# IndoToD: A Multi-Domain Indonesian Benchmark For End-to-End Task-Oriented
Dialogue Systems

Muhammad Dehan Al Kautsar1, Rahmah Khoirussyifa’ Nurdini1, Samuel
Cahyawijaya3,
Genta Indra Winata2, Ayu Purwarianti1
1Institut Teknologi Bandung 2Bloomberg
3The Hong Kong University of Science and Technology
<EMAIL_ADDRESS>
###### Abstract

Task-oriented dialogue (ToD) systems have been mostly created for high-
resource languages, such as English and Chinese. However, there is a need to
develop ToD systems for other regional or local languages to broaden their
ability to comprehend the dialogue contexts in various languages. This paper
introduces IndoToD, an end-to-end multi-domain ToD benchmark in Indonesian. We
extend two English ToD datasets to Indonesian, comprising four different
domains by delexicalization to efficiently reduce the size of annotations. To
ensure a high-quality data collection, we hire native speakers to manually
translate the dialogues. Along with the original English datasets, these new
Indonesian datasets serve as an effective benchmark for evaluating Indonesian
and English ToD systems as well as exploring the potential benefits of cross-
lingual and bilingual transfer learning approaches.

##  1 Introduction

Figure 1: Illustration of the dialogues collection pipeline: a) Delexicalize
dialogue conversations to create English templates; b) Translate English
conversation templates to Indonesian; c) Retrieve KB to retrieve corresponding
entities; d) Lexicalize the dialogue conversations to collect Indonesian
dialogues.

Task-oriented dialogue (ToD) systems are conversational agents designed to
interact with users and assist them in various domains, such as restaurant
Bordes et al. (2016); Wen et al. (2017), public transport Budzianowski et al.
(2018); Lin et al. (2021b), and in-car assistance Eric et al. (2017). This
system also serves as the base for many commercial products that operate using
the dialogue systems approach because of its ability to operate and understand
the dialogue context without using hand-crafted rules Lin et al. (2021b).
Despite the recent growing interest in developing end-to-end ToD systems due
to their simplicity, ToD systems are mostly created using monolingual datasets
in high-resource languages such as English and Chinese. Moreover, building a
ToD system can be challenging due to the limited availability of datasets for
training and evaluating the system, which has been identified as the most
critical factor in preventing the creation of bilingual/multilingual ToD
systems Wen et al. (2017); Razumovskaia et al. (2021).

Developing ToD systems in additional underrepresented languages is essential
to expand their capabilities to understand dialogue contexts in diverse
languages Kanakagiri and Radhakrishnan (2021). One of them is Indonesian, a
language spoken by many people worldwide yet still considered an
underrepresented language in the end-to-end ToD system. Indonesian is ranked
as the fourth most internet users in the world based on the latest data per
country Aji et al. (2022), with around 212 million internet
users.111https://www.internetworldstats.com/stats3.htm However, the language
itself is still categorized as an underrepresented language in the natural
language processing (NLP) community because of problems such as scattered
datasets and minimum community engagement Wilie et al. (2020); Cahyawijaya et
al. (2021, 2022). To the best of our knowledge, there is only one publicly
available end-to-end Indonesian ToD dataset222https://indonlp.github.io/nusa-
catalogue/ which is COD Majewska et al. (2023), a multilingual ToD dataset
that is solely used for evaluation – only test set available. It has a very
limited number of samples for Indonesian with 194 dialogues across 11 domains
($\sim$18 dialogues per domain) and no training data provided. This emphasizes
the need to create larger end-to-end Indonesian ToD datasets to expand the
capabilities to build and evaluate localized Indonesian ToD systems.

To address the aforementioned issues, we propose IndoToD, an end-to-end multi-
domain ToD benchmark in Indonesian. IndoToD comprises the collection of two
parallel Indonesian end-to-end ToD datasets covering four different domains by
manually translating two existing English datasets: CamRest676 Wen et al.
(2017) and SMD Eric et al. (2017) using delexicalization and lexicalization
processes, as well as the evaluation of existing end-to-end ToD frameworks in
various settings such as monolingual, cross-lingual, and bilingual. IndoToD
provides more dialogue samples compared to Majewska et al. (2023) that can be
utilized for training and evaluation.

This paper’s contributions are summarized three-fold as follows:

  * •

We introduce IndoToD, the multi-domain benchmark for Indonesian ToD systems.
The benchmark comprises two datasets: IndoCamRest and IndoSMD, with four
different domains that serve as resources for training and evaluation.

  * •

We establish baselines on monolingual, bilingual, and cross-lingual settings
on existing end-to-end ToD frameworks.

  * •

We analyze the effectiveness of training bilingual datasets to leverage more
training data and handle tasks in both languages in building ToD systems.

##  2 IndoToD Benchmark

Statistics | COD-id | IndoCamRest (ours) | IndoSMD (ours)
---|---|---|---
# domains |  11: Flights, Weather, Alarm, Ride Sharing, Movies, Music, Media, Homes, Banks, Travel, Payment |  1: Restaurant Search |  3: POI navigation, Calendar Scheduling, Weather Information
# dialogues |  194 |  676 |  323
# dialogues/domain |  17.64 |  676 |  107.67
# distinct entities |  N/A |  110 |  325
# turns/dialogue |  12.83 |  4.06 |  2.63
# tokens/utterance |  N/A |  9.54 |  8.14
# vocab (lexicalized) |  N/A |  1,103 |  883
# vocab (delexicalized) |  N/A |  829 |  577

Table 1: Comparison of our IndoToD benchmark with the Indonesian subset of COD
Majewska et al. (2023).

IndoToD benchmark333https://github.com/dehanalkautsar/IndoToD is created to
develop the Indonesian ToD system. The benchmark covers four different
domains, i.e., restaurant search, point-of-interest (POI) navigation, calendar
scheduling, and weather information. IndoToD extends the existing English
datasets, and we follow a streamlined process to conduct dialogue collection.
The dialogues are multi-turn conversations that involve two speakers (i.e.,
user $U$ and system $S$). For each conversation, there is a knowledge base
(KB) for the system to generate the correct entity for the user.

###  2.1 Dataset Collection

We use CamRest676 Wen et al. (2017) and SMD Eric et al. (2017) as the original
English datasets that will go through several stages before becoming datasets
that can be used in various Indonesian Task-oriented Dialogue (ToD) system
experiments. CamRest676 is a ToD dataset that focuses on restaurant search
queries collected via Wizard-of-Oz (WoZ) framework Kelley (1984). It consists
of a collection of dialogues between the user and the system, where each
dialogue has a task-specific goal (e.g., finding a restaurant). In this
dataset, the user acts as a client who requests restaurant information and the
system acts as an information provider that guides and answers the user’s
request.

Meanwhile, SMD is an in-car assistant multi-domain ToD dataset. It covers
several domains such as POI navigation, calendar scheduling, and weather
information. This dataset was also created by using the same WoZ framework to
get a high-quality dataset that imitates a conversation between two
individuals in a way that resembles a real-life interaction between a driver
and an in-car assistant. In the WoZ framework, the source of conversation is
created through a human-to-human dialogue conversation which is collected
through crowd-sourcing. Because of that, the conversations between the user
and the system are more natural.

These datasets then will be used as a reference for generating a ToD dataset
that mimics natural conversation between two speakers and includes relevant
knowledge base information for generating accurate responses. Both two
datasets provide a diverse variety of dialogues since they have a different
set of domains that complement the IndoToD benchmark.

###  2.2 Dataset Construction

The datasets have been created through several steps, as shown in Figure 1.
First, existing English datasets are delexicalized into several user and
system template sentences using Madotto et al. (2020) approach. Then, these
template sentences are translated by native Indonesian annotators into
Indonesian sentences. Lastly, new Indonesian datasets are built through the KB
retrieval and lexicalization process.

#### Delexicalization

Initially, the datasets are pre-processed by delexicalizing the entities. The
process is carried out using the source code implemented by Madotto et al.
(2020) to remove the entities in the dataset’s dialogue. The aim of this step
is to reduce the number of dialogues that require annotation (translation) by
replacing all entities with a common, pre-defined value based on the type of
entity. This pre-processing method is effective because the number of
sentences that need to be translated is reduced by 30% and 34% for CamRest676
and SMD, respectively. The output of this step is a list of pre-processed,
entity-less sentences for each dataset.

#### Translation

The list of pre-processed sentences is translated into Indonesian by
annotators. In total, there are 3,834 and 1,126 sentences translated from
CamRest676 and SMD, respectively. This process is followed by a cross-
validation process where an annotator has to check the work of other
annotators to maintain the dataset’s quality. Note that for the SMD dataset,
we only use 11% (323 dialogues) of its original dialogues due to the
limitations on the annotator side.

#### KB Retrieval and Lexicalization

After the translation process, to build each dialogue, a collection of
Indonesian sentences and their corresponding entities are needed to begin the
lexicalization process. We retrieve the Indonesian entities by querying the KB
for the required slots to construct the dialogue. The lexicalization process
is conducted by using Indonesian entities and sentences that have been
retrieved earlier. This process ends with an evaluation of 100 randomly
selected newly created Indonesian dialogues by human evaluators to detect any
errors to ensure the quality of the resulting dialogues. If there are any
errors, we ask them to manually edit that utterance. The sample of corrected
Indonesian utterances is presented in Appendix Table 1.

Statistics | IndoCamRest | IndoSMD
---|---|---
Restaurant Search |  POI Navigation |  Calendar Scheduling |  Weather Information
# dialogues |  676 |  100 |  114 |  109
# slot types |  6 (area, food, price range, postcode, phone, address) |  5 (POI name, traffic info, POI category, address, distance) |  6 (event, time, date, party, room, agenda) |  4 (location, weekly time, temperature, weather attr.)
# distinct slot values |  88 |  131 |  79 |  78

Table 2: Per-domain statistics of IndoCamRest and IndoSMD.

###  2.3 Dataset Statistics

We collect a total of 999 dialogues with 6,338 utterances from two datasets,
named as IndoCamRest and IndoSMD, which are derived from CamRest676 and SMD
datasets, respectively. We compare several statistic attributes between our
datasets (IndoCamRest and IndoSMD) and COD that are shown in Table 1.
Furthermore, the per-domain statistics of IndoCamRest and IndoSMD are
presented in Table 2. IndoCamRest and IndoSMD share some of the same
statistical characteristics as the previous datasets, including the number of
domains, intents, and slot types. It is worth noting that both of our datasets
have a substantially larger number of dialogues per domain compared to COD. In
addition, IndoCamRest and IndoSMD comprise both the lexicalized and
delexicalized forms in the dialogue. The inclusion of the delexicalized form
is intended to facilitate the generation of dialogues by lexicalizing the
delexicalized sentences.

