# Developers’ Perception: Fixed Bugs Often Overlooked as Quality Contributions

Vitaly Alifanov<EMAIL_ADDRESS>0009-0007-5226-9887 Innopolis
UniversityInnopolisRussia ,  Kamil Almetov<EMAIL_ADDRESS>0009-0005-8174-1613 Innopolis UniversityInnopolisRussia ,  Ivan Kornienko
<EMAIL_ADDRESS>0009-0000-8013-3839 Innopolis
UniversityInnopolisRussia ,  Arsen Mutalapov<EMAIL_ADDRESS>0009-0000-4578-8944 Innopolis UniversityInnopolisRussia and  Yegor Bugayenko
0000-0001-6370-0678<EMAIL_ADDRESS>HuaweiMoscowRussia

###### Abstract.

High-quality software products rely on both well-written source code and
timely detection and thorough reporting of bugs. However, some programmers
view bug reports as negative assessments of their work, leading them to
withhold reporting bugs, thereby detrimentally impacting projects. Through a
survey of 102 programmers, we discovered that only a third of them perceive
the quantity of bugs found and rectified in a repository as indicative of
higher quality. This finding substantiates the notion that programmers often
misinterpret the significance of testing and bug reporting.

##  1\. Introduction

Programmers contribute to the creation of a software product by writing code,
while testers contribute by finding and reporting bugs (Bugayenko, 2018b). The
importance of testing is hard to underestimate, especially when the ultimate
quality of service and reliability of a product are important (Black, 2016;
Beizer, 1995). The primary output of testing is bug reports, while their
quantity and quality may be used as indicators of the effectiveness of testing
(Gelperin and Hetzel, 1988; Beizer, 2003; Bugayenko, 2018a).

However, not every programmer understands the importance of testing and the
value that a large number of high-quality reports may bring to a project
(Pettichord et al., 2013). Some programmers may perceive bug reports as a
negative opinion about their work (Bugayenko, 2018a, 2015). Because of this,
they may refrain from submitting them, trying not to hurt the feelings of
other programmers (Zhang et al., 2014). Some programmers may believe that the
absence of bug reports in their projects is a sign of high quality of coding
(Hutcheson, 2003). Such an attitude of programmers may hurt software projects
(Cohen et al., 2004). Even though some authors raised this concern, there were
no empirical studies conducted so far that would help understand how many
programmers share this attitude.

In order to shed light on this problem, we interviewed 102 programmers with a
two-phased survey. In the first phase, we asked them to choose a subjectively
“better” software product out of five available, showing them ten metrics for
each project: the “Closed Bug Reports” was one of the metrics. We also asked
them which metric they pay attention to. The target metric appeared to not
even be in the top three. In the second phase, we showed them a sample low-
quality bug report and asked what they would do with it, giving them six
options to choose from: three options were in favor of fixing the report,
while another three were in favor of not reporting it at all. More than 60% of
answers were in favor of improving and reporting, while 40% suggested not
reporting it at all.

The results demonstrate that programmers are not afraid of reporting bugs but
don’t consider a larger number of bug reports in a project as an indicator of
its higher quality. While the first finding is positive, the second one may be
interpreted as a confirmation of the belief that programmers don’t understand
the value of testing.

The rest of the paper is organized as following. Section 2 shows a few
previous studies related to the problem we reseach. Section 3 explains our
method of study. Section 4 presents the results we obtained. Section 5
discusses our findings and highlights limitations of our method.

##  2\. Related Work

Software quality is being researched thoroughly, and the research-related
studies are divided into several parts such as testing, debugging, and
developers’ opinions.

Straubinger and Fraser (2023) studied whether developers, consultants,
testers, and managers want to test their software. The results show that
developers think of testing as mundane and tend to prioritize other tasks.
Thus, one’s opinion on testing does not only rely on software quality but
rather on personal motivation to improve the software.

Winter et al. (2023) considered 386 software developers’ opinions on automatic
program repair tools (APR). One of the results was that programmers tend to
use APR in testing to find bugs faster. Still, developers want to have either
choice or control over the bug-fixing process because accuracy is a
significant concern for most developers. Despite that, analysis of APR does
not cover the perceptions toward software quality, but rather, a portion of
it. Thus, a comprehensive study is still needed.

Souza et al. (2022) asked 41 participants to answer three research questions
regarding developers’ viewpoints about bug-introducing changes, reasons to
agree with an assumption, and which assumptions are consensus/dissensus among
viewpoints. The results about bug-introducing changes drew five opinions, and
the most popular of them was “no matter the software history, only tests, and
developers’ experience are important” with the belief that experienced
developers introduce fewer bugs. However, the survey did not include the study
of views about bugs, rather only about the consequences of bugs and low
software quality.

To conclude, the research community produces papers and surveys on software
quality, testing, and debugging frequently. However, to the best of our
knowledge, there was no research on the perception of programmers on the bug
reporting process and its relation to the software quality.

##  3\. Method

The main objective of the research was to determine how programmers perceive
bug reports. It led us to the following research questions:

RQ1:

Do programmers consider software quality being positively correlated with the
number of bugs reported?

RQ2:

Do programmers feel comfortable when they report bugs?

We created a two-phased survey, where each phased helped us answer one RQ.

In the first phase of the survey, we asked programmers to choose one out of
six projects. We asked them to select the project that, according to their
subjective understanding, has the highest quality. For each project, we showed
them ten metrics, as shown in Table 1. We did show them the names of the
projects but did not provide links to GitHub repositories. Thus, they were
unable to study the source code of the projects and had to rely solely on the
provided metrics while making their decisions.

We also asked programmers to explain their choice by identifying the most
important metric out of ten available. The answers were intended to reveal the
importance of the number of closed bug reports (one of the metrics on the
list), although this was not explicitly announced to the respondents. Those
programmers who chose this metric would be considered as those who perceive
the number of reported and fixed bugs as being positively correlated with
software quality.

Metric | xgen | SWXMLHash | Fuzi | quick-xml | libexpat
---|---|---|---|---|---
Closed bug reports | 2 | 29 | 1 | 64 | 79
GitHub Stars | 276 | 1,345 | 1,049 | 982 | 918
Lines of code | 3,777 | 4,407 | 7,027 | 17,837 | 652,333
Release date (days ago) | 1,040 | 553 | 1,062 | 2,840 | 382
Contributors | 11 | 38 | 21 | 86 | 83
Closed issues | 18 | 140 | 57 | 264 | 286
Primary programming language | Go | Swift | Swift | Rust | C
Last commit (days ago) | 126 | 179 | 176 | 2 | 10
GitHub Forks | 66 | 196 | 149 | 205 | 422
Closed pull requests | 22 | 114 | 45 | 333 | 466
Table 1. This is a comparison table that contains XML parsing frameworks and
metrics for them taken from GitHub. Respondents should put these libraries in
the order from the most preferable to the least. Then, they should explain
their choice and provide key metrics developers were based on.

At the second phase of the survey, we asked programmers to study a sample bug
report, which intentionally contained a number of quality issues, as
demonstrated in Fig. 1. We gave programmers six options, which were implicitly
split into two groups:  1) real problems with the bug report, and  2)
“emotional“ barriers that a programmer could and should ignore.  Choosing
answers from the second group means that they feel uncomfortable while
reporting bugs. The options sound like this (the group names were not visible
to programmers):

  1. Op1:

(G1) Better explain how to reproduce it

  2. Op2:

(G2) Don’t report, since it’s a primitive bug

  3. Op3:

(G1) Get rid of the redundant 3–minute video

  4. Op4:

(G2) Instead, post to project chat, for a faster fix

  5. Op5:

(G1) Specify its priority and severity

  6. Op6:

(G2) RTFM, maybe you do something wrong

Error during login:
Message with some errors is displayed when attempting to log in. No
authentication token is obtained. I tried to log in with a valid username and
password, but no auth token was received. I used logs to check the token
To reproduce try to log in with:
Login="login123"
Password="password123"
System information:
Browser=Google Chrome
Attachments:
3-minute video with authorization process
Additional information:
This issue started occurring after the latest system update. I have tried
logging in on multiple devices and browsers, and the issue persists. Figure 1.
This is the bug report about some authentication problems. Respondents were
asked to choose a few options out of six: what would they do in order to
improve this report?

Our intention was to completely conceal our research objective from the
respondents. The question in the first phase included nine metrics besides bug
reporting. The question in the second phase didn’t directly inquire about
feelings—it asked about the most harmful aspects of the particular bug report.
We believe that concealing the research questions from the respondents
enhanced the objectivity of the results. Moreover, the questions are
situational, as they engage and interest the respondent more effectively in
answering.

In compiling the questions, we were guided by the criteria in the related book
(Gillham, 2008). In particular, we were guided by the brevity and simplicity
of the questions to take the least time for the interviewee to reach as large
an audience as possible.

