an intrinsic capability of using objects around
them or creating new objects to extend the robot’s capability will also
relieve engineers from the complex task of modeling all environmental
potential affordances. Furthermore, this deeper understanding of the relation
between the physical realization of the robot and the task that the robot
shall perform will also enable robots to better cooperate with other robots or
humans.

##  8 SUST(AI)N: Smart building sensitive to daily sentiment

The SUST(AI)N project aims to provide awareness in smart buildings by means of
precision-sensing AI processing shared across devices (i.e., distributed
intelligence) and leveraging the flexibility and implementation efficiency of
reconfigurable hardware. The integration of these interconnected, self-
monitoring systems will enable probabilistic reasoning and, ultimately, the
achievement of awareness. Furthermore, zero-energy buildings are sought
through energy autonomy in the devices and nodes using energy-efficient
sensing, processing, and communication, as well as energy harvesting circuits.
In this project, awareness is defined as the capability of the distributed
system, which can be seen as a form of connected superorganism, to reach C2
consciousness through distributed, probabilistic reasoning over possible
choices of the learning model, data sources, and node configuration. In
particular, we will follow a hierarchical learning model in which the sensor
sources, context classes, node configurations, and individual splits in the
distributed learning are optimized. Via this hierarchical meta-learning, the
system flexibly adapts to the demands and constraints of a building context,
achieving self-monitoring and self-healing capabilities and accurate sensing
of the group sentiment.

This project will have a direct impact on tertiary buildings, aiming to make
them compliant with the regulatory decree 2010/31/EU (European Parliament),
which stipulates the necessity for automation control systems by 2025. The
outputs of SUST(AI)N may also have an impact in the field of
collaborative/swarm robotics and other instances of multi-agent systems where
a distributed form of self-aware intelligence may enhance performance,
reliability, and adaptation of the whole system. Particularly, the multi-
layered node intelligence, including a node-level intelligence to gather and
process local information and a meta-level intelligence layer to decide which
information to consider and which machine learning model structure to utilize,
could be applied to robots to perform collaborative tasks in an optimal way.
Another possible application would be the distributed control of multi-body
robots, where each node can be associated with a body part, and the meta-level
intelligence can optimally decide how to aggregate the information coming from
the various body parts. Energy harvesting can add some degree of energy
autonomy to the robots.

To date, the project has already developed a RISC-V processor (see Fig. 4),
exploiting probabilistic circuits that will be utilized for the control of the
node, an accelerator, and federated learning schemes leveraging various forms
of social learning, including both opaque (i.e., neural networks) and
transparent models (i.e., based on decision trees) that can also be executed
on devices with severely limited computational resources. In addition, indoor
light energy harvesting has been explored to power the nodes using several
types of solar cells.

Figure 4: The adaptive awareness that will be developed in SUST(AI)N combines
local information across distributed nodes in an aware building scenario.

##  9 VALAWAI: Value-aware AI

The VALAWAI project is developing tools and methodologies to make AI
applications more ethically guided and, hence, trustworthy. More specifically,
it aims to infuse AI applications with mechanisms that make them able to abide
by human values and be value-aware so that they can recommend or explain
behavior in terms of its underlying values. Values are principles of behavior,
such as not harming others, speaking the truth, or respecting other opinions.
They are reflected in norms that are behavioral patterns designed to satisfy
the group’s values. Examples of norms are the community rules of social media,
the medical protocols adopted by physicians in a hospital for making end-of-
life decisions, or the constraints built into a social robot to avoid
unacceptable behavior by the robot or its user. General norms for the use of
AI are now being translated into legal frameworks, such as the AI Act,
although shaping and applying these frameworks to concrete cases is still a
work in progress. Obviously, the behavior of robots, particularly social
robots, is expected to abide by the values and norms of our society, and so
the issue of how to deal with values is critical for the acceptable deployment
of robots, particularly in a home context.

Value-awareness goes beyond value-alignment. Generally speaking, awareness
pertains to systems that construct representations of the relevant aspects of
a situation and can access these representations in order to deliberatively
reason about a course of action and explain why a particular behavior or
decision was chosen or why not. For example, situation awareness in the
context of firefighting means that firefighters develop a clear understanding
of the layout of the building, the inhabitants, the materials on fire, ways to
extinguish the fire, etc. Based on these representations, they decide on the
best course of action and are able to explain why they followed that course.
By analogy, value-aware AI means that the AI system has a representation of
the values and norms relevant to a particular situation and uses that to make
or recommend decisions and justify them.

To achieve value-aware AI, the VALAWAI project develops a toolbox with
components for representing values and norms and for linking them to concrete
situations and decision-making processes. The project valides this toolbox in
three use cases: medical protocols, social media, and social robots. The
toolbox is generic and can support a variety of functions related to values
and norms: alert users when they transgress norms, help a group to communicate
and coordinate norms, determine whether certain norms help to achieve certain
values and explain behavior in terms of norms. The VALAWAI toolbox,
particularly the cognitive architecture contained in it, can also be used for
achieving awareness of aspects other than values and norms and has, therefore,
a broad utility for robotics in general.

##  10 Discussion and Conclusions

This paper summarized ongoing and diverse activities around the concept of
synthetic awareness that have been driven by the EIC Pathfinder Challenge
“Awareness Inside” call funded by the European Union through the Horizon
Europe program. This diversity not only enriches our understanding but also
poses significant challenges in defining and achieving a unifying concept of
awareness that is applicable across different robotic applications. For
example, the very definition of awareness varies strongly from project to
project, with its position towards consciousness ranging from direct
opposition (EMERGE) to almost a synonym of hard consciousness (ASTOUND). While
in the context of SymAware, awareness is emphasized as the ability of agents
in a MAS to recognize and comprehend external stimuli, especially in scenarios
involving the presence of human beings, METATOOL awareness focuses on the
neuroscience-derived idea of uncertainty monitoring. In both projects,
awareness involves perceiving, adapting, coordinating, communicating, and
making informed decisions while ensuring safety and resilience in a dynamic
environment.

With this paper, we represented this blooming field of research with the goal
of, on the one hand, making a step towards converging on a common framework or
language and, on the other involving the whole robotics and AI community in
this discussion.

Of the eight projects, five are specifically focused on robotics-related
experiments. Namely, METATOOL investigates robots capable of self-evaluation
and tool invention; SymAware holds relevance for various applications,
encompassing both industrial and domestic robotic systems, with a specific
emphasis on autonomous vehicles and air-traffic control systems; CAVAA
utilizes social robots like MiRo-e for its tests; EMERGE explores various
robotic applications; and VALAWAI showcases its ideas through domestic social
robots. Interestingly, the authors of this paper who are involved in the
remaining three projects also believe in applying their innovation to
robotics. More specifically, the SYMBIOTIK and ASTOUND innovation could
transform social robotics. The SYMBIOTIK project could enhance human-robot
interactions by integrating awareness and emotion-sensing capabilities into
robotics, making these interactions more natural and user-friendly. Developing
artificial consciousness based on the Attention Schema Theory, the ASTOUND
project will enhance the performance of natural language understanding and
provide collaborative robots with the capability to effectively and safely
interact with human partners by predicting their intentions and objectives.
Finally, in line with the goals of EMERGE, the SUST(AI)N project could advance
collaborative and swarm robotics, as well as multi-agent systems, by
developing a distributed form of self-aware intelligence, enhancing system
performance, reliability, and adaptation through a hierarchical learning model
and energy-efficient, self-sufficient technologies.

Ultimately, all the authors of this paper believe that incorporating awareness
components in technology could make systems more resilient, adaptable, and
human-centric. We believe that understanding awareness will allow AI systems
to grasp better and respond to various situations. Synthetic awareness will
advance AI towards more coherent, adaptive, and self-evolving behavior.

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# Geometric local systems on the projective line minus four points

Yeuk Hay Joshua Lam and Daniel Litt

###### Abstract.

Let $J(m)$ be an $m\times m$ Jordan block with eigenvalue $1$. For
$\lambda\in\mathbb{C}\setminus\\{0,1\\}$, we explicitly construct all rank $2$
local systems of geometric origin on
$\mathbb{P}^{1}\setminus\\{0,1,\lambda,\infty\\}$, with local monodromy
conjugate to $J(2)$ at $0,1,\lambda$ and conjugate to $-J(2)$ at $\infty$. The
construction relies on Katz’s middle convolution operation. We use our
construction to prove two conjectures of Sun-Yang-Zuo (one of which was proven
earlier by Lin-Sheng-Wang; the other was proven independently from us by Yang-
Zuo).

##  1\. Introduction

It was known (in some sense) to Riemann [Kat96, Introduction], in his work on
hypergeometric functions, that all rank two local systems on
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$ with trivial determinant and quasi-
unipotent local monodromy are _of geometric origin_ — that is, they arise in
the cohomology of a family of varieties over
$\mathbb{P}^{1}\setminus\\{0,1,\infty\\}$. The case of $\mathbb{P}^{1}$ minus
$4$ points is more complicated; Beauville [Bea82] famously classified the
$D\subset\mathbb{P}^{1},|D|=4$ such that $\mathbb{P}^{1}\setminus D$ carries a
family of elliptic curves with stable reduction along $D$, or equivalently,
carries a rank $2$ $\mathbb{Z}$-local system of geometric origin with
unipotent local monodromy. In general, rank $2$ motivic local systems on
$\mathbb{P}^{1}$ minus four points are poorly understood (and the situation
when one increases the rank or the number of points deleted is completely
mysterious).