##  3 Experimental Setup

###  3.1 Experiment Settings

We set up a benchmark for both Indonesian and English ToD to evaluate the
performance of the current ToD systems. We explore three different training
and evaluation settings:

  * •

Monolingual – id/en. We train the end-to-end ToD systems using monolingual
corpus in each language independently.

  * •

Cross-lingual. We train the systems using the English end-to-end ToD dataset
before testing it using an Indonesian test set to analyze the effectiveness of
the cross-lingual approach in understanding the context of the dialogue.

  * •

Bilingual – id+en. We train the systems by combining English and Indonesian
datasets, utilizing parallel corpora from CamRest and SMD. This allows us to
examine how the system’s performance is affected by exposure to bilingual
languages.

For monolingual and bilingual settings, we test each ToD framework in English
and Indonesian test sets separately to analyze the impact of monolingual and
bilingual training in each language. We also split the data into 3:1:1 and
8:1:1 for the CamRest and SMD, respectively.

Baselines | IndoCamRest | IndoSMD
---|---|---
BLEU | Match | Success | Combined | BLEU | Match | Success | Combined
monolingual
Sequicity Lei et al. (2018) | 16.74 | 92.31 | 81.29 | 103.54 | 8.75 | 44.83 | 61.04 | 61.69
LABES Zhang et al. (2020a) | 17.59 | 70.00 | 78.49 | 91.83 | 1.01 | 0.00 | 34.82 | 18.42
MinTL Lin et al. (2020) | 14.61 | 80.33 | 80.31 | 94.93 | 0.45 | 24.14 | 23.70 | 24.37
GALAXY He et al. (2022) | 17.89 | 96.24 | 84.62 | 108.32 | - | - | - | -
cross-lingual
Sequicity Lei et al. (2018) | 0.08 | 8.46 | 31.58 | 20.10 | 0.00 | 15.52 | 29.46 | 22.50
LABES Zhang et al. (2020a) | 0.02 | 0.00 | 60.96 | 30.50 | 0.00 | 15.79 | 55.84 | 35.82
MinTL Lin et al. (2020) | 1.61 | 16.67 | 68.81 | 44.35 | 0.00 | 13.33 | 56.75 | 35.04
GALAXY He et al. (2022) | 0.05 | 0.00 | 47.24 | 23.67 | 0.00 | 0.00 | 37.11 | 18.56
bilingual
Sequicity Lei et al. (2018) | 15.84 | 89.23 | 84.01 | 102.46 | 10.95 | 58.33 | 75.66 | 77.94
LABES Zhang et al. (2020a) | 18.77 | 96.36 | 81.68 | 107.79 | 8.78 | 47.37 | 71.11 | 68.02
MinTL Lin et al. (2020) | 18.65 | 87.30 | 79.63 | 102.12 | 11.59 | 48.33 | 77.86 | 74.68
GALAXY He et al. (2022) | 17.09 | 94.74 | 83.60 | 106.26 | 0.08 | 45.00 | 9.17 | 27.17
Table 3: Experiment settings result on Indonesian test set. bold denotes the
best score per metric.

###  3.2 ToD Baselines

As baselines, We evaluate four ToD frameworks: Sequicity Lei et al. (2018),
LABES Zhang et al. (2020a), MinTL Lin et al. (2020), and GALAXY He et al.
(2022). Since the four ToD system frameworks are designed for English, we have
made adjustments to the frameworks to suit the requirements of our
experiments. We largely adopt the configuration settings used in the original
paper, including batch size, decoding method, early stop count, and learning
rate.

To adapt the Sequicity and LABES frameworks for our experiments, we utilize
fastText Bojanowski et al. (2017) instead of GloVe Pennington et al. (2014).
This was necessary because fastText can accommodate English, Indonesian, and
bilingual Indonesian-English word vectors Conneau et al. (2017). Specifically,
we incorporate common crawl444https://fasttext.cc/docs/en/crawl-vectors.html
English and Indonesian fastText word vectors for monolingual English and
Indonesian experiment settings respectively. Meanwhile, we use bilingual
English-Indonesian word vectors Conneau et al. (2017) for other settings.

Unlike the other frameworks, MinTL utilizes T5 Raffel et al. (2020) and BART
Lewis et al. (2020) as its backbone, which sets it apart regarding approach
and potential performance. Note that for this experiment, we only use T5
Raffel et al. (2020) and mT5 Xue et al. (2021) as the backbone models, and use
this framework without utilizing _Levenshtein Belief Spans (Lev)_ resulting in
an inference process similar to Sequicity. In all of the experiment settings,
mT5-small is implemented in the framework and used as the backbone of the
framework. We also try to use T5 for monolingual and bilingual settings to
analyze the impact of using two different kinds of pre-trained language models
(LMs) that will be discussed later in the Subsection 4.2. We implement reader
and evaluator modules for both CamRest and SMD datasets in MinTL, as their
paper only included the reader and evaluator for the MultiWOZ Budzianowski et
al. (2018) dataset.

Baselines | CamRest | SMD
---|---|---
BLEU | Match | Success | Combined | BLEU | Match | Success | Combined
monolingual
Sequicity Lei et al. (2018) | 21.17 | 90.30 | 80.99 | 106.81 | 21.04 | 82.12 | 78.21 | 101.20
LABES Zhang et al. (2020a) | 19.98 | 93.64 | 72.93 | 103.26 | 1.94 | 80.81 | 60.07 | 72.39
MinTL Lin et al. (2020) | 20.34 | 83.21 | 83.64 | 103.77 | 18.53 | 83.90 | 78.60 | 99.78
GALAXY He et al. (2022) | 20.53 | 96.24 | 84.62 | 110.96 | 18.25 | 80.00 | 77.32 | 96.91
bilingual
Sequicity Lei et al. (2018) | 22.53 | 91.04 | 82.06 | 109.08 | 21.04 | 75.56 | 83.48 | 100.56
LABES Zhang et al. (2020a) | 22.58 | 97.27 | 79.33 | 110.88 | 22.35 | 83.83 | 74.02 | 101.28
MinTL Lin et al. (2020) | 22.88 | 86.26 | 82.26 | 107.14 | 18.39 | 80.23 | 79.10 | 98.05
GALAXY He et al. (2022) | 20.31 | 95.49 | 84.67 | 110.39 | 16.58 | 77.37 | 76.84 | 93.68
Table 4: Experiment settings result on English test set. bold denotes the best
score per metric.

###  3.3 Hyper-parameters

We conduct the learning rate search [1e-4, 6e-4, 1e-5] to optimize the
performance of MinTL on the new datasets, and the best learning rate is 6e-4.
For other ToD frameworks, we adopt their proposed learning rate in our
experiments. We use a learning rate of 3e-3 for Sequicity, 1e-4 for GALAXY,
and, for LABES, we set the learning rate to 3e-4 and 1e-4 in the CamRest and
SMD experiments, respectively. We follow the inference sampling strategies
recommended by the original papers that introduce the frameworks. We use a
beam size of 5 for generating inference responses in GALAXY, while in the
other frameworks, we employ greedy search. We run each experiment once per
experiment setting for uniformity with an A100 40GB GPU.

###  3.4 Evaluation Metrics

We use four evaluation metrics for end-to-end ToD tasks to evaluate the
quality of responses. There are 1) BLEU score Papineni et al. (2002) to assess
how fluent the generated response is, 2) Match rate Lin et al. (2020), checks
whether the ToD system can produce the entity constraints specified by the
user, thereby measuring task completion, 3) Success F1 score Lei et al.
(2018), which determines whether the system has successfully provided all of
the information that the user has requested, such as the address or phone
number, and 4) Combined score Lin et al. (2020); Mehri et al. (2019) using the
equation Combined = (Match + Success) $\times$ 0.5 + BLEU. We assess the
performance of all frameworks using the CamRest and SMD datasets, along with
the designated experiment settings.

##  4 Results & Analysis

###  4.1 Results

Tables 3 and 4 show the experiment results for Indonesian and English test
sets, respectively. We conclude that GALAXY has a decent score in both
monolingual settings for CamRest, and Sequicity with fastText word vector has
the best model in monolingual settings for SMD. Furthermore, MinTL outperforms
other frameworks in zero-shot cross-lingual settings for both datasets. The
non-contextual word-embedding frameworks, such as Sequicity and LABES achieve
the highest combined scores for bilingual settings. Furthermore, it was
observed that GALAXY does not perform as well as other frameworks when dealing
with Indonesian, as we denote its results by using ’-’ in Table 3.

Additionally, we present the combined metric results of the ToD frameworks on
the IndoSMD test set, categorized by domain, in Figure 2. We discover that POI
navigation is relatively easier than other domains to learn, while calendar
scheduling is the hardest domain for ToD frameworks on SMD dataset. Fine-
grained results can be found in Appendix Table 2.

Figure 2: ToD framework results on IndoSMD per domain. Left: POI navigation
domain, Center: Calendar scheduling domain, and Right: Weather information
domain.

###  4.2 Analysis

#### Monolingual English vs. Indonesian Performance.

In general, the monolingual model trained and tested in English performs
better than in Indonesian across different frameworks and datasets. We
hypothesize that the pre-trained LMs used in the experiments are better
trained to handle English data. GALAXY outperforms the monolingual setting on
both Indonesian and English CamRest datasets. However, we find that the GALAXY
framework fails to adapt to the IndoSMD dataset and the model is not
converged. We further observe that the experiments using GALAXY in Indonesian
are not as effective as English, as evidenced by the superior performance on
the English CamRest dataset, despite having the same amount of dialogue and
utterance as IndoCamRest. Furthermore, it is worth noting on the IndoToD
benchmark that we are not using Indonesian T5 models. This is because there is
no paper working on releasing the Indonesian T5 models that are comparable to
the English T5 models we use in the experiment.

#### Transferability from English to Indonesian.

Based on the cross-lingual results, it can be concluded that existing ToD
frameworks are unable to handle the task when trained in a different language.
While not all training data needs to be in the same language as the test set,
it is still essential for the framework to have some knowledge of the target
language in order to understand the task at hand. Based on the inference
results, we observe that all of the ToD frameworks generate responses in
English despite Indonesian user inputs. This can be correlated to the fact
that the training data used in the cross-lingual setting is in English. Among
all the ToD frameworks that are evaluated in this setting, MinTL performs the
best in the cross-lingual setting (due to mT5 as its pre-trained model,
trained on multilingual datasets), while GALAXY and Sequicity do not perform
well.