##  4\. Results

Using our existing professional connections and social networks, we invited
programmers to participate in the survey, through Google Forms. 102 people
agreed to participate, knowing that the results of this study will be
published111The entire CSV dataset with all answers provided is available in a
public GitHub repository: https://github.com/system205/QuestionnaireData. 80
of them filled up the form in English, while 22 in Russian.

Table 2 demonstrates the distribution of positions of respondents in their
companies, as they reported in the survey. The majority (55%) of them
identified themselves as engineers (“Developer,” “Engineer,” and “Senior”),
which is suitable for the goals of our research.

Position | Responses |
---|---|---
Developer | 15 | 37%
Lead/Manager/Head/Chief | 6 | 15%
Senior | 4 | 10%
Engineer | 3 | 8%
Intern | 2 | 5%
Student | 2 | 5%
Junior | 1 | 3%
Others | 7 | 17%
Table 2. A summary of work positions of respondents, as they reported in the
survey, where “Others” include responses like “Freelance,” “PM,” “Lab,”
“Unknown” and no responses at all. The total count of responses is not equal
to the total number of respondents because not everybody answered to this
question in the survey.

Table 3 demonstrates how experienced are the respondents, according to their
own information provided while conducting the survey. Almost the majority of
respondents (46%) are rather experienced software engineers (three or more
years), which is suitable for the goals of our research.

Years of experience | Responses |
---|---|---
0 $\leq$ n ¡ 1 | 13 | 27%
1 $\leq$ n ¡ 3 | 9 | 19%
3 $\leq$ n ¡ 5 | 6 | 13%
5 $\leq$ n ¡ 10 | 10 | 20%
n $\geq$ 10 | 6 | 13%
Others | 4 | 8%
Table 3. A summary of years of experience, as they were claimed by the
respondents. The total count of responses is not equal to the total number of
respondents because not everybody answered to this question in the survey.

Table 4 demonstrates the distribution of the amount of bugs respondents report
per year, according to their own evaluation. The majority of them (73%)
reported less than 15 bugs, which is one bug per month. This number looks
reasonably low and may confirm the intuition that programmers tend to avoid
formal reporting of bugs.

Number of bugs | Responses |
---|---|---
0 | 6 | 17%
0 ¡ n ¡ 5 | 8 | 23%
5 $\leq$ n ¡ 15 | 12 | 33%
15 $\leq$ n ¡ 40 | 3 | 8%
40 $\leq$ n ¡ 100 | 4 | 11%
Others | 3 | 8%
Table 4. A summary of how many bugs respondents reported during the last
calendar year. The total count of responses is not equal to the total number
of respondents because not everybody answered to this question in the survey.

###  4.1. RQ1: Do programmers consider software quality being positively
correlated with the number of bugs reported?

Survey respondents were asked to place different XML parsing libraries from
most preferable to the least. The resulting distribution is shown in Table 5.
The results demonstrate that libraries with more closed bug reports and closed
issues (“libexpat” and “quick-xml”) are preferable. However, when we asked
respondents to answer “What are the metrics you were based on?,” we revealed
that the main metrics are the “number of stars,” “primary programming
language,” and “number of days from the last commit” (“quick-xml” and
“libexpat” have the smallest numbers: two and ten days respectively).

The distribution of chosen metrics is shown in Table 6, indicating how many of
them were reported by respondent as the first or primary choice. Even though
34 respondents considered bug reports and issues when choosing the library
with the higher quality, these metrics are not determinant taking only the 4th
and the 7th places among primary factors.

Framework | Top1 | Top2 | Top3 | Top4 | Top5
---|---|---|---|---|---
libexpat | 28 | 28 | 13 | 9 | 24
quick-xml | 36 | 22 | 20 | 17 | 7
SWXMLHash | 25 | 26 | 26 | 15 | 10
xgen | 8 | 10 | 17 | 20 | 47
Fuzi | 5 | 16 | 26 | 41 | 14
Table 5. This table demonstrates metrics and their priorities. Top1 indicates the highest priority, and Top5 means the least priority. Metric | Top priority | Rest
---|---|---
Stars | 28 | 26
Primary language | 21 | 21
Last commit (days ago) | 11 | 34
Release date (days ago) | 8 | 23
Closed bug reports | 8 | 26
Lines of code | 6 | 26
Contributors | 5 | 29
Closed issues | 5 | 29
Forks | 4 | 19
Closed pull requests | 1 | 10
Table 6. This table produces how a specific metric is chosen as the highest priority and the rest kinds of priorities denoted as “Rest.” Option | Votes |
---|---|---
Op1: Explain how to reproduce it | 60 | 25%
Op2: Don’t report a primitive bug | 4 | 2%
Op3: Remove the 3–minute video | 42 | 18%
Op4: Post to project chat | 20 | 8%
Op5: Specify priority and severity | 66 | 27%
Op6: RTFM, maybe you are wrong | 49 | 20%
Table 7. A summary of votes our respondents provided for each option. The
total count of votes is larger than the total number of respondents because
every respondent was allowed to chose more than one option.

As an answer to RQ1, we conclude that programmers consider software quality to
be positively correlated with the number of bugs reported but the correlation
between quality and number of bugs is not strong. Exactly 33% of respondents
can answer “yes” to RQ1. The respondents’ answers demonstrate that developers
pay more attention to other metrics such as “programming language” and “number
of days since the last commit.”

###  4.2. RQ2: Do programmers feel comfortable when they report bugs?

We showed a bug report related to an authentication problem (Fig. 1) and
provided six possible options to choose from. Three of them were focused on
specific improvements and further reporting, and another three were related to
the avoidance of reporting.

Table 7 shows the distribution of chosen options on the request “What would
you do to improve the bug report?” In total, we got 2.5 times more votes about
technical improvements (167) rather than mental avoidance (68). Thus, the
majority of voters (60%) are okay with the reporting of this bug. In addition,
40% of respondents chose only technical options.

Summarizing these results, we can state that programmers are not afraid of bug
reporting. However, about half of them feel uncertain and prefer to read a
manual before submitting a report.

##  5\. Discussion

Why do most programmers choose GitHub stars as a metric of quality? Indeed,
even though we emphasized quality, some developers referred to the term
“popularity.” The choice of popularity and GitHub stars might be because
developers shift the responsibility of choice to others, even though we asked
them to make their selection based on quality.

Is 102 respondents a large enough study group? While other researchers have
asked about respondents’ opinions and viewpoints, our method included
situational questions which elicited more truthful answers. Due to this
methodology, respondents treat a question as a real-life situation, which is a
key strength of our study. However, there are limitations. One of them is the
number of participants. 102 people represent a relatively small sample to
describe the common thinking of software developers. In future empirical
studies, we plan to interview a larger group of engineers.

How reliable is the statistics collected through the opinions of the bug
report? Indeed, the list of options we provided to the respondents is rather
limited. People may rely on different behaviors not listed in the specified
options, and we do not provide an opportunity for them to write their own
opinion. We also found that about half of the programmers did not feel any
ambiguity when dealing with external bug reports.

##  6\. Conclusion

We conducted a survey and studied the attitude of 102 software professionals
towards bug reporting and its impact on software quality. Along with our
initial hypothesis, only a third of all respondents suggested considering the
amount of reported and fixed bug reports in a repository as an indicator of
its quality. However, contrary to our initial hypothesis, the survey
demonstrated that programmers do not feel bad about reporting bugs. Thus, we
conclude that most programmers do not fully understand the importance and the
value that testing and bug reporting provide to their projects.