The goal of this paper is to explicitly write down all local systems of
geometric origin on $\mathbb{P}^{1}\setminus\\{0,1,\infty,\lambda\\}$, with
local monodromies conjugate to

$\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$

at $0,1,\lambda$ and with local monodromy conjugate to

$\begin{pmatrix}-1&1\\\ 0&-1\end{pmatrix}$

at $\infty$. In so doing we give a short proof of two conjectures of Sun-Yang-
Zuo [SYZ21, Conjectures 4.8 and 4.10] on motivic local systems on
$X=\mathbb{P}^{1}\setminus\\{0,1,\infty,\lambda\\}$. [SYZ21, Conjecture 4.8]
was earlier proven in [LSW22] by different methods.

######  Remark 1.0.1.

While this paper was in preparation Yang and Zuo [YZ23] independently claimed
a fascinating proof of [SYZ21, Conjecture 4.10], via completely different
techniques from ours111Our work began in December of 2022, whereas the work of
loc.cit. had been ongoing for several years, as we were informed by the
authors of [YZ23]. Their 128-page proof relies on $p$-adic Hodge theory and
the Langlands correspondence; by contrast our proof is only a few pages and we
_explicitly_ write down the relevant local systems and the families in whose
cohomology they appear.

###  1.1. Main results

Let $X$ be a smooth complex curve, and in the case when $X$ is not proper, we
pick a smooth compactification $\overline{X}$ of $X$. Recall that a
$\mathbb{C}$-local system $\mathbb{V}$ on $X$ is said to be of geometric
origin if there exists a dense open $U\subset X$, and a family of smooth
proper algebraic varieties $\pi:\mathcal{Y}\to U$ such that $\mathbb{V}$
appears as a subquotient of $R^{i}\pi_{*}\mathbb{C}$ for some $i\geq 0$. (The
appearance of $U$ in the definition is a slightly technical point which the
reader may ignore at first pass.)

Motivated by the previous work of Faltings [Fal83], Deligne showed that, for a
fixed $X$, there are only finitely many $\mathbb{Q}$-local systems of fixed
rank $n$ which are of geometric origin. One of our main motivations was
whether a strengthened form of Deligne’s theorem [Del87] could hold:

######  Question 1.1.1.

Can there be infinitely many $\mathbb{C}$-local systems of rank $n$ on $X$
which are of geometric origin, and moreover whose local monodromies at the
boundary $\overline{X}-X$ are fixed?

######  Remark 1.1.2.

The condition on local monodromies is certainly necessary for the above
question to be interesting: indeed, the hypergeometric local systems of rank
two on $X=\mathbb{P}^{1}-\\{0,1,\infty\\}$ are all of geometric origin.
However, the local monodromies at $0,1,\infty$ have eigenvalues being $m$-th
roots of unity, and upon bounding $m$ there are only finitely many such
hypergeometric local systems. We find the monodromy condition natural since it
implies that we are considering points of the _relative character varieties_ ,
which are the natural replacement of character varieties in the case of non-
proper $X$.

Note that [Lit21] shows that there are only finitely many $K$-local systems of
geometric on $X$ if $K$ embeds in any finite extension of $\mathbb{Q}_{p}$,
for any prime $p$.

We now specialize to our case of interest. For any $\lambda\neq 0,1$, let $X$
be the curve $\mathbb{P}^{1}-\\{0,1,\lambda,\infty\\}$. We consider rank two
local systems on $X$ satisfying the following condition $(\star)$: namely with
local monodromy conjugate to

$\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$

at $0,1,\lambda$ and local monodromy conjugate to

$\begin{pmatrix}-1&1\\\ 0&-1\end{pmatrix}$

at $\infty$. We recall the following:

######  Definition 1.1.3.

A local system $\mathbb{V}$ on a manifold $Y$ is said to be MCG-finite if the
set of isomorphism classes of local systems $\\{f^{*}\mathbb{V}\\}$ obtained
by acting on $\mathbb{V}$ by elements $f$ of the mapping class group of $Y$ is
finite.

We may now state our first result.

######  Theorem 1.1.4.

For any $\lambda\neq 0,1$, there exists infinitely many rank two local systems
of geometric origin on $X$ satisfying condition $(\star)$. Moreover, any rank
two local system of geometric origin on $X$ satisfying $(\star)$ is MCG-
finite.

######  Remark 1.1.5.

Theorem 1.1.4 answers 1.1.1 in the positive. It would be very interesting to
further investigate 1.1.1 in the case of $X$ proper and to understand whether
there is a qualitative difference between the two cases.

This result was claimed independently, though inexplicitly, by Yang-Zuo
[YZ23]; they rephrase MCG-finiteness in terms of algebraic solutions to the
Painlevé VI equation. In that language, these solutions were, we believe,
discovered originally by Hitchin [Hit95]; Lysovyy-Tykhyy [LT14], in their
classification of algebraic solutions to Painlevé VI, refer to them as the
“Cayley solutions”. The reason for this name is that, under condition
$(\star)$, the equation of the relative character variety is the _Cayley
cubic_ : see [LT14, Equation (61)].

In fact, we classify all rank two local systems of geometric origin and
satisfying $(\star)$. The following is the explicit form of our
classification:

######  Theorem 1.1.6.

Let $f:E\to\mathbb{P}^{1}$ be the $2:1$ cover branched over
$\\{0,1,\infty,\lambda\\}$. Consider the fiber square

$\textstyle{Z^{\prime}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E\times\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[m]\times\operatorname{id}}$$\textstyle{E\times\mathbb{P}^{1}}$

where $W^{\prime}$ is the double cover of $E\times\mathbb{P}^{1}$ branched
along the graph of $f$ and $E\times\\{\infty\\}$. For any rank two local
system $\mathbb{V}$ of geometric origin on $X$ satisfying $(\star)$,
$\mathbb{V}$ appears as a subquotient of the cohomology of the family

$Z^{\prime}_{m}\to E\times\mathbb{P}^{1}\to\mathbb{P}^{1}$

for some $m$.

######  Example 1.1.7.

We write down explicitly all representations corresponding to rank two motivic
local systems with the above specified local monodromies. Our formulas are
obtained by plugging the values of the Cayley solutions [LT14, p.145] into
[Boa05, Equation (24)]222in the following, $\alpha,\beta$ correspond
$r_{Y},r_{Z}$, respectively, of [LT14]. Since the fundamental group
$\pi_{1}(X)$ is the free group generated by loops around $0,1,\lambda$, it
suffices to write down three matrices $M_{0},M_{1},M_{\lambda}$, corresponding
to the images of the three generators:

$M_{0}=\begin{pmatrix}1+x_{2}x_{3}/x_{1}&-x_{2}^{2}/x_{1}\\\
x_{3}^{2}/x_{1}&1-x_{2}x_{3}/x_{1}\end{pmatrix},M_{1}=\begin{pmatrix}1&-x_{1}\\\
0&1\end{pmatrix},\ M_{\lambda}=\begin{pmatrix}1&0\\\ x_{1}&1\end{pmatrix},\ $

where

$x_{1}=2\cos\bigg{(}\frac{\pi(\alpha+\beta)}{2}\bigg{)},\
x_{2}=2\sin\bigg{(}\frac{\pi\alpha}{2}\bigg{)},\
x_{3}=2\sin\bigg{(}\frac{\pi\beta}{2}\bigg{)}$

for $\alpha,\beta\in\mathbb{Q}$.

The idea of the proofs of Theorem 1.1.4 and Theorem 1.1.6 is to analyze local
systems satisfying $(\star)$ via Katz’s _middle convolution_ operation
[Kat96]. The key is to observe that local systems of geometric origin
satisfying $(\star)$ may be obtained via middle convolution from certain
finite monodromy local systems on $X$. Namely, if $f:E\to\mathbb{P}^{1}$ is
the double cover of $\mathbb{P}^{1}$ branched at $\\{0,1,\infty,\lambda\\}$,
then any motivic local system satisfying $(\star)$ arises via middle
convolution from $f_{*}\mathbb{L}|_{X}$, where $\mathbb{L}$ is a rank one
local system on $E$ with finite monodromy; moreover this construction produces
all local systems of geometric origin satisfying $(\star)$. Moreover, since
$f_{*}\mathbb{L}|_{X}$ itself is MCG-finite, the same is true of its middle
convolution.

######  Remark 1.1.8 (Comparison with the work of Yang-Zuo).

  1. (1)

As mentioned above, our main tool is Katz’s middle convolution functor. Our
approach gives new proofs of all the main results of [YZ23]. Indeed, [YZ23,
Theorem 1.7] is a corollary of Theorem 1.1.6, and [YZ23, Theorem 1.5] follows
immediately from 3.2.2.

  2. (2)

An important invariant of a local system of geometric origin is the _trace
field_ , i.e. the smallest field generated by the traces of elements of
$\pi_{1}(X)$. From the description as middle convolution of rank one torsion
local systems on $E$, it is straightforward to see that the trace fields of
our local systems are $\mathbb{Q}(\zeta_{m}+\zeta_{m}^{-1})$ for $\zeta_{m}$ a
primitive $m$-th root of unity, with $m$ being the order of the rank one local
system. It does not seem possible to derive this information with the
techniques of [YZ23]; Zuo has informed us that he conjectured these trace
fields in 2018; see also [YZ23, Appendix A], where this conjecture is alluded
to.

This also explains a technical point of loc.cit.: indeed, there, the authors
assume that $E$ has supersingular reduction at $p$, and use crucially the fact
that the trace fields of all irreducible $\operatorname{SL}_{2}$-local systems
on $X$ (satisfying $(\star)$) are unramified at $p$. The latter is certainly
not true if $E$ has ordinary reduction, in which case there will be rank two
local systems with trace field $\mathbb{Q}(\zeta_{p^{k}}+\zeta_{p^{k}}^{-1})$
for any $k\geq 1$, which ramifies at $p$; correspondingly, there are pre-
periodic Higgs bundles which are not periodic, in contrast with the case when
$E$ has supersingular reduction (see [YZ23, Corollary 3.4.1]).

