# Spatial Data Mining of Public Transport Incidents reported in Social Media

Kamil Raczycki, Marcin Szymański,
Yahor Yeliseyenka<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Artificial
Intelligence,
Wrocław University of Science and TechnologyWrocławPoland and  Piotr
Szymański, Tomasz Kajdanowicz<EMAIL_ADDRESS>Department of Artificial Intelligence,
Wrocław University of Science and TechnologyWrocławPoland

###### Abstract.

Public transport agencies use social media as an essential tool for
communicating mobility incidents to passengers. However, while the short term,
day-to-day information about transport phenomena is usually posted in social
media with low latency, its availability is short term as the content is
rarely made an aggregated form. Social media communication of transport
phenomena usually lacks GIS annotations as most social media platforms do not
allow attaching non-POI GPS coordinates to posts. As a result, the analysis of
transport phenomena information is minimal. We collected three years of social
media posts of a polish public transport company with user comments. Through
exploration, we infer a six-class transport information typology. We
successfully build an information type classifier for social media posts,
detect stop names in posts, and relate them to GPS coordinates, obtaining a
spatial understanding of long-term aggregated phenomena. We show that our
approach enables citizen science and use it to analyze the impact of three
years of infrastructure incidents on passenger mobility, and the sentiment and
reaction scale towards each of the events. All these results are achieved for
Polish, an under-resourced language when it comes to spatial language
understanding, especially in social media contexts. To improve the situation,
we released two of our annotated data sets: social media posts with incident
type labels and matched stop names and social media comments with the
annotated sentiment. We also opensource the experimental codebase.

datasets, neural networks, gaze detection, text tagging

††conference: SIGSPATIAL ’21; November 02–-05, 2021; Beijing,
China††copyright: none††ccs: Applied computing Transportation††ccs:
Information systems Specialized information retrieval

##  1\. Introduction

Public transport companies have embraced social media in the recent decade to
perform multiple communicative functions. This rise of communication
concerning spatial phenomena has been followed by releases of public transport
schedules in standardized GTFS formats across the world. As social media
communication is very short-term oriented, often it is up to citizen data
science to retrieve information across longer periods of time, aggregate and
analyze them to and make wider communities, affected by transport phenomena,
aware of long-term changes in how public transport operates. These citizen
science challenges affect multiple areas of spatial language understanding,
geographic information mining, and aligning language to transport network
structures and more sophisticated data sets.

One such case can be made for a Wrocław, where following two decades of under-
investment, the local public transport service quality is in critical
condition as the city public transport operator - MPK Wrocław - struggles to
maintain a 10-day streak without tram derailment. As service disruptions occur
with high frequency, MPK Wrocław provides information about anomalies in the
trips, using information boards, dedicated applications, and social media
(Facebook and Twitter). MPK Wrocław carries about 200 million passengers a
year, about 550 thousand a day. It’s capacity to inform about service outages
is not only critical to maintaining a standard of life quality in daily
activities in the city but also indirectly concerns political decisions,
mobility behavior, carbon emissions, and many other areas of citizen science
activity.

While day-to-day communication is maintained regularly with low latency, it is
never released in aggregated form. More and more often agencies remove old
social media posts, to avoid citizens reacting to comments made under outdated
content. The grass-root ability to improve local quality of life depends on
inhabitants being able to make data-driven arguments. Citizen science
therefore, heavily relies on the automatic spatial understanding of social
media information related to public transport phenomena. Without an aggregated
data set released in an open data fashion, it is crucial to pinpoint where
traffic disruption happens in both GIS and mobility network perspectives.
Citizens need to be able to map social posts concerning service disruptions to
a GPS coordinate and a relevant stop or lines in the transportation network.

We present a method and a study of a public transport operator’s service
disruption communication in the situation of a persistent crisis. We analyze
posts and comments from the official social media channels of MPK Wrocław
concerning traffic obstructions and contribute the following results:

  * •

a systematic typology of communique and discuss the magnitude of social
response,

  * •

an approach for classification and spatial understanding of service disruption
alerts,

  * •

use cases to analyze the aggregated impact on the transportation network and
mobility and automatic examination of the sentiment of citizens’ feedback.

We built a robust method for automatic spatial language processing to allow
processing social media communication into a spatial database and show its use
cases. Our work concerns the Polish language, which is an under-resourced
language in this regard. Our method can work with any source of social media
posts, provided a list of stop names mapped to GPS coordinates, or open street
map data for the evaluated location.

##  2\. Background

###  2.1. How do public traffic operators handle social media?

Bregman (2012) discusses five ways traffic operators engage in social media
communication activities:

  1. (1)

sharing information about real-time service changes and advisories (timely
updates),

  2. (2)

providing information about services, fares, and long-range planning projects
(public information),

  3. (3)

taking advantage of the interactive aspects of social media to connect with
their customers in an informal way (citizen engagement),

  4. (4)

recognizing current workers and recruiting new employees (employee
recognition),

  5. (5)

displaying a personal touch and entertaining their riders pleasurable brand-
related content (entertainment).

Bregman finds that 77% of surveyed transit agencies used Twitter and 49% used
Facebook (multiple answers allowed) to inform about real-time service alerts
and 66% of agencies updated their relevant channels more than once a day with
a service update. Liu et al. (2016) designed an online survey of the top 50
transport agencies around the USA to collect general information about their
purpose and social media usage. They show that public transport agencies are
extremely interested in measuring their social media success: more than half
of public transport agencies currently measure social media outcomes through a
number of “friends and followers”, “likes” and “retweets”. Only a few of them
measure their social media program effectiveness with positive perception
similar to sentiment analysis. Wojtowicz and Wallace (2016) describes how the
Washington State DOT (WSDOT) was chosen as an example of well-executed social
media communication capable of limiting uncertainty in high disaster
situations and proving to be a recognizable source of reliable information.

Pender et al. (2014a) propose 6 stages of the disruption and recovery process
on the role of information delivery. Authors state that social media can be a
very efficient tool to inform passengers about disruptions but it can exclude
a certain group of society that doesn’t use smartphones. In another review on
(Pender et al., 2014b) present how transit agencies use social media during
unplanned transit network disruptions. Gal-Tzur et al. (2014) focuses on
categorizing live feed of tweets on an example of 3 mln tweets tweeded during
and before Liverpool soccer matches. Using bag-of-words models, authors
propose 2 models: one to recognize if the tweet was tweeted by an authority
(city, transport company) and second to recognize if the tweet was about
transport. Authors propose a hierarchical approach to categorize tweets
further in 3 main branches: information about transport needs, detection of
irregular events that impact on mobility and traveler opinions about transport
services.

Further studies include evaluating how transit agencies communicated during
large events (Cottrill et al., 2017) or how crowd sourced data can complement
agency communication of disruptions (Congosto et al., 2015). An important area
of contemporary social media and public transport research includes using
social media check-ins for mobility research (Lv et al., 2017; Tu et al.,
2017; Rashidi et al., 2017; Serna-Nocedal et al., 2017), planning decisions
(Ni et al., 2016; Nikolaidou and Papaioannou, 2018), transport performance
(Haghighi et al., 2018), user activities (Chaniotakis et al., 2017) and
satisfaction (Alshehri and O’Keefe, 2019). Recently new technologies emerged
to push service disruption information such as bots and chat bots (Gault et
al., 2019).

###  2.2. Social media meets citizen science

Social media profiles of public transport agencies are also important for
citizen science. Perriam (2019) describes how non-academic organisations can
use data from social media on the example of the public sector organization
”Transport for London”. He focuses on the ethnographic research into the
workflow of the organisation and shows that TfL uses social media (here tweets
in particular) not only for customer service but also for staff management,
productivity observation and decision making.

Li et al. (2019) discuss how one can collect public opinions from social media
to extract from them potential improvements proposals or use them as a support
in decision making. The researchers describe the case of the subway in the
city of Nanjing in China using the data collected from the Sina microblog.
Three types of analysis have been made (sentiment analysis, spatiotemporal
analysis and topic/subject analysis) to localize the most important topics
raised in public space. They note that passengers comment mainly on the Metro
congestion, air-conditioning temperature, environment in the carriage,
equipment failure, and so on. Other comments have been summed up to formulate
some example prescriptions: what should be improved/done according to
citizens.

Raghothama et al. (2016) describe the analysis of delays using the data
(mainly GTFS data, besides that for example timetables) from the cities of
Rome and Stockholm. For Stockholm, the focus was placed primarily on
recognizing sources and factors of delays, for Rome the ability to predict
future delays was more important. Generally, Ragothama et al. presented the
process of working with geospatial and real-time data and they pointed out
potential issues in such analysis.

##  3\. Data collection

MPK Wrocław publishes information about impediments and changes in a public
transport schedule using a variety of available communication channels: a
dedicated mobile app - iMPK; timetable’s display boards located near bus and
tram stops; social media - official Facebook and Twitter profiles.

Figure 1. Example Facebook post with highlighted information extracted from
it.

Between 2017 and 2019 we collected 4161 tweets, 1680 Facebook posts with MPK
communique concerning various topics of public transport operations.
Additional metadata from social media was acquired from the content’s
publication date and reactions, including 24809 comments collected. Because
MPK Wrocław posts diverse messages on Facebook, hashtag #AlertMPK was used to
filter posts that reported incidents and other information concerning schedule
changes. Figure 1 shows an example of a post and data acquired from it. After
filtering there were 622 posts left for an analysis111We were unable to assure
the deanonymization of our GitHub repository, however in the final version of
the paper we will provide a link to the repository with relevant code, data
sets, trained models and a streamlit dashboard for exploring current twitter
data stream..

These textual data points contain spatial information we were interested in
extracting, the two main aspects We were interested in obtaining the spatial
understanding of incident locations and being able to relate them to passenger
mobility characteristics. We obtained bus and tram stops positions as well as
schedules from open data GTFS repositories and mobility information from the
recent survey of the metropolitan area mobility (Wrocław, 2020), which divides
the Wrocław area into regions and includes SIM-card based approximation of
passenger exchange between regions.

##  4\. Spatial Understanding

One can identify in the posts of MPK the specific language they use. The posts
with a similar content type tend to use the same template or to have a similar
structure (Figure 2). There was additional information derived from acquired
raw data: type of incident, the position of the incident, and lines affected
by the change in a schedule.

Figure 2. Example tweets of type fix \- they use the same template, there is
only a difference in the street name. Tweets translated by authors.

Maintaining a template-based information system allows the public transport
operator to decrease the cognitive load needed to understand the post, which
is usually received by the user in a stressful context related to service
disruption. The structured form also builds the foundation for trust as the
users get used to the communication style and identify it with the operator’s
brand. Such posts are also easier to parse for citizen sciences tasks.
Finally, when an irregular situation is reported, its non-template form
increases the alertness of the reader.

Following manual exploration we distinguished multiple types of the content in
the MPK’s posts. As good categorization of posts is essential in the context
of future applications we decided to propose the following typology of
transport information:

  * •

accident \- traffic accidents (including accidents without MPK’s vehicles)

  * •

event \- events affecting urban traffic, e.g. matches, marathons, special
(e.g. holiday) timetable

  * •

fix \- information about restoring the state from before the failure

  * •

incident \- unwanted, random events, e.g. police intervention, fire brigade
action, aggressive passenger, medical intervention, etc.