## References

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# Common envelope jets supernova r-process yields can reproduce $\rm[Eu/Fe]$
abundance evolution in the Galaxy

Aldana Grichener  Department of Physics, Technion, Haifa, 3200003, Israel;
<EMAIL_ADDRESS><EMAIL_ADDRESS>Chiaki Kobayashi
Centre for Astrophysics Research, Department of Physics, Astronomy and
Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK;
<EMAIL_ADDRESS>Noam Soker Department of Physics, Technion, Haifa,
3200003, Israel<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We use a numerical Galactic chemical evolution model and find that the common
envelope jets supernova (CEJSN) r-process scenario can account for both the
very early average ratio of europium to iron and its evolution at later times
in the Milky-Way (MW) Galaxy. In the CEJSN scenario a neutron star (NS)
spirals-in inside a red supergiant (RSG) star all the way to the core and
destroys it. According to this scenario r-process isotopes are
nucleosynthesized inside neutron-rich jets that the accretion disk around the
NS launches inside the core. The merger of a NS with an RSG core already takes
place in the very young Galaxy. The wide variety of these mergers properties
can explain the scatter in the ratio of europium to iron around the mean at
low metalicites. We conclude that CEJSNe can be a major contributor to
r-process nucleosynthesis.

stars: jets – stars: massive – stars: neutron – binaries: close – stars:
supergiants

##  1 INTRODUCTION

The formation sites of heavy elements is an unsolved problem in nuclear
astrophysics. Various astrophysical objects and transient events have been
studied as possible sources of heavy r-process elements over the past decades.
Among the possible candidates are magnetorotational supernovae (e.g., Winteler
et al. 2012; Nishimura et al. 2015; Halevi & Mösta 2018; Reichert et al. 2021;
Yong et al. 2021), binary neutron star (NS) mergers (e.g., Goriely et al.
2011; Wanajo et al. 2014; Metzger 2017; van de Voort et al. 2021; Dvorkin et
al. 2021) and collapsars (e.g., Siegel et al. 2019; Brauer et al. 2021; Siegel
et al. 2021). These astronomical sites have been tested with Galactic chemical
evolution models that lead to different conclusions. Recent studies suggest
that multiple r-process sites are required to explain the r-process abundances
in the Galaxy (e.g., Haynes & Kobayashi 2019; Yamazaki et al. 2021; Farouqi et
al. 2021; Tsujimoto 2021; Naidu et al. 2021). Molero et al. (2021) argue that
the same applies for ultra faint dwarf galaxies.

Another r-process site that might contribute to the formation of heavy
elements is the common envelope jets supernova (CEJSN) r-process scenario
(Papish et al. 2015; Grichener & Soker 2019a, b), in which r-process
nucleosynthesis occurs in NS-red super giant (RSG) common envelope systems
where a cold NS is engulfed by a RSG and orbits inside it. The NS accretes
mass from the envelope of the RSG through an accretion disk and eventually
reaches the CO core, where it accretes mass at a higher rate. The gravity of
the NS destroys the core forming a thicker accretion disk around the NS, and
the massive thick disk launches neutron-rich jets in which r-process
nucleosynthesis can occur.

The physics of CEJSNe and their possible outcomes are a topic of ongoing
research (e.g., Moreno Méndez et al. 2017; Gilkis et al. 2019; López-Cámara et
al. 2019, 2020; Grichener et al. 2021). Due to their wide diversity CEJSNe
events can account for various phenomena in astrophysics, such as peculiar
supernovae whose light curves might be shaped by the operation of jets at
different episodes during the explosion (e.g., Soker & Gilkis 2018; Soker et
al. 2019). Grichener & Soker (2021) argue that the ultra-relativistic jets
that a black hole (BH) launches as it orbits in the envelope of a RSG can emit
the high energy neutrinos ($\approx 10^{15}~{}\rm{eV}$) that the IceCube
neutrino observatory have been detecting since 2013 (e.g., Aartsen et al.
2013).

The evolution towards CEJSNe with a NS companion is almost identical to the
evolution of binary systems that result in NS-NS mergers. In both cases the
scenario begins with two massive stars in a binary system. The more massive
star (of $M\gtrsim 15M_{\odot}$) explodes first as a core collapse supernova
(CCSN) leaving a NS behind, while its companion (of $M\gtrsim 10M_{\odot}$)
evolves slower, and eventually reaches the RSG phase. The RSG star expands and
swallows the NS due to tidal forces, initiating a common envelope evolution
(CEE) phase in which the NS spirals in-inside the envelope of the giant star.
If the entire envelope is ejected before the NS reaches the CO core, then the
NS remains outside the core, and a NS-core binary system is formed. The core
eventually explodes in a second CCSN event forming another NS. Both NSs might
merge by emitting gravitational waves as their distance decreases (e.g.,
Vigna-Gómez et al. 2018, Fragos et al. 2019). However, if the NS enters the
core of the RSG before the core explodes the system will result in a CEJSN
event (e.g., Grichener & Soker 2019a).

In this letter we study the contribution of CEJSNe to europium time evolution
in the Galaxy. In section 2 we describe our chemical evolution model along the
parameter space of the CEJSN r-process scenario. In section 3 we present our
results and compare them to observations. We summarize in section 4.

##  2 The numerical scheme

The Galactic chemical evolution code we use in this paper is the same as in
Kobayashi et al. (2020) but includes CEJSNe as the r-process site instead of
magnetorotational supernovae and neutron star mergers. The slow neutron
capture process and electron capture supernovae are included, but these do not
contribute to the production of europium (see Fig. 32 of Kobayashi et al.
2020). The code provides the production of elements following gas inflow and
star formation in the solar neighborhood. The timescales of the inflow and
star formation are determined from independent observational constraints,
e.g., the metallicity distribution function (see Fig. 2 of Kobayashi et al.
2020), and the results do not depend so much on these parameters (see Fig. A1
of Kobayashi et al. 2020). The resultant star formation history shows a peak
at a few Gyrs ago. The natal kick of the NS is not included.

As in Kobayashi et al. (2020), Kroupa IMF is adopted for $0.01M_{\odot}$ to
$50M_{\odot}$, and a significant amount of Fe is produced by CCSNe
(hypernovae) in addition to Type Ia supernovae (SNe Ia). The nucleosynthesis
yields and progenitor model of SNe Ia are taken from Kobayashi, Leung & Nomoto
(2020), which is based on Chandrasekhar explosions of C+O white dwarfs in the
single degenerate scenario. As a result the observed $\rm[O/Fe]-[Fe/H]$
relation is very well reproduced including a sharp decreasing trend from
[Fe/H]. We note though that other SN Ia scenarios exist (e.g., table 1 in
Soker 2019).

The time-metallicity evolution of r-process elements depends strongly on the
typical delay time from star formation to r-process nucleosynthesis by the
events. For CEJSNe, this delay is mostly due to the time it takes to form the
RSG-NS binary system, as the timescale of the CEE is much shorter. This delay
time resembles the typical time to form a binary NS system since they both
depend mainly on the stellar evolution time of massive stars (few to tens of
$~{}\rm{Myr}$; e.g., Crowther 2012). However, since the merger of the NSs is
driven by radiation of gravitational waves, the additional delay until the
merger is usually much longer (tens of $~{}\rm{Myr}$ to millions of
$~{}\rm{Gyr}$; e.g., Beniamini & Piran 2019). According to Grichener & Soker
(2019b) the typical delay time from star formation to r-process enrichment by
CEJSNe is $t_{\rm d}\simeq 10-30~{}\rm{Myr}$.

To find the europium time evolution in the Milky-Way (MW) due to CEJSNe we
estimate the europium mass from a single event. The total mass of r-process
yield per one CEJSN r-process event is $\rm 0.01-0.03M_{\rm\odot}$ (Grichener
& Soker, 2019a). We assume the solar abundance distribution of r-process
elements. According to table 2 in Côté et al. (2018) the ratio of the mass
fraction of europium to the total mass fraction of r-process elements in the
solar r-process residuals is $\simeq 10^{-3}$. Therefore, for the CEJSN
r-process yield estimated in Grichener & Soker (2019a) the europium mass that
one typical CEJSN event forms is $M_{\rm Eu}\simeq 1\times\rm 10^{-5}-3\times
10^{-5}M_{\rm\odot}$.

##  3 The ${\rm[Eu/Fe]}$-${\rm[Fe/H]}$ evolution plane

###  3.1 Observations and possible sites

Observations show that the ratio of europium to iron in the MW changes with
metalicity. In Figs. 1 and 2 we present the observed values of $\rm[Eu/Fe]$
versus $\rm[Fe/H]$ for stars in the MW (black dots). We use the standard
notation $\rm[X/Y]=log_{10}(N_{\rm X}/N_{\rm Y})-log_{10}(N_{\rm X}/N_{\rm
Y})_{\rm\odot}$, where $N_{\rm X}$ and $N_{\rm Y}$ are the column density of
element X and element Y, respectively. For older stars with lower metalicities
of ${\rm[Fe/H]}\lesssim-1$ there is a large scatter in the values of
${\rm[Eu/Fe]}$ around about the same average value of $\rm[Eu/Fe]\simeq 0.4$,
while for stars with ${\rm[Fe/H]\gtrsim-1}$ the ratio $\rm[Eu/Fe]$ decreases
with increasing $\rm[Fe/H]$. This decrease is commonly known as the evolution
‘knee’. The transition between the two behaviours occurs for
${\rm[Fe/H]}\simeq-1$, corresponding to a Galaxy age of $\simeq 2~{}\rm{Gyr}$.