###  1.2. Higgs bundles and Higgs-de Rham flow

As before, let $D=\\{0,1,\infty,\lambda\\}\subset\mathbb{P}^{1}$,
$X=\mathbb{P}^{1}\setminus D$, and let $\mathbb{V}$ be a local system of
geometric origin on $X$ of rank $2$, satisfying $(\star)$. By e.g. [LL22b,
Proposition 4.1.4],
$(\mathbb{V}\otimes\mathscr{O}_{X},\operatorname{id}\otimes d)$ canonically
extends to a filtered flat vector bundle on $\mathbb{P}^{1}$ with logarithmic
flat connection,

$(F^{1},\mathscr{E},\nabla:\mathscr{E}\to\mathscr{E}\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log
D)),$

where $F$ restricts to the Hodge filtration on $X$, and moreover where the
residues of $\nabla$ have real parts lying in $[0,1)$.

One can show (see 4.1.1) that if $\mathbb{V}$ satisfies $(\star)$, then
$F^{1}\simeq\mathscr{O}_{\mathbb{P}^{1}}$ and
$\mathscr{E}/F^{1}\simeq\mathscr{O}_{\mathbb{P}^{1}}(-1)$. The connection
$\nabla$ yields a natural map

$\theta:F^{1}\to\mathscr{E}/F^{1}\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log D),$

which, being a non-zero map
$\mathscr{O}_{\mathbb{P}^{1}}\to\mathscr{O}_{\mathbb{P}^{1}}(1)$, vanishes at
a unique point $w(\mathbb{V})\in\mathbb{P}^{1}$. Moreover $w(\mathbb{V})$
determines $\mathbb{V}$ up to isomorphism (4.1.2) .

The following is a slight strengthening of a conjecture of Sun-Yang-Zuo:

######  Conjecture 1.2.1 (Slight strengthening of [SYZ21, Conjecture 4.10]).

Let $f:E\to\mathbb{P}^{1}$ be the double cover branched over $D$, viewed as an
elliptic curve with the identity the point over $\infty$. As
$\\{\mathbb{V}\\}$ ranges over all local systems of geometric origin
satisfying $(\star)$, the $w(\mathbb{V})$ are precisely the image under $f$ of
the torsion points of $E$.

We give a short proof of this conjecture in 4.2.2, and explain how to deduce
[SYZ21, Conjecture 4.10] in § 7.2.

The proof has two parts. The first is to show that those $\mathbb{V}$ with
$w(\mathbb{V})=f(x)$, for $x\in E$ torsion, are of geometric origin; this will
follow from a more precise form of Theorem 1.1.6.

The second is to show that these are are the only points of geometric origin.
We give two approaches. The first is to again use the middle convolution,
which preserves the property of geometric origin and sends $\mathbb{V}$ with
$w(\mathbb{V})$ not the image of a torsion point to a local system manifestly
not of geometric origin.

The second approach relies on Sun-Yang-Zuo’s theory of the Higgs-de Rham flow,
as we now explain; as part of it we give a simple proof of [SYZ21, Conjecture
4.8]. One may view $F^{1}\oplus\mathscr{E}/F^{1}$, with the Higgs field

$\begin{pmatrix}0&\theta\\\ 0&0\end{pmatrix}$

as a graded nilpotent Higgs bundle. Sun-Yang-Zuo, using Ogus-Vologodsky’s non-
abelian Hodge theory in positive characteristic [OV07], produce an
endomorphism of the moduli of nilpotent parabolic Higgs bundles $\psi_{p}$
which they call the Higgs-de Rham flow. They show that for Higgs bundles
associated to motivic local systems, almost all of their reductions mod $p$
are $f$-_periodic_ under the Higgs-de Rham flow for some fixed $f$ independent
of $p$; the converse is expected to hold (see e.g. [KS20] or [LSW22,
Conjecture 1.7]). The moduli of stable nilpotent graded parabolic Higgs
bundles on $(\mathbb{P}^{1},D)$ whose underlying graded vector bundle is
$\mathscr{O}_{\mathbb{P}^{1}}\oplus\mathscr{O}_{\mathbb{P}^{1}}(-1)$ is
isomorphic to $\mathbb{P}^{1}$ via the map sending a Higgs bundle to the
vanishing locus of its Higgs field.

Sun-Yang-Zuo conjecture [SYZ21, Conjecture 4.8] and Lin-Sheng-Wang prove
[LSW22]:

######  Theorem 1.2.2 ([SYZ21, Conjecture 4.8], [LSW22, Theorem 1.6]).

Viewing $\mathbb{P}^{1}$ as the moduli of graded nilpotent Higgs bundles as
above, the map $\psi_{p}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ fits in the
following commutative diagram:

$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[p]}$$\scriptstyle{f}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathbb{P}^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{p}}$$\textstyle{\mathbb{P}^{1}}$

We give a more formal statement in § 5.

This immediately implies that any point whose reduction is $f$-periodic under
almost all $\psi_{p}$ is torsion. We will conclude the paper by giving a
simple new proof of Theorem 1.2.2 and a proof of 1.2.1, also via the middle
convolution. Briefly, we relate the moduli space of Higgs bundles in question
(which is isomorphic to $\mathbb{P}^{1}$) to the moduli space of rank one
Higgs bundles with zero Higgs field on $E$, and observe that the middle
convolution commutes with the Higgs-de Rham flow. But the Higgs-de Rham flow
for Higgs bundles of rank one with zero Higgs field on $E$ is simply $[p]$.

###  1.3. Acknowledgments

We are grateful for useful conversations with Bruno Klingler, Raju
Krishnamoorthy, Mao Sheng, and Kang Zuo. Lam was supported by a Dirichlet
Postdoctoral Fellowship at Humboldt University. Litt was supported by the
NSERC Discovery Grant, “Anabelian methods in arithmetic and algebraic
geometry.”

##  2\. Preliminaries on convolution in the Betti and $\ell$-adic settings

Let $k$ be a field and let $D\subset\mathbb{P}^{1}_{k}$ be a reduced effective
divisor containing $\infty$. Let $X=\mathbb{P}^{1}\setminus D$. There are
natural projection maps

$\pi_{i}:X\times X\setminus\Delta\to X,$

for $i=1,2$ as well as a map

$m:X\times X\setminus\Delta\to\mathbb{G}_{m}$

given by

$m:(x,y)\mapsto x-y.$

Let

$j:X\times X\setminus\Delta\hookrightarrow\mathbb{P}^{1}_{k}\times X$

be the evident inclusion.

######  Definition 2.0.1.

Let $\chi$ be a rank one local system on $\mathbb{G}_{m}$, and let
$\mathbb{V}$ be a local system on $X$. The _middle convolution_
$MC_{\chi}(\mathbb{V})$ is defined to be

$MC_{\chi}(\mathbb{V})=R^{1}(\pi_{2})_{*}j_{*}(\pi_{1}^{*}\mathbb{V}\otimes
m^{*}\chi).$

######  Remark 2.0.2.

In the $\ell$-adic or Betti setting, the middle convolution has a nice
geometric interpretation. If $\mathbb{V}$ is _pure of weight zero_ and $\chi$
has finite order, then $MC_{\chi}(\mathbb{V})$ is precisely the weight one
part of $R^{1}(\pi_{2})_{*}(\pi_{1}^{*}\mathbb{V}\otimes m^{*}\chi).$

We will also make use of the following variant of middle convolution. Let $Y$
be a smooth proper curve, and let $f:Y\to\mathbb{P}^{1}$ be a finite,
generically étale map, with branch divisor $D$. Let $I=f^{-1}(\infty)$. Let
$\Gamma$ be the graph of $f$. Then again there are natural projections

$\pi_{1}:Y\times X\setminus(\Gamma\cup I\times X)\to Y,\pi_{2}:Y\times
X\setminus(\Gamma\cup I\times X)\to X$

and a natural map

$m:Y\times X\setminus(\Gamma\cup I\times X)\to\mathbb{G}_{m}$ $(y,x)\mapsto
f(y)-x.$

Let

$i:Y\times X\setminus(\Gamma\cup I\times X)\hookrightarrow Y\times X$

be the evident inclusion.

######  Definition 2.0.3.

Let $\chi$ be a rank one local system on $\mathbb{G}_{m}$, and let
$\mathbb{V}$ be a local system on $Y$. The _middle convolution_
$MC_{\chi}(f,\mathbb{V})$ is defined to be

$MC_{\chi}(f,\mathbb{V})=R^{1}(\pi_{2})_{*}i_{*}(\pi_{1}^{*}\mathbb{V}\otimes
m^{*}\chi).$

It follows from e.g. proper base change that:

######  Proposition 2.0.4.

There is a natural isomorphism $MC_{\chi}(f,\mathbb{V})\simeq
MC_{\chi}(f_{*}\mathbb{V})$.

The definitions are above are convenient for computing in the Betti and
$\ell$-adic settings; in the positive characteristic de Rham and Higgs setting
we will use slightly different formalisms.

##  3\. Computing the convolution

Suppose $D$ has even degree with $\deg(D)\geq 4$ (with the case $\deg(D)=2$
being uninteresting) and let $f:Y\to\mathbb{P}^{1}$ be the double cover
branched over $D$. Let $\mathbb{L}$ be a rank one local system on $Y$. Suppose
$k$ has characteristic different from $2$, and let $\chi$ be the unique non-
trivial rank one local system on $\mathbb{G}_{m}$ with $\chi^{2}=\text{triv}.$
We write $MC_{-1}$ instead of $MC_{\chi}$ to emphasize this particular choice
of $\chi$.