#### Monolingual vs. Bilingual.

The results demonstrate that, in most cases, the bilingual setting yields
higher scores than the monolingual (i.e., Indonesian and English monolingual
setting), especially in the Indonesian test set. The bilingual setting has the
advantage of using a larger amount of training data and performing tasks in
both languages Lin et al. (2021b), increasing both the overall metric scores
and the individual metric components. We also compare and analyze the
framework’s performance closely based on the test set we used. The results
show that most of the bilingual settings scores are higher than monolingual
settings in both the Indonesian and English test sets showing the effective
usage of language data counterpart for domain adaptation.

However, based on the inference result in Table 5, we observe that the
responses generated between bilingual and monolingual settings are not
substantially different in terms of the semantics, confirmed by the fact that
the difference in BLEU scores between these two settings is not significant.
However, we see some improvements in belief spans generation in the bilingual
setting. ToD frameworks can understand the context from the user better when
they are trained with a larger amount of training data, although the training
data are a mixture of English and Indonesian. This highlights the significance
of leveraging cross-lingual data, in this case, using English data on top of
Indonesian data, in the training process of the ToD system to enhance the
model performance and achieve better metric scores in the target language.

Figure 3: Monolingual and bilingual MinTL combined score using T5 and mT5
models on CamRest676, SMD, IndoCamRest, and IndoSMD as test sets.

Furthermore, we delve deeper into analyzing the potential of bilingual
training using the MinTL framework. Interestingly, this framework is flexible
since we can modify the pre-trained transformer LM unlike other frameworks. We
explore not only English T5, but also multilingual version of T5. We compare
the performance of T5-small and mT5-small as the backbone of the MinTL
framework, shown in Figure 3 and Appendix Table 3 for per metric details.
Based on our findings, we observe that bilingual training positively impacts
the BLEU score, match rate, and combined metrics for the MinTL framework,
either using T5 or mT5. The results indicate that the bilingual setting can
yield benefits on the efficient training process of T5 and mT5, especially in
Indonesian, that is considered as underrepresented language. Moreover, we
notice that mT5-small is more effective than T5-small in end-to-end ToD tasks.
Overall, the bilingual training is beneficial for training end-to-end ToD,
especially to improve the performance on non-English languages.

#### Error Analysis.

We conduct an error analysis to investigate the limitations of existing
frameworks. In general, most issues occur in the form of a “template
response”. The end-to-end ToD systems sometimes generate responses that appear
to be template-like despite varying user inputs which might be because of the
limited trainable parameters and dataset used in the training process. Despite
its goal to complete the user’s task, the ToD must also be able to maintain
the user’s satisfaction whenever using the ToD as a product. Users may become
disinterested if they receive repeated responses, potentially leading to
reduced usage. Another issue we notice is the occurrence of repeated tokens in
the generated output by the framework during the inference phase.
Specifically, we find that the Sequicity and MinTL frameworks tend to produce
responses with repeated tokens more than the other ToD frameworks. This could
be due to the decoding method applied by these frameworks in our research
which relied on a greedy search approach. The other issue that is worth
mentioning in inference examples is the hallucination in the generated
outputs. Some of the outputs do not have the correct combination of slots that
are found in the training set.

Input |  Belief spans |  Response
---|---|---
"carikan saya restoran mahal yang menyediakan makanan internasional."
(find me an expensive restaurant that serves international food) |  gold: "restaurant-food": "internasional", "restaurant-pricerange": "mahal"
monolingual: "restaurant-food": "internasional", "restaurant-pricerange":
"menengah"
bilingual: "restaurant-food": "internasional", "restaurant-pricerange": "mahal" |  gold: "maafkan saya. saya tidak bisa menemukan restoran apapun yang menyediakan makanan internasional di kisaran harga yang mahal"
(I am sorry. I couldn’t find any restaurant that provides international food
at such an expensive price range)
monolingual: "value_name adalah restoran mahal yang menyediakan makanan
internasional. apakah kamu ingin tahu lokasinya?"
(value_name is an expensive restaurant that serves international food. Would
you like to know its location?)
bilingual: "value_name menyediakan makanan internasional. apakah kamu ingin
informasi lebih lanjut?"
(value_name serves international foods. Would you like other further
information?)

Table 5: Example of the LABES response on IndoCamRest using monolingual and
bilingual settings. The English translation is in italic.

##  5 Related Work

#### ToD Datasets.

Multiple studies have contributed to creating datasets for ToD using various
approaches. bAbI Bordes et al. (2016) is an English synthetic dataset
combining five subtasks into ToD system datasets. SMD Eric et al. (2017) uses
the WoZ scheme to build an English car assistant dialogue system that has
knowledge of weather and places. Other English datasets using the WoZ scheme
are CamRest Wen et al. (2017), a dataset for a restaurant reservation,
MultiWoZ Budzianowski et al. (2018), a multi-domain WoZ dataset containing
five domains collected using the human-to-human scheme, and OpenDialKG Moon et
al. (2019), a synthetic dataset containing four domains. On non-English
datasets. such as Chinese, there are CrossWoZ Zhu et al. (2020), a large-scale
Chinese WoZ dataset with five domains, and RiSAWoZ Quan et al. (2020), a
large-scale multidomain Chinese WoZ dataset with rich semantic annotations up
to 12 domains. MetaLWOz Lee et al. (2019) is a goal-oriented dialogue corpus
containing 51 domains and collected using WoZ. Other datasets include
bilingual languages, such as BiToD Lin et al. (2021b), a dataset for tourism
assistants focusing on English and Chinese languages, and COD Majewska et al.
(2023), a dataset containing Russian, Arabic, Indonesian, and Swahili. Based
on the aforementioned datasets, we conclude that the ToD dataset with
contextual knowledge focusing on Indonesian is not yet available.

#### Indonesian Dialogue Systems Datasets.

Research for the ToD system in the Indonesian language is still underway.
IndoNLG Cahyawijaya et al. (2021), a benchmark for natural language generation
in low-resource languages, focuses on summarization, question answering, chit-
chat, and machine translation for Indonesian, Javanese, and Sundanese
languages. An Indonesian language subset of the XPersona Lin et al. (2021a)
dataset, which consists of approximately 17 thousand dialogues, is used for
the chit-chat task. It is worth noting that although IndoNLG is making
significant progress, it has not yet explored the ToD system. Furthermore,
NusaCrowd Cahyawijaya et al. (2022), a large pool of hundreds of Indonesian
NLP data, only listed one Indonesian ToD dataset, i.e., COD Majewska et al.
(2023), which is a multilingual ToD dataset that is solely used for evaluation
with only 194 dialogues spanning across 11 domains. Therefore, the Indonesian
ToD dataset is still very limited, and urgent action is required to increase
the coverage of Indonesian ToD datasets.

#### Frameworks for End-to-End ToD.

GALAXY He et al. (2022) is a pre-trained dialogue model that uses semi-
supervised learning to learn dialogue policy from limited labeled dialogues
and large-scale unlabeled dialogue corpora. LABES Zhang et al. (2020a),
conversely, represents belief states as discrete latent variables and models
them jointly with system responses given user inputs. MinTL Lin et al. (2020)
efficiently uses pre-trained LMs in developing task-oriented dialogue systems,
eliminating the need for ad hoc modules in studies that use this framework.
DAMD Zhang et al. (2020b) is a network that accommodates state-action pair
structure in generation and can leverage the proposed multi-action data
augmentation framework to address the multidomain response generation problem.
Sequicity Lei et al. (2018) uses the Seq2Seq model for dialogue state tracking
and generating a response to the user. Some other recent works Bang et al.
(2023); Hudeček and Dušek (2023) explore the potential of using large LMs for
performing zero-shot ToD. In this paper, we utilize GALAXY, LABES, MinTL, and
Sequicity to evaluate the performance of various end-to-end ToD frameworks.

##  6 Conclusion

We introduce IndoToD, an end-to-end multi-domain Indonesian ToD benchmark. We
extend the existing two English ToD datasets using an efficient framework by
delexicalizing the conversation into templates and reconstruction using KB. We
evaluate our benchmark in monolingual, cross-lingual, and bilingual settings.
We also show the benefits of having English data to improve Indonesian
performance. Furthermore, we conduct some analysis from multiple viewpoints
that shows the existing ToD system’s inability to understand dialogue context
from unseen language and the current ToD system’s behavior in handling
underrepresented language such as Indonesian using our datasets. We also
investigate some errors that usually occur in ToD system responses to explore
its limitation.

##  7 Limitations

There are several limitations that hinder our progress in creating the first
Indonesian end-to-end ToD benchmark. Firstly, we encountered a limitation in
translating the English dataset. Due to the constraints of our resources,
specifically having only three annotators, we could only translate the CamRest
dataset and a mere 10% of the SMD dataset. Expanding the dataset size would
significantly contribute to advancing the Indonesian ToD system. Second, we
only use publicly available end-to-end ToD system framework repositories as
our baselines. Based on MultiWoZ
benchmark555https://github.com/budzianowski/multiwoz, there are several end-
to-end ToD frameworks with slightly superior metric scores compared to our
baselines Budzianowski et al. (2018). But because of their unavailability, we
could not evaluate them and only focused on evaluating the best available ToD
frameworks possible.

##  8 Ethics Statement

Our work spotlights a need to develop ToD systems for underrepresented
languages such as Indonesian through our new ToD datasets: IndoCamRest and
IndoSMD. During our study, we commit to the ethical principles of NLP research
and are well aware of its impact on the language community. We strongly
believe that there is no ethical issue within this work, including the data
collection process, annotation, existing dataset usage, and experiments.
Within our work, annotators are well-rewarded and give us informed consent as
they understand and agree to provide their annotations to build new ToD
datasets. We uphold our annotator’s privacy and follow the data protection and
privacy regulation for releasing the datasets. The dataset itself is free from
abusive language and personal information as our ultimate goal in this work is
to contribute and make an impact on society with provide useful NLP task
resources and more linguistic diversity in the NLP field.

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##  Appendix A Annotator Recruitment

We recruit two Indonesian native speakers that can speak English to translate
the delexicalized English utterance template to Indonesian. The age of the
annotators ranged from 19 to 25, and they were college students by profession.
They work around 18 hours in total, and they are compensated approximately
22,583 IDR per hour for their work, 18,62% bigger than the worker minimum wage
in West Java (19,038 IDR per
hour)666https://www.bps.go.id/indicator/19/1172/1/upah-rata---rata-per-jam-
pekerja-menurut-provinsi.html.

##  Appendix B Sample Evaluation Correction by Annotators

Appendix Table 1 shows several examples of evaluation by annotators. The
evaluation process is intended to maintain high-quality translation from the
annotator. We report five sampled English utterances, incorrect Indonesian
utterances, and corrected Indonesian utterances by annotators.