Figure 1: The ${\rm[Eu/Fe]}$ over ${\rm[Fe/H]}$ distribution in the Milky-Way
due to the contribution of common envelope jets supernovae that produce
europium mass of $M_{\rm Eu}=2\times 10^{-5}M_{\rm\odot}$ and have delay times
of $t_{\rm d}=10~{}\rm{Myr}$ (orange large dots), $t_{\rm d}=20~{}\rm{Myr}$
(yellow large dots) and $t_{\rm d}=30~{}\rm{Myr}$ (purple large dots). The
black dots represent observational data taken from Cayrel et al. (2004), Honda
et al. (2004), Hansen et al. (2012), Hansen et al. (2014), Roederer et al.
(2014) and Zhao et al. (2016).  Figure 2: Same as Fig. 1 for common envelope
jets supernovae with a delay time of $t_{\rm d}=20~{}\rm{Myr}$ that produce
europium mass of $M_{\rm Eu}=1\times 10^{-5}M_{\rm\odot}$ (orange large dots),
$M_{\rm Eu}=2\times 10^{-5}M_{\rm\odot}$ (yellow large dots) and $M_{\rm
Eu}=3\times 10^{-5}M_{\rm\odot}$ (purple large dots).

The large europium scatter observed at low metalicites suggests that r-process
nucleosynthesis sites are rare (e.g., François et al. 2007), and that the
$\rm[Eu/Fe]$ chemical evolution pattern might have been produced by dilution
of the gas that contained the heavy elements before the beginning of extensive
iron production (e.g., Shen et al. 2015). This implies that at least one of
the sites that contribute to r-process production should be spatially
uncorrelated with iron sources. We note, however, that total un-correlation
with iron results in too large scatter of $\rm[Eu/Fe]$ (e.g., Haynes &
Kobayashi 2019).

The high natal kick velocities of the binary NS systems can bring them far
from their birthplace before they merge, away from the location of their
progenitor iron-producing CCSNe. However, Safarzadeh et al. (2019) found that
binary NS mergers cannot account for r-process enrichment in the MW at early
times given their long delay times (for another view see Beniamini & Piran
2019). Collapsars, on the other hand, do not have natal kicks and therefore
the r-process elements they produce are likely to mix with the iron from the
CCSN explosion, resulting in overproduction of r-process elements relative to
iron in metal poor stars in the MW (e.g., Macias & Ramirez-Ruiz 2019; Fraser &
Schönrich 2021).

Binary NS mergers have a problem accounting for the ${\rm[Eu/Fe]}$ trend in
the MW for higher metalicities as well. The commonly accepted delay time
distribution expected from binary NS mergers (of $\propto t^{-1}$) cannot
explain the evolution ‘knee’, leading to the need of either a modification of
the delay time distribution (e.g., Beniamini & Piran 2019) or addition of
other sources of heavy elements (e.g., Kobayashi et al. 2020). Banerjee et al.
(2020) argue that taking the natal kicks of binary NSs and the formation
history of the MW into consideration can naturally reproduce the evolution
‘knee’ without the need for neither of them due to the effect of kick induced
migration on the frequency and effective delay time distribution of binary NS
mergers. Due to all the opposing views (see also Côté et al. 2019; Wehmeyer et
al. 2019), further research is needed to fully resolve the contribution of
binary NSs to r-process enrichment in the Galaxy.

Our study focuses on CEJSNe as a possible r-process site. CEJSNe experience a
natal kick from the CCSN that forms the NS in the binary system, driving the
binary system away from the explosion site. Since the newly born NS has a
massive companion, the natal kick of the binary system is moderate (see
Grichener & Soker 2019b), leading to some correlation with iron production. In
addition, the propagation of the jets dilute the r-process in a large volume.
These two effects prevent spatial correlation between the iron production and
the later r-process nucleosynthesis. Moreover, the relatively short delay time
of CEJSNe allows them to account for r-process production in the early Galaxy
and for the evolution ‘knee’ at higher metalicites, as we show next.

###  3.2 The contribution of the CEJSN-r process scenario

We present our results of the Galactic chemical evolution model of the CEJSN
r-process scenario (section 2) by the large-colored dots in Figs. 1 and 2. Our
calculations assume that the CEJSNe r-process scenario alone accounts for the
formation of europium in the MW.

In Fig. 1 we assume that a single CEJSN r-process event produces europium mass
of $M_{\rm Eu}=2\times 10^{-5}M_{\rm\odot}$. The three different curves
correspond to three different delay times as we indicate in the figure. We
find that for ${\rm[Fe/H]}\gtrsim-1$ all curves coincide and reproduce the
observed evolution ‘knee’. Moreover, the possible delay times of the CEJSN
r-process scenario (Grichener & Soker 2019b) are sufficiently short to produce
europium at lower metalicites. The different delay times cover a range of
average values of $\rm[Eu/Fe]$ at very low metalicities. The nature of the
Galactic chemical evolution models does not allow to study individual events,
and therefore we do not expect to cover the entire range of stars in the
figure.

In Fig. 2 we present the evolution on the same plane for CEJSNe with a delay
time of $t_{\rm d}=20~{}\rm{Myr}$ and three possible values of europium mass
per event as we indicate in the figure. All three curves have the same trend.
Each curve reproduces the descending evolution ‘knee’ at higher metalicites of
${\rm[Fe/H]}\gtrsim-1$ , and has a more or less constant average value for
older stars. We learn that when keeping other parameters unchanged, on average
one CEJSN r-process event cannot produce less than $M_{\rm Eu}\simeq 1.5\times
10^{-5}M_{\rm\odot}$ of europium nor more than $M_{\rm Eu}\simeq 2.5\times
10^{-5}M_{\rm\odot}$ to reproduce the observations.

##  4 Summary

We used a Galactic chemical evolution model (section 2) to find the
contribution of the CEJSN r-process scenario to the europium abundances in the
MW. We adopted the parameters of the CEJSN r-process scenario from previous
studies (Grichener & Soker, 2019a, b) and found that this scenario can account
for the observations (Figs. 1 and 2). In particular, we can reproduce the
evolution ‘knee’ together with the delayed iron production from SNe Ia and
also the average europium abundance at early times because of the short
timescales.

The chemical evolution model studies a large population of events. Therefore,
the results in the figures present the average behavior. We note that
individual CEJSN r-process events might have a diverse range of properties,
which could explain the observed scatter of r-process elements. For instance,
the NS-RSG merger that initiates the CEJSN event can take place at different
time delays and with different RSG properties, both of which influence the
accreted mass onto the NS and therefore the r-process yield per event.

Some of the parameters that make the events diverse are the amount of iron
that the NS progenitor has synthesised, the mass of CO core of the RSG at the
time of the event, and the NS mass. Moreover, the jets that the NS launches
mix with the interstellar medium that surrounds the binary system. The
distance the jets propagate depends on the density of the material around the
jets and the presence of dense clouds. This leads the r-process material
synthesized in the jets to mix with gas that contains different amounts of
iron. All these properties influence both the amount of europium and the
amount of iron it mixes with, explaining the scatter in the observed
${\rm[Eu/Fe]}$.

Overall, our study demonstrates the advantages of the CEJSN r-process scenario
in accounting for the r-process at very early times and in explaining the
later evolution. The next step would be to perform hydrodynamical simulations
of the NS-core merger, including the launching of jets by the NS and the
nucleosynthesis inside the jets including detailed neutrino transport.

## Acknowledgments

This research was supported by a grant from the Israel Science Foundation
(769/20). CK acknowledges funding from the UK Science and Technology Facility
Council (STFC) through grant ST/R000905/1 & ST/V000632/1. The work was also
funded by a Leverhulme Trust Research Project Grant on “Birth of Elements”.

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on the space $N$, and hence on $\operatorname{Teich}(\Sigma)$.

Consider the symplectic structure on
$\widetilde{N}=\mathbb{R}_{>0}\times\widetilde{\operatorname{Diff}}_{\mathsf{o}}(S^{1})$,
given by (60) with ${\mathsf{F}}$ replaced by a lift
$\widetilde{{\mathsf{F}}}$ to the universal cover. We may regard
$\widetilde{{\mathsf{F}}}$ as $\mathbb{Z}$-equivariant function on
$\mathbb{R}$, that is,
$\widetilde{{\mathsf{F}}}(x+1)=\widetilde{{\mathsf{F}}}(x)+1$. The expression

(63)
$u(x)=\log(\widetilde{{\mathsf{F}}}^{\prime}(x))+{\ell}(\widetilde{{\mathsf{F}}}(x)-x);$

is a periodic function on $\mathbb{R}$. Taking into account the dependence on
$\ell,\widetilde{{\mathsf{F}}}$, this is a function on $\widetilde{N}$ with
values in $|\Omega|^{0}_{S^{1}}$. Thus
${\mathsf{d}}u\wedge{\mathsf{d}}u^{\prime}$ is a 2-form on $N$ with values in
$|\Omega|^{1}_{S^{1}}$; integrating over $S^{1}$ it is a 2-form on
$\widetilde{N}$.