###  3.1. Betti and $\ell$-adic computations

######  Proposition 3.1.1.

The rank of $MC_{-1}(f,\mathbb{L})$ is $\deg(D)-2$. For $\mathbb{L}$ non-
trivial, the local monodromies of $MC_{-1}(f,\mathbb{L})$ are given by the
following. Let $J(\alpha,\ell)$ denote the Jordan block given by a matrix of
size $\ell$ and generalized eigenvalues $\alpha$.

  * •

At a point $P\neq\infty$, there is a Jordan block $J(1,2)$, and all other
Jordan blocks are $J(1,1)$.

  * •

At $P=\infty$, there is a Jordan block $J(-1,2)$, and all other Jordan blocks
are $J(-1,1)$.

###### Proof.

Given $x\in X$, the fiber $U_{x}$ of $\pi_{2}:Y\times X\setminus(\Gamma\cup
I\times X)\to X$ over $x$ is isomorphic to $Y\setminus
f^{-1}(\\{x\cup\infty\\})$, which has Euler characteristic $1-\deg(D)$. The
restriction $\pi_{1}^{*}\mathbb{L}\otimes m^{*}\chi|_{U_{x}}$ has non-trivial
monodromy at the two points of $f^{-1}(x)$ and trivial monodromy at
$f^{-1}(\infty)$. Hence the Grothendieck-Ogg-Shafarevich formula yields

$\chi(j_{*}(\pi_{1}^{*}\mathbb{L}\otimes m^{*}\chi|_{U_{x}}))=2-\deg(D).$

As $\pi_{1}^{*}\mathbb{L}\otimes m^{*}\chi|_{U_{x}}$ is a non-trivial rank one
local system, $H^{0}(\pi_{1}^{*}\mathbb{L}\otimes
m^{*}\chi|_{U_{x}})=H^{0}(\pi_{1}^{*}\mathbb{L}\otimes m^{*}\chi|_{U_{x}})=0$.
Hence $MC_{-1}(f,\mathbb{L})$ has rank equal to $\deg(D)-2$.

For the computation of local monodromies, we refer the reader to [DR07, Lemma
5.1]. Indeed, $MC_{-1}(f,\mathbb{L})$ is equivalent to the standard middle
convoluiton applied to (the restriction to $X$ of) $f_{*}\mathbb{L}$, and the
latter satisfies the conditions of [DR07, Lemma 5.1] by our assumptions that
$\mathbb{L}$ is non-trivial. ∎

The upshot of this proposition is that if $D=(0)+(1)+(\lambda)+(\infty)$, so
$Y=E$ is an elliptic curve, and $\mathbb{L}$ is a rank one local system on
$E$, then $MC_{-1}(f,\mathbb{L})$ has rank $2$ and satisfies $(\star)$. If
$\mathbb{L}$ is unitary (hence underlies a $\mathbb{C}$-VHS), then
$MC_{-1}(f,\mathbb{L})$ underlies a $\mathbb{C}$-VHS as well.

###  3.2. de Rham computations

In the special case when $D=(0)+(1)+(\lambda)+(\infty)$, $Y$ is an elliptic
curve with origin $e=f^{-1}(\infty)$, and we denote it by $E$ instead. Let
$(\mathscr{L},\nabla)$ be a locally free sheaf on $E$ of rank one and degree
zero with integrable connection. If $k=\mathbb{C}$, the Riemann-Hilbert
correspondence gives $\mathbb{L}:=\ker(\nabla)$ a rank one local system on
$E$. We may write $\mathscr{L}=\mathcal{O}(e-q)$ for some $q\in E$. In this
section we compute the parabolic Higgs bundle on $(\mathbb{P}^{1},D)$ obtained
from $(\mathscr{L},\nabla)$ via middle convolution.

Let $x\in\mathbb{P}^{1}\setminus D$ be a point and $x_{1},x_{2}$ its preimages
in $E$. Consider the rank one locally free sheaf with flat connection on
$(E-\\{x_{1}\\}-\\{x_{2}\\})\times\\{x\\}$ corresponding to the character
$m^{*}\chi$ under the Riemann-Hilbert correspondence, and let
$(\mathscr{M}_{\star},\nabla)$ denote its Deligne canonical extension to $E$,
where we view $\mathscr{M}_{\star}$ as a parabolic bundle. The degree of
$\mathscr{M}_{0}$ is $-1$; it is $\mathscr{O}_{E}(-p)$ where
$2p=x_{1}+x_{2}=e$. (To see this, note that $m^{*}\chi$ is trivialized on the
double cover of $E$ branched at $x_{1},x_{2}$, which is
$\operatorname{Spec}_{E}\mathscr{O}_{E}\oplus\mathscr{O}_{E}(-p),$ where the
multiplication is induced by the map $\mathscr{O}_{E}(-2p)\to\mathscr{O}_{E}$
vanishing at $x_{1},x_{2}$.)

The following is obtained by specializing [LL22a, Theorem 5.1.6] to our
setting: 333we match up the notation with loc.cit. for the reader’s
convenience. Let $E^{\circ}$ denote $E-E[2]$. In the notation of loc.cit., we
set $\mathscr{B}\subset E^{\circ}$ to be a contractible neighborhood of
$x_{1}$, $\mathscr{C}$ the constant family $E\times\mathscr{B}$,
$s_{1}:\mathscr{B}=E-E[2]\rightarrow\mathscr{C}$ the diagonal map and
$s_{2}=-s_{1}$, and $\mathbb{V}$ the unitary local system on $\mathscr{C}$
whose restriction to each fiber $\mathscr{C}_{y}$ is
$\mathbb{L}\otimes(m^{*}\chi)|_{\mathscr{C}_{y}}$

######  Proposition 3.2.1.

The Higgs field on $MC_{-1}(f,\mathbb{L})$ at $x$ is computed via the map

$H^{0}(E,\mathscr{L}\otimes\mathscr{M}_{0}\otimes\omega_{E}(x_{1}+x_{2}))\to
H^{1}(E,\mathscr{L}\otimes\mathscr{M}_{0})\otimes T^{*}_{x}X$

given as adjoint (via Serre duality) to the multiplication map

$H^{0}(E,\mathscr{L}\otimes\mathscr{M}_{0}\otimes\omega_{E}(x_{1}+x_{2}))\otimes
H^{0}(E,\mathscr{L}^{\vee}\otimes\mathscr{M}_{0}^{\vee}\otimes\omega_{E})\to
H^{0}(\omega_{E}^{\otimes 2}(x_{1}+x_{2}))\to T_{x}^{*}X,$

where the map $H^{0}(\omega_{E}^{\otimes 2}(x_{1}+x_{2}))\to T_{x}^{*}X$ is
the one given by pullback on cotangent spaces along the map
$E\rightarrow\mathcal{M}_{1,2}$ given by

$x_{1}\mapsto(E,x_{1},-x_{1}).$

######  Lemma 3.2.2.

The Higgs bundle corresponding to $MC_{-1}(f,\mathbb{L})$ has Higgs field
vanishing only at $f(q)$.

###### Proof.

We first show that the Higgs field vanishes at $x=f(q)$. The Higgs field
factors through

$H^{0}(\omega_{E}(-q+x_{1}+x_{2}))\otimes H^{0}(\omega_{E}(q))\rightarrow
H^{0}(\omega_{E}^{\otimes 2}(x_{1}+x_{2})).$

On the other hand, when $x_{1}=q$, we have the following commutative diagram

(3.1) ${H^{0}(\omega_{E})\otimes
H^{0}(\omega_{E})}$${H^{0}(\omega_{E}^{\otimes
2})}$${H^{0}(\omega_{E}(-q+x_{1}+x_{2}))\otimes
H^{0}(\omega_{E}(q))}$${H^{0}(\omega_{E}^{\otimes
2}(x_{1}+x_{2})).}$$\scriptstyle{\simeq}$

Indeed, the natural map $H^{0}(\omega_{E})\rightarrow H^{0}(\omega(x_{2}))$ is
an isomorphism, and the same is true of the map $H^{0}(\omega_{E})\rightarrow
H^{0}(\omega_{E}(q))$. Finally, the composition $H^{0}(\omega_{E}^{\otimes
2})\rightarrow H^{0}(\omega^{\otimes 2}(x_{1}+x_{2}))\rightarrow T^{*}_{x}X$
is zero, since this is the map induced by pullback of differentials along the
composite map

$E\xrightarrow{x\mapsto(E,x,-x)}\mathcal{M}_{1,2}\rightarrow\mathcal{M}_{1,1},$

which is constant. ∎

##  4\. Proofs of the main theorems in characteristic zero

We may now give a short proof of some of our main theorems. We assume standard
material about complex variations of Hodge structure; for a brief primer see
[LL22a, §3] and [LL22b, §4]. Before giving proofs we need some brief
preliminaries.

###  4.1. Preliminaries

######  Proposition 4.1.1.

Let $X=\mathbb{P}^{1}\setminus\\{0,1,\infty,\lambda\\}$ and let $\mathbb{V}$
be an irreducible complex local system on $X$ satisfying $(\star)$ and
underlying a $\mathbb{C}$-VHS. Then the Deligne canonical extension
$(\mathscr{E},F^{1},\nabla)$ of
$(\mathbb{V}\otimes\mathscr{O}_{X},\operatorname{id}\otimes d)$ satisfies
$\mathscr{E}\simeq\mathscr{O}_{\mathbb{P}^{1}}\oplus\mathscr{O}_{\mathbb{P}^{1}}(-1)$,
with $F^{1}=\mathscr{O}_{\mathbb{P}^{1}}.$.