##  Appendix C Per Domain Experiment Settings Result on IndoSMD Test Set

We report per domain results for all of the baselines on IndoSMD dataset on
Appendix Table 2. We can conclude that Sequicity has the best score on
monolingual Indonesian setting, meanwhile MinTL outperforms other frameworks
on cross-lingual and bilingual settings by having the most biggest metric
score compared to others when handling Indonesian test set.

##  Appendix D MinTL Result Comparisons

We present MinTL result comparisons by using two pre-trained LM: T5 and mT5 on
Appendix Table 3. The table compares monolingual and bilingual approach on
English test set. The result shows that bilingual setting have a beneficial
impact on MinTL to learn the task well compares to monolingual setting.

##  Appendix E Annotator Instruction and Consent

We report instructions for annotators to translate several English CamRest and
SMD utterance templates, as well as annotator informed consent templates on
Appendix Figure 1, Appendix Figure 2, and Appendix Figure 3. The instructions
are in Indonesian as all of the annotators are Indonesian native speakers.

English Utterance |  Incorrect Indonesian utterance |  Corrected Indonesian utterance
---|---|---
"remind me that i have a doctors appointment at 5 pm" |  "ingatkan saya bahwa saya akan janji temu dengan dokter pada pukul 5 sore" |  "ingatkan saya bahwa saya memiliki janji temu dengan dokter pada pukul 5 sore"
"that you, that’s all I need to know." |  "itu kamu, hanya itu saja yang perlu saya ketahui" |  "terimakasih, hanya itu saja yang perlu saya ketahui" (there is a typo on the English utterance. It suppose to be ’thank you’, not ’that you’)
"are there ant jamaican restaurants in any part of town?" |  "apakah ada restoran semut jamaika di bagian manapun dari kota ini?" |  "apakah ada restoran jamaika di bagian manapun dari kota ini?" (there is a typo on the English utterance. It suppose to be ’any’, not ’ant’)
"la tasca is located at 14-16 Bridge Street and the postcode is C.B 2, 1 U.F. Their phone number is 1223 464630. May I help you with anything else?" |  "la tasca berlokasi di 14 -16 bridge street dan kode pos nya adalah c.b 2, 1 u.f. nomornya adalah 1223 464630. bisakah saya membantu kamu dengan hal yang lain?" |  "la tasca berlokasi di 14 -16 bridge street dan kode pos nya adalah c.b 2, 1 u.f. nomornya adalah 01223 464630. bisakah saya membantu kamu dengan hal yang lain?" (we check in to the KB and we find out the correct entity for its phone number is ’01223 464630’)
"it is located at Quayside Off Bridge street. The phone number is is 01223 301030." |  "alamatnya adalah quayside off bridge street. nomor teleponnya adalah 01223 301030." |  "alamatnya adalah quayside off bridge street. nomor teleponnya adalah 01223 307030." (we check in to the KB and find out that the correct entity is 01223 307030)

Table 1: Sample evaluation correction by annotators.

Baselines | Domain
---|---
POI Navigation | Calendar Scheduling | Weather Information
BL | Ma | Su | Co | BL | Ma | Su | Co | BL | Ma | Su | Co
monolingual
Sequicity | 7.90 | 25.00 | 89.05 | 64.93 | 6.85 | 61.90 | 46.91 | 61.26 | 10.57 | 33.33 | 38.71 | 46.59
LABES | 1.30 | 0.00 | 40.50 | 21.55 | 0.00 | 0.00 | 32.30 | 16.15 | 0.00 | 14.30 | 20.00 | 17.15
MinTL | 0.02 | 50.00 | 22.92 | 36.48 | 0.34 | 19.05 | 30.30 | 25.02 | 0.004 | 4.76 | 26.51 | 15.64
GALAXY | - | - | - | - | - | - | - | - | - | - | - | -
cross-lingual
Sequicity | 0.00 | 12.50 | 50.53 | 31.52 | 0.00 | 14.29 | 4.82 | 9.56 | 0.00 | 19.05 | 25.45 | 22.25
LABES | 0.00 | 13.30 | 75.50 | 44.40 | 0.00 | 16.70 | 38.30 | 27.50 | 0.00 | 42.90 | 54.50 | 48.70
MinTL | 0.00 | 16.67 | 35.29 | 25.98 | 0.004 | 14.29 | 59.54 | 36.92 | 0.00 | 4.76 | 59.41 | 32.09
GALAXY | 0.01 | 11.76 | 33.66 | 22.72 | 0.00 | 0.00 | 26.17 | 13.09 | 0.00 | 0.00 | 49.54 | 24.77
bilingual
Sequicity | 8.92 | 44.44 | 77.47 | 69.88 | 11.73 | 61.90 | 67.37 | 76.37 | 11.72 | 57.14 | 54.05 | 67.32
LABES | 4.50 | 60.00 | 77.90 | 73.45 | 6.90 | 33.30 | 53.70 | 50.40 | 9.70 | 42.90 | 76.30 | 69.30
MinTL | 4.75 | 55.56 | 74.24 | 69.65 | 15.94 | 47.62 | 80.00 | 79.75 | 13.85 | 52.38 | 81.54 | 80.81
GALAXY | 0.06 | 47.06 | 11.76 | 29.47 | 0.06 | 50.00 | 7.23 | 28.68 | 0.12 | 42.86 | 8.96 | 26.03

Table 2: Per domain experiment settings result on Indonesian test set using
SMD dataset. bold denotes the best score per metric. BL, Ma, Su, and Co denote
the BLEU score, Match rate, Success F1, and Combined score respectively.

Experiment Setting | CamRest | SMD
---|---|---
BLEU | Match | Success | Combined | BLEU | Match | Success | Combined
T5-small
Monolingual | 17.57 | 83.21 | 83.25 | 100.80 | 15.31 | 71.54 | 76.22 | 89.19
Bilingual | 18.50 | 85.50 | 80.35 | 101.42 | 16.23 | 75.29 | 79.52 | 93.63
mT5-small
Monolingual | 20.34 | 83.21 | 83.64 | 103.77 | 18.53 | 83.90 | 78.60 | 99.78
Bilingual | 22.88 | 86.26 | 82.26 | 107.14 | 18.39 | 80.23 | 79.10 | 98.05

Table 3: MinTL Lin et al. (2020) result comparisons using pre-trained LM: T5
and mT5 as backbone model on English test set. bold denotes the best score per
metric.

Figure 1: Instruction for annotators to translating English CamRest templates.

Figure 2: Instruction for annotators to translating English SMD templates.

Figure 3: Annotator informed consent template.


#  Moduli spaces of BPS lumps with holomorphic impurities

J. M. Queiruga<EMAIL_ADDRESS>Department of Applied Mathematics,
University of Salamanca, 37008, Salamanca, Spain  Institute of Fundamental
Physics and Mathematics, University of Salamanca, 37008 Salamanca, Spain
Department of Physics, UPV/EHU, 48080, Bilbao, Spain

(03-12-2021)

###### Abstract

A self-dual generalization of the lump-impurity system is introduced. This
model possesses lump-antilump-like pairs as static solutions of the pertinent
Bogomolny equations. This allows for a moduli space approximation analysis of
the BPS solutions which are identified as lump-antilump configurations. Some
geometrical properties of the resulting moduli are analyzed. In addition, it
is argued that, this type of impurity models can be interpreted as a limit of
certain non-impurity theories.

##  1 Introduction

The study of soliton-antisoliton annihilation processes in nonlinear field
theories entails many difficulties. One of the main reasons for this is that,
in general, a soliton-antisoliton configuration does not belong to any BPS
sector. This implies, in particular, that there is always a non-vanishing
static interaction between solitons of opposite charges and as a consequence,
the canonical moduli space approximation moduli-1 , is not available. In
general, the study of such processes requires a full numerical analysis, which
often does not allow to extract sufficiently general properties.

It was realized hook ; tong  that it is possible to preserve (part of) the BPS
structure in models where a non-dynamical field (impurity) is introduced (for
recent advances see vortex-imp-2 ; BPS-imp-phi4 ; vortex-imp-3 ; no-force ;
solvable-imp ; queiruga_skyrme_imp ; kim ). As in standard BPS theories, there
is a moduli space of solutions, such that trajectories over this internal
space connect isoenergetic configurations. If one considers only translational
degrees of freedom, the shape of the soliton does not change through the
moduli space in the standard case. Once the impurity is introduced, the shape
of the soliton gets distorted close to the impurity, in such a way that the
combination of this deformation with the translation still preserves the BPS
property. Moreover, each point in the moduli space is associated to different
spectral structures, and this feature leads to very interesting phenomena,
like for example the so-called spectral wall spec-wall . In addition, the fact
that the full system preserves a part of the BPS structure implies the absence
of static soliton-impurity interaction and this allows for a clean study of
internal modes in multikink collisions.

It was soon understood in one-dimensional models that for a specific choice,
the impurity can generate a sort of kink solutions which can be interpreted as
kink-antikink configurations. The moduli space of such configurations is still
one-dimensional, as the only degree of freedom is associated to the
generalized translational symmetry (the impurity is non-dynamical and does not
contribute with degrees of freedom to the moduli). In contrast to standard
situations, even the moduli associated to a single soliton in impurity models
is nontrivial and effectively describes the soliton-impurity interaction.
Beyond one-dimensional models, there is a rather poor understanding of
soliton-antisoliton processes, both at analytical and numerical level. In this
sense, this work presents a new approach to study some properties of soliton-
antisoliton configurations that preserve the BPS structure in a $2+1$
dimensional model.

We will focus on a modification of the $\mathbb{C}P^{1}$ model, whose BPS
solutions are rational maps between two-dimensional Riemann spheres, either
holomorphic or antiholomorphic. The impurity field will be introduced in such
a way that it can be interpreted as a frozen lump with opposite topological
charge with respect to the dynamical lump. Of course, if one wants to preserve
part of the BPS structure (and therefore the canonical moduli space), the
coupling between impurity and dynamical field (the lump) needs to have a
particular form, dictated by SUSY. As it is well-known, the underlying
supersymmetric structure can be used to obtain the Bogomolny equations
necessary to build the moduli space. We will study some properties of such BPS
solutions and the moduli spaces obtained from them.