######  Proposition 8.4.

$\omega_{\widetilde{N}}=-{\frac{1}{2}}{\mathsf{d}}\ell\wedge{\mathsf{d}}u_{0}+\frac{1}{4}\int_{S^{1}}\
{\mathsf{d}}u\wedge{\mathsf{d}}u^{\prime}$

where $u_{0}=\int_{S^{1}}\ u$ (a scalar function of
$(\ell,\widetilde{\mathsf{F}})\in\widetilde{N}$)

###### Proof.

We work out the terms appearing in

$\frac{1}{4}\int_{S^{1}}\
{\mathsf{d}}u\wedge{\mathsf{d}}u^{\prime}=\frac{1}{4}\int_{S^{1}}{\mathsf{d}}\big{(}\log(\widetilde{{\mathsf{F}}}^{\prime})+{\ell}(\widetilde{{\mathsf{F}}}-\operatorname{Id})\big{)}\wedge{\mathsf{d}}\big{(}\log(\widetilde{{\mathsf{F}}}^{\prime})+{\ell}(\widetilde{{\mathsf{F}}}-\operatorname{Id})\big{)}^{\prime}$

according to their homogeneity with respect to $\ell$, and compare to the
corresponding terms in (61). The term of homogeneity $0$ is
$\frac{1}{4}(\widetilde{F^{\prime}})^{-2}{\mathsf{d}}\widetilde{F}^{\prime}\wedge{\mathsf{d}}\widetilde{F}^{\prime\prime}$,
matching that in (61). The terms of homogeneity $1$ are (using integration by
parts to combine two terms)

${\frac{1}{2}}\int_{S^{1}}{\mathsf{d}}\widetilde{F}^{\prime}\wedge{\mathsf{d}}\ell+{\frac{1}{2}}\int_{S^{1}}{\mathsf{d}}\ell\wedge{\mathsf{d}}\log(\widetilde{F}^{\prime})$

The first integral is zero, by the fundamental theorem of calculus, while the
second integral gives one of the terms of
${\frac{1}{2}}{\mathsf{d}}\ell\wedge{\mathsf{d}}u_{0}$. The terms of
homogeneity 2 are

$\displaystyle=\frac{1}{4}\int_{S^{1}}\ell^{2}{\mathsf{d}}\widetilde{{\mathsf{F}}}\wedge{\mathsf{d}}\widetilde{{\mathsf{F}}}^{\prime}+{\frac{1}{2}}\int_{S^{1}}(\widetilde{{\mathsf{F}}}^{\prime}-1){\mathsf{d}}\widetilde{{\mathsf{F}}}\wedge\ell{\mathsf{d}}\ell$

(again we used an integration by parts to combine two terms). The integral

$-\int_{S^{1}}{\mathsf{d}}\widetilde{F}\wedge\ell{\mathsf{d}}\ell={\mathsf{d}}\ell\wedge{\mathsf{d}}\int_{S^{1}}\ell\widetilde{F}={\mathsf{d}}\ell\wedge{\mathsf{d}}\int_{S^{1}}\ell(\widetilde{F}-\operatorname{Id})$

gives one of the terms in ${\mathsf{d}}\ell\wedge{\mathsf{d}}u_{0}$. The
remaining terms match the corresponding terms in (61). ∎

Having $\omega_{\widetilde{N}}$ in this form, it is straightforward to
introduce Darboux coordinates, by its Fourier expansion:

$u(x)=\sum_{n\in\mathbb{Z}}u_{n}e^{2\pi inx}.$

We obtain:

######  Proposition 8.5.

The symplectic form on $\widetilde{N}$ is given by

$\omega_{\widetilde{N}}=-{\frac{1}{2}}{\mathsf{d}}\ell\wedge{\mathsf{d}}u_{0}+\pi
i\sum_{m>0}m\ {\mathsf{d}}u_{-m}\wedge{\mathsf{d}}u_{m}$

Rewriting the second sum in terms of real and imaginary parts of $u_{m}$, one
obtains a Darboux normal form.

##  Appendix A Affine moment maps, central extensions

###  A.1. Central extensions

Let $H$ be a Lie group with Lie algebra $\mathfrak{h}$. Given a central
extension

$0\to\mathbb{R}\to\widehat{\mathfrak{h}}\to\mathfrak{h}\to 0,$

let $\mathcal{E}\subseteq\widehat{\mathfrak{h}}^{*}$ be the affine space of
linear functionals taking $1\in\mathbb{R}$ to $1$. The adjoint action of $H$
on $\widehat{\mathfrak{h}}$ defines an affine $H$-action on this space, with
linear part the coadjoint action on $\mathfrak{h}^{*}$. Conversely, suppose
$\mathcal{E}$ is an affine space with an affine $H$-action, with underlying
linear $H$-space the coadjoint representation. For $\mu\in\mathcal{E}$, the
map $\mathfrak{h}\to\mathfrak{h}^{*},\ \xi\mapsto\xi.\mu$, defined by the
infinitesimal action, is a Lie algebra cocycle. Suppose this cocycle is skew-
symmetric:

(64) $\langle\xi.\mu,\ \eta\rangle=-\langle\eta.\mu,\ \xi\rangle$

for all $\xi,\eta$. (This condition does not depend on the choice of $\mu$.)
Then one obtains a central extension: Take $\widehat{\mathfrak{h}}$ to be the
vector space of affine-linear functions on $\mathcal{E}$, with bracket

$[\widehat{\xi},\widehat{\eta}](\mu)=\langle\xi.\mu,\ \eta\rangle.$

Here $\xi,\eta$ are the linear functionals underlying
$\widehat{\xi},\widehat{\eta}\in\widehat{\mathfrak{h}}$. The affine space
$\mathcal{E}$ is a Poisson submanifold of $\widehat{\mathfrak{h}}^{*}$, with
symplectic leaves the coadjoint orbits. The action groupoid
$H\times\mathcal{E}\rightrightarrows\mathcal{E}$ has a canonical symplectic
structure making it into a symplectic groupoid. See [3, Example A.12] for
further discussion.

###  A.2. Affine moment maps

Let $\mathcal{E}$ be an affine $H$-space as above, with underlying linear
$H$-space the coadjoint representation. Given an $H$-manifold $M$ with a
closed invariant 2-form $\omega$, we may consider $\mathcal{E}$-valued affine
moment maps

$\Phi\colon M\to\mathcal{E};$

that is, $\Phi$ is $H$-equivariant and satisfies the moment map
condition444Generating vector fields are defined in terms of their action on
functions as $(\xi_{M}f)(m)=\frac{d}{dt}|_{t=0}f\big{(}\exp(-t\xi).m\big{)}$.
With this convention, $\xi\mapsto\xi_{M}$ is a Lie algebra morphism. Note that
if $M=\mathcal{E}$ is an affine space, then $\xi_{M}(\mu)=-\xi.\mu$.

$\omega(\xi_{M},\cdot)=-\langle{\mathsf{d}}\Phi,\xi\rangle.$

Note that the differential ${\mathsf{d}}\Phi$ is a 1-form with values in the
linear space $\mathfrak{h}^{*}$, hence its pairing with $\xi$ is defined.
Examples of Hamiltonian $H$-spaces with $\mathcal{E}$-valued moment maps are
the coadjoint orbits in $\mathcal{E}$, with the KKS symplectic structure.

A necessary condition for the existence of an affine moment map (for some
$\mathcal{E}$) is that the 1-forms $\alpha\in\Omega^{1}(M,\mathfrak{h}^{*})$,
given as $\langle\alpha,\xi\rangle=-\omega(\xi_{M},\cdot)$, are exact. In
fact, this condition is sufficient as well:

######  Proposition A.1.

Let $M$ be a connected $H$-manifold, with an invariant closed 2-form
$\omega\in\Omega^{2}(M)$. Suppose that the 1-form

$\alpha\in\Omega^{1}(M,\mathfrak{h}^{*})$

given as $\langle\alpha,\xi\rangle=\omega(\xi_{M},\cdot)$ is exact. Let
$\mathcal{E}$ be the affine space of all primitives of $\alpha$, and let

$\Phi\colon M\to\mathcal{E}$

be the map taking any point of $M$ to the unique primitive vanishing at that
point. Then $\mathcal{E}$ is an affine $H$-space, with underlying linear
action the coadjoint action, satisfying the skew-symmetry (64). Furthermore,
$\Phi$ is an affine moment map.