  * •

malfunction \- failures of infrastructure (e.g. power outages, derailments)
and vehicles (e.g. broken pantograph, another vehicle malfunction)

  * •

renovation \- planned infrastructure repairs

  * •

unknown \- traffic difficulties with no specified cause

We manually tagged 3532 posts with the above classes and build a classifier
using the FastText model (Joulin et al., 2017). 3000 of those were used in a
training process and the rest was used for testing. Given the structured
language of the posts, even a simple embedding based model was able to achieve
the 87% accuracy for the test set. We use the model to classify the Facebook
posts, which are often lengthier but include the structured information as one
of the paragraphs.

We excluded three of the proposed classes from further analysis: event and
renovation, because they were dedicated to planned changes in schedules, not
crisis management; and the fix class because it informed about the restoration
to normal and not about disruptions. Other classes were further analyzed as
unplanned events that caused disruptions in traffic. After filtering there
were 482 posts left. Distribution of those posts types can be seen in a figure
3.

Figure 3. Distribution of classified #AlertMPK posts.

It is important to mention, that although some of the posts (especially
Facebook ones) seem to be little equivocal, ambiguous, typically one can
identify the correct class quite easily and, as it was measured with Cohen’s
kappa at 0.75 for each pair of annotators (3 annotators altogether), the
multi-class approach turned out to be inferior compared to the single class
division approach.

The spatial understanding was performed by rule-based regexp matching, due to
the lack of character-level embeddings for the polish language and important
impact of inflection on stop and street names in posts. We analyzed word
sequences appearing before the introduction or enumeration of spatial terms
such as street names or stop names. Next, we matched word sequences that
followed these prefixes to the street and stop name lists available in GTFS
and OpenStreetMap data, using Levenshtein distance and thresholding.

Positions of disruptions were obtained using street names mentioned in posts.
Official stops positions were used whenever possible. Intersections were found
using Google Maps and the rest of the streets were approximated using Open
Street Maps. Figure 4 shows map of disruptions with color-coded types.

To detect most affected bus and tram lines, every occurrence of a line’s name
or a number among different tweets with #AlertMPK hashtag were found and the
number of different tweets in which the line was mentioned was counted. The
results are shown in the figure 5.

##  5\. Applications and use cases

###  5.1. Citizen science: understanding the impact on the quality of life

On average there one disruption happened every 2.28 days and almost half of
all disruptions were caused by malfunctioning infrastructure or vehicles. As
shown in Figure 4 the spatial distribution of incidents spanned the entire
tram network in the city and had an important impact on the quality of life.
Having extracted the spatial characteristics of each of the incidents we can
evaluate how both lines and passengers were affected in more detail.

Figure 4. Map of all disruptions from obtained data set with color coded
types.

The most frequently mentioned lines are the tramlines. Even the least
mentioned tram lines appeared in tweets much more often than the bus lines. A
probable reason for that is that any malfunction or incident related to a tram
is far more critical for the traffic infrastructure than the bus’s one. A
damaged tram blocks the tracks along which other vehicles move, so even an
accident of a single tram may cause exclusion of many other trams from traffic
or at least change their routes. And the changes are noted in the #AlertMPK
tweets. Buses are more resistant to traffic changes, thus they are mentioned
far less frequently.

Figure 5. The number of #AlertMPK posts in which the tram and bus lines were
mentioned.

The perspective on the passenger mobility impact can be drawn from relating
the data to the 2018 mobility survey. The survey provides raw data which
includes SIM-card based trips counts between regions. As public transport and
car mobility are intertwined by the Downs–Thomson paradox (Mogridge et al.,
1987) we can treat the passenger mobility data jointly, following the
assumption that tram derailment, bus accidents and other public transport
incidents negatively impact all means of travel on the affected mobility route
and its alternatives in the surround region. We mapped the points of incidents
to mobility regions in the study and aggregating the passenger counts for the
hour at which the incident occurred. This allows us to estimate that at least
732 618 passengers may have been impacted by service disruptions in the 244
weekdays in which incidents were noted, which is an average of 3000 per day.
This is a lower bound estimation as it is calculated under the assumption that
disruptions are fixed in at most an hour, which is usually not the case.

Figure 6. Map of total passengers impacted per region during service
disruption on weekdays in the evaluated data set.

####  5.1.1. Sentiment of reactions

Another use case of our approach is to understand how citizens and passengers
react to the incidents. The sentiment of the text data differs when comparing
MPK posts and users’ comments referring to the posts. The Facebook comments
section, although moderated in the case of MPK profile, consists of texts of
multiple styles, tones, lengths, which are expressing different attitudes
towards the incidents and events described in the posts as well as towards the
public transport company (MPK). Thus we found it interesting to evaluate the
sentiment of the comments.

Detecting sentiment of passengers proved to be an arduous task due to the
presence of misspelling, grammatically incorrect sentences, and contextualized
writing style, that assume possessing the appropriate knowledge to interpret
the content. Regular incidents caused completely different reactions among
passengers, starting from specifying the details of the event and ending with
sarcasm and humor. There is a convention that passengers use positive words to
carry a negative message. It varies from person to person and depends on many
aspects such as age, gender and others.

The difficulty of the task may be evidenced by the fact, that during the
process of manual labeling, the annotators were relatively often conflicted
about the right class label for a comment, which was measured by the kappa
score between the pairs of annotators (the value ranged from 0.3 to 0.45 and
was much lower than the same measure for the post type classification - on
average 0.77). What made it even more difficult, besides sarcasm and
ambiguity, was the length of the comments. Most of them are relatively short.
Hence, in the preprocessing, comments containing no more than three words were
removed. Comments which consisted only of emojis or pictures were also
filtered.

As it is very hard to identify sarcasm and humor in the content. These kinds
of human communication are strongly dependent on context and comment context
plays an important role in sarcasm and humor polarity detection. For example,
comments: “MPK did not disappoint my expectations” and “The new week is
starting nicely” could be perceived in two ways depending on the context:
sarcastic and humorous or ordinary. If the topic relates to an accident -
negatively, and if it relates to innovation or infrastructure improvement -
positively. Finally, the root of sarcasm lies in frustration (Filik et al.,
2016). Thus there is a very high probability that a sarcastic comment has
negative polarity, which is the opposite of humorous content.

Figure 7. The general sentiment of the comments. The comments were labeled
automatically by the fast-text model trained on the set of manually tagged
observations.

The basic sentiment analysis classifying comments whether they are positive,
neutral or negative resulted in a very imbalanced data set. As figure 7 shows,
most of the comments were labeled as neutral. Neutral comments are the
comments that are emotionally neutral and the ones that are too short or too
ambiguous to assign them as positive or negative. Because of that and the fact
that most of the comments are relatively short and thus hard to classify, the
neutral class is the most numerous. The least numerous category was a positive
class. Since the main topic of the analysis was the comments under the posts
about problems in motion, positive comments should not appear often.

Figure 8. The polarity of the comments towards MPK, where the comments were
manually labeled.

When another approach was tested, which measured the sentiment towards MPK
rather than the general emotional attitude expressed in the commentary,
positive reactions occurred much more often 8. Unfortunately, the percent of
negative comments was also higher.

(a) Malfunctions (a) Random incidents (a) Traffic accidents Figure 11. Maps
showing disruptions positions and reactions bound to them as circles. From the
top: malfunctions, incidents, accidents. Unknown type was left out, because
there was almost zero interactions under those posts.

####  5.1.2. Reactions and spread

User’s reactions can help approximate social sentiment towards certain types
of disruptions. First of all, ”Like” reaction is most apparent in the list
probably because it’s the easiest and most popular reaction for Facebook
users. Additionally, there is a significant difference between ”Haha” reaction
and the rest of the reactions. Figure 12 shows full comparison of number of
reactions to #AlertMPK posts.

Figure 12. Distribution of reactions to #AlertMPK posts per type.

Maps in figure 11 show disruption positions marked as circles where the radius
of a circle is bound to a total amount of passengers’ interactions (reactions
+ number of comments) with a post. The visualization shows that people react
most to malfunctions and least to traffic accidents. Few outliers with
significantly more reactions were found, especially among malfunction type of
posts. We describe in more detail three posts with the highest number of
interactions.

First disruption (Figure 13) blocked south part of a city on tram rails where
7 tram lines run every day. Diverted trams increased delay times of other tram
lines and replacement buses hugely impacted traffic jams because Powstańców
Śląskich street is one of the busiest streets in the city. Cause of disruption
was a broken tram pantograph and people were mad, that’s why there are more
comments than reactions under this post.

Figure 13. Post of type Malfunction with 82 reactions and 144 comments (226
interactions total). Can be seen in the south of the map 9(a).

Second disruption (Figure 14) disabled the only tram track leading to one of
the furthest districts in the city - Leśnica. Replacement buses had to keep
waiting in a long traffic jam and many people who depended on a public
transport system were even hours later at home after work. Same as first
disruption, the cause of it was a malfunction and again passengers’
frustration was reflected in a high number of comments demanding to improve
the condition of public transport.

Figure 14. Post of type Malfunction with 91 reactions and 109 comments (200
interactions total). Can be seen in the north-west of the map 9(a).

The third disruption (Figure 15) was independent of MPK and was depended on
poorly performed road works. That’s why there is a big difference between the
number of reactions (105) and comments (35). Additionally, most of these
reactions were ”Haha” reactions (77). The disruption amused passengers more
than upset them.

Figure 15. Post of type Incident with 105 reactions and 35 comments (140
interactions total). Can be seen in the south-east of the map 10(a).

##  6\. Conclusions

To address the problem of spatial understanding of social media posts that
convey transport information, we collected a three-year data set of public
posts from official channels of a polish public transport company. After
exploration, we proposed a systematic typology of communique and discussed the
magnitude of social response, which may be useful for building similar tools
for other agencies. For this typology, we build a successful classifier using
language models at the time available for the Polish language. We used lists
of street names and stop names from open data sources such as GTFS and
OpenStreetMap. We used distance-based matching of word sequences in a manually
selected context to provide spatial understanding. We then managed to
successfully use the obtained spatial understanding to relate information from
social posts to external data sets such as the mobility survey. We applied our
approach to discussing the aggregated impact on the transportation network and
mobility and automatic examination of the sentiment of citizens’ feedback. We
release both the data sets and experimental codebase under a permissive
opensource license to encourage further research in the topic, especially in
under-resourced languages and to empower data scientists worldwide with a tool
for performing citizen science.