###### Proof.

From $(\star)$, the residue matrices of $\nabla$ at $\\{0,1,\lambda\\}$ have
trace zero, and the residue matrix at $\infty$ has trace $1$. This means that
$\mathscr{E}$ and hence $F^{1}\oplus\mathscr{E}/\mathscr{F}^{1}$ has degree
$-1$, by e.g. [EV86, B.3]. Let

$\theta:F^{1}\to\mathscr{E}/F^{1}\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log D),$

be the non-zero $\mathscr{O}$-linear map obtained from $\nabla$. As in the
introduction we consider the Higgs bundle

$F^{1}\oplus\mathscr{E}/\mathscr{F}^{1}$

with the Higgs field $\begin{pmatrix}0&\theta\\\ 0&0\end{pmatrix}.$

By Simpson’s theory [Sim90, Theorem 8], this Higgs bundle is parabolically
stable of parabolic degree zero. Parabolic stability implies the parabolic
degree of $F^{1}$ is positive, and hence the honest degree satisfies $\deg
F^{1}>-1$. The fact that $\theta$ is non-zero implies that $\deg
F^{1}\leq\deg\mathscr{E}/F^{1}+\deg\Omega^{1}_{\mathbb{P}^{1}}(\log D)$, and
hence that $2\deg F^{1}\leq\deg\mathscr{E}+2=1$.

Hence $\deg F^{1}=0$, and $\deg\mathscr{E}/F^{1}=-1$. Finally, we must have
that the extension

$0\to F^{1}\to\mathscr{E}\to\mathscr{E}/F^{1}\to 0$

splits as
$\operatorname{Ext}^{1}(\mathscr{O}_{\mathbb{P}^{1}}(-1),\mathscr{O}_{\mathbb{P}^{1}})=0.$
∎

Let $D=\\{0,1,\infty,\lambda\\}$. Given $\mathbb{V}$ an irreducible local
system satisfying $(\star)$ and underlying a (necessarily unique, by
irreducibility) $\mathbb{C}$-VHS, the above proposition shows that we may
canonically associate to $\mathbb{V}$ a nonzero map
$\theta:\mathscr{O}_{\mathbb{P}^{1}}\to\mathscr{O}_{\mathbb{P}^{1}}(-1)\otimes\Omega^{1}_{\mathbb{P}^{1}}(\log
D)=\mathscr{O}_{\mathbb{P}^{1}}(1),$ which vanishes at a unique point
$w(\mathbb{V})$ of $\mathbb{P}^{1}$, as in the introduction.

######  Proposition 4.1.2.

With assumptions as in 4.1.1, the point $w(\mathbb{V})$ determines
$\mathbb{V}$ up to isomorphism.

###### Proof.

The point $w(\mathbb{V})$ determines $\theta$ up to multiplication by a non-
zero scalar, which by 4.1.1 above determines the stable parabolic Higgs bundle
associated to $\mathbb{V}$ up to isomorphism. Now Simpson’s theory [Sim90,
Theorem 8] tells us that this parabolic Higgs bundle determines $\mathbb{V}$.
∎

###  4.2. Proofs

We now begin with the proofs of our main theorems in characteristic zero.

######  Lemma 4.2.1.

Let $X$ be a smooth variety. A rank one local system $\mathbb{L}$ on $X$ is of
geometric origin if and only if it is torsion.

###### Proof.

Torsion evidently implies geometric origin, as all torsion rank one local
systems arise as summands of $\pi_{*}\mathbb{C}$ for $\pi:Y\to X$ some cyclic
étale cover, so we prove the converse.

Suppose $\mathbb{L}$ is rank one and of geometric origin. Then the monodromy
representation of $\mathbb{L}$ is defined over the ring of integers
$\mathscr{O}_{K}$ of some number field $K$. Moreover for each embedding
$\mathscr{O}_{K}\hookrightarrow\mathbb{C}$,
$\mathbb{L}\otimes_{\mathscr{O}_{K}}\mathbb{C}$ is unitary, as it underlies a
polarizable $\mathbb{C}$-VHS of rank one. Hence if $\gamma\in\pi_{1}(X)$ is a
loop, the scalar in $\mathscr{O}_{K}^{\times}$ given by the monodromy of
$\gamma$ is an algebraic integer all of whose Galois conjugates have absolute
value one, hence a root of unity. ∎

######  Proposition 4.2.2 (1.2.1).

Suppose $\mathbb{V}$ is an irreducible rank $2$ local system on $X$ satisfying
$(\star)$. $\mathbb{V}$ is of geometric origin if and only if $\mathbb{V}$
underlies a $\mathbb{C}$-VHS with $w(\mathbb{V})=f(x)$, for $x$ a torsion
point of $E$.

###### Proof.

Let $q$ be a point of $E$. Then by Narasimhan-Seshadri [NS65], the unique
unitary connection $\nabla$ on $\mathscr{L}=\mathscr{O}(q-e)$ has torsion
monodromy if and only if $q$ itself is torsion; in particular the sheaf
$\mathbb{L}$ of flat sections of $(\mathscr{L},\nabla)$ is of geometric origin
if and only if $q$ is torsion, by 4.2.1.

Suppose $q$, and hence $\mathbb{L}$, is torsion. Now
$\mathbb{V}=MC_{-1}(f,\mathbb{L})=MC_{-1}(f_{*}\mathbb{L})$ is evidently of
geometric origin (as middle convolution preserves the property of being of
geometric origin), is irreducible [Kat96, Theorem 2.9.8(2)], and
$w(\mathbb{V})=q$ by 3.2.2. As $w(\mathbb{V})$ uniquely determines
$\mathbb{V}$ by 4.1.2, we have shown that if $w(\mathbb{V})$ is torsion, then
$\mathbb{V}$ is of geometric origin.

Conversely, suppose $\mathbb{V}$ is of geometric origin and $w(\mathbb{V})=q$
is not torsion; we have that
$\mathbb{V}=MC_{-1}(f,\mathbb{L})=MC_{-1}(f_{*}\mathbb{L})$ for $\mathbb{L}$
the sheaf of flat sections to the unitary flat line bundle
$(\mathscr{O}(q-e),\nabla)$ as above, by 3.2.2 and 4.1.2. Hence [Kat96,
Theorem 2.9.8(1)] $f_{*}\mathbb{L}=MC_{-1}(\mathbb{V})$. So if $\mathbb{V}$
was of geometric origin, the same would be true for $f_{*}\mathbb{L}$, and
hence for $f^{*}f_{*}\mathbb{L}=\mathbb{L}\oplus\mathbb{L}^{\vee}$. Hence
$\mathbb{L}$ itself would be of geometric origin, hence have finite monodromy,
by 4.2.1. But this contradicts our assumption that $q$ is not torsion. ∎

###### Proof of Theorem 1.1.4 and Theorem 1.1.6.

Theorem 1.1.4 is immediate from what we’ve proven above; we may simply
consider $MC_{-1}(f,\mathbb{L})$ as $\mathbb{L}$ varies over all rank one
local systems on $E$ with finite monodromy. MCG-finiteness follows from the
fact that the middle convolution is equivariant for the action of the mapping
class group, and local systems with finite monodromy are evidently MCG-finite.

To prove Theorem 1.1.6, note that the proof above tells us that each
$\mathbb{V}$ of geometric origin satisfying $(\star)$ is of the form
$R^{1}(\pi_{2})_{*}i_{*}(\pi_{1}^{*}\mathbb{L}\otimes m^{*}\chi)$ for $\chi$
of order $2$ and $\mathbb{L}$ a finite order rank one local system on $E$, say
of order $m$. But the local system $\pi_{1}^{*}\mathbb{L}\otimes m^{*}\chi$ on
$U=E\times\mathbb{P}^{1}\setminus(\Gamma_{f}\cup E\times\\{\infty\\})$ is
trivialized on the preimage of $U$ in the variety $Z^{\prime}_{m}$ defined in
the statement of Theorem 1.1.6, by construction. Hence its cohomology appears
in the cohomology of $Z^{\prime}_{m}\to\mathbb{P}^{1}$, as desired. ∎

##  5\. Recollections on parabolic bundles and Higgs-de Rham flows

For the rest of the paper we will not in general be working over $\mathbb{C}$.
We prepare to state and prove a precise form of [SYZ21, Conjecture 4.8],
stated earlier as Theorem 1.2.2.

###  5.1. Parabolic de Rham and Higgs bundles

Let $\mathcal{C}/k$ be a smooth proper curve over a field $k$, and
$\mathcal{D}=\sum\mathcal{D}_{i}\subset\mathcal{C}$ a reduced effective
divisor of degree $l$. We refer the reader to [KS20, Definition 2.3] for the
definition of parabolic de Rham and Higgs bundles; roughly this consists of
bundles $(V_{\alpha})_{\alpha}$ indexed by $(\alpha_{1},\cdots,\alpha_{l})$
with $\alpha_{i}\in\mathbb{R}$, such that

  * •

each $V_{\alpha}$ is itself a de Rham (respectively Higgs) bundle, and

  * •

they are equipped with maps of de Rham (respectively Higgs) bundles
$V_{\alpha}\xhookrightarrow{}V_{\beta}$ for all $\alpha\geq\beta$ (where we
write $\alpha\geq\beta$ whenever $\alpha_{i}\geq\beta_{i}$ for all $i$).

The bundles $(V_{\alpha})_{\alpha}$ are required to satisfy several more
conditions, for which we refer the reader to loc.cit..