###### Proof.

Since $M$ is connected, a primitive of $\alpha$ is unique up to a constant
function with values in $\mathfrak{h}^{*}$. This shows that $\mathcal{E}$ is
an affine space over $\mathfrak{h}^{*}$. The group $H$ acts on
$\mathcal{E}\subseteq C^{\infty}(M,\mathfrak{h}^{*})$ by
$(h.f)(m)=\operatorname{Ad}_{h}\big{(}f(h^{-1}.m)\big{)}$; hence the
difference of two elements transforms under the coadjoint action. For
$f\in\mathcal{E}$ we have

$\langle\xi.f,\eta\rangle=\langle\mathcal{L}_{\xi_{M}}f+\operatorname{ad}_{\xi}f,\eta\rangle=-\omega(\xi_{M},\eta_{M})-\langle
f,[\xi,\eta]\rangle,$

which is skew-symmetric in $\xi,\eta$. To verify that $\Phi$ (as in the
proposition) is a moment map, fix $m_{0}\in M$. Then
$\Phi(m_{0})(m)=\int_{m_{0}}^{m}\alpha=-\Phi(m)(m_{0})$. Hence, the map
$m\mapsto\Phi(m)|_{m_{0}}$ is a primitive for $-\alpha$. This shows
$\langle{\mathsf{d}}\Phi,\xi\rangle=-\alpha(\xi)=-\omega(\xi_{M},\cdot)$. ∎

Note that $\mathcal{E}$, and hence the central extension of $\mathfrak{h}$, is
determined by the pullback of $\omega$ to any $G$-invariant submanifold of
$M$. For example, if the action has a fixed point, or more generally if it
admits an invariant isotropic submanifold, then the central extension is
trivial, and the action admits an ordinary $\mathfrak{h}^{*}$-valued moment
map.

The constructions above apply to infinite-dimensional settings, provided that
one has a reasonable notion of smooth dual. In particular, the affine action
of $\operatorname{Diff}_{+}(\mathsf{C})$ on the space
$\operatorname{Hill}(\mathsf{C})$ of Hill operators (Section 7.1) defines a
central extension of $\operatorname{Vect}(\mathsf{C})$, the Virasoro algebra.

##  Appendix B Gauge theory constructions

In this appendix, we review the Atiyah-Bott construction for principal
$G$-bundles $P\to\Sigma$ over oriented surfaces with boundary. Here $G$ is any
Lie group with an invariant inner product on its Lie algebra (we are mainly
interested in the case $G=\operatorname{PSL}(2,\mathbb{R})$).

###  B.1. Atiyah algebroid, connections

Let $P\to M$ be a principal $G$-bundle. The groups of gauge transformations
and automorphisms are denoted
$\operatorname{Gau}(P)\subseteq\operatorname{Aut}(P)$. The _Atiyah algebroid_
$\operatorname{At}(P)=TP/G\to M$ is the Lie algebroid whose sections are the
$G$-invariant vector fields on $P$, that is, infinitesimal automorphisms. It
fits into the exact sequence of Lie algebroids

(65)
$0\longrightarrow\mathfrak{g}(P)\longrightarrow\operatorname{At}(P)\stackrel{{\scriptstyle\mathsf{a}}}{{\longrightarrow}}TM\longrightarrow
0,$

with the adjoint bundle $\mathfrak{g}(P)=P\times_{G}\mathfrak{g}$. On the
level of sections, this is the exact sequence
$0\to\mathfrak{gau}(P)\to\mathfrak{aut}(P)\to\operatorname{Vect}(M)\to 0$. A
principal connection $\theta\in\Omega^{1}(P,\mathfrak{g})^{G}$ is equivalent
to a vector bundle splitting of the Atiyah sequence. This may be described by
either of the bundle maps

$s^{\theta}\colon\operatorname{At}(P)\to\mathfrak{g}(P),\ \ \ j^{\theta}\colon
TM\to\operatorname{At}(P)$

called _vertical projection_ and _horizontal lift_ ; thus
$s^{\theta}|_{\mathfrak{g}(P)}=\operatorname{id}_{\mathfrak{g}(P)}$,
$\mathsf{a}\circ j^{\theta}=\operatorname{id}_{TM},\ s^{\theta}\circ
j^{\theta}=0$. The section $s^{\theta}(v)$ corresponds to $\iota_{v}\theta$
under the identification $\mathfrak{g}(Q)\cong\Omega^{0}(Q,\mathfrak{g})^{G}$.
We denote by

${\mathsf{d}}^{\theta}\colon\Omega^{p}(M,\mathfrak{g}(P))\to\Omega^{p+1}(M,\mathfrak{g}(P))$

the covariant derivative; in terms of the identification of forms
$\beta\in\Omega^{\bullet}(M,\mathfrak{g}(P))$ with $G$-basic forms
$\widetilde{\beta}\in\Omega^{\bullet}(P,\mathfrak{g})$ we have
$\widetilde{{\mathsf{d}}^{\theta}\beta}={\mathsf{d}}\widetilde{\beta}+[\theta,\widetilde{\beta}]$.
The curvature $F^{\theta}\in\Omega^{2}(M,\mathfrak{g}(P))$ corresponds to the
basic form
$\widetilde{F}^{\theta}={\mathsf{d}}\theta+{\frac{1}{2}}[\theta,\theta]$.

The set $\mathcal{A}(P)$ of principal connections is an affine space over
$\Omega^{1}(M,\mathfrak{g}(P))$: given a smooth family of connections
$\theta_{t}$ with $\theta_{0}=\theta$, the corresponding tangent vector
$\beta\in\Omega^{1}(M,\mathfrak{g}(P))$ is determined by either one of the
equations

$\frac{d}{dt}|_{t=0}\theta_{t}=\widetilde{\beta},\ \ \
\frac{d}{dt}|_{t=0}s^{\theta_{t}}=\beta\circ\mathsf{a},\ \ \
\frac{d}{dt}|_{t=0}j^{\theta_{t}}=-\beta.$

######  Lemma B.1.

The generating vector fields for the $\operatorname{Aut}(P)$-action on
$\mathcal{A}(P)$ are given by

(66)
$v_{\mathcal{A}(P)}|_{\theta}=-{\mathsf{d}}^{\theta}(s^{\theta}(v))-\iota_{\mathsf{a}(v)}F^{\theta},\
\ v\in\mathfrak{aut}(P)=\Gamma(\operatorname{At}(P)).$

In particular, for
$v\in\Gamma(\mathfrak{g}(P))=\Omega^{0}(M,\mathfrak{g}(P))$, the generating
vector field is $-{\mathsf{d}}^{\theta}v$.

###### Proof.

By our sign convention for generating vector fields,
$v_{\mathcal{A}(P)}|_{\theta}=-v.\theta$. Regarding $v$ as a $G$-invariant
vector field $\widetilde{v}$ on $P$, we have

$\widetilde{v.\theta}=\mathcal{L}_{\widetilde{v}}\theta={\mathsf{d}}\iota_{\widetilde{v}}\theta+\iota_{\widetilde{v}}{\mathsf{d}}\theta=({\mathsf{d}}+[\theta,\cdot])\iota_{\widetilde{v}}\theta+\iota_{\widetilde{v}}\widetilde{F}^{\theta}=\widetilde{{\mathsf{d}}^{\theta}(s^{\theta}(v))}+\widetilde{\iota_{\mathsf{a}(v)}F^{\theta}}.\qed$

###  B.2. Trivializations

Suppose $P\to M$ admits a section $\iota\colon M\to P$, defining a
trivialization $P\cong M\times G$ such that $\iota(m)=(m,e)$ with the
principal action $a.(m,g)=(m,ga^{-1})$. Connections $\theta\in\mathcal{A}(P)$
are described in terms of their connection 1-forms $A=\iota^{*}\theta$ by

$\theta=\operatorname{Ad}_{g^{-1}}(A)+\operatorname{pr}_{2}^{*}\theta^{L}$

The trivial connection (given by $\theta=\operatorname{pr}_{2}^{*}\theta^{L}$)
gives a splitting of the Atiyah algebroid $\operatorname{At}(P)\cong
TM\oplus\mathfrak{g}(P)$. The trivialization of $P$ determines a
trivialization of all its associated bundles, and in particular gives an
isomorphism

(67) $\mathfrak{g}(P)\cong M\times\mathfrak{g}.$

We have

$s^{\theta}(X,\xi)=\xi+\iota_{X}A,\ \ j^{\theta}(X)=(X,-\iota_{X}A).$

The covariant derivative is the operator on
$\Omega^{\bullet}(M,\mathfrak{g}(P))\cong\Omega^{\bullet}(M,\mathfrak{g})$
given by ${\mathsf{d}}_{A}={\mathsf{d}}+[A,\cdot]$; the curvature is
$F_{A}=\iota^{*}\widetilde{F}^{\theta}={\mathsf{d}}A+{\frac{1}{2}}[A,A]$.