## References

  * (1)
  * Alshehri and O’Keefe (2019) Alanawd Alshehri and Robert O’Keefe. 2019.  Analyzing Social Media to Assess User Satisfaction with Transport for London’s Oyster.  _International Journal of Human–Computer Interaction_ 35, 15 (2019), 1378–1387.
  * Bregman (2012) Susan Bregman. 2012\.  _Uses of social media in public transportation_.  Number 99. Transportation Research Board.
  * Chaniotakis et al. (2017) Emmanouil Chaniotakis, Constantinos Antoniou, Georgia Aifadopoulou, and Loukas Dimitriou. 2017\.  Inferring activities from social media data.  _Transportation research record_ 2666, 1 (2017), 29–37.
  * Congosto et al. (2015) Mariluz Congosto, Damaris Fuentes-Lorenzo, and Luis Sánchez. 2015\.  Microbloggers as sensors for public transport breakdowns.  _IEEE Internet Computing_ 19, 6 (2015), 18–25.
  * Cottrill et al. (2017) Caitlin Cottrill, Paul Gault, Godwin Yeboah, John D Nelson, Jillian Anable, and Thomas Budd. 2017\.  Tweeting Transit: An examination of social media strategies for transport information management during a large event.  _Transportation Research Part C: Emerging Technologies_ 77 (2017), 421–432.
  * Filik et al. (2016) Ruth Filik, Alexandra Turcan, Dominic Thompson, Nicole Harvey, Harriet Davies, and Amelia Turner. 2016\.  Sarcasm and emoticons: Comprehension and emotional impact.  _Quarterly Journal of Experimental Psychology_ 69 (2016), 2130 – 2146.  https://doi.org/10.1080/17470218.2015.1106566
  * Gal-Tzur et al. (2014) Ayelet Gal-Tzur, Susan M. Grant-Muller, Tsvi Kuflik, Einat Minkov, Silvio Nocera, and Itay Shoor. 2014\.  The potential of social media in delivering transport policy goals.  _Transport Policy_ 32 (2014), 115 – 123.  https://doi.org/10.1016/j.tranpol.2014.01.007
  * Gault et al. (2019) Paul Gault, Caitlin D Cottrill, David Corsar, Peter Edwards, John D Nelson, Milan Markovic, Mujtaba Mehdi, and Somayajulu Sripada. 2019\.  TravelBot: Utilising social media dialogue to provide journey disruption alerts.  _Transportation Research Interdisciplinary Perspectives_ 3 (2019), 100062\.
  * Haghighi et al. (2018) N Nima Haghighi, Xiaoyue Cathy Liu, Ran Wei, Wenwen Li, and Hu Shao. 2018\.  Using Twitter data for transit performance assessment: a framework for evaluating transit riders’ opinions about quality of service.  _Public Transport_ 10, 2 (2018), 363–377.
  * Joulin et al. (2017) Armand Joulin, Edouard Grave, Piotr Bojanowski, and Tomas Mikolov. 2017.  Bag of Tricks for Efficient Text Classification. In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers_. Association for Computational Linguistics, 427–431.
  * Li et al. (2019) Dawei Li, Yujia Zhang, and Cheng Li. 2019.  Mining Public Opinion on Transportation Systems Based on Social Media Data.  _Sustainability_ 11, 15 (2019), 1–15.  https://doi.org/10.3390/su11154016
  * Liu et al. (2016) Jenny Liu, Wei Shi, Sam Elrahman, Ban Xuegang, and Jack Reilly. 2016.  Understanding social media program usage in public transit agencies.  _International Journal of Transportation Science and Technology_ 2 (2016), 83–92.  https://doi.org/10.1016/j.ijtst.2016.09.005
  * Lv et al. (2017) Yisheng Lv, Yuanyuan Chen, Xiqiao Zhang, Yanjie Duan, and Naiqiang Li Li. 2017.  Social media based transportation research: The state of the work and the networking.  _IEEE/CAA Journal of Automatica Sinica_ 4, 1 (2017), 19–26.
  * Mogridge et al. (1987) Martin John Henry Mogridge, DJ Holden, J Bird, and GC Terzis. 1987.  The Downs/Thomson paradox and the transportation planning process.  _International Journal of Transport Economics/Rivista internazionale di economia dei trasporti_ (1987), 283–311.
  * Ni et al. (2016) Ming Ni, Qing He, and Jing Gao. 2016.  Forecasting the subway passenger flow under event occurrences with social media.  _IEEE Transactions on Intelligent Transportation Systems_ 18, 6 (2016), 1623–1632.
  * Nikolaidou and Papaioannou (2018) Anastasia Nikolaidou and Panagiotis Papaioannou. 2018.  Utilizing social media in transport planning and public transit quality: Survey of literature.  _Journal of Transportation Engineering, Part A: Systems_ 144, 4 (2018), 04018007.
  * Pender et al. (2014a) Brendan Pender, Graham Currie, Alexa Delbosc, and Nirajan Shiwakoti. 2014a.  Social Media Use during Unplanned Transit Network Disruptions: A Review of Literature.  _Transport Reviews_ 34, 4 (2014), 501–521.  https://doi.org/10.1080/01441647.2014.915442
  * Pender et al. (2014b) Brendan Pender, Graham Currie, Alexa Delbosc, and Nirajan Shiwakoti. 2014b.  Social media use during unplanned transit network disruptions: A review of literature.  _Transport Reviews_ 34, 4 (2014), 501–521.
  * Perriam (2019) Jessamy Perriam. 2019\.  A Tweet is Not Just a Tweet: Public Sector Understandings and Analysis of Social Media Customer Service Data.  _SMSociety ’19: Proceedings of the 10th International Conference on Social Media and Society_ , 33–40.  https://doi.org/10.1145/3328529.3328542
  * Raghothama et al. (2016) Jayanth Raghothama, Vinutha Magal Shreenath, and Sebastiaan Meijer. 2016.  Analytics on public transport delays with spatial big data. 28–33.  https://doi.org/10.1145/3006386.3006387
  * Rashidi et al. (2017) Taha H Rashidi, Alireza Abbasi, Mojtaba Maghrebi, Samiul Hasan, and Travis S Waller. 2017.  Exploring the capacity of social media data for modelling travel behaviour: Opportunities and challenges.  _Transportation Research Part C: Emerging Technologies_ 75 (2017), 197–211.
  * Serna-Nocedal et al. (2017) Ainhoa Serna-Nocedal, Jon Kepa Gerrikagoitia, Unai Bernabé, and Tomás Ruiz Sánchez. 2017\.  Sustainability analysis on Urban Mobility based on Social Media content.  _Transportation Research Procedia_ 24 (2017), 1–8.
  * Tu et al. (2017) Wei Tu, Jinzhou Cao, Yang Yue, Shih-Lung Shaw, Meng Zhou, Zhensheng Wang, Xiaomeng Chang, Yang Xu, and Qingquan Li. 2017.  Coupling mobile phone and social media data: A new approach to understanding urban functions and diurnal patterns.  _International Journal of Geographical Information Science_ 31, 12 (2017), 2331–2358.
  * Wojtowicz and Wallace (2016) Jeffrey Wojtowicz and William Wallace. 2016.  Use of social media by transportation agencies for traffic management.  _Transportation Research Record_ 2551 (2016), 82–89.  https://doi.org/10.3141/2551-10
  * Wrocław (2020) Municipal Office Wrocław. 2018 (accessed August 21, 2020).  _Kompleksowe Badania Ruchu we Wrocławiu i otoczeniu - KBR 2018_.  https://bip.um.wroc.pl/artykul/565/37499/kompleksowe-badania-ruchu-we-wroclawiu-i-otoczeniu-kbr-2018


$\displaystyle-\text{Tr}\left(\left\\{f,\phi_{\kappa}(\sigma,z)_{\mathbf{1}}\right\\},\text{Tr}\left(\left\\{\phi_{\kappa}(\sigma,z)_{\mathbf{1}},\Phi(\sigma^{\prime},z^{\prime})_{\mathbf{2}}\right\\}^{-1},\left\\{\Phi(\sigma^{\prime},z^{\prime})_{\mathbf{2}},g\right\\}\right)_{\mathbf{2}(z^{\prime},\sigma^{\prime})}\right)_{\mathbf{1}(z,\sigma)}$
$\displaystyle-\text{Tr}\left(\left\\{f,\Phi(\sigma,z)_{\mathbf{1}}\right\\},\text{Tr}\left(\left\\{\Phi(\sigma,z)_{\mathbf{1}},\phi_{\kappa}(\sigma^{\prime},z^{\prime})_{\mathbf{2}}\right\\}^{-1},\left\\{\phi_{\kappa}(\sigma^{\prime},z^{\prime})_{\mathbf{2}},g\right\\}\right)_{\mathbf{2}(z^{\prime},\sigma^{\prime})}\right)_{\mathbf{1}(z,\sigma)},$

where the labels 1,2 instruct us to take the trace along the corresponding
tensor factors. It reduces to

$\displaystyle\left\\{f,g\right\\}^{\ast}$
$\displaystyle=\left\\{f,g\right\\}$ (4.138)
$\displaystyle-\text{Tr}\Big{(}\left\\{f,\phi_{\kappa}(\sigma,z)_{\mathbf{1}}\right\\},\left\\{\Phi(\sigma,z)_{\mathbf{1}},g\right\\}\Big{)}_{\mathbf{1}(z,\sigma)}+\text{Tr}\Big{(}\left\\{f,\Phi(\sigma,z)_{\mathbf{1}}\right\\},\left\\{\phi_{\kappa}(\sigma,z)_{\mathbf{1}},g\right\\}\Big{)}_{\mathbf{1}(z,\sigma)}$

and equals the canonical Poisson bracket if we restrict to phase space
functionals that are $\kappa$-shift invariant. Step I is then accomplished.

The partially gauge fixed theory is no longer quadratic in the velocity
$\partial_{\tau}A$ but linear and this restructures completely the whole set
of Hamiltonian constraints allowing to ‘restart’ the Dirac algorithm again. To
see this, consider a theory with a Lagrangian that can be decomposed in the
form $L=L_{0}+L_{1}+L_{2}$, where $L_{0}$, $L_{1}$ and $L_{2}$ are the terms
independent, linear and quadratic in the velocities. The Hamiltonian energy
function is $H=L_{2}-L_{0}$, i.e. the quadratic term is preserved, the linear
term is absent and the velocity-independent term flips sign. As the gauge
fixed theory is now linear in the velocities, the Hamiltonian energy function
and the canonical Hamiltonian coincide when restricted to the constraint
surface defined by (4.98), after taking $\Phi=0$. Thus, $H=-L_{0}$ and (4.98)
leads now to three new ‘primary’ constraints.

Explicitly, the gauge fixed action is invariant under the remaining
$\Omega$-shifts, it is given by

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
CS\left(\mathbb{A}\right)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right)$
(4.139)

and has the following Lagrangian

$L=ic\bigg{\\{}\mathop{\displaystyle\int}\nolimits_{\text{M}}\tilde{\Omega}\wedge\text{Tr}\big{(}A\wedge\partial_{\tau}A-2A_{\tau}F_{A}\big{)}+\Omega_{\tau}\mathop{\displaystyle\int}\nolimits_{\text{M}}CS(A)+\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\Big{(}A_{\tau}A+\left(\kappa_{\tau}A-\tilde{\kappa}A_{\tau}\right)i_{\mathcal{R}}\mathbb{A}\Big{)}\bigg{\\}}.$
(4.140)

To verify the invariance of (4.139) under $\Omega$-shifts it is useful to
notice that

$0=i_{\mathcal{R}}\Big{(}d_{\mathbb{M}}\Omega\wedge\Omega\wedge\kappa\wedge\text{Tr}(s\mathbb{A})\Big{)}=-d_{\mathbb{M}}\Omega\wedge\Omega\wedge\text{Tr}(s\mathbb{A})+d_{\mathbb{M}}\Omega\wedge\Omega\wedge\kappa\text{Tr}(si_{\mathcal{R}}\mathbb{A}).$
(4.141)

We restart the Dirac procedure by taking (4.139) as the new action functional.
For the canonical momentum, we have now