This paper is organized as follows. In Sec. 2 we construct the BPS impurity
Lagrangian and compute BPS equations and BPS bound using SUSY techniques. In
Sec. 3, general properties of the solutions as well as some connection to
other models found in the literature as discussed. In Sec. 4 we describe
certain properties of the moduli space and present some examples. In Sec. 5,
geometrical properties of the impurity moduli are analyzed, namely, the Kähler
structure. Sec. 6 contains some considerations about the relation between
impurity models and some multifield systems. Finally, Sec. 7 is devoted to our
conclusions and further discussion. We have also added an Appendix with the
explicit calculation of the degree formula of the BPS solutions for general
impurities.

##  2 Lagrangian and BPS equations

Perhaps, the simplest nonlinear $\sigma$-model in two dimensions is the $O(3)$
model MS . It can be formulated in terms of a unit 3-vector and its Lagrangian
is given by a Dirichlet term plus a constraint

$\mathcal{L}_{O(3)}=\frac{1}{4}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a},\,\,\phi^{a}\phi^{a}=1,\,\,a=1,2,3.$
(2.1)

In two dimensions, the solutions of the model are maps from the one point
compactification of $\mathbb{R}^{2}$ ($\simeq\mathbb{S}^{2}$ ) to
$\mathbb{S}^{2}$. There is a geometrically meaningful formulation of the model
written in terms of the stereographic projection

$\phi^{a}=\left(\frac{u+\bar{u}}{1+|u|^{2}},\frac{-i(u-\bar{u})}{1+|u|^{2}},\frac{1-|u|^{2}}{1+|u|^{2}}\right).$
(2.2)

This model is usually called the $\mathbb{C}P^{1}$ model, for obvious reasons.
In terms of the complex field $u$ the Lagrangian is given by

$\mathcal{L}_{CP^{1}}=\frac{1}{2}\frac{\partial_{\mu}u\partial^{\mu}\bar{u}}{(1+|u|^{2})^{2}},$
(2.3)

where we have introduced a $1/2$ factor for later convenience. The addition of
a complex background field $\sigma$ to the field $u$, say in the form $\sigma
V(u)$, breaks in general the BPS structure enjoyed by the original
$\mathbb{C}P^{1}$ Lagrangian. The idea is to look for specific couplings such
that a fraction of the BPS property is preserved. In order to do that, we can
use the underlying SUSY structure of the $\sigma$-model. In the off-shell SUSY
formulation we have

$\mathcal{L}_{\mathbb{C}P^{1}}=\frac{1}{2}\frac{\partial_{\mu}u\partial^{\mu}\bar{u}+F\bar{F}}{(1+|u|^{2})^{2}}+\text{fermions},$
(2.4)

where the fermionic terms are not displayed explicitly and $F$ is a complex
auxiliary field. It is possible to add (in the SUSY formulation of the model)
a BPS preserving impurity term of the form

$\mathcal{L}_{\text{impurity}}=\frac{1}{(1+|u|^{2})^{2}}\sigma(z)(F-2\partial_{\bar{z}}u)+\text{h.c.}+\text{fermions}..$
(2.5)

It is easy to show that this type of terms preserve half of the fermionic SUSY
transformations and, as a consequence, half of the BPS structure BPS-imp-susy
. Although $\sigma$ may have any dependence on the spatial coordinates without
spoiling the BPS structure, we have chosen a holomorphic form. We will see
later that, with this choice, the impurity can be interpreted as a sort of
frozen lump. By imposing the field equation of the auxiliary field we get

$F=-2\bar{\sigma},\,\bar{F}=-2\sigma.$ (2.6)

Finally, after substituting the auxiliary field equation into the Lagrangian
we have

$\mathcal{L}=\mathcal{L}_{\mathbb{C}P^{1}}+\mathcal{L}_{\text{impurity}}=\frac{1}{(1+|u|^{2})^{2}}\left(\frac{1}{2}\partial_{\mu}u\partial^{\mu}\bar{u}-2\sigma\bar{\sigma}-2\sigma\partial_{\bar{z}}u-2\bar{\sigma}\partial_{z}\bar{u}\right).$
(2.7)

We can use the Bogomolny rearrangement to decompose the static energy as
follows

$E=\int
d^{2}x\frac{1}{(1+|u|^{2})^{2}}\left(2|\partial_{\bar{z}}u+\bar{\sigma}|^{2}+\left(\partial_{z}u\partial_{\bar{z}}\bar{u}-\partial_{\bar{z}}u\partial_{z}\bar{u}\right)\right).$
(2.8)

Therefore we have

$E\geq\pi|Q|,\,Q=\frac{1}{\pi}\int
d^{2}x\frac{\partial_{z}u\partial_{\bar{z}}\bar{u}-\partial_{\bar{z}}u\partial_{z}\bar{u}}{\left(1+|u|^{2}\right)^{2}}$
(2.9)

As one can see from (2.8), the (standard) topological bound is saturated when
$u$ satisfies the BPS equation

$\partial_{\bar{z}}u+\bar{\sigma}=0.$ (2.10)

It is a straightforward exercise to verify that (2.10) also satisfies the
second order Euler-Lagrange equations. This equation can be also obtained
directly from the kernel of the SUSY transformation of the fermions in the
chiral superfield together with (2.6). The general form of solutions of (2.10)
with finite energy is given by

$u=\frac{p(z)}{q(z)}-\int\bar{\sigma}d\bar{z},$ (2.11)

where $p(z),q(z)$ are holomorphic polynomials. The important feature about
(2.11) is that it is a combination of a holomorphic function (the lump
solution of the original $\mathbb{C}P^{1}$ model) plus an antiholomorphic
function defined by the impurity. This form resembles that of a superposition
of a lump and an antilump, which unlike the $\mathbb{C}P^{1}$ case, belongs to
the BPS sector. Even though for concreteness we have focussed our calculations
on the $\mathbb{C}P^{1}$ model, previous results are straightforwardly
generalized to any nonlinear $\sigma$-model in two dimensions.

##  3 Some properties of the solutions

As it is well-known, the topological degree of a lump is given by the maximum
degree of the pair of polynomials that enter in the rational map (assuming
that the rational maps are given in the irreducible form). However, the
topological degree of the soliton in the presence of the antiholomorphic
impurity depends nontrivially on $\sigma$ itself. We will focus our analysis
on impurities of the form

$\int
d\bar{z}\sigma(\bar{z})=\frac{\sigma_{1}(\bar{z})}{\sigma_{2}(\bar{z})},$
(3.1)

where $\sigma_{1,2}(\bar{z})$ are polynomials. We have chosen to write the
impurity as in (3.1) in order to write the BPS solution (2.11) in a simple way

$u=\frac{p(z)}{q(z)}+\frac{\sigma_{1}(\bar{z})}{\sigma_{2}(\bar{z})}.$ (3.2)

The rational map $R(z)=p(z)/q(z)$ is an arbitrary holomorphic function while
the antiholomorphic part is fixed by the impurity.

###  3.1 Negative degree impurities

We now choose the following boundary condition at infinity

$\lim_{|z|\rightarrow\infty}u=0.$ (3.3)

This condition constrains the relative degree of the polynomials
$p,q,\sigma_{1},\sigma_{2}$ as follows

$\deg p<\deg q,\,\,\deg\sigma_{1}<\deg\sigma_{2}\,.$ (3.4)

i.e., both lump and impurity have a negative degree at infinity. The analysis
performed in Appendix A shows that the topological degree of a solution of the
form (3.2) with the conditions (3.4) is

$N=\deg q-\deg\sigma_{2}.$ (3.5)

Therefore, it seems that one can interpret a charge N solution in the presence
of this kind of impurities as a configuration of $\deg q$ lumps determined by
the rational map and $\deg\sigma_{2}$ antilumps determined by the impurity. In
order to confirm this interpretation one can compute the sign of the
topological charge density at the zeros of $q$ and $\sigma_{2}$. Let us assume
for now, that the polynomials $q$ and $\sigma_{2}$ have respectively $n_{1}$
and $n_{2}$ different roots and no common roots with $p$ and $\sigma_{1}$,

$\displaystyle q$ $\displaystyle=$
$\displaystyle(z-a_{1})(z-a_{2})...(z-a_{n_{1}}),$ (3.6)
$\displaystyle\sigma_{2}$ $\displaystyle=$
$\displaystyle(\bar{z}-b_{1})(\bar{z}-b_{2})...(\bar{z}-b_{n_{2}}).$ (3.7)

The topological charge density is given by

$n=\frac{1}{\pi}\frac{|\partial_{z}u|^{2}-|\partial_{\bar{z}}u|^{2}}{(1+|u|^{2})^{2}},$
(3.8)

which in our case takes the form

$n=\frac{1}{\pi}\frac{|\Delta_{pq}|^{2}|\sigma_{2}|^{4}-|\Delta_{\sigma_{1}\sigma_{2}}|^{2}|q|^{4}}{\left(|q|^{2}|\sigma_{2}|^{2}+|\Sigma|^{2}\right)^{2}},$
(3.9)

where

$\displaystyle\Delta_{pq}$ $\displaystyle=$ $\displaystyle
q\partial_{z}p-p\partial_{z}q,$ (3.10)
$\displaystyle\Delta_{\sigma_{1}\sigma_{2}}$ $\displaystyle=$
$\displaystyle\sigma_{2}\partial_{\bar{z}}\sigma_{1}-\sigma_{1}\partial_{\bar{z}}\sigma_{2},$
(3.11) $\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\sigma_{2}p+\sigma_{1}q.$ (3.12)

Now, close to one of the roots of $q$, say $a_{i}$ we have

$\displaystyle n=\frac{1}{\pi}\Pi_{j\neq
i}|a_{i}-a_{j}|^{2}\Pi_{j=1}^{n_{2}}|\bar{a}_{i}-b_{j}|^{2}>0,$ (3.13)

while close to a root of $\sigma_{2}$, $b_{i}$ we have

$\displaystyle n=-\frac{1}{\pi}\Pi_{j\neq
i}|b_{i}-b_{j}|^{2}\Pi_{j=1}^{n_{1}}|\bar{b}_{i}-a_{j}|^{2}<0.$ (3.14)

This confirms the interpretation of the solutions (3.2) as a configuration of
$n_{1}$ lumps and $n_{2}$ antilumps. The argument can be extended for other
base point conditions, but still the topological charge density will be
concentrated about the poles of the rational functions (the zeroes of $q(z)$
and $\sigma_{2}(\bar{z})$). Let us go back for a moment to the ordinary
$\mathbb{C}P^{1}$ model. Let us consider a lump described by the following
polynomial

$u=(z-a_{1})(z-a_{2})...(z-a_{n}).$ (3.15)

If the zeroes of $u$, $a_{1},a_{2},....,a_{n}$ are well separated, this
solution can be interpreted as a system of $n$ lumps, located at the position
of the zeroes. In addition, the $\mathbb{C}P^{1}$ action is invariant under
the inversion $u\rightarrow 1/u$, and more importantly, also the topological
charge density, therefore

$n(u)=n(1/u).$ (3.16)

As a consequence, the solution $1/u$ has the same interpretation: it is a
system consisting on $n$ lumps, but now located at the poles of $1/u$. The
impurity model discussed here does not possess inversion symmetry, that is, in
general if $u$ is a solution, $1/u$ is not. This implies, in particular, that
the zeroes and poles of the holomorphic and impurity components of the
solution do not have a symmetric role. The properties of impurities with
$\deg\sigma_{1}>\deg\sigma_{2}$ will be studied in Sec. 4.2.