######  Remark B.2 (Signs, I).

We stress that the isomorphism

(68) $\mathfrak{gau}(P)=C^{\infty}(M,\mathfrak{g})$

given by (67) differs by sign from the ‘standard’ identification as the Lie
algebra of $\operatorname{Gau}(P)=C^{\infty}(M,G)$ (with pointwise
multiplication). In fact, (68) takes a function $\xi\in
C^{\infty}(M,\mathfrak{g})$ to the vertical vector field whose restriction to
$\\{m\\}\times G$ is $\xi(m)^{R}$. In particular, (68) induces _minus_ the
pointwise Lie bracket on $C^{\infty}(M,\mathfrak{g})$. More generally,
$\mathfrak{aut}(P)=C^{\infty}(M,\mathfrak{g})\rtimes\operatorname{Vect}(M)$
with the bracket

$[(\xi,X),(\eta,Y)]=(-[\xi,\eta],\mathcal{L}_{X}\eta-\mathcal{L}_{Y}\xi).$

###  B.3. Central extensions

Suppose $Q\to\mathsf{C}$ is a principal $G$-bundle over a compact oriented
1-manifold, and that the Lie algebra $\mathfrak{g}$ carries an invariant
metric (denoted by a dot). Then $\mathfrak{g}(Q)$ inherits a bundle metric.
The affine space $\mathcal{A}(Q)$ of connections has
$\mathfrak{gau}(Q)^{*}=\Omega^{1}(\mathsf{C},\mathfrak{g}(Q))$ as its
underlying linear space, where the pairing with
$\mathfrak{gau}(Q)=\Omega^{0}(\mathsf{C},\mathfrak{g}(Q))$ is given by the
metric followed by integration. This identification takes the linear part of
the gauge action to the coadjoint action. Furthermore,
$\langle\xi.\theta,\eta\rangle=-\langle{\mathsf{d}}^{\theta}\xi,\eta\rangle$
satisfies the skew-symmetry condition (64). By the discussion of Appendix A.1,
this defines a central extension of $\mathfrak{gau}(Q)$.

######  Remark B.3 (Signs, II).

For trivial bundles $Q=\mathsf{C}\times G$, one often uses the ‘standard’
identification $\mathfrak{gau}(Q)\cong C^{\infty}(\mathsf{C},\mathfrak{g})$ to
define the pairing. As explained in Remark B.2, this is opposite to the
identification coming from $\mathfrak{g}(Q)\cong\mathsf{C}\times\mathfrak{g}$,
and hence results in the opposite pairing.

We may also consider the larger group $\operatorname{Aut}(Q)$ of all principal
bundle automorphisms, with Lie algebra
$\mathfrak{aut}(Q)=\Gamma(\operatorname{At}(Q))$; its smooth dual is
$\mathfrak{aut}(Q)^{*}=\Omega^{1}(\mathsf{C},\operatorname{At}(Q)^{*})$. Let

(69)
$\mathcal{E}(Q)\subseteq\Gamma(\operatorname{Sym}^{2}\operatorname{At}(Q)^{*})$

be the affine space of all fiberwise quadratic forms on $\operatorname{At}(Q)$
whose restriction to $\mathfrak{g}(Q)$ is given by
$\xi\mapsto{\frac{1}{2}}\xi\cdot\xi$. The underlying linear space consists of
quadratic forms on $\operatorname{At}(Q)$ whose restriction to
$\mathfrak{g}(Q)$ is zero; it is identified with the space
$\Omega^{1}(\mathsf{C},\operatorname{At}(Q)^{*})$, where an element $\gamma$
of this space defines the quadratic form
$w\mapsto\langle\gamma(\mathsf{a}(w)),w\rangle$. The group
$\operatorname{Aut}(Q)$ acts on $\mathcal{E}(Q)$, and the underlying linear
action is coadjoint action. One may also check that it satisfies the skew-
symmetry property (64); hence one obtains a central extension of
$\mathfrak{aut}(Q)$.

######  Remark B.4.

There is a natural map

$\mathcal{E}(Q)\to\mathcal{A}(Q),\ \ \phi\mapsto\theta$

given by $s^{\theta}(v)\cdot\xi=-2\phi(v,\xi)$ for
$v\in\Gamma(\operatorname{At}(Q),\ \xi\in\mathfrak{g}(Q)$ (where we think of
$\phi$ as a symmetric bilinear form). This map is affine with respect to the
quotient map
$\Omega^{1}(\mathsf{C},\operatorname{At}(Q)^{*})\to\Omega^{1}(\mathsf{C},\mathfrak{g}(Q)^{*})$,
and determines a Lie algebra morphism
$\widehat{\mathfrak{gau}}(Q)\to\widehat{\mathfrak{aut}}(Q)$ lifting the
natural inclusion.

######  Remark B.5.

There is a natural $\operatorname{Aut}(Q)$-equivariant section section

(70) $\mathcal{A}(Q)\to\mathcal{E}(Q),\ \
\theta\mapsto{\frac{1}{2}}s^{\theta}\stackrel{{\scriptstyle.}}{{\vee}}s^{\theta}.$

Here $\vee$ denotes the product in the symmetric algebra. The map (70) is the
moment map for the $\operatorname{Aut}(Q)$-action on $\mathcal{A}(Q)$, in the
sense that it restricts to moment maps on the coadjoint orbits
$\mathcal{O}\subseteq\mathcal{A}(Q)$.

For a trivial bundle $Q=\mathsf{C}\times G$, we have
$\mathcal{A}(Q)=\Omega^{1}(\mathsf{C},\mathfrak{g})$ and
$\mathcal{E}(Q)=\Omega^{1}(\mathsf{C},\mathfrak{g})\times|\Omega|^{2}_{\mathsf{C}}$;
in terms of these identifications the map is

$A\mapsto(A,{\frac{1}{2}}A\cdot A).$

It is gauge equivariant for the action $h.(A,q)=(h\bullet A,\,q+A\cdot
h^{*}\theta^{L}-{\frac{1}{2}}h^{*}\theta^{L}\cdot h^{*}\theta^{L}$).

###  B.4. Atiyah-Bott

We now assume that $\Sigma$ is a compact oriented surface, possibly with
boundary, and that the Lie algebra $\mathfrak{g}=\operatorname{Lie}(G)$ comes
with an invariant metric. Let $P\to\Sigma$ be a principal $G$-bundle.

######  Definition B.6.

The _Atiyah-Bott form_ on $\mathcal{A}(P)$ is given by

$\omega_{AB}(a,b)=\int_{\Sigma}a\cdot b,\ \ \ \
a,b\in\Omega^{1}(\Sigma,\mathfrak{g}(P)).$

The Atiyah-Bott form $\omega_{AB}$ is closed, by translation invariance, and
nondegenerate in the weak sense that $\omega_{AB}(a,\cdot)=0\Leftrightarrow
a=0$. The 2-form is invariant under the action of the group
$\phantom{\cdot}{}^{b}\\!\\!\operatorname{Aut}_{+}(P)$ of principal bundle
automorphisms whose base map lies in
$\phantom{\cdot}{}^{b}\operatorname{Diff}_{+}(\Sigma)$. We are interested in a
moment map for this action. The Lie algebra
$\phantom{\cdot}{}^{b}\\!\\!\mathfrak{aut}(P)$ consists of $G$-invariant
vector fields on $P$ that are tangent to $\partial P$; these are the sections
of a Lie algebroid $\phantom{\cdot}{}^{b}\\!\\!\operatorname{At}(P)$. Let
$\partial P=P|_{\partial\Sigma}$ the boundary restriction of $P$. For a
connection $\theta$, denote by $\partial\theta$ its pullback to a connection
on $P$.

######  Proposition B.7.

The action of $\phantom{\cdot}{}^{b}\\!\\!\operatorname{Aut}_{+}(P)$ on
$\mathcal{A}(P)$ has moment map

$\mathcal{A}(P)\to\Omega^{2}(\Sigma,\operatorname{At}(P)^{*})\times\mathcal{E}(\partial
P),\ \theta\mapsto(-F^{\theta}\cdot
s^{\theta},{\frac{1}{2}}s^{\partial\theta}\stackrel{{\scriptstyle.}}{{\vee}}s^{\partial\theta}).$

For the action of $\operatorname{Gau}(P)$, the moment map takes values in
$\Omega^{2}(\Sigma,\mathfrak{g}(P))\times\mathcal{A}(\partial P)$, and is
given by $(-F^{\theta},\partial\theta)$.