$P_{\tau}=0,\text{ \ \ }P=ic\ \tilde{\Omega}\wedge A,$ (4.142)

or in components,

$P_{z}=ic\big{(}\Omega_{\overline{z}}A_{\sigma}-\Omega_{\sigma}A_{\overline{z}}\big{)},\text{
\ \
}P_{\overline{z}}=ic\big{(}\Omega_{\sigma}A_{z}-\Omega_{z}A_{\sigma}\big{)},\text{
\ \
}P_{\sigma}=ic\big{(}\Omega_{z}A_{\overline{z}}-\Omega_{\overline{z}}A_{z}\big{)},$
(4.143)

leading to the existence of four primary constraints, given by

$\displaystyle P_{\tau}$ $\displaystyle\approx 0,$ (4.144)
$\displaystyle\phi_{z}$
$\displaystyle=P_{z}-ic\big{(}\Omega_{\overline{z}}A_{\sigma}-\Omega_{\sigma}A_{\overline{z}}\big{)}\approx
0,$ $\displaystyle\phi_{\overline{z}}$
$\displaystyle=P_{\overline{z}}-ic\big{(}\Omega_{\sigma}A_{z}-\Omega_{z}A_{\sigma}\big{)}\approx
0,$ $\displaystyle\phi_{\sigma}$
$\displaystyle=P_{\sigma}-ic\big{(}\Omega_{z}A_{\overline{z}}-\Omega_{\overline{z}}A_{z}\big{)}\approx
0.$

The total Hamiltonian is now

$H_{T}=H+\text{Tr}\left(u_{\tau},P_{\tau}+u_{z},\phi_{z}+u_{\overline{z}},\phi_{\overline{z}}+u_{\sigma},\phi_{\sigma}\right)_{(z,\sigma)},$
(4.145)

where

$H=ic\bigg{\\{}2\mathop{\displaystyle\int}\nolimits_{\text{M}}\tilde{\Omega}\wedge\text{Tr}\big{(}A_{\tau}F_{A}\big{)}-\Omega_{\tau}\mathop{\displaystyle\int}\nolimits_{\text{M}}CS(A)-\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\Big{(}A_{\tau}A+\left(\kappa_{\tau}A-\tilde{\kappa}A_{\tau}\right)i_{\mathcal{R}}\mathbb{A}\Big{)}\bigg{\\}}.$
(4.146)

We expect the first class constraint associated to the $\Omega$-shift symmetry
to re-emerge as a particular linear combination of the four primary
constraints (4.144).

From (4.146), we get the functional derivatives

$\displaystyle\frac{\delta H}{\delta A_{\tau}}$
$\displaystyle=2ic\Big{(}\Omega_{z}F_{\overline{z}\sigma}+\Omega_{\sigma}F_{z\overline{z}}+\Omega_{\overline{z}}F_{\sigma
z}-\Omega_{z\overline{z}}\left(A_{\sigma}-\kappa_{\sigma}i_{\mathcal{R}}\mathbb{A}\right)\Big{)},$
(4.147) $\displaystyle\frac{\delta H}{\delta A_{z}}$
$\displaystyle=2icQ_{\overline{z}\sigma},\text{ \ \ }\frac{\delta H}{\delta
A_{\overline{z}}}=2icQ_{\sigma z},\text{ \ \ }\frac{\delta H}{\delta
A_{\sigma}}=2ic\Big{(}Q_{z\overline{z}}+\Omega_{z\overline{z}}\left(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A}\right)\Big{)},$

where this time we have defined

$\displaystyle Q$ $\displaystyle=-\tilde{\Omega}\wedge
d_{A}A_{\tau}-\Omega_{\tau}F_{A}.$ (4.148)

Similar as done before, we must verify the time preservation of the
constraints (4.144) under the new time flow defined by the total Hamiltonian
(4.145). From the variations (4.147), we find

$\big{\\{}\text{Tr}\left(\eta,P_{\tau}\right)_{(z,\sigma)},H_{T}\big{\\}}=\gamma(\eta)\approx
0$ (4.149)

and obtain a secondary constraint given by

$\gamma(\eta)=-2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\tilde{\Omega}\wedge\text{Tr}\left(\eta
F_{A}\right)+2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\tilde{\Omega}\wedge\text{Tr}\Big{(}\eta\left(A-\tilde{\kappa}i_{\mathcal{R}}\mathbb{A}\right)\Big{)},$
(4.150)

which is nothing but (4.126) with $\Phi=0$. We also find that

$\displaystyle\left\\{H_{T},\phi_{z}\right\\}$
$\displaystyle=2ic\Big{(}Q_{\overline{z}\sigma}+\left(\Omega_{\overline{z}}u_{\sigma}-\Omega_{\sigma}u_{\overline{z}}\right)\Big{)},$
(4.151) $\displaystyle\left\\{H_{T},\phi_{\overline{z}}\right\\}$
$\displaystyle=2ic\Big{(}Q_{\sigma
z}+\left(\Omega_{\sigma}u_{z}-\Omega_{z}u_{\sigma}\right)\Big{)},$
$\displaystyle\left\\{H_{T},\phi_{\sigma}\right\\}$
$\displaystyle=2ic\Big{(}Q_{z\overline{z}}+\Omega_{z\overline{z}}\left(A_{\tau}-\kappa_{\tau}i_{\mathcal{R}}\mathbb{A}\right)+\left(\Omega_{z}u_{\overline{z}}-\Omega_{\overline{z}}u_{z}\right)\Big{)},$

are actually conditions over the Lagrange multipliers
$u_{\sigma},u_{z},u_{\overline{z}}$. Concerning the time preservation of the
secondary constraint (4.150), a similar argument leading to (4.134) holds,
hence no new constraints are produced.

By introducing the constraint 2-form

$\phi=\phi_{z}d\overline{z}\wedge d\sigma+\phi_{\overline{z}}d\sigma\wedge
dz+\phi_{\sigma}dz\wedge d\overline{z},$ (4.152)

we recover the $\Omega$-shift symmetry generator

$\mathcal{S}_{\Omega}(s)=\text{Tr}(s,\Omega_{\tau}P_{\tau}+\tilde{\Omega}\wedge\phi)_{(z,\sigma)},$
(4.153)

which obeys

$\\{H_{T},\mathcal{S}_{\Omega}(s)\\}=\\{\gamma(\eta),\mathcal{S}_{\Omega}(s)\\}=0.$
(4.154)

At this point, as follows from the Dirac procedure, it is necessary to
classify the set of constraints found so far as first or second class
constraints. However, we will do this only after implementing the step II
mentioned above, see section (3), which corresponds to taking the degenerate
limit $\zeta\rightarrow 0$. The reason for this is that here we are mainly
interested in recovering the conventional 4d CS theories from the Hamiltonian
theory point of view and not in pursuing a thorough Hamiltonian analysis of
the generalized 4d CS theory.

In the $\zeta\rightarrow 0$ limit, we have that

$\Omega_{\tau}=0,\text{ \ \ }\tilde{\Omega}=\omega=\Omega_{z}dz=\varphi dz,$
(4.155)

where the component $\Omega_{z}=\varphi$ is identified with the twist function
of the underlying integrable field theory. We also supplement the limit with
the condition (3.25). The action functional (4.139), the Lagrangian (4.140)
and the canonical Hamiltonian (4.146) are those of the 4d CS theory and are
given, respectively, by (3.26) and

$\displaystyle L$
$\displaystyle=ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\omega\wedge\text{Tr}\big{(}A\wedge\partial_{\tau}A-2A_{\tau}F_{A}\big{)}+ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d{\omega}\wedge\text{Tr}\big{(}A_{\tau}A\big{)},$
(4.156) $\displaystyle H$
$\displaystyle=2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\omega\wedge\text{Tr}\big{(}A_{\tau}F_{A}\big{)}-ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\omega\wedge\text{Tr}\big{(}A_{\tau}A\big{)}.$

Step II is then accomplished.

Concerning the Hamiltonian constraints, we have from (4.144) and (4.150), that

$\displaystyle P_{\tau}\approx 0,\text{ \ \ }\phi_{z}=P_{z}\approx 0,\text{ \
\ }\phi_{\overline{z}}=P_{\overline{z}}+ic\varphi A_{\sigma}\approx 0,\text{ \
\ }\phi_{\sigma}=P_{\sigma}-ic\varphi A_{\overline{z}}\approx 0$ (4.157)

and

$\gamma(\eta)=-2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}\omega\wedge\text{Tr}\left(\eta
F_{A}\right)+2ic\mathop{\displaystyle\int}\nolimits_{\text{M}}d\omega\wedge\text{Tr}\big{(}\eta
A\big{)}.$ (4.158)

The constraint $\phi_{z}\approx 0$, is actually an identity reflecting the
fact that the field component $A_{z}$ completely decouples from the theory.
The rest of the Hamiltonian analysis follows exactly the lines considered in
Vicedo-PCM ; me-PCM , to which the reader is referred for further details.
Thus, we will not repeat their results here, but instead gather some relevant
facts to be used later.

The constraint $P_{\tau}\approx 0$ is first class and can be gauged fixed by
choosing a gauge fixing condition of the form

$A_{\tau}=A_{\tau}(A_{\sigma}|_{\mathfrak{p}}),$ (4.159)

i.e. the component $A_{\tau}$ is chosen to be a function of the gauge field
component $A_{\sigma}$ evaluated at the set of poles $\mathfrak{p}$ of the
twist 1-form $\omega$. The constraint $\phi_{z}$ together with the component
$A_{z}$, can be ignored. The constraints $\phi_{\overline{z}}$ and
$\phi_{\sigma}$ form a second class pair and (4.158) is a first class
constraint, provided we restrict the gauge parameters to satisfy the condition
$\eta|_{\mathfrak{p}}=0$, which is a possible way to cancel the obstruction
(4.29). The latter constraint, i.e. $\gamma(\eta)\approx 0$, then reduces to
$F_{A}\approx 0$ and as a gauge fixing condition, we can choose

$A_{\overline{z}}\approx 0,$ (4.160)

together with the restriction $\partial_{\overline{z}}g=0$ over the gauge
parameters.

The 4d CS theory is completely recovered but apparently it is not gauge
invariant because the WZ-type term is now proportional to
$\omega\wedge\chi(g)$, as is well known in the literature.

###  4.4 Recovering the lambda-PCM Lax connection

Here, we quickly explore the implications of the condition (3.25) in
determining the analytic structure of the lambda-PCM Lax connection.

In the gauge $\Phi=0$, the equations of motion (2.89) reduce to

$\Omega\wedge F_{\mathbb{A}}=d_{\mathbb{M}}\Omega\wedge\left(\mathbb{A}-\kappa
i_{\mathcal{R}}\mathbb{A}\right)$ (4.161)

and (2.90) becomes a trivial identity. The gauge fixed action functional is
(4.139) and in the $\zeta\rightarrow 0$ limit, it becomes (3.24), i.e.

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS\left(\mathbb{A}\right)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right).$
(4.162)

The eom (4.161) reduce to

$\omega\wedge F_{\mathbb{A}}=d\omega\wedge\left(\mathbb{A}-\kappa
i_{\mathcal{R}}\mathbb{A}\right).$ (4.163)

The usual 4d Chern-Simons theory is recovered by imposing the condition
(3.25), i.e.

$\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d\omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right)=0.$
(4.164)

In this case, the action and the eom are given, respectively, by

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS\left(\mathbb{A}\right),\text{ \ \ }\omega\wedge
F_{\mathbb{A}}=d\omega\wedge\mathbb{A}.$ (4.165)

Notice that, despite of the fact that we are imposing the condition (4.164),
both expressions in (4.165) are invariant under the residual $\omega$-shift
symmetry given by ${}^{\omega}\mathbb{A}=\mathbb{A}+s\omega$. This follows
from the fact that
$d\omega\wedge\omega=\underline{\pi}^{\ast}(d_{C}\omega_{C}\wedge\omega_{C})=0$,
where $d_{C}$ is the exterior derivative on the base manifold $C$. Locally, we
have $\omega=\varphi(z)dz$, thus the gauge field component $A_{z}dz$ decouples
from the theory, a prominent characteristic of the action (1.1).