###  3.2 Connection with Magnetic Skyrmions

It was pointed out in the literature BPS-imp-susy ; Schroers ; Schroers1  that
there is a connection between the $\mathbb{C}P^{1}$ model with impurities and
the Magnetic Skyrmion at critical coupling. The Magnetic Skyrmion is a two
dimensional soliton stabilized by a first order term called Dzyaloshinskii-
Moriya interaction. This reminds the first order coupling between the field
and the impurity which ensures the BPS property. It was shown in Schroers
that at critical coupling, the BPS Magnetic Skyrmion in terms of the
stereographic coordinate $w$ has the following form

$w_{MS}=\frac{1}{-\frac{i}{2}\kappa e^{i\alpha}\bar{z}+f(z)},$ (3.17)

where $\kappa$ and $\alpha$ are real constants of the model and $f(z)$ is a
general holomorphic map from the plane to the Riemann sphere. From our
previous discussion we have learnt that for a constant impurity of the form

$\sigma=-\frac{i}{2}\kappa e^{i\alpha},$ (3.18)

the BPS solutions of the $\mathbb{C}P^{1}$ impurity model have the form

$u_{i\mathbb{C}P^{1}}=-\frac{i}{2}\kappa e^{i\alpha}\bar{z}+f(z),$ (3.19)

which implies the following relation between the solutions of both models

$u_{i\mathbb{C}P^{1}}=\frac{1}{w_{MS}}.$ (3.20)

In addition, taking into account (3.16), the topological charge density of
both solutions coincides and, as a consequence, they describe de same
configuration. It is not difficult to envisage an impurity model in the spirit
of (2.7) for which its BPS solution coincide exactly with that of the critical
Magnetic Skyrmion. The underlying SUSY structure of the model simplifies again
the design. First of all, the BPS equation in the off-shell SUSY formulation
of the model can be written as

$F-2\partial_{\bar{z}}u=0$ (3.21)

The second step is to look for field-impurity couplings such that the
algebraic field equation for $F$ gives the correct contribution. For the
critical Magnetic Skyrmion this should be

$F=-2\sigma u^{2}.$ (3.22)

This value of the auxiliary field is given by the following terms

$\displaystyle\mathcal{L}_{\mathbb{C}P^{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial_{\mu}u\partial^{\mu}\bar{u}+F\bar{F}}{(1+|u|^{2})^{2}}+\frac{u^{2}}{(1+|u|^{2})^{2}}\sigma(x)(F-\partial_{\bar{z}}u)+\text{h.c.}+\text{fermions}..$
(3.23)

After solving the $F$ equation, the on-shell Lagrangian has the form

$\mathcal{L}=\frac{1}{(1+|u|^{2})^{2}}\left(\frac{1}{2}\partial_{\mu}u\partial^{\mu}\bar{u}-2|u|^{4}\sigma\bar{\sigma}-2u^{2}\sigma\partial_{\bar{z}}u-2\bar{u}^{2}\bar{\sigma}\partial_{z}\bar{u}\right).$
(3.24)

For a constant impurity $\sigma=A\in\mathbb{C}$ the general solution of the
BPS equation is given by

$\partial_{\bar{z}}u+Au^{2}=0\Rightarrow
u_{i\mathbb{C}P^{1}}=\frac{1}{A\bar{z}+f(z)},$ (3.25)

which is exactly the BPS solution of the critical Magnetic Skyrmions.

##  4 The Moduli space

The manifold defined by the BPS solutions of a given charge is often called
the moduli space and denoted by $\mathcal{M}_{N}$. The low velocity dynamics
of solitons belonging to a BPS sector can be approximated by a geodesic motion
in $\mathcal{M}_{N}$. In the $\mathbb{C}P^{1}$ case it is well-known that, the
coordinates associated to the soliton size cannot be described in
$\mathcal{M}_{1}$ since the associated metric diverges, but it is still
possible to restrict the moduli space to describe, for example, only
translational degrees of freedom. For $N=2$, scattering of solitons can be
described in $\mathcal{M}_{2}$, even though, due to the conformal invariance,
solitons have tendency to collapse or expand indefinitely Lesse ; Ward ;
Zakrzewski . The moduli space of the impurity model shares some similarities
with that of the standard $\mathbb{C}P^{1}$, but it has a reacher structure.
In this case, the moduli does not depend only on the topological sector, but
also on the choice of the impurity. We will denote the impurity moduli space
by $\mathcal{M}_{N}^{\sigma}$. We will not give here a detailed classification
of the possible moduli spaces, instead we will analyze the main novelties with
respect to the standard case and illustrate them through some examples.

###  4.1 Examples of geodesics

Let us focus now on the charge $N=0$ configuration with the boundary condition
(3.3). As in the usual case without impurity, this base point condition
implies a one-to-one correspondence between the charge N lump and the rational
holomorphic maps satisfying the condition $\deg p<\deg q$. With this base
point condition, the charge 1 solution of the ordinary $\mathbb{C}P^{1}$ model
is given by

$u_{s}=\frac{\beta}{z-\gamma},$ (4.1)

where $\beta\in\mathbb{C}^{\star}$ and $\gamma\in\mathbb{C}$. It is clear that
the moduli space associated to this solution in the standard lump
configuration is $\mathcal{M}_{1}=\mathbb{C}^{\star}\times\mathbb{C}$. Let us
now choose now $\sigma_{2}=\bar{z}$, which corresponds to a antilump centered
at the origin. The BPS solution in the impurity model corresponding to (4.1)
is

$u=\frac{\beta}{z-\gamma}+\frac{1}{\bar{z}}.$ (4.2)

According to the degree formula (A.10) this solution corresponds to an $N=0$
charge in the impurity model and it still satisfies (3.3). Moreover, it
describes a (dynamical) lump located at $z=\gamma$ and a (non-dynamical)
untilump located at the origin. Note that also we have to impose that
$\gamma\neq 0$ (so, $q$ and $\sigma_{2}$ do not have common roots). It is
clear that the moduli space of the zero charge solutions of the impurity model
$\mathcal{M}_{0}^{\sigma}$ verifies
$\mathcal{M}_{0}^{\sigma}=\mathbb{C}^{\star}\times\mathbb{C}^{\star}$. If we
allow $\beta$ and $\gamma$ to depend on $t$, the kinetic energy associated
with $u$ is given by

$T[u]=\int d^{2}x\frac{1}{2(1+|u|^{2})^{2}}|\dot{u}|^{2}.$ (4.3)

As in the standard $\mathbb{C}P^{1}$ model, the kinetic energy diverges unless
$\dot{\beta}=0$. This effectively reduces the moduli space to
$\mathbb{C}^{\star}$. The configuration (4.2) corresponds to a lump of
(constant) width $\beta$ and centered at $\gamma(t)$ and a antilump fixed at
the origin and width $1$. Since (4.2) is an exact BPS solution, the geodesic
motion on the moduli space is a good approximation for small velocities
$\dot{\gamma}\rightarrow 0$ (becoming exact for $\gamma$ constant). The metric
on the moduli space can be computed immediately from (4.3). Expanding (4.3) we
get

$T[u]=\int
d^{2}x\frac{|\beta|^{2}|\dot{\gamma}|^{2}}{|z-\gamma|^{4}(1+|u|^{2})^{2}}.$
(4.4)

The metric is therefore given by

$g_{\gamma\bar{\gamma}}=\int
d^{2}x\frac{|\beta|^{2}}{|z-\gamma|^{4}(1+|u|^{2})^{2}},$ (4.5)

where $u$ is given by (4.2). It is worth noting that, even the moduli space of
the charge $N=0$ configuration is nontrivial due to the presence of the
impurity, see for example Fig. 1, for some examples of geodesics in the moduli
with $\beta=2$. The impurity tends to bend the geodesics towards the origin
for small impact parameter in the x-direction, while in the y-direction has
the opposite effect. As the impact parameter grows, the lump does not feel the
impurity and the moduli space becomes trivial. In this situation the geodesics
tend to straight lines.

Figure 1: Geodesics in the regular moduli space of the N=0 charge solutions
for $\beta=2$.

There is another fundamental feature that affects our maps. Let us consider a
general solution of the form

$u=R(z)+\Sigma(\bar{z}),$ (4.6)

where $R$ and $\Sigma$ are rational maps. Let us assume that $\deg
R=\deg\Sigma=n$. We have

$\lim_{z\rightarrow\infty}u=AR^{n}e^{in\theta}+BR^{n}e^{-in\theta}=AR^{n}e^{in\theta}\left(1+\frac{B}{A}e^{-2in\theta}\right)$
(4.7)

Let us define $B/A=re^{it}$. If $n>0$ we may chose the base space condition
$u(\infty)=\infty$ almost everywhere, except for $r=1,\,\theta=\frac{t}{2n}$,
where the field vanishes at infinity. As pointed out in Schroers  in the
context of Magnetic Skyrmions, this can be traced back to the fact that such
maps do not extend to smooth maps between spheres. Equivalently, this can be
also seen from the topological degree jump in the formula (A.10), when the
ratio of the leading coefficients of the holomorphic and antiholomorphic parts
has modulus one. These solutions, even belonging to the BPS sector, do not
have a well-defined moduli. On the other hand, solutions with boundary
condition $u(\infty)=0$ (vanishing impurities at infinity) have a well-defined
topological degree for all moduli space values.