We make the following definition following [KS20, Example 2.5]

######  Definition 5.1.1.

Let $V$ be an arbitrary vector bundle on $\mathcal{C}$. For
$(\alpha_{1},\cdots,\alpha_{l})\in\mathbb{Q}^{l}$, we define the parabolic
bundle $V(-\sum_{i=1}^{l}\mathcal{D}_{i})$ by

$V(-\sum_{i=1}^{l}\alpha_{i}\mathcal{D}_{i})_{\beta}=V(\sum_{i=1}^{l}-\lceil\alpha_{i}+\beta_{i}\rceil\mathcal{D}_{i}).$

######  Remark 5.1.2.

When $\alpha_{i}=0$, the above gives the trivial parabolic structure along
$\mathcal{D}_{i}$.

The following gives a description of parabolic bundles on curves; see [Bis97]
for the analogous result in characteristic zero:

######  Proposition 5.1.3 ([KS20, Proposition 2.14]).

Suppose $\mathcal{C}^{\prime}$ and $\mathcal{C}$ are smooth curves over
$\mathbb{F}_{q}$ and $\mathcal{C}^{\prime}\rightarrow\mathcal{C}$ is a
$\mathbb{Z}/N$-covering, branched at the divisor
$\mathcal{D}\subset\mathcal{C}$. Then there is an equivalence of categories
between $\mathbb{Z}/N$-equivariant connections (resp. Higgs bundles) on
$\mathcal{C}^{\prime}$ and the category of parabolic connections on
$\mathcal{C}$ whose parabolic structure is supported on $\mathcal{D}$ with
weights in $\frac{1}{N}\mathbb{Z}$.

###  5.2. Conjecture of Sun-Yang-Zuo

We now recall the statement of [SYZ21, Conjecture 4.8]. We consider
logarithmic graded semistable Higgs bundles on
$(\mathbb{P}^{1},D=0+1+\lambda+\infty)$ such that the underlying graded vector
bundle is isomorphic to $\mathcal{O}\oplus\mathcal{O}(-1)$. By [SYZ21], the
moduli space $\mathcal{M}_{HIG}$ of such is isomorphic to $\mathbb{P}^{1}$, by
taking the unique zero of the Higgs field; they then consider the _twisted
Higgs-de Rham flow_ on $\mathcal{M}_{HIG}$, which in general depends on a lift
of $(\mathbb{P}^{1},D)$ to $W_{2}$, inducing a self-map
$\psi_{p}:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$.

######  Conjecture 5.2.1 (Sun-Yang-Zuo).

For $p\neq 2$ and any lifting of $(\mathbb{P}^{1},D)$ to $W_{2}$, the
following diagram commutes

(5.1)
${E}$${E}$${\mathbb{P}^{1}}$${\mathbb{P}^{1}}$$\scriptstyle{[p]}$$\scriptstyle{f}$$\scriptstyle{f}$$\scriptstyle{\psi_{p}}$

where $f:E\to\mathbb{P}^{1}$ is the elliptic curve double cover branched over
$D$, with $f^{-1}(\infty)$ being the identity.

###  5.3. Translation to parabolic Higgs bundles

Following [KS20, LSW22], we translate 5.2.1 into the language of parabolic
Higgs-de Rham flows, which is more natural for our purposes. Let $k$ denote an
algebraic closure of $\mathbb{F}_{p}$, with $p$ different from $2$, and let
$F_{k}:{\rm Spec}(k)\rightarrow{\rm Spec}(k)$ denote absolute Frobenius.

######  Definition 5.3.1.

  1. (1)

Let $HIG^{par}_{p-1,N}(\mathcal{C},\mathcal{D})$ denote the category of
parabolic logarithmic Higgs bundles on $(\mathcal{C},\mathcal{D})$ which are
nilpotent of exponent $\leq p-1$, whose parabolic structures are supported on
$\mathcal{D}$, with weights lying in $\frac{1}{N}\mathbb{Z}$.

  2. (2)

Let $MIC^{par}_{p-1,N}(\mathcal{C},\mathcal{D})$ be the category of adjusted
(see [KS20, Definition 2.9]) logarithmic parabolic flat bundles on
$(\mathcal{C},\mathcal{D})$, whose $p$-curvatures and nilpotent part of the
residues are nilpotent of exponent $\leq p-1$, and whose parabolic structures
are supported on $\mathcal{D}$ with weights lying in $\frac{1}{N}\mathbb{Z}$.

From [KS20, Theorem 2.10], we have the parabolic version of the inverse
Cartier transform:

$C^{-1}_{par}:HIG_{p-1,N}^{par}(\mathcal{C},\mathcal{D})\rightarrow
MIC_{p-1,N}^{par}(\mathcal{C},\mathcal{D}).$

This in general depends on a chosen lift of $(\mathcal{C},\mathcal{D})$ to
$W_{2}$. By taking the associated graded of the Harder-Narasimhan filtration,
we also have the functor
$\operatorname{Gr}:MIC_{p-1,N}^{par}(\mathcal{C},\mathcal{D})\rightarrow
HIG_{p-1,N}^{par}(\mathcal{C},\mathcal{D})$.

######  Definition 5.3.2.

Let $\mathcal{M}_{\frac{1}{2}\infty}$ denote the moduli of graded semistable
parabolic Higgs bundles on $(\mathbb{P}^{1},D)$, with underlying the graded
parabolic bundle $(\mathcal{O}\oplus\mathcal{O}(-1))(-\frac{1}{2}\infty)$ in
the notation of 5.1.1.

######  Proposition 5.3.3 ([LSW22, Proposition 2.7]).

The functor $\operatorname{Gr}\circ C^{-1}_{par}$ induces a self-map on
$\mathcal{M}_{\frac{1}{2}\infty}$. Moreover there is a natural isomorphism
$\mathcal{M}_{HIG}\simeq\mathcal{M}_{\frac{1}{2}\infty}$, identifying
$\psi_{p}$ with $\operatorname{Gr}\circ C^{-1}_{par}$.

##  6\. Proof of Theorems in positive characteristic

We now let $\lambda\in\mathbb{F}_{q}$ be distinct from $0$ and $1$, and denote
by $X$ the curve $\mathbb{P}^{1}\setminus\\{0,1,\lambda,\infty\\}$. As before,
let $E\rightarrow\mathbb{P}^{1}$ be the elliptic curve double cover branched
at $0,1,\lambda,\infty$, and let $E^{\circ}$ denote the curve $E-E[2]$.

###  6.1. Middle convolution, again

We now reinterpret the constructions of § 2 to define de Rham and Higgs
analogues of the local systems $MC_{-1}(f_{*}\mathbb{L})$ constructed earlier,
in positive characteristic.

We fix for the rest of this section a cyclic étale cover
$\eta:\tilde{E}\rightarrow E$ of degree $m$, for some $m$ odd. We denote by
$S$ the surface $E\times\mathbb{P}^{1}$, and by $p_{1},p_{2}$ the projection
from $S$ to $E$ and $\mathbb{P}^{1}$ respectively. Let $D_{S}$ be the divisor
$\Gamma+E\times{\infty}$ on $S$, where $\Gamma$ denotes the graph of
$f:E\rightarrow\mathbb{P}^{1}$.

######  Definition 6.1.1.

Let $w:W^{\prime}\rightarrow S$ be the double cover branched along $D_{S}$.
Define the fiber product
$Z^{\prime}\vcentcolon=(\tilde{E}\times\mathbb{P}^{1})\times_{E\times\mathbb{P}^{1}}W^{\prime}$.

For $x\in X$, let $Z_{x},W_{x}$ denote the fibers of $Z^{\prime},W^{\prime}$
at $x$; these fit into the following Cartesian square:

(6.1)
${Z_{x}}$${W_{x}}$${\tilde{E}}$${E}$$\scriptstyle{2:1}$$\scriptstyle{2:1}$$\scriptstyle{\eta}$

######  Notation 6.1.2.

Denote by $Z_{X},W_{X}$ the fiber products
$Z^{\prime}\otimes_{\mathbb{P}^{1}}X,W^{\prime}\otimes_{\mathbb{P}^{1}}X$
respectively, and by $J_{Z,X},J_{W,X}$ the relative Jacobians of $Z_{X},W_{X}$
over $X$. Also we write $z_{X}:Z_{X}\rightarrow X,w_{X}:W_{X}\rightarrow X$
for the natural maps.

######  Proposition 6.1.3.

The pullbacks
$W_{E^{\circ}}\vcentcolon=W^{\prime}\times_{\mathbb{P}^{1}}E^{\circ}$,
$Z_{E^{\circ}}\vcentcolon=Z^{\prime}\times_{\mathbb{P}^{1}}E^{\circ}$,
equipped with maps $w_{E^{\circ}}:W_{E^{\circ}}\rightarrow E^{\circ}$,
$z_{E^{\circ}}:Z_{E^{\circ}}\rightarrow E^{\circ}$, may be compactified to
semistable families $w_{E}:W_{E}\rightarrow E$, $z_{E}:Z_{E}\rightarrow E$
over $E$. Furthermore we may choose $Z_{E}$ and $W_{E}$ such that

  * •

the natural $(\mathbb{Z}/m\times\mathbb{Z}/2)\times\mathbb{Z}/2$-action 444the
$\mathbb{Z}/m\times\mathbb{Z}/2$-action comes from the construction of
$Z^{\prime}$ as a fiber product in Definition 6.1.1, whereas the
$\mathbb{Z}/2$-action one acting on
$Z_{E^{\circ}}=Z\times_{\mathbb{P}^{1}}E^{\circ}$ via the $-1$-map on the
second factor on $Z_{E^{\circ}}$ extends,

  * •

the natural $\mathbb{Z}/2\times\mathbb{Z}/2$-action on $W_{E^{\circ}}$
extends, and

  * •

there is a commutative diagram

(6.2)
${Z_{E}}$${W_{E}}$${E}$$\scriptstyle{z_{E}}$$\scriptstyle{\theta}$$\scriptstyle{w_{E}}$

extending the natural one with $Z_{E^{\circ}},W_{E^{\circ}},E^{\circ}$, and
such that $\theta$ is equivariant for the first $\mathbb{Z}/2$-action, and the
entire diagram is equivariant for the second $\mathbb{Z}/2$-action.