In components, the eom in (4.165) are equivalent to the set of equations

$\varphi F_{\overline{z}\mu}=\omega_{z\overline{z}}A_{\mu},\text{ \ \ }\varphi
F_{\tau\sigma}=0,$ (4.166)

where $\mu=\tau,\sigma$ and $d\omega=\omega_{z\overline{z}}dz\wedge
d\overline{z}$. The lambda deformed PCM is specified by the twist function
k-def

$\varphi(z)=b\frac{z^{2}-1}{z^{2}-a^{2}},\text{ \ \ }\omega=\varphi(z)dz,$
(4.167)

where $a,b\in\mathbb{R}.$ The zeroes $\mathfrak{z}$ and the poles
$\mathfrak{p}$ of $\omega$ on the chart $\mathcal{U}_{1}$ are located at
$z=\pm 1$ and $z=z_{\pm}=\pm a$, respectively141414There is an order 2 pole at
$\infty$ covered by an analysis on the chart $\mathcal{U}_{0}$. We will not
consider this pole here, due to the fact that the lambda deformed PCM Lax
connection to be re-derived below, is known unifying  to satisfy all the
analytic properties at $\infty$. and all are real numbers. In local
coordinates around each pole in the chart $\mathcal{U}_{1}$, we have that

$d\omega=i\pi
b\frac{(z_{+}^{2}-1)}{2z_{+}}\Big{(}\delta_{zz_{+}}-\delta_{zz_{-}}\Big{)}dz\wedge
d\overline{z},$ (4.168)

where we have used the expression (4.111). Then,

$\omega_{z\overline{z}}=i\pi
b\frac{(z_{+}^{2}-1)}{2z_{+}}\Big{(}\delta_{zz_{+}}-\delta_{zz_{-}}\Big{)}.$
(4.169)

The first equation in (4.166), in the gauge $A_{\overline{z}}=0$ (cf.
(4.160)), implies

$\frac{(z^{2}-1)}{(z^{2}-z_{+}^{2})}\partial_{\overline{z}}A_{\mu}(z)=-\frac{(z_{+}^{2}-1)}{2z_{+}}\partial_{\overline{z}}\left(\frac{A_{\mu}(z_{+})}{z-z_{+}}-\frac{A_{\mu}(z_{-})}{z-z_{-}}\right).$
(4.170)

Thus, we have

$(z^{2}-1)A_{\mu}(z)=H_{\mu}(z)+\frac{(z_{+}^{2}-1)}{2z_{+}}\Big{(}(z-z_{+})A_{\mu}(z_{-})-(z-z_{-})A_{\mu}(z_{+})\Big{)},$
(4.171)

with $H_{\mu}(z)$ holomorphic. Consistency, i.e.
$A_{\mu}(z)|_{z=z_{\pm}}=A_{\mu}(z_{\pm})$, fixes $H_{\mu}(z)$ and we end up
with the result

$A_{\mu}(z)=f_{+}(z)A_{\mu}(z_{-})+f_{-}(z)A_{\mu}(z_{+}),\text{ \ \
}f_{\pm}(z)=\pm\frac{(1-z_{+}^{2})}{2z_{+}}\frac{(z-z_{\pm})}{(z^{2}-1)}.$
(4.172)

This connection interpolates between the set of poles $\mathfrak{p}$.

Now we solve the condition (4.164), which is equivalent to

$\mathop{\displaystyle\int}\nolimits_{\mathcal{U}_{1}}d\omega\text{Tr}\left(A_{\tau}^{2}-A_{\sigma}^{2}\right)=0,$
(4.173)

or to

$\text{Tr}\Big{(}\left(A_{\tau}(z_{+})^{2}-A_{\sigma}(z_{+})^{2}\right)-\left(A_{\tau}(z_{-})^{2}-A_{\sigma}(z_{-})^{2}\right)\Big{)}=0.$
(4.174)

This equation can be solved by taking the following linear combination (cf.
(4.159))

$A_{\tau}(z_{\pm})=p(z_{\pm})A_{\sigma}(z_{+})+q(z_{\pm})A_{\sigma}(z_{-}).$
(4.175)

Then, (4.174) imply that

$p(z_{+})^{2}-p(z_{-})^{2}=1,\text{ \ \ }q(z_{+})^{2}-q(z_{-})^{2}=-1,\text{ \
\ }p(z_{+})q(z_{+})=p(z_{-})q(z_{-}).$ (4.176)

The solutions are

$p(z_{\pm})=\frac{1}{2}s\left(z_{\pm}\pm z_{\pm}^{-1}\right),\text{ \ \
}q(z_{\pm})=\frac{1}{2}s^{\prime}\left(z_{\pm}\mp z_{\pm}^{-1}\right),$
(4.177)

where $s=\pm 1$ and $s^{\prime}=\pm 1$ are sign functions. Thus, (4.172) for
$\mu=\tau$, becomes

$A_{\tau}(z)=g_{+}(z)A_{\sigma}(z_{-})+g_{-}(z)A_{\sigma}(z_{+}),$ (4.178)

where

$g_{+}(z)=-\frac{s^{\prime}}{2}\left(1-z_{+}^{2}\right)\frac{\left(z-z_{+}^{-1}\right)}{\left(z^{2}-1\right)},\text{
\ \
}g_{-}(z)=-\frac{s}{2}\left(1-z_{+}^{2}\right)\frac{\left(z-z_{-}^{-1}\right)}{\left(z^{2}-1\right)}.$
(4.179)

In the light-cone coordinates the Lax connection $\mathscr{L}_{\pm}(z)$, now
identified with $A_{\pm}(z)$, has the following analytic structure

$A_{\pm}(z)=\frac{1}{2}\Big{(}A_{\tau}(z)\pm
A_{\sigma}(z)\Big{)}=\frac{I_{\pm}}{1\pm z},$ (4.180)

for some currents $I_{\pm}$, i.e. the Lax connection has poles at the zeroes
of the twist 1-form $\omega$. Using the solutions (4.172) with $\mu=\sigma$,
(4.178) and comparing with (4.180), we find that $s=s^{\prime}=-1$ and
identify

$I_{\pm}=\frac{(z_{+}-z_{+}^{-1})}{4}\Big{(}(1\mp
z_{+})A_{\sigma}(z_{+})-(1\pm z_{+})A_{\sigma}(z_{-})\Big{)}.$ (4.181)

The latter are nothing but the lambda deformed PCM currents Sfetsos ; lambda-
bos . Finally, the last equation in (4.166) boils down to

$\partial_{+}I_{-}+\partial_{-}I_{+}=0,\text{ \ \
}\partial_{+}I_{-}-\partial_{-}I_{+}+\left[I_{+},I_{-}\right]=0,$ (4.182)

which are the lambda deformed PCM eom. They also coincide, formally, with the
conventional PCM eom.

It is interesting to notice the instrumental rôle played by the condition
(4.164) in deriving the Lax connection of the associated integrable field
theory within our approach. We want to emphasize that the condition (4.164) is
new and never used in conventional 4d CS theories, where the Lax connection of
the integrable model associated to the CS theory is constructed without making
any reference to it. See for instance CY  for the original construction for
integrable field theories with order and disorder surface defects and section
§5 of Lacroix  for a quick review of the Lax pair/CS theory relation.

Finally, let us summarize the main results accomplished in this section. We
have performed a thorough study of the generalized theory in the particular
example $(\text{M}=S^{3},\alpha)$, covering integrable theories of the PCM
type. We constructed explicitly the contact form $\alpha$, applied the
Hamiltonian analysis (implementing the steps I and II introduced in section
(3)), found the important result (4.73) to be used later and constructed the
PCM lambda deformed Lax connection.

In the next section we will consider a complementary approach that allows to
introduce the quadratic action (2.81) in a very intuitive and simple way.
Then, in section (6) we move to the final part of the present work, where we
present the main result (1.3), which is valid for the PCM type models
considered in this section.

##  5 Quadratic action from a duality approach

This time, by using a duality approach, we recover the generalization of the
4d Chern-Simons theory introduced above in section (2). The argument follows
the same logic used in NA loc CS  but adapted now to the present case, where
we have two shift transformations in $\mathcal{S}$. The original 4d CS theory
needs to be modified first in an specific way in order for the theory to be
dualized in a consistent manner. We also show how the quadratic action (2.83)
emerges naturally as a dual model.

We start with the 4d CS theory on $\mathbb{M}=\Sigma\times C$, with action
(1.1)

$S=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\omega\wedge
CS(\mathbb{A}).$ (5.1)

The difficulty in dualizing (5.1) comes from the structure of the ‘twist’ form
$\omega$ and the 4d manifold $\mathbb{M}$, which turns the action ill-defined
for implementing a duality transformation via gauging. In what follows, we
shall break the strategy for constructing the dual model in the correct way
into two simple steps.

The first step consist in regularizing the theory. Instead of (5.1) we
consider its $\Omega$-shift invariant extension (4.139), which is defined now
on the manifold $\mathbb{M}=\mathbb{R}\times\text{M}$, with a non-trivial
circle bundle space M and with an action functional defined by

$S(\mathbb{A})=ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge
CS\left(\mathbb{A}\right)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\mathbb{A}i_{\mathcal{R}}\mathbb{A}\right).$
(5.2)

A simpler way to deduce the action (5.2) without invoking the Hamiltonian
formalism and so on, starts by replacing $\omega\rightarrow\Omega$ in (5.1)
and follows by adding the necessary terms that make the new action invariant
under $\Omega$-shifts. To see this, we simply use the result

$\Omega\wedge CS\left({}^{\Omega}\mathbb{A}\right)=\Omega\wedge
CS(\mathbb{A})+d_{\mathbb{M}}\Omega\wedge\Omega\wedge\text{Tr}(s\mathbb{A})$
(5.3)

and the identity (4.141) in order to replace

$d_{\mathbb{M}}\Omega\wedge\Omega\wedge\text{Tr}(s\mathbb{A})=-d_{\mathbb{M}}\Omega\wedge\kappa\wedge\Omega\text{Tr}(si_{\mathcal{R}}\mathbb{A}).$
(5.4)

Thus, after combining both expressions, using
$s\Omega=\\!^{\Omega}\mathbb{A}-\mathbb{A}$ and the fact that
$i_{\mathcal{R}}(^{\Omega}\mathbb{A})=i_{\mathcal{R}}\mathbb{A}$, we quickly
realize that the action (5.2) raises as the obvious $\Omega$-shift invariant
extension of the action (5.1). Notice that, it is also invariant under the
rescalings (2.82). At this stage, it is interesting to compare (5.2) with
(2.81) for $\Phi=0$. Notice that $S(\mathbb{A}=s\Omega)=0$, thus the
functional (5.2) descends to the quotient $\mathcal{A}/\mathcal{S}_{\Omega}$,
where one component of $\mathbb{A}$ manifestly decouples. Furthermore, under
finite gauge transformations, we find that

$S(^{g}\mathbb{A})=S(\mathbb{A})+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\chi(g)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(Y^{\prime}\mathbb{J}\right),$
(5.5)

where we have used (2.91), (2.92) and defined

$Y^{\prime}=2i_{\mathcal{R}}\mathbb{A}+i_{\mathcal{R}}\mathbb{J}.$ (5.6)

The last contribution on the rhs of (5.5) is related to the obstruction
(2.26), while the second term is the WZ-type term found before.