###  4.2 The regularized moduli. Positive degree impurities.

One of the limitations of the moduli space approximation for lump
configurations comes from the existence of moduli space coordinates with
infinite inertia (see Sutcliffe_bound  for a recent treatment of this problem
using boundary metrics). This is clear in the standard situation even for the
$N=1$ lump, where a time dependent radius or phase leads to infinite value of
the moduli metric for these coordinates. A similar situation arises in the
presence of the impurities discussed in Sec. 3. In order to analyze the origin
of this divergence we can expand the BPS solution at infinity

$u=\frac{a_{1}(t)}{z}+\frac{a_{2}(t)}{z^{2}}+...+b_{1}\bar{z}^{m}+...,$ (4.8)

where we are assuming that the holomorphic part vanishes at infinity and the
antiholomorphic contribution with $m\in\mathbb{Z}$ comes from the presence of
the impurity. The kinetic energy density associated to $\dot{a}_{1}(t)$ at
infinity is given by

$T|_{|z|\rightarrow\infty}\propto\begin{cases}\frac{|\dot{a}_{1}(t)|^{2}}{|z|^{2}},\,m\leq
0\\\ \frac{|\dot{a}_{1}(t)|^{2}}{|z|^{2+4m}},\,m>0\end{cases}$ (4.9)

Taking into account the integration measure, if $m\leq 0$ the kinetic integral
gives a logarithmic singularity. However, for $m>0$, the kinetic energy
integral associate to the leading term $a_{1}(t)$ behaves as $1/r^{4m+2}$, and
therefore, it converges.

We conclude that an impurity lump can collapse or expand within the geodesic
approximation if the impurity has positive degree. A detailed analysis of this
moduli space for low charge solutions is left for a future work.

###  4.3 Some solutions and their topological charge densities

As we have discussed, the interpretation of the impurity depends crucially on
its form. One cannot immediately say that any combination of a holomorphic
function (the lump) of degree $n_{1}$ with a holomorphic function (the
impurity) of degree $n_{2}$ with well-separated zeros (poles) can be seen as a
system of $n_{1}$ lumps and $n_{2}$ antilumps.

(a) $u_{1}$, $A=1$, $\gamma=5$ and $\epsilon=-5$ (b) $u_{3}$, $A=1$,
$\gamma=5$ and $\epsilon=-5$

(c) $u_{2}$, $A=2$ and $\gamma=0$ (d) $u_{2}$, $A=1/2$ and $\gamma=0$

Figure 2: Behavior of the topological charge density for different solutions.
The color palette indicates the topological charge density.

The first example we consider is the following BPS solution

$u_{1}=\frac{A}{z-\gamma}+\frac{1}{\bar{z}-\epsilon}.$ (4.10)

Our degree formula gives

$N(u_{1})=0.$ (4.11)

Intuitively, this configuration should correspond to a well-separated lump-
antilump system for $|\gamma-\epsilon|$ large. The lump is centered at
$\gamma$, while the antilump is centered at $\epsilon$. This is consistent
with the vanishing degree of the configuration and with the considerations of
Sec. 3 (see Fig. 2 (a)). The second BPS solution we would like to consider is
given by

$u_{2}=Az-\gamma+\bar{z}$ (4.12)

Despite that it may seem, this solution is more subtle. First, our degree
formula gives the following

$N(u_{2})=\begin{cases}1,\,|A|<1\\\ 0,\,|A|=1\\\ -1,\,|A|>1\end{cases}$ (4.13)

When the holomorphic part “dominates” ($|A|>1$) the total degree of the
solution is $1$, and the full solution can be interpreted as a lump deformed
in the $y$-direction (see Fig. 2 (c)). In the opposite situation ($|A|<1$),
the total charge is $-1$, and as a consequence we have a sort of antilump also
stretched in the $y$-direction (see Fig. 2 (d)). The intermediate situation
($|A|=1$), has degree zero. The interpretation of this case is simple, the map
$u_{2}$ with $A=1$ is actually real, and cannot cover the Riemann sphere, it
falls therefore in the topological sector of zero charge.

In order to show the rich variety of solutions we consider finally

$u_{3}=\frac{Az}{(z-\gamma)(z-\epsilon)}+\frac{1}{\bar{z}}$ (4.14)

Once again, the intuitive picture that one can get from the solution is a
system of two lumps (at $\gamma$ and $\epsilon$) and one antilump at the
origin (see Fig. 2 (b)). This is also consistent with the total charge of the
configuration

$N(u_{3})=1.$ (4.15)

One can say, in general, that when the total degree associated to the impurity
(understood as the difference of degrees between numerator and denominator) is
negative, the topological degree of the solution is well-defined. That is, if
one considers vanishing impurities at infinity, the total degree does not
change through the moduli space. Of course, as in the standard lump case, one
has to remove also the common root points. On the other hand, if the total
impurity degree is positive, the moduli space splits into regions of different
degrees.

##  5 Kähler property of the regular moduli.

It is the purpose of this section to show an important geometric property of
the moduli space of the BPS impurity lump. In the standard situation, the
moduli space $\mathcal{M}_{N}$ of $N$-lump configuration has real dimension
$4N$, since it is defined by the set of based rational maps of complex
dimension $2N$ . As we have mentioned, 2 coordinates in the moduli space
manifold have infinite inertia. It is known that the restriction to the finite
$4N-2$ submanifold, from which we have eliminated the infinite inertia
coordinates, is Kähler Din . Despite that it may seem, due to the appearance
of holomorphic and non-holomorphic coordinates in the BPS solution, we will
show here that the moduli space restricted to the finite $4N-2$ submanifold is
also Kähler. First, following Din , we use the base point condition
$u(\infty)=1+B$. The BPS impurity lump takes the form

$u=\frac{p(z)}{q(z)}+B\sigma(\overline{z})=\frac{\prod_{i=1}^{N}(z-a_{i})}{\prod_{i=1}^{N}(z-b_{i})}+B\frac{\prod_{i=1}^{m}(\overline{z}-\alpha_{i})}{\prod_{i=1}^{m}(\overline{z}-\beta_{i})}.$
(5.1)

According to (A.10) this corresponds to a charge $N-m$ solution, and it is
well-defined whenever numerators and denominators do not have common roots. We
assume now that $a_{i}\rightarrow a_{i}(t)$ and $b_{i}\rightarrow b_{i}(t)$
(note that $\alpha_{i}$ and $\beta_{i}$ are constant coefficients determined
by the choice of the impurity) and compute the kinetic energy

$T=\int d^{2}x\frac{|\partial_{t}u|^{2}}{(1+|u|^{2})^{2}}.$ (5.2)

If we introduce

$\widetilde{p}=p+B\sigma(\bar{z})q(z),$ (5.3)

and taking into account (5.1) we have

$T=\int
d^{2}x\frac{|\dot{\widetilde{p}}q-\widetilde{p}\dot{q}|^{2}}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}}=\int
d^{2}x\frac{|\dot{p}q-p\dot{q}|^{2}}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}},$
(5.4)

where the term proportional to the impurity has disappeared from the
numerator. In order to make explicit the moduli coordinates, the time
derivatives can be rewritten as follows

$T=\int
d^{2}x\frac{1}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}}|\dot{a}_{i}\partial_{a_{i}}-\dot{b}_{i}\partial_{b_{i}}|^{2}|p|^{2}|q|^{2}.$
(5.5)

As in the standard case, the idea is to find a Kähler potential from which the
metric can be derived. If such potential exist, the Kähler property of the
metric is automatically satisfied. After some simple algebraic manipulations
we propose the following potential

$T=\dot{w}_{\mu}\dot{\overline{w}}_{\nu}\int
d^{2}x\partial_{w_{\mu}}\partial_{\overline{w}_{\nu}}\log\left(|p+B\sigma(\bar{z})q|^{2}+|q|^{2}\right),$
(5.6)

where $u_{\mu}=a_{\mu}$ for $\mu=1,..,N$ and $w_{\mu}=b_{\mu-N}$ for
$\mu=N+1,...,2N$. In order to prove the equality between (5.5) and (5.6) we
expand the square in (5.5), split the integral in four terms and compare them
with (5.6). We have

$\displaystyle\partial_{a_{i}}\partial_{\bar{a}_{i}}\log\left(|p+B\sigma(\bar{z})q|^{2}+|q|^{2}\right)=\frac{|q|^{2}\partial_{a_{i}}p\partial_{\bar{a}_{i}}\bar{p}}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}},$
(5.7)
$\displaystyle\partial_{a_{i}}\partial_{\bar{b}_{i}}\log\left(|p+B\sigma(\bar{z})q|^{2}+|q|^{2}\right)=\frac{\bar{p}q\partial_{a_{i}}p\partial_{\bar{b}_{i}}\bar{q}}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}},$
(5.8)
$\displaystyle\partial_{b_{i}}\partial_{\bar{a}_{i}}\log\left(|p+B\sigma(\bar{z})q|^{2}+|q|^{2}\right)=\frac{p\bar{q}\partial_{b_{i}}q\partial_{\bar{a}_{i}}\bar{p}}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}},$
(5.9)
$\displaystyle\partial_{b_{i}}\partial_{\bar{b}_{i}}\log\left(|p+B\sigma(\bar{z})q|^{2}+|q|^{2}\right)=\frac{|p|^{2}\partial_{b_{i}}q\partial_{\bar{b}_{i}}\bar{q}}{(q\overline{q}+\widetilde{p}\overline{\widetilde{p}})^{2}}.$
(5.10)

By comparing (5.7)-(5.10) with (5.5) it is immediate to see the announced
equivalence. The expression (5.6) provides automatically the real Kähler
potential

$\mathcal{K}=\int d^{2}x\log\left(|p+B\sigma(\bar{z})q|^{2}+|q|^{2}\right).$
(5.11)

Finally, the Kähler metric has the standard Kähler representation in terms of
the potential

$g_{\mu\overline{\nu}}=\partial^{2}_{w_{\mu}\overline{w}_{\nu}}\mathcal{K}.$
(5.12)

Several comments are in order. Notice that the only differences w.r.t. the
standard case is the addition of extra terms containing the impurity. Also in
this case the Kähler potential is strictly divergent, but this divergence does
not affect the metric. We have focussed on the $\mathbb{C}P^{1}$ models, but,
at least for two dimensional sigma models with impurities, the Kähler property
seems to hold (for a complete proof in general sigma models see Ruback ). For
the type of impurity we have chosen here (which tends to a nonzero value at
infinity) the restriction to the regular moduli is translated to the
(standard) constraint $\sum_{j}(\dot{a}_{i}-\dot{b}_{i})=0$. In principle, if
one does not care about the behaviour of the impurity at infinity, while
keeping the finite base point condition for the holomorphic part, the form of
the Kähler potential is the same (note that we have not used any property of
the impurity in our derivation). In addition, if
$\lim_{\bar{z}\rightarrow\infty}\sigma(\bar{z})=\infty$, no further
constraints are necessary in order to keep the kinetic energy convergent.