###### Proof sketch.

The desired compactifications are obtained by normalizing
$W^{\prime},Z^{\prime}$ in the function fields of
$W_{E^{\circ}},Z_{E^{\circ}}$. The three bullets are clear. One may prove
semistability in a number of ways, e.g. by computing with local models; we now
sketch a proof of semistability by analyzing the monodromies of the local
systems
$R^{1}w_{E^{\circ},*}\overline{\mathbb{Q}}_{\ell},R^{1}z_{E^{\circ},*}\overline{\mathbb{Q}}_{\ell}$
about the points of $E[2]$. As a curve has semistable reduction if and only if
its Jacobian does, it suffices to show, by the Néron-Ogg-Shafarevich
criterion, that these local monodromies are unipotent.

Recall that we have the map $z_{X}:Z_{X}\rightarrow X$. It suffices to show
(as $E\to\mathbb{P}^{1}$ is ramified to order $2$ at $0,1,\infty,\lambda$)
that the local monodromies of $R^{1}z_{X,*}\overline{\mathbb{Q}}_{\ell}$
around all punctures have generalized eigenvalues $1$ or $-1$. This is true
for
$R^{1}z_{X,*}\overline{\mathbb{Q}}_{\ell}/R^{1}w_{X,*}\overline{\mathbb{Q}}_{\ell}$,
since this is a sum of middle convolutions as in Theorem 1.1.6, so it remains
to show the same for $R^{1}w_{X,*}\overline{\mathbb{Q}}_{\ell}$.

Denote the map $E^{\circ}\rightarrow X$ by $\pi^{\circ}$, and let $\tau$
denote the non-trivial summand of
$R^{1}\pi^{\circ}_{*}\overline{\mathbb{Q}}_{\ell}$. Then
$R^{1}w_{X*}\overline{\mathbb{Q}}_{\ell}$ is a direct sum of the local system
associated to the constant family $E\times X$, and $MC_{-1}(\tau)$. For
$MC_{-1}(\tau)$, the local monodromies at the punctures satisfy $(\star)$,
again by [DR07, Lemma 5.1]. ∎

######  Definition 6.1.4.

  1. (1)

For each character $\chi:\mathbb{Z}/m\rightarrow{\overline{\mathbb{F}}_{p}}$,
we define $\widetilde{MC}_{-1}^{dR}(\chi)$ to be the
$\chi\otimes(-1)$-isotypic555here $\chi\otimes(-1)$ denotes the character of
$(\mathbb{Z}/m\times\mathbb{Z}/2)$, the first factor of the group
$(\mathbb{Z}/m\times\mathbb{Z}/2)\times\mathbb{Z}/2$ from 6.1.3, given by
tensoring $\chi$ with the $-1$-character of $\mathbb{Z}/2$ part of the
relative log-de Rham cohomology of $(Z_{E},z_{E}^{-1}(E[2])$ over $(E,E[2])$.

  2. (2)

Moreover, the additional $\mathbb{Z}/2$-action in Proposition 6.1.3 implies
that each $\widetilde{MC}_{-1}^{dR}(\chi)$ has a $\mathbb{Z}/2$-equivariant
structure with respect to the covering $E\rightarrow\mathbb{P}^{1}$, and so
descends to a parabolic logarithmic de Rham bundle on $(\mathbb{P}^{1},D)$ by
5.1.3, which we denote by $MC_{-1}^{dR}(\chi)$; moreover the latter is
equipped with a Hodge filtration descended from $E$. Finally, let
$MC_{-1}(\chi)$ denote the parabolic, graded, logarithmic, Higgs bundle on
$(\mathbb{P}^{1},D)$ given by the associated graded of $MC_{-1}^{dR}(\chi)$
with respect to the Hodge filtration.

For each $x\in\mathbb{P}^{1}$ with $x\neq 0,1,\lambda,\infty$, let $W_{x}$ be
the fiber of $W^{\prime}$ above $x$; $W_{x}$ is a smooth curve and is moreover
equipped with a $\mathbb{Z}/2$-covering map $h_{W,x}:W_{x}\rightarrow
E_{x}=E\times\\{x\\}$.

By 5.1.3, the $\mathbb{Z}/2$-equivariant bundle with connection
$\mathcal{O}_{W_{x}}$ descends to a parabolic logarithmic connection $J_{x}$
on $E_{x}$. Let $L_{\chi,x}$ denote the $\chi$-isotypic component of
$\eta_{*}\mathcal{O}_{\tilde{E}}$. Note that the isomorphism classes of
$E_{x}$ and $L_{\chi,x}$ are independent of the choice of $x$, and we simply
denote them by $E,L_{\chi}$ respectively. The line bundle $L_{\chi}$ has
degree zero, and so is isomorphic to $\mathcal{O}_{E}(e-\tilde{y})$ for a
unique $\tilde{y}\in E$; we say that $y\vcentcolon=\pi(\tilde{y})$ is the
point corresponding to $L_{\chi}$.

######  Proposition 6.1.5.

The fiber of $MC_{-1}(\chi)$ at $x$ is given by

(6.3) $H^{1}(L_{\chi}\otimes J_{x})\oplus H^{0}(L_{\chi}\otimes
J_{x}\otimes\Omega^{1}_{E}(f^{-1}(x))),$

with the first (respectively second) factor being the degree one (zero) piece.

The Higgs field vanishes only at the point $y$ corresponding to $L_{\chi}$.
Moreover the graded vector bundle underlying $MC_{-1}(\chi)$ is isomorphic to
$\mathcal{O}\oplus\mathcal{O}(-1)$.

###### Proof.

Since the maps $Z_{X},W_{X}\rightarrow X$ are smooth, the restriction
$MC^{dR}_{-1}(\chi)|_{X}$ is given by the quotient of the relative de Rham
cohomologies of $Z_{X}$ and $W_{X}$ over $X$. Therefore the fiber
$MC_{-1}^{dR}(\chi)|_{x}$ is canonically identified with the
$\chi\otimes(-1)$-isotypic component of $H^{1}_{dR}(Z_{x})$.

Let $h_{Z,x}:Z_{x}\rightarrow E$ be the natural map. To take the
$\chi\otimes(-1)$ isotypic component, we can first take the
$\chi\otimes(-1)$-component of $h_{Z,x,*}^{dR}\mathcal{O}_{Z_{x}}$; the latter
is a logarithmic connection on the line bundle $L_{\chi}\otimes J_{x}$, with
poles along $f^{-1}(x)$. The fiber of $MC_{-1}^{dR}(\chi)$ at $x$ is then
given by taking $\mathbb{H}^{1}$ of the associated de Rham complex

$L_{\chi}\otimes J_{x}\rightarrow L_{\chi}\otimes
J_{x}\otimes\Omega^{1}_{E}(f^{-1}(x)),$

with the Hodge filtration given by the naive filtration. Therefore the fiber
of the Higgs cohomology is given by (6.3).

Denote the graded vector bundle underlying $MC_{-1}(\chi)$ by $\mathcal{F}$;
we now prove that $\mathcal{F}\simeq\mathcal{O}\oplus\mathcal{O}(-1)$ as a
graded bundle.

Suppose $\lambda_{W}\in W(\mathbb{F}_{q})$ is a lift of $\lambda$, so that the
divisor $D$ lifts canonically to a divisor $D_{W}\subset\mathbb{P}^{1}_{W}$,
and we also have the pair $(E_{W},E_{W}[2])$ lifting $(E,E[2])$. Fixing a lift
$\chi_{W}:\mathbb{Z}/m\rightarrow W({\overline{\mathbb{F}}_{p}})^{\times}$ of
$\chi$, we can then define a graded logarithmic Higgs bundle
$MC_{-1,\mathbb{P}^{1}_{W}}(\chi_{W})$ on $(\mathbb{P}^{1}_{W},D_{W})$ exactly
as in 6.1.4, which lifts $MC_{-1}(\chi)$.

Let us write the generic fiber $MC_{-1,\mathbb{P}^{1}_{W}}(\chi_{W})[1/p]$ as
$M\oplus N$, with $M$ (respectively $N$) being the degree one (resp. zero)
piece for the Hodge grading; this is a stable Higgs bundle of degree zero on
$\mathbb{P}^{1}_{W[1/p]}$, with non-vanishing Higgs field. Under the usual
Simpson correspondence, there is some $\mathbb{L}$ such that this Higgs bundle
corresponds to the local system $MC_{-1}(f,\mathbb{L})$ in the notation of
2.0.3, which has unipotent monodromy around $0,1,\lambda_{W}$ by 3.1.1;
therefore the parabolic weights are zero at these points. This forces
$\deg(M)=0,$ $\deg(N)=-1$ by 4.1.1; since Chern classes are invariant under
specialization, the same is true of $\mathcal{F}$, as required.

Finally, let $L_{\chi_{W}}$ denote the $\chi_{W}$-isotypic component of
$\eta_{*}\mathcal{O}_{\tilde{E}_{W}}$, which is a line bundle lifting
$L_{\chi}$. By 3.2.2, $\theta$ vanishes at the point
$\tilde{x}\in\mathbb{P}^{1}_{W[1/p]}$ corresponding to $L_{\chi_{W}}$, and
hence the Higgs field of $MC_{-1}(\chi)$ vanishes at the point $x$
corresponding to $L_{\chi}$, as claimed. ∎

######  Corollary 6.1.6.

For all choices of $\eta:\tilde{E}\rightarrow E$ and $\chi$, $MC_{-1}(\chi)$
corresponds to a point in
$\mathcal{M}_{HIG}\simeq\mathcal{M}_{\frac{1}{2}\infty}$, where these spaces
are defined in 5.2, 5.3.