The second step consists of introducing the $\kappa$-shift symmetry. To do so,
we consider a Lie algebra valued field
$\Phi\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$, which transforms like

${}^{\kappa}\Phi=\Phi+s,$ (5.7)

for an arbitrary $s\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$ under
$\kappa$-shifts and like $\Phi\rightarrow t^{-1}\Phi$ under the arbitrary
rescalings (2.82). We also demand that it is invariant under $\Omega$-shifts,
i.e. ${}^{\Omega}\Phi=\Phi$. The key idea now NA loc CS  is to notice that the
combination $\mathbb{A}-\kappa\Phi$ is invariant under $\kappa$-shifts and
rescalings. Thus, a double-shift invariant action functional is obtained from
(5.2) after making the substitution
$\mathbb{A}\rightarrow\mathbb{A}-\kappa\Phi$. We find that151515Notice how the
top-form $\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa$ appears naturally
from this point of view. This result actually motivated the definition of the
inner product (2.42) introduced above.

$S(\mathbb{A},\Phi)=S(\mathbb{A})+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\Phi^{2}\right)-2ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\text{Tr}\Big{(}\Phi\left(\Omega\wedge\kappa\wedge
F_{\mathbb{A}}+d_{\mathbb{M}}\Omega\wedge\kappa\wedge\mathbb{A}\right)\Big{)}.$
(5.8)

We have not specified the behavior of $\Phi$ under gauge transformations.
However, in order to be consistent with the transformation of $\Phi$, as
defined in (2.52), under gauge transformations (2.97) we demand that

${}^{g}\Phi=g^{-1}\left(\Phi+B(\mathbb{J})\right)g.$ (5.9)

Then, we get

$S(^{g}\mathbb{A},^{g}\Phi)=S(\mathbb{A},\Phi)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\chi(g)+ic\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}d_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(W\mathbb{J}\right),$
(5.10)

for some $W\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$. Again, there is a
term related to (2.26) and a WZ-type contribution. Even if we set
${}^{g}\Phi=g^{-1}\Phi g$, we obtain the same type of expression with a change
$W\rightarrow W^{\prime}$ for another
$W^{\prime}\in\Omega_{\mathbb{M}}^{0}\otimes\mathfrak{g}$, whose explicit form
is not relevant in what follows.

Now we consider the duality manipulations. Using the $\kappa$-shift symmetry
we can reach a gauge where $\Phi=0$, in this case we recover the regularized
theory (5.2). Alternatively, if we calculate the $\Phi$ field eom and put them
back into the action (5.8), we find a dual action that is classically
equivalent to (2.83). Indeed, the $\Phi$ eom is nothing but the first
expression defined in (2.52), i.e.

$\Phi=\frac{\Omega\wedge\kappa\wedge
F_{\mathbb{A}}+d_{\mathbb{M}}\Omega\wedge\kappa\wedge\mathbb{A}}{\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa}$ (5.11)

and fulfills all the required transformation properties, see for instance
(2.55) and (2.56). At this point we can see an advantage of the regularized
theory, as the denominator in the expression right above never vanishes,
making the variational problem for the quadratic field $\Phi$ well-defined
over the manifold $\mathbb{M}$. Otherwise, we would have a contribution of the
form $\omega\wedge d\alpha=0$, see (3.14), in the second term on the rhs of
(5.8) turning the theory linear in the field $\Phi$. The effective action
obtained after replacing (5.11) in (5.8) gives the dual action

$S_{\text{dual}}=ic(\mu,\mu),$ (5.12)

which is precisely the $\mathcal{S}$-invariant quadratic action constructed in
(2.83). Thus, the actions (5.2) and (5.12) are dual to each other. Once in the
form (5.12), we can recover (5.2) if we gauge fix the $\kappa$-shift symmetry
with the gauge fixing condition $\Phi=0$. The original theory (5.1) is finally
recovered by taking the degenerate limit $\zeta\rightarrow 0$ and by imposing
the boundary conditions (4.164) on the gauge connection $\mathbb{A}$. This
further clarifies the why of the two step strategy introduced before in
section (3). From (5.11), we find that

$\Phi(\mathbb{A}=s\Omega)=0,\text{ \ \ }\Phi(\mathbb{A}=s\kappa)=s$ (5.13)

and from this result we obtain $S(\mathbb{A}=s\Omega)_{\text{dual}}=0$ and
$S(\mathbb{A}=s\kappa)_{\text{dual}}=0$. This confirms that (5.12) descends to
the quotient $\overline{\mathcal{A}}$, where a second component of
$\mathbb{A}$ manifestly decouples from the theory. The number of components of
$\mathbb{A}$, as a Lie algebra valued 1-form is two, see (4.66) and (4.84).
Recall that we still have the action of the gauge group $\mathcal{G}$ on
$\overline{\mathcal{A}}$ to be taken into account.

The following diagram roughly summarizes our findings:

$\begin{array}[]{ccc}S=\small\text{\\{4d CS theory
\eqref{1.1}\\}}&\overset{\Omega\text{-shift extension}}{\xrightarrow{\hskip
85.35826pt}}&S(\mathbb{A})=\small\text{\\{regularized action \eqref{reg
action}\\}}\\\ &&\\\
\;\;\Bigg{\uparrow}\begin{array}[]{c}\footnotesize\text{step I:
}\Phi_{\eqref{Phi eom}}=0\\\
\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\footnotesize\text{step II: }\zeta\rightarrow
0\end{array}&&\Bigg{\downarrow}\kappa\footnotesize\text{-shift extension}\\\
&&\\\ S_{\text{dual}}=\small\text{\\{}\text{generalized 4d CS theory
\eqref{last}\\}}&\overset{\Phi\text{-integration}}{\xleftarrow{\hskip
68.28644pt}}&S(\mathbb{A},\Phi)=\small\text{\\{}\text{extended action
\eqref{T-dual action}\\}}.\end{array}$

The dual pair of action functionals is formed by (5.2) and (5.12), i.e.
$S(\mathbb{A})$ and $S_{\text{dual}}$ right above. Steps I and II are to be
supplemented with a solution to the condition (3.25).

In the previous sections we have studied systematically several classical
aspects of the generalized 4d CS theory. Now, it is time to consider the path
integral formulation of it. After all, as mentioned in the introduction, our
main goal is to show that the usual 4d CS theories (1.1) can be embedded into
a more general theory whose path integral formulation takes the canonical form
(1.3). This is precisely the topic of the next and last section.

##  6 Path integral and non-Abelian localization

Here we comment on the second ingredient (1.5) involved in the formula of non-
Abelian localization, i.e. the path integral symplectic measure. The first
one, discussed extensively above, being the quadratic form (1.4) of the action
functional. As announced, we will show that the path integral for the
generalized theory, at least for the main example considered in section (4),
takes the form (1.3). See NA loc CS , for the original 3-dimensional CS theory
formulation.

Consider the theory defined formally by the path integral

$Z=\mathcal{N}^{\prime}\int\nolimits_{\mathcal{A}\times\mathcal{S}_{\kappa}}\mathcal{D}\mathbb{A}\mathcal{D}\Phi\;\text{exp}\left[\frac{i}{\hbar}S(\mathbb{A},\Phi)\right],$
(6.1)

where the action in the exponential is given by (5.8), $\mathcal{N}^{\prime}$
is defined by

$\mathcal{N}^{\prime}=\mathcal{N}\times\frac{1}{\text{Vol}(\mathcal{S}_{\kappa})\times\text{Vol}(\mathcal{S}_{\Omega})\times\text{Vol}(\mathcal{G})}$
(6.2)

and $\mathcal{N}$ is a normalization constant. As we have shown in (2.55),
(2.56), (5.13), the field $\Phi$ belongs to the orbit generated by the action
of the $\kappa$-shift group $\mathcal{S}_{\kappa}$, while it is a fixed point
under the action of the $\Omega$-shift group $\mathcal{S}_{\Omega}$. Then, the
integral associated to the measure $\mathcal{D}\Phi$ is over
$\mathcal{S}_{\kappa}$, while the integral associated to
$\mathcal{D}\mathbb{A}$ is, as usual, over the whole space $\mathcal{A}$ of
gauge connections.

The translation-invariant measure $\mathcal{D}\Phi$ is defined independently
of any metric on $\mathbb{M}$ by the invariant, quadratic form

$\left(\Phi,\Phi\right)_{\mathbb{M}}=-\int\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\Phi^{2}\right).$ (6.3)

This quadratic form is, up to scale, used to formally define the volume of
$\mathcal{S}_{\kappa}$. Similar expressions are used to define
$\mathcal{S}_{\Omega}$ and $\mathcal{G}$, as anticipated before in (2.42) and
(4.50). This is to be complemented with the formal definition of the
translation-invariant measure $\mathcal{D}\mathbb{A}$ induced by the norm
(4.72) and its orthogonal decomposition between the spaces
$\overline{\mathcal{A}}$ and $\mathcal{S}$.

On the one hand, using the $\kappa$-shift symmetry, we can fix $\Phi=0$
trivially with unit Jacobian, and the resulting integral over
$\mathcal{S}_{\kappa}$ produces a formal factor of
$\text{Vol}(\mathcal{S}_{\kappa})$. Hence, the theory is equivalent to (5.2),
i.e. to the $\Omega$-shift invariant extension of the 4d CS theory (1.1). The
resulting action functional is valued in the quotient
$\mathcal{A}/\mathcal{S}_{\Omega}$ and $\mathcal{D}\mathbb{A}$ integrates over
$\mathcal{A}/\mathcal{S}_{\Omega}\times\mathcal{S}_{\Omega}$, where the
integral over $\mathcal{S}_{\Omega}$ produces a formal factor or
$\text{Vol}(\mathcal{S}_{\Omega})$. Then, in principle, we get

$Z=\mathcal{N}\times\frac{1}{\text{Vol}(\mathcal{G})}\int\nolimits_{\mathcal{A}/\mathcal{S}_{\Omega}}\mathcal{D}\mathbb{A}\;\text{exp}\left[\frac{i}{\hbar}S(\mathbb{A})\right].$
(6.4)

On the other hand, because of the field $\Phi$ appears only quadratically in
the action (5.8), we can perform the path integral over $\Phi$ directly.
Integrating out $\Phi$, produces a contribution

$I=\int\nolimits_{\mathcal{S}_{\kappa}}\mathcal{D}\Phi\;\text{exp}\left[-\frac{c}{\hbar}\int\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\Phi^{2}\right)\right].$ (6.5)

The resulting action functional is given by the dual quadratic expression
(5.12) and is valued in $\overline{\mathcal{A}}$. Thus $\mathcal{D}\mathbb{A}$
integrates over $\overline{\mathcal{A}}\times\mathcal{S}$. Here, we make use
of the results (4.72), (4.73) related to the fact that the quotient space
$\overline{\mathcal{A}}$ is symplectic and equipped with a Kähler metric in
order to write the measure along $\overline{\mathcal{A}}$ in the form (see
(1.5))

$\mathcal{D}\mathbb{A}|_{\overline{\mathcal{A}}}=\text{exp}\;\hat{\underline{\Omega}}.$
(6.6)

As we showed above, this measure is to be taken over elements
$\Pi({\mathbb{A}})\in\overline{\mathcal{A}}$ of the form (4.64).