###### Proof.

Let
$v\in\phantom{\cdot}^{b}\mathfrak{aut}(P)=\Gamma(\phantom{\cdot}^{b}\\!\\!\operatorname{At}(P))$,
with boundary restriction $\partial v$. Given $b\in
T_{\theta}\mathcal{A}(P)=\Omega^{1}(\Sigma,\mathfrak{g}(P))$, we have, using
(66),

$\displaystyle\omega_{AB}(v^{\sharp}|_{\theta},\,b)$
$\displaystyle=\int_{\Sigma}v^{\sharp}|_{\theta}\cdot b$
$\displaystyle=-\int_{\Sigma}\iota_{\mathsf{a}(v)}F^{\theta}\cdot
b-\int_{\Sigma}{\mathsf{d}}^{\theta}(s^{\theta}(v))\cdot b$
$\displaystyle=\int_{\Sigma}\iota_{\mathsf{a}(v)}b\cdot
F^{\theta}+\int_{\Sigma}s^{\theta}(v)\cdot{\mathsf{d}}^{\theta}b-\int_{\partial{\Sigma}}s^{\partial\theta}(\partial
v)\cdot\partial b.$

where $\partial b\in T_{\partial\theta}\mathcal{A}(\partial P)$ is the
pullback of $b$ to the boundary. Suppose $b\in T_{\theta}\mathcal{A}(P)$ is
realized as the velocity vector for a family of connections $\theta_{t}$ with

$\theta_{0}=\theta,\ \ \ \frac{d}{dt}\Big{|}_{t=0}\theta_{t}=\widetilde{b}.$

Then $\frac{d}{dt}\big{|}_{t=0}s^{\theta_{t}}=b\circ\mathsf{a},\
\frac{d}{dt}\big{|}_{t=0}F^{\theta_{t}}={\mathsf{d}}^{\theta_{t}}b$, and
therefore

$\frac{d}{dt}\Big{|}_{t=0}F^{\theta_{t}}\cdot
s^{\theta_{t}}=(b\circ\mathsf{a})\cdot
F^{\theta}+s^{\theta}\cdot{\mathsf{d}}^{\theta}b.$

Here $b\circ\mathsf{a},\ s^{\theta}$ are regarded as sections of
$\operatorname{At}(P)^{*}\otimes\mathfrak{g}(P)$, while $F^{\theta},\
{\mathsf{d}}^{\theta}b$ are elements of $\Omega^{2}(\Sigma,\mathfrak{g}(P))$.
For the boundary term, we note

$\frac{d}{dt}\Big{|}_{t=0}\left({\frac{1}{2}}s^{\partial\theta_{t}}\stackrel{{\scriptstyle.}}{{\vee}}s^{\partial\theta_{t}}\right)=s^{\partial\theta}\stackrel{{\scriptstyle.}}{{\vee}}(\partial
b\circ\mathsf{a})=s^{\partial\theta}\cdot\partial b$

where the last equality comes from the inclusion
$\Omega^{1}(\partial{\Sigma},\operatorname{At}(\partial
P)^{*})\hookrightarrow\Gamma(\operatorname{Sym}^{2}(\operatorname{At}(\partial
P)^{*}))$. This gives

$\omega_{AB}(v^{\sharp}|_{\theta},\,b)=\frac{d}{dt}\Big{|}_{t=0}\int_{\Sigma}F^{\theta_{t}}\cdot
s^{\theta_{t}}(v)-\int_{\partial{\Sigma}}\Big{(}\frac{d}{dt}\Big{|}_{t=0}{\frac{1}{2}}s^{\partial\theta_{t}}\vee
s^{\partial\theta_{t}}\Big{)}(\partial v,\cdot)\Big{)}.\qed$

######  Remark B.8 (Signs, III).

The sign in the moment map for $\operatorname{Gau}(P)$ depends on the
identifications $\mathfrak{gau}(P)\cong\Omega^{2}(P,\mathfrak{g}(P))$ and
$\mathfrak{gau}(\partial P)\cong\Omega^{1}(\partial P,\mathfrak{g}(\partial
P))$. For trivial bundles, one often uses the opposite pairing, resulting in a
sign change of the moment map. See Remarks B.2, B.3.

##  Appendix C The trumpet moduli space as a symplectic cross-section

Our starting point will be the interpretation of $N$ as a symplectic slice in
a symplectic groupoid
$\mathcal{G}\rightrightarrows\operatorname{Hill}(S^{1})$, see [3]. As a
groupoid,
$\mathcal{G}\cong\operatorname{Hill}(S^{1})\times\operatorname{Diff}_{\mathsf{o}}(S^{1})$
(an action groupoid), with source and target maps
$\mathsf{s}(T,{\mathsf{F}})=T,\ \mathsf{t}(T,{\mathsf{F}})={\mathsf{F}}.T$ and
groupoid multiplication
$(T_{1},{\mathsf{F}}_{1})\circ(T_{2},{\mathsf{F}}_{2})=(T_{2},{\mathsf{F}}_{1}\circ{\mathsf{F}}_{2})$.
In terms of this ‘left trivialization’, the symplectic structure
$\omega_{\mathcal{G}}$ is given by [3, Equation 47]

(71)
$\omega_{\mathcal{G}}=\int_{S^{1}}\Big{(}{\mathsf{d}}T\wedge\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}+T\,\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\wedge\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}^{\prime}-\frac{1}{4}\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}^{\prime\prime\prime}\wedge\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}\Big{)}.$

For our description of the 2-form for the trumpet, it will be more convenient
to work with ‘right trivialization’, expressing the 2-form in terms of
$(T_{0},{\mathsf{F}})$ where

(72)
$T={\mathsf{F}}^{-1}.T_{0}=({\mathsf{F}}^{\prime})^{2}T_{0}+{\frac{1}{2}}\mathcal{S}({\mathsf{F}}).$

######  Proposition C.1.

In right trivialization,

$\omega_{\mathcal{G}}={\mathsf{d}}\int_{S^{1}}\ \
\Big{(}T_{0}{\mathsf{F}}^{\prime}-\frac{1}{4}({\mathsf{F}}^{\prime})^{-1}{\mathsf{d}}{\mathsf{F}}^{\prime\prime}\Big{)}$

###### Proof.

This is based on a straightforward but lengthy calculation, substituting (72)
and simplifying. Here are some relevant steps. Using the formula for the
exterior differential of the Schwarzian derivative ([1, Lemma A.2]) one shows

$\int_{S^{1}}\
{\mathsf{d}}\mathcal{S}({\mathsf{F}})\wedge\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}=\int_{S^{1}}\
\frac{{\mathsf{d}}{\mathsf{F}}^{\prime}\wedge{\mathsf{d}}{\mathsf{F}}^{\prime\prime}}{(\mathsf{F}^{\prime})^{2}}$

This then implies

$\int_{S^{1}}\
{\mathsf{d}}T\wedge\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}=\int_{S^{1}}\
\Big{(}{\mathsf{F}}^{\prime}{\mathsf{d}}T_{0}\wedge{\mathsf{d}}\mathsf{F}+2T_{0}\,{\mathsf{d}}{\mathsf{F}}^{\prime}\wedge{\mathsf{d}}{\mathsf{F}}+\frac{1}{2}\frac{{\mathsf{d}}{\mathsf{F}}^{\prime}\wedge{\mathsf{d}}{\mathsf{F}}^{\prime\prime}}{(\mathsf{F}^{\prime})^{2}}\Big{)}$

Furthermore,

$\int_{S^{1}}\
T\,\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\wedge\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}^{\prime}=\int_{S^{1}}\
\Big{(}T_{0}\,{\mathsf{d}}{\mathsf{F}}^{\prime}\wedge{\mathsf{d}}{\mathsf{F}}+\frac{1}{2}\mathcal{S}({\mathsf{F}})\frac{{\mathsf{d}}{\mathsf{F}}\wedge{\mathsf{d}}{\mathsf{F}}^{\prime}}{(\mathsf{F}^{\prime})^{2}}\Big{)}$

and

$-\frac{1}{4}\int_{S^{1}}\
\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}^{\prime\prime\prime}\wedge\big{(}\frac{{\mathsf{d}}\mathsf{F}}{\mathsf{F}^{\prime}}\big{)}=\int_{S^{1}}\
\Big{(}-\frac{1}{4}\frac{{\mathsf{d}}{\mathsf{F}}^{\prime}\wedge{\mathsf{d}}{\mathsf{F}}^{\prime\prime}}{(\mathsf{F}^{\prime})^{2}}-\frac{1}{2}\mathcal{S}({\mathsf{F}})\frac{{\mathsf{d}}{\mathsf{F}}\wedge{\mathsf{d}}{\mathsf{F}}^{\prime}}{(\mathsf{F}^{\prime})^{2}}\Big{)}.$

Adding these three contributions, the formula for $\omega_{\mathcal{G}}$
follows. ∎

Passage to the slice
$N=\mathbb{R}_{>0}\times\operatorname{Diff}_{\mathsf{o}}(S^{1})$ amounts to
putting $T_{0}(x)=-\frac{\ell^{2}}{4}$, resulting in the formula (60).

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