### 6.2.

Since $z_{E}:Z_{E}\rightarrow E$ and $w_{E}:W_{E}\rightarrow E$ are semistable
families of curves, we may define their relative log Higgs cohomologies (i.e.
the associated graded of relative log de Rham cohomology with respect to the
Hodge filtration), and by another application of Proposition 5.1.3 we may
descend to obtain parabolic logarithmic Higgs bundles on $\mathbb{P}^{1}$,
which we denote by $R^{1}z_{*}^{HIG}\mathcal{O}_{Z}$ and
$R^{1}w_{*}^{HIG}\mathcal{O}_{W}$ respectively. Let $(\mathcal{E},\theta)$
denote the quotient Higgs bundle
$R^{1}z_{*}^{HIG}\mathcal{O}_{Z}/R^{1}w_{*}^{HIG}\mathcal{O}_{W}$. Similarly
we have the logarithmic flat bundles
$R^{1}z_{*}^{dR}\mathcal{O}_{Z},R^{1}w_{*}^{dR}\mathcal{O}_{W},(\mathcal{E}_{dR},\nabla)\vcentcolon=R^{1}z_{*}^{dR}\mathcal{O}_{Z}/R^{1}w_{*}^{dR}\mathcal{O}_{W}$.

######  Proposition 6.2.1.

$C^{-1}(\mathcal{E},\theta)|_{X}\simeq(\mathcal{E}_{dR},\nabla)|_{X}$, and
therefore $\operatorname{Gr}\circ
C^{-1}(\mathcal{E},\theta)|_{X}\simeq(\mathcal{E},\theta)|_{X}$. Here $C^{-1}$
denotes the usual Cartier transform, and $\operatorname{Gr}$ the associated
graded with respect to the Hodge filtration on de Rham cohomology.

###### Proof.

We have $(\mathcal{E},\theta)|_{X}\simeq
R^{1}z_{X*}^{HIG}\mathcal{O}_{Z_{X}}/R^{1}w_{X*}^{HIG}\mathcal{O}_{W_{X}}$.
The first statement now follows from [OV07, Theorem 3.4] since we have lifts
of $Z_{X},W_{X}$ to $W_{2}$, and the second is immediate. ∎

###### Proof of Conjecture 5.2.1.

Let $g:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ be the map making (5.1)
commute. Then $g$ has degree $p^{2}$, and the same is true of
$\operatorname{Gr}\circ C^{-1}_{par}$ by [SYZ21, § 4.3]; therefore it suffices
to show that $\operatorname{Gr}\circ C^{-1}_{par}$ and $g$ agree on infinitely
many closed points. For any $m\geq 1$ odd, let $\tilde{E}\rightarrow E$ be a
non-trivial étale $\mathbb{Z}/m$-covering, and for any character
$\chi:\mathbb{Z}/m\rightarrow{\overline{\mathbb{F}}_{p}}$ we have the Higgs
bundles $MC_{-1}(\chi)$ on $(\mathbb{P}^{1},D)$. By 6.1.6, by tensoring with a
rank one Higgs bundle if necessary, these Higgs bundles correspond to points
in $\mathcal{M}_{HIG}\simeq\mathbb{P}^{1}$ and we will show that $g$ and
$\operatorname{Gr}\circ C^{-1}_{par}$ agree on all of these points.

Recall that $F_{k}:k\rightarrow k$ denotes absolute Frobenius. The Higgs-de
Rham functor $\operatorname{Gr}\circ C^{-1}$ is $F_{k}$-linear, and by
Proposition 6.2.1 $\operatorname{Gr}\circ
C^{-1}(\mathcal{E},\theta)|_{X}\simeq(\mathcal{E},\theta)|_{X}$. Now
$MC_{-1}(\chi)|_{X}$ is the $\chi$-isotypic component for the
$\mathbb{Z}/m$-action on $(\mathcal{E},\theta)|_{X}$, and therefore
$\operatorname{Gr}\circ C^{-1}(MC_{-1}(\chi))|_{X}$ is the $\chi^{p}$-isotypic
component. On the other hand, by definition of the $MC_{-1}(\chi)$’s the
$\chi^{p}$-isotypic component is given by $MC_{-1}(\chi^{p})|_{X}$. Therefore
$\operatorname{Gr}\circ C^{-1}(MC_{-1}(\chi))|_{X}\simeq
MC_{-1}(\chi^{p})|_{X}$, and hence $\operatorname{Gr}\circ
C_{par}^{-1}(MC_{-1}(\chi))$ and $MC_{-1}(\chi^{p})$ are isomorphic up to
tensoring with a rank one Higgs bundle with trivial Higgs field 666note that
$\operatorname{Gr}$, which is a priori the Hodge filtration restricted to
$MC_{-1}(\chi)$, is also the Harder-Narasimhan filtration, which is what Sun-
Yang-Zuo use to formulate their conjecture. By the explicit description of
$MC_{-1}(\chi)$ given in Proposition 6.1.5, in particular the claim about the
vanishing locus of the Higgs field, we may conclude. ∎

##  7\. Some loose ends

### 7.1.

For each $\lambda\in\mathbb{F}_{q}$, suppose we have a rank two
$\overline{\mathbb{Q}}_{\ell}$-local system $\mathbb{V}$ on
$X=\mathbb{P}^{1}-\\{0,1,\lambda,\infty\\}$ satisfying $(\star)$. Let
$\lambda_{W}\in W(\mathbb{F}_{q})$ be a lift of $\lambda$, and let
$X_{W}=\mathbb{P}^{1}-\\{0,1,\lambda_{W},\infty\\}$ be the corresponding lift
of $X$, and $\tilde{f}$ the lift of $f$. By definition $\mathbb{V}$
corresponds to a representation of the tame fundamental group
$\pi_{1}^{t}(X)$; the specialization isomorphism
$\pi_{1}^{t}(X_{W})\simeq\pi_{1}^{t}(X)$ gives a local system on $X_{W}$,
which we denote by $\mathbb{V}_{W}$.

######  Proposition 7.1.1.

The local system $\mathbb{V}_{W}$ is motivic, i.e. there exists an abelian
scheme $\pi:\mathcal{A}\rightarrow X_{W}$ such that $\mathbb{V}$ appears as a
direct summand of $R^{1}\pi_{*}\overline{\mathbb{Q}}_{\ell}$.

###### Proof.

Note that, by using the Weil restriction of abelian schemes, it suffices to
show that the pullback of $\mathbb{V}_{W}$ to $X_{W}\otimes{\rm
Spec}(W(\mathbb{F}_{q^{2}}))$ appears as a direct summand of an abelian
scheme. This now follows by applying $MC_{-1}$ to $\mathbb{V}$. Indeed, by
[DR07, Lemma 5.1], $MC_{-1}(\mathbb{V})$ has rank two, and the local
monodromies at $0,1,\lambda,\infty$ are each semisimple with eigenvalues $1$
and $-1$. As in the rest of the paper, let $f:E\rightarrow\mathbb{P}^{1}$ be
the elliptic curve branched at $0,1,\lambda,\infty$, and let
$f^{\circ}:E^{\circ}=E-E[2]\rightarrow X$ denote the restriction to $X$.
Therefore $f^{\circ*}MC_{-1}(\mathbb{V})$ extends to a local system on $E$,
and let us denote this by $\mathbb{W}$. Then either
$\mathbb{W}\simeq\mathbb{L}\oplus[-1]^{*}\mathbb{L}$ for a rank one local
system $\mathbb{L}$ on $E$, or it is of the form $h_{*}\mathbb{L}$ for a rank
one local system $\mathbb{L}$ on $E_{\mathbb{F}_{q^{2}}}$, and $h$ denotes the
map $E_{\mathbb{F}_{q^{2}}}\rightarrow E$. Now let
$E_{W}\rightarrow\mathbb{P}^{1}$ denote the elliptic curve branched at
$0,1,\lambda_{W},\infty$. In the first case above, $\mathbb{L}$ lifts
canonically to a rank one local system $\widetilde{\mathbb{L}}$ on $E_{W}$,
and we have $MC_{-1}(\tilde{f},\widetilde{\mathbb{L}})\simeq\mathbb{V}_{W}$,
and therefore the latter appears in an abelian scheme. In the second case, we
similarly have that $\mathbb{V}_{W(\mathbb{F}_{q^{2}})}$, the pullback of
$\mathbb{V}_{W}$ to $X_{W}\otimes W(\mathbb{F}_{q^{2}})$, is isomorphic to
$MC_{-1}(\tilde{f},\widetilde{\mathbb{L}})$, and again we may conclude. ∎

###  7.2. Another conjecture of Sun-Yang-Zuo

###### Proof of [SYZ21, Conjecture 4.10].

The only if direction is immediate from Conjecture 5.2.1, proven above, or
from 4.2.2. We now check the if direction.

Suppose we have $(E,\theta)\in\mathcal{M}_{HIG}$ with $(\theta)_{0}=f(x)$ for
some torsion point $x$ of order prime to $p$. Then we have shown that
$(E,\theta)$ appears in a family of abelian varieties which lifts to
$W(\mathbb{F}_{q})$, and therefore lifts to a periodic Higgs bundle in
$\mathcal{M}_{W(\mathbb{F}_{q})}$, by [LSZ19, Theorem 1.4]. ∎

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