An important consequence of the fact that the metric on
$\overline{\mathcal{A}}$ is Kähler is that the Riemannian measure
$\mathcal{D}\mathbb{A}$ on $\overline{\mathcal{A}}$ is actually the same as
the symplectic measure defined by $\hat{\underline{\Omega}}$. Indeed, if $X$
is a symplectic manifold of dimension $2n$ with symplectic form
$\hat{\Omega}$, then the symplectic measure on $X$ is given by the top-form
$\hat{\Omega}^{n}/n!$. This measure can be represented by the expression
$\text{exp}\;\hat{\Omega}$, where we implicitly pick out from the series
expansion of the exponential the term which is of top degree on $X$.
Consequently, because of the Riemannian and the symplectic measures on
$\overline{\mathcal{A}}$ agree, we can formally replace
$\mathcal{D}\mathbb{A}$ over $\overline{\mathcal{A}}$ in the path integral by
the expression (6.6) above and write instead

$Z(\epsilon)=\mathcal{N}\times
I\times\frac{1}{\text{Vol}(\mathcal{G})}\mathop{\displaystyle\int}\nolimits_{\overline{\mathcal{A}}}\
\text{exp}\left[\hat{\underline{\Omega}}-\frac{1}{2\epsilon}(\mu,\mu)\right],$
(6.7)

where we have used
$\text{Vol}(\mathcal{S})=\text{Vol}(\mathcal{S}_{\kappa})\times\text{Vol}(\mathcal{S}_{\Omega})$
and defined $\epsilon=\hbar/2c$. This integral takes the canonical form (1.3)
with $X=\overline{\mathcal{A}}$, as required by the non-Abelian localization
method. The normalization constant $\mathcal{N}$ can be adjusted to an
specific valued if needed.

The main consequence of an expression like (6.7) is that it suggests an
interesting relationship between the quantum integrable structure of the 4d CS
theory and the geometry of the symplectic quotient space
$\overline{\mathcal{A}}$. In principle, the 4d CS theories and their
associated integrable models and field theories could be explored via standard
localization techniques.

##  7 Concluding remarks

In this concluding section, we make some comments, provide further
explanations concerning the results presented along the text and touch on some
topics we judge interesting to be considered in the future.

Clearly, the pre-symplectic form (2.1) plays a crucial rôle in the
construction of the path integral (1.3), as it specifies the moment map $\mu$
used to define the quadratic action $S\sim(\mu,\mu)$, as well as the path
integral symplectic measure $e^{\underline{\Omega}}$ over the quotient space
$\overline{\mathcal{A}}$. Thus, some comments on what inspired its definition
are in order.

Consider the original pre-symplectic form defined in NA loc CS , which in the
present notation takes the form

$\hat{\Omega}=-\frac{1}{2}\mathop{\displaystyle\int}\nolimits_{\text{M}}\alpha\wedge\text{Tr}\left(\hat{\delta}\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right).$
(7.1)

The circle fibers, correspond to the integral curves of the Reeb vector field
$R$ satisfying the normalization condition $\alpha(R)=1$. After introducing
the time direction, we extend M to $\mathbb{M}=\mathbb{R}\times\text{M}$.
Locally, the manifold $\mathbb{M}$ looks like $\mathbb{M}=\Sigma\times C$ and
the light-cone tangent vectors to the Minkowskian cylinder $\Sigma$, are given
by $\partial_{\pm}\sim\partial_{\tau}\pm\partial_{\sigma}$. It is then natural
to extend them, respectively, to their global counterparts (3.10), (4.38)

$\mathcal{R}\sim\frac{1}{\alpha_{\tau}}\partial_{\tau}+R,\text{ \ \
}\mathcal{R}^{\prime}\sim\frac{1}{\alpha_{\tau}}\partial_{\tau}-R$ (7.2)

and to introduce two 1-forms $\kappa$ and $\Omega^{\prime}$ such that
$\kappa(\mathcal{R})=1$ and $\Omega^{\prime}(\mathcal{R}^{\prime})=1$, see
(3.2), (4.39). Because of $R$ defines vertical and horizontal directions in
the tangent space of the total space M, we can add any horizontal 1-form
$\omega$ with no $d\tau$ term to $\kappa$ or to $\Omega^{\prime}$, without
spoiling the normalization conditions and this is because
$i_{\mathcal{R}}\omega=i_{\mathcal{R}^{\prime}}\omega=0$. We choose to add it
to $\Omega^{\prime}$ and define $\omega$ as the pull-back, by the projection
map $\underline{\pi}$, of the twist 1-form $\omega_{C}$ on $C$ that specifies
the associated integrable field theory. There is some room to introduce an
arbitrary parameter, which we call $\zeta$. Then, a natural generalization of
(7.1) to four dimensions, in which we include the time direction, is given by

$\hat{\Omega}=-\frac{1}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\hat{\delta}\mathbb{A}\wedge\hat{\delta}\mathbb{A}\right).$
(7.3)

Now we show how the 1-forms $\Omega$, $\kappa$ of (3.2) used in (7.3) are
constructed. Let us start with the interpolating expressions

$\kappa=s\alpha_{\tau}d\tau+(1-s)\alpha,\text{ \ \
}\Omega=\omega+\zeta^{\prime}\left((1-s)d\tau-\frac{s}{\alpha_{\tau}}\alpha\right),$
(7.4)

and

$\mathcal{R}=\frac{1}{2}\left(\frac{1}{s\alpha_{\tau}}\partial_{\tau}+\frac{1}{1-s}R\right),\text{
\ \ \
}\mathcal{R}^{\prime}=\frac{1}{2\zeta^{\prime}}\left(\frac{1}{1-s}\partial_{\tau}-\frac{\alpha_{\tau}}{s}R\right),$
(7.5)

where $s\in(0,1)$ and $\zeta^{\prime}\in\mathbb{R}$. They satisfy the
normalization conditions
$i_{\mathcal{R}}\kappa=i_{\mathcal{R}^{\prime}}\Omega=1.$ By demanding that
$\Omega\wedge\kappa\wedge d_{\mathbb{M}}\kappa=\zeta d\tau\wedge\alpha\wedge
d\alpha$, we find

$\zeta^{\prime}=\frac{\zeta}{(1-s)\left[s^{2}+(1-s)^{2}\right]}.$ (7.6)

However, the conditions
$i_{\mathcal{R}}\Omega=i_{\mathcal{R}^{\prime}}\kappa=0$ require that $s$ take
the specific value $s=1/2$, corresponding to the ‘light-cone’ solutions
introduced above in (3.2), (3.3). It is also possible to make the changes
$(\alpha,R)\rightarrow(-\alpha,-R)$ in all formulae.

The path integral (1.3) is specified by the symplectic data associated to
$\hat{\Omega}$ and the Hamiltonian action of $\mathcal{H}$ on
$\overline{\mathcal{A}}$. Also notice that the generalized 4d CS theory does
not require a twist 1-form $\omega$ to be well-defined. Actually, for
$\omega=0$, we have that $\Omega\wedge\kappa=\Omega_{\tau}d\tau\wedge\alpha$
and this case can be seen as the simplest canonical 4d extension of (7.1).
Then, the generalized theory allows, in principle, to embed any 4d CS theory,
regardless of the analytic structure of $\omega_{C}$, into a quantization
framework based on the non-Abelian localization method. In particular, it may
offer an approach for quantizing non-ultralocal integrable field theories
Maillet  from a more geometric perspective. These type of theories all have
twist 1-forms $\omega_{C}$ with zeroes Vicedo-PCM , like the PCM type models
considered above, and quantization is problematic because of the non-
ultralocality prevents a straightforward use of techniques coming from the
quantum inverse scattering method. Furthermore, as argued heuristically in
CWY1 , the zeroes of $\omega_{C}$ corresponds to points where
$\hbar\rightarrow\infty$, hence an approach based on localization may provide
a 4d CS theories description even in this regime.

Let us notice that the combined shift symmetries in $\mathcal{S}$ dictate the
very form of the quadratic action (5.12). Thus, a natural question to be asked
is what is the rôle played by the vector field $\mathcal{R}^{\prime}$ and how
its associated moment map modifies the quadratic action.
Let us start with the complete induced vector field on $\mathcal{A}$, cf.
(2.39), which is given by

$V(\tilde{p},\eta,a)=d_{\mathbb{A}}\eta+p\pounds_{\mathcal{R}}\mathbb{A+}p^{\prime}\pounds_{\mathcal{R}^{\prime}}\mathbb{A},$
(7.7)

where161616Alternatively, we may introduce a four entry notation
$(p^{\prime},p,\eta,a)$.  $\tilde{p}=p+p^{\prime}$. For the moment map $\mu$
associated to $V(p^{\prime},0,0)$, we find that

$\left\langle\mu,(p^{\prime},0,0)\right\rangle=\frac{p^{\prime}}{2}\mathop{\displaystyle\int}\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge\text{Tr}\left(\pounds_{\mathcal{R}^{\prime}}\mathbb{A\wedge
A}\right),$ (7.8)

which descends to the quotient space $\overline{\mathcal{A}}$. We also find
the Poisson bracket, cf. (2.36),

$\left\\{\left\langle\mu,(p^{\prime},0,0)\right\rangle,\left\langle\mu,(0,\lambda,0)\right\rangle\right\\}=\left\langle\mu,(0,-p^{\prime}\pounds_{\mathcal{R}^{\prime}}\lambda,0)\right\rangle.$
(7.9)

Thus, the bracket (2.38) is replaced by

$\Big{[}(\tilde{p},\eta,a),(\tilde{q},\lambda,b)\Big{]}=\Big{(}0,[\eta,\lambda]-p\pounds_{\mathcal{R}}\lambda-p^{\prime}\pounds_{\mathcal{R}^{\prime}}\lambda+q\pounds_{\mathcal{R}}\eta+q^{\prime}\pounds_{\mathcal{R}^{\prime}}\eta,c(\eta,\lambda)\Big{)},$
(7.10)

where $\tilde{q}=q+q^{\prime}$. Now, we extend the inner product (2.42) to

$\big{(}\left(\tilde{p},\eta,a\right),\left(\tilde{q},\lambda,b\right)\big{)}=-\int\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\left(\eta\lambda\right)-\tilde{p}b-\tilde{q}a.$
(7.11)

The final step is to verify if the inner product just defined is invariant,
which is equivalent to having, cf. (2.44),

$\Big{(}\left[\left(\tilde{p},\eta,a\right),\left(\tilde{q},\lambda,b\right)\right],(\tilde{r},\phi,c)\Big{)}=\Big{(}\left(\tilde{p},\eta,a\right),\left[\left(\tilde{q},\lambda,b\right),(\tilde{r},\phi,c)\right]\Big{)},$
(7.12)

where $\tilde{r}=r+r^{\prime}$. This conditions boils down to

$\displaystyle\tilde{r}d(\eta,\lambda)+r^{\prime}\int\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}$
$\displaystyle\Big{(}\eta\big{(}\pounds_{\mathcal{R}}\lambda-\pounds_{\mathcal{R}^{\prime}}\lambda\big{)}\Big{)}$
(7.13)
$\displaystyle=\tilde{p}d(\lambda,\phi)+p^{\prime}\int\nolimits_{\mathbb{M}}\Omega\wedge\kappa\wedge
d_{\mathbb{M}}\kappa\text{Tr}\Big{(}\lambda\big{(}\pounds_{\mathcal{R}}\phi-\pounds_{\mathcal{R}^{\prime}}\phi\big{)}\Big{)}.$

We solve this by taking $d(\ast,\ast)=0$ and
$p^{\prime}=q^{\prime}=r^{\prime}=0$. Thus, the vector field (7.7), the
bracket (7.10) and the inner product (7.11) reduce to the ones considered
before. As a consequence, the quadratic action remains unaltered. The vector
fields $\mathcal{R}$ and $\mathcal{R}^{\prime}$ have different uses in the
formulation of the generalized 4d CS theory, at least as implied by the
solutions to (7.13) chosen in the present paper. It would be interesting to
consider other possible solutions and their implications.

For integrable field theories on (semi)-symmetric spaces, the formulation
presented here requires to consider non-trivial $S^{1}$ bundles over the base
space $C=\mathbb{CP}^{1}/\mathbb{Z}_{4}$. Coset spaces of the form
$\mathbb{CP}^{1}/\mathbb{Z}_{T}$ were first considered in dihedral  on an
approach devised to reformulate $\mathbb{Z}_{T}$-graded coset $\sigma$-model
as dihedral affine Gaudin models. The particular case $T=4$, was also studied
in me def coset , where the symmetric-space $\lambda$-model exchange algebra
was recovered from the point of view of the conventional 4d CS theory. It is
then desirable to study the generalized 4d CS theory on non-trivial circle
bundles over spaces of this type, due to their relation to important non-
ultralocal integrable field theories like the $\sigma$-models on
(semi)-symmetric spaces and their integrable deformations eta-def bos ; eta-
def fer ; Rivelles-Hector ; lambda-bos ; lambda-fer ; Hybrid lambda ; PS
lambda  too. We expect to consider this in the near future.

The generalized theory presents a behavior, under the action of finite gauge
transformations, that is similar to the conventional 4d CS theory. Thus, the
last couple of terms in (2.95) must be properly handled first in order for the
expression (6.7) to make perfect sense at the quantum level. In this work we
have adopted the strategy of imposing restrictions over the gauge elements
$g\in\mathcal{G}$ in order to cancel both contributions, making the generalize
theory gauge invariant. We do not know if this approach is the only way to do
it or if there is some gauge group structure that can be exploited instead. We
expect to consider this subtle issue in a more systematic way elsewhere.

The present construction relies on having a compact direction in the
2-dimensional space-time $\Sigma$, which we chose to be a cylinder
$\Sigma=\mathbb{R}\times S^{1}$. An interesting problem would be to consider
instead a strip $\Sigma=\mathbb{R}\times[-L,L]$. This case would cover
integrable field theories defined on the real line or a finite segment,
depending on the choice of the parameter $L$. It is a well-known fact that
Chern-Simons theories and WZW models are closely related and that WZW models
with open string boundary conditions require an specific set of D-brane
configurations on group manifold or subsets of it. For example and just to
name a few, D-branes are considered in Schomerus ,NA Kinks ,Driezen  in the
context of WZW models, symmetric space sine-Gordon theories and lambda
deformed integrable field theories, respectively. How our generalized
4-dimensional Chern-Simons theory is related to integrable models defined on a
segment is not clear at this moment and remains as an open problem, but a
sensible starting point could be to first explore the CS/WZW relation for open
string configurations.

## Acknowledgements

The author thanks the referee for valuable comments and suggestions.

## References

  * (1) K. Costello. Supersymmetric gauge theory and the Yangian. [e-Print: arXiv:1303.2632].
  * (2) K. Costello. Integrable lattice models from four-dimensional field theories. Proc.Symp.Pure Math. 88 (2014) 3-24. https://doi.org/10.1090/pspum/088. [e-Print: arXiv:1308.0370].
  * (3) K. Costello, E. Witten and M. Yamazaki. Gauge theory and integrability, I. ICCM Not. 6, 46-191 (2018). https://dx.doi.org/10.4310/ICCM.2018.v6.n1.a6. [e-Print: arXiv:1709.09993].
  * (4) K. Costello, E. Witten and M. Yamazaki. Gauge theory and integrability, II. ICCM Not. 6, 120-149 (2018). https://dx.doi.org/10.4310/ICCM.2018.v6.n1.a7. [e-Print: arXiv:1802.01579].
  * (5) K. Costello and M. Yamazaki. Gauge theory and integrability, III. [e-Print: arXiv:1908.02289].
  * (6) S. Lacroix. Four-dimensional Chern–Simons theory and integrable field theories. J.Phys.A 55 (2022) 8, 083001. https://doi.org/10.1088/1751-8121/ac48ed. [e-Print: 2109.14278 [hep-th]]
  * (7) E. Witten and C. Beasley. Non-Abelian localization for Chern-Simons theory. J.Diff.Geom. 70 (2005) 2, 183-323. DOI: 10.4310/jdg/1143642932. [e-Print: hep-th/0503126 [hep-th]].
  * (8) P. Orlik. Seifert Manifolds. Lecture Notes in Mathematics 291, Ed. by A. Dold and B. Eckmann, Springer-Verlag, Berlin, 1972.
  * (9) E. Witten. Two dimensional gauge theories revisited. J.Geom.Phys. 9 (1992) 303-368. https://doi.org/10.1016/0393-0440(92)90034-X. [e-Print: hep-th/9204083 [hep-th]]
  * (10) C. Beasley. Localization for Wilson loops in Chern-Simons theory. Adv.Theor.Math.Phys. 17 (2013) 1, 1-240. DOI: https://dx.doi.org/10.4310/ATMP.2013.v17.n1.a1. [e-Print: 0911.2687 [hep-th]].
  * (11) A. Pressley and G. Segal. Loop Groups. Clarendon Press, Oxford, 1986.
  * (12) F. Delduc, S. Lacroix, M. Magro and B. Vicedo. A unifying 2d action for integrable $\sigma$-models from 4d Chern-Simons theory. Lett.Math.Phys. 110 (2020) 1645-1687. https://doi.org/10.1007/s11005-020-01268-y. [e-Print: 1909.13824 [hep-th]].
  * (13) J. Martinet. Formes de contact sur les varietétés de dimension 3. Springer Lecture Notes in Math 209 (1971) 142–163.
  * (14) John B. Etnyre. Introductory Lectures on Contact Geometry. arXiv:math/0111118
  * (15) D. E. Blair. Riemannian Geometry of Contact and Symplectic Manifolds. Birkhäuser, Boston, 2002.
  * (16) H. Geiges. An introduction to Contact Topology. Cambridge studies in advanced Mathematics. Cambridge University Press, 2008.
  * (17) W. M. Boothby and H. C. Wang. On Contact Manifolds. Annals of Mathematics Second Series, Vol. 68, No. 3 (Nov., 1958), 721-734. https://doi.org/10.2307/1970165
  * (18) R. Bott and L. W. Tu. Differential Forms in Algebraic Topology. Springer-Verlag, 1982.
  * (19) M. Henneaux and C. Teitelboim. Quantization of Gauge Systems. Princeton University Press. 1992
  * (20) J. M. Maillet. New integrable canonical structures in two-dimensional models. Nucl.Phys.B 269 (1986) 54-76. DOI: 10.1016/0550-3213(86)90365-2
  * (21) B. Vicedo. 4D Chern-Simons theory and affine Gaudin models. Lett.Math.Phys. 111 (2021) 1, 24. https://doi.org/10.1007/s11005-021-01354-9. [e-Print: arXiv:1908.07511].
  * (22) D. M. Schmidtt. Holomorphic Chern-Simons theory and lambda models: PCM case. JHEP 04 (2020) 060. https://doi.org/10.1007/JHEP04(2020)060. [e-Print: 1912.07569 [hep-th]].
  * (23) T. J. Hollowood, J. L. Miramontes and D. M. Schmidtt. S-matrices and quantum group symmetry of k-deformed sigma models. J.Phys.A 49 (2016) 46, 465201. https://doi.org/10.1088/1751-8113/49/46/465201. [e-Print: 1506.06601 [hep-th]].
  * (24) K. Sfetsos. Integrable interpolations: From exact CFTs to non-Abelian T-duals. Nucl.Phys.B 880 (2014) 225-246. https://doi.org/10.1016/j.nuclphysb.2014.01.004. [e-Print: 1312.4560 [hep-th]].
  * (25) O. Babelon and D. Bernard. Dressing symmetries. Commun.Math.Phys. 149 (1992) 279-306. https://doi.org/10.1007/BF02097626. [e-Print: hep-th/9111036 [hep-th]].
  * (26) B. Vicedo. On integrable field theories as dihedral affine Gaudin models. Int. Math. Res. Not. rny128 (2018). https://doi.org/10.1093/imrn/rny128. [e-Print: arXiv:1701.04856].
  * (27) D. M. Schmidtt. Symmetric space $\lambda$-model exchange algebra from 4d holomorphic Chern-Simons theory. JHEP12(2021)004. https://doi.org/10.1007/JHEP12(2021)004. [e-Print: 2109.05637 [hep-th]].
  * (28) F. Delduc, M. Magro and B. Vicedo. On classical q-deformations of integrable sigma-models. JHEP 1311 (2013) 192. https://doi.org/10.1007/JHEP11(2013)192. [e-Print: 1308.3581 [hep-th]].
  * (29) F. Delduc, M. Magro and B. Vicedo. Integrable deformation of the $AdS_{5}\times S^{5}$ superstring action. Phys.Rev.Lett. 112 (2014) no.5, 051601. https://doi.org/10.1103/PhysRevLett.112.051601. [e-Print: arXiv:1309.5850].
  * (30) H. Benítez and V. Rivelles. Yang-Baxter deformations of the $AdS_{5}\times S^{5}$ pure spinor superstring. JHEP 02 (2019) 056. https://doi.org/10.1007/JHEP02(2019)056. e-Print: 1807.10432 [hep-th].
  * (31) T. J. Hollowood, J. L. Miramontes and D. M. Schmidtt. Integrable deformations of strings on symmetric spaces. JHEP 1411 (2014) 009. https://doi.org/10.1007/JHEP11(2014)009. [e-Print: arXiv:1407.2840].
  * (32) T. J. Hollowood, J. L. Miramontes and D. M. Schmidtt. An integrable deformation of the $AdS_{5}\times S^{5}$ superstring. J.Phys. A47 (2014) no.49, 495402. https://doi.org/10.1088/1751-8113/47/49/495402. [e-Print: arXiv:1409.1538].
  * (33) D. M. Schmidtt. Exploring the lambda model of the hybrid superstring. JHEP 10 (2016) 151. https://doi.org/10.1007/JHEP10(2016)151. e-Print: 1609.05330 [hep-th].
  * (34) H. Benítez and D. M. Schmidtt. $\lambda$-deformation of the $AdS_{5}\times S^{5}$ pure spinor superstring. JHEP 10 (2019) 108. https://doi.org/10.1007/JHEP10(2019)108. e-Print: 1907.13197 [hep-th].
  * (35) A. Yu. Alekseev and V. Schomerus. D-branes in the WZW model. Phys.Rev.D 60 (1999) 061901. https://doi.org/10.1103/PhysRevD.60.061901. e-Print: hep-th/9812193 [hep-th].
  * (36) T. J. Hollowood, J. L. Miramontes and D. M. Schmidtt. The Structure of Non-Abelian Kinks. JHEP 10 (2013) 058. https://doi.org/10.1007/JHEP10(2013)058. e-Print: 1306.6651 [hep-th].
  * (37) S. Driezen, A. Sevrin and D. C. Thompson. D-branes in $\lambda$-deformations. JHEP 09 (2018) 015. https://doi.org/10.1007/JHEP09(2018)015. e-Print: 1806.10712 [hep-th].