##  6 Holomorphic impurities as multifield systems.

The interpretation of impurities as frozen lumps pursued throughout this work
can be realized by means of a multifield model. Let us consider the following
Lagrangian

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{1}}{(1+|u|^{2})^{2}}\left(\frac{1}{2}\partial_{\mu}u\partial^{\mu}\bar{u}-2w\bar{w}-2w\partial_{\bar{z}}u-2\bar{w}\partial_{z}\bar{u}\right)+\frac{\Lambda_{2}}{(1+|w|^{2})^{2}}\partial_{\mu}w\partial^{\mu}\bar{w}.$
(6.1)

This describes two complex fields coupled in the same way as the BPS impurity
model. For $\Lambda_{2}>>\Lambda_{1}$ the fields $u$ and $w$ decouple (for
similar results in one-dimensional theories see multifield ). In this limit,
$w$ behaves as a standard lump, therefore

$w=\frac{p(z)}{q(z)}\,\,\text{or}\,\,w=\frac{p(\bar{z})}{q(\bar{z})}.$ (6.2)

This gives two branches defined by the holomorphic or antiholomorphic
character of $w$. If we choose one antiholomorphic solution and plug it back
into the Lagrangian (6.1), the resulting model in the $\Lambda_{1}$ sector
reproduces the impurity model discussed in the paper, where the impurity now
is determine by the $w$-lump. Physically, one may interpret this model as a
system of two lump fields such that the energy scale of one of them is much
bigger than the other. This effectively freezes one of them, which acts like
an impurity which does not back-react to the evolution of the other.

##  7 Conclusions

We have studied the $\mathbb{C}P^{1}$ model coupled to impurities in such a
way that part of the original self-dual structure is preserved. The Bogomolny
bound remains invariant, but the Bogomolny equations are a sort of
inhomogenous Cauchy-Rimenann equations augmented by the impurity. Although any
impurity coupled to the $\mathbb{C}P^{1}$ field in the form (2.7) preserves
the (half of the) BPS structure, we have restricted our analysis to
(anti)holomorphic impurities. This constraint provides a natural
interpretation for the impurity: it can be considered as a frozen lump of
negative charge, in whose background the positive charge lumps move. This
characteristic allows to obtain in a clean way these impurity models as
certain limits of theories with two complex fields.

We have analyzed in detail the topological charge of the BPS solutions and
proved that depends crucially on the form of the impurity. Under some
circumstances, the topological charge may change the sign through a trajectory
in the moduli space, pointing that, certain maps (BPS solutions) do not extend
to smooth maps between 2-spheres. This peculiarity was already glimpsed in the
context of critical Magnetic Skyrmions Schroers . Elaborating on this point,
we have clarified the connection between holomorphic BPS impurity models and
critical Magnetic Skyrmions, resulting that, in the constant impurity case,
they are related through an inversion transformation of the field. This
relation implies, in particular, that the topological charge densities of two
solutions (related by the inversion) coincide in both models. We have also
shown that it is straightforward to build an impurity model whose BPS solution
coincide with that of the critical Magnetic Skyrmion case in the limit of
constant impurities.

One feature that moves this family of models away from the standard
$\mathbb{C}P^{1}$ case, is the moduli space structure. It is possible to
choose the impurity in such a way that even the moduli coordinate associated
to the lump size is well-defined in the geodesic approximation. In the
standard case, coordinates with infinite inertia can be related to the
conformal invariance of the model. In the impurity model, it does not seem to
be the case as conformal symmetry depends on the specific form of the impurity
and not only on its behavior at infinity. Remarkably, we have shown that,
despite these differences, the geometry of the moduli space is still Kähler.

Another relevant point is the connection of this family of impurity models
with non-impurity models at certain limits. It has been pointed out in
Schroers1  that, at the level of BPS solutions, the $\mathbb{C}P^{1}$ impurity
model can be obtained from a gauge non-abelian ($SU(2)$) $\sigma$-model, where
the gauge field is chosen such that the BPS equations of the model mimics
those of the BPS impurity. For the rest of the discussion, it is important to
emphasize that, in this case, the gauge field itself is taken to be the
impurity, i.e., it has no dynamics. A different approach was taken in tong .
Here, certain types of (magnetic) impurity models were obtained as certain
limits of gauge theories with product group $U(1)\times U(1)$. The impurities
are interpreted as frozen vortices associated to one of the gauge groups. As a
consequence, the impurities are effectively obtained at some limit (in this
case the limit of infinite mass of the vortices associated to one of the gauge
groups). This is the point of view we take in Sec. 6. We have chosen to couple
the would be dynamical lump to an extra complex field in the same fashion as
the field-impurity BPS coupling. In this case, both fields are originally
dynamical. Then, the energy scale associated to the second field is taken to
be large compared to the other. At this limit, both fields decouple and the
BPS equation of the “heavier” one forces the holomorphic character of the
solution. Once this solution is reintroduced in the action, acts as an
impurity, in whose background the other moves. This provides a natural
interpretation of impurities as frozen (anti)lumps.

As we have mentioned the impurity models discussed here possess an underlying
SUSY structure. This structure can be used to build new impurity models even
in the higher-derivative case queiruga_susy_1 ; queiruga_susy_2 ;
queiruga_susy_3 ; nita_1 ; nita_2 ; nita_3 . Further research, regarding the
study of the fermionic sectors of the SUSY impurity models as well as
generalizations to other spacetime dimensions, are left for a future work.

Acknowledgements.\- This work is supported by the Spanish Ministry
MCIU/AEI/FEDER grant (PGC2018-094626-B-C21) and the Basque Government grant
(IT-979-16). The author is grateful to A. Wereszczynski for useful comments.

## Appendix A: Subtleties in the definition of the impurity topological degree

The topological degree of a map $u$ between Riemann spheres can be written as
follows

$N=\frac{i}{2\pi}\int\frac{du\wedge d\bar{u}}{\left(1+|u|^{2}\right)^{2}},$
(A.1)

where $u$ is the map given by

$u=\frac{p(z)}{l(z)}+\frac{\sigma_{1}(\bar{z})}{\sigma_{2}(\bar{z})}=\frac{p(z)\sigma_{2}(\bar{z})+\sigma_{1}(\bar{z})l(z)}{l(z)\sigma_{2}(\bar{z})}\equiv\frac{r(z,\bar{z})}{q(z,\bar{z})}$
(A.2)

We assume that the polynomials $p(z),\,l(z)$ and
$\sigma_{1}(\bar{z}),\sigma_{2}(\bar{z})$ do not have common roots and their
degrees are $n,k,s_{1}$ and $s_{2}$ respectively. It can be shown easily that

$\frac{du\wedge
d\bar{u}}{\left(1+|u|^{2}\right)^{2}}=\frac{1}{2}\,d\left(\frac{ud\bar{u}-\bar{u}du}{1+|u|^{2}}\right).$
(A.3)

We follow the procedure of Schroers . The expression in the brackets of the
r.h.s of (A.3) can be expanded as follows

$\frac{ud\bar{u}-\bar{u}du}{1+|u|^{2}}=\frac{rd\bar{r}-\bar{r}dr+qd\bar{q}-\bar{q}dq}{r\bar{r}+q\bar{q}}+d\ln
q-d\ln\bar{q}.$ (A.4)

Since $p$ and $r$ do not have common roots, it follows that the first term in
the r.h.s. of (A.4) is non-singular. The second term, however, has poles at
the zeros of $q(z,\bar{z})$. We can use now the Stokes’ theorem for forms. If
$\omega$ is a one-form and $\Omega$ is a region in the complex plane we have

$\int_{\partial\Omega}\omega=\int_{\Omega}d\omega.$ (A.5)

Let $\Gamma_{\eta}^{i}$ be a circle of radius $\eta$ surrounding the zero
$\beta_{i}$ of $q(z,\bar{z})$ and $\Gamma_{R}$ a circle of radius $R$ centered
at the origin. Taking into account (A.3), (A.4) and (A.5), the integral (A.1)
over the complex plane can be written as

$N=\frac{i}{4\pi}\lim_{\eta\rightarrow
0}\sum_{i=1}^{k}\int_{\Gamma_{\eta}^{i}}\left(d\ln
q-d\ln\bar{q}\right)+\frac{i}{4\pi}\lim_{R\rightarrow\infty}\int_{\Gamma_{R}}\frac{ud\bar{u}-\bar{u}du}{1+|u|^{2}}\equiv
I_{1}+I_{2}$ (A.6)

The first term in the r.h.s. of (A.6) ($I_{1}$) can be computed using the
residue theorem. It produces a positive contribution from the holomorphic
zeroes of $l(z)$ and a negative contribution from the antiholomorphic zeroes
of $\sigma_{2}$, that is $I_{1}=k-s_{2}$. For the second term ($I_{2}$) we
define $z=Re^{i\theta}$, expand for $R\rightarrow\infty$ and integrate. We
have three cases:

Case i): $n-k>s_{1}-s_{2}$. The integral $I_{2}$ gives

$I_{2}=\begin{cases}n-k,\,n>k\\\ 0,\,n\leq k\end{cases}$ (A.7)

Case ii): $n-k<s_{1}-s_{2}$. The integral $I_{2}$ gives

$I_{2}=\begin{cases}s_{2}-s_{1},\,s_{1}>s_{2}\\\ 0,\,s_{1}\leq
s_{2}\end{cases}$ (A.8)

Case iii): $n-k=s_{1}-s_{2}$. Assume that the leading coefficient of the
rational function $p(z)/l(z)$ at $z$ is $A\in\mathbb{C}$ ,and the leading
coefficient of $s_{1}(\bar{z})/s_{2}(\bar{z})$ is $B\in\mathbb{C}$. Let us
assume that $s_{1}>s_{2},$, we have three possibilities depending on the
modulus of $B/A$. If we define $B/A=re^{it}$ and $m=s_{1}-s_{2}$ we have

$I_{2}(r,t)=-\frac{m(r^{2}-1)}{2\pi}\int_{0}^{2\pi}\frac{d\theta}{1+r^{2}-2r\cos(t-2m\theta)}$
(A.9)

For $m>0$ and $r=1$ we have $I_{2}(r,t)=0$. For $m>0$ and $r>1$,
$I_{2}(r,t)=-m$. Now, since $I_{1}(r,t)=-I_{2}(1/r,t)$ we have for $m>0$ and
$r<1$, $I_{2}(r,t)=m$. For $m\leq 0$, $I_{2}=0$. Therefore, finally

$N=\begin{cases}n-s_{2},\,n-k>s_{1}-s_{2},\,n>k\\\
k-s_{2},\,n-k>s_{1}-s_{2},\,n\leq k\\\
k-s_{1},\,n-k<s_{1}-s_{2},\,s_{1}>s_{2}\\\
k-s_{2},\,n-k<s_{1}-s_{2},\,s_{1}\leq s_{2}\\\
k-s_{1},\,n-k=s_{1}-s_{2},\,s_{1}>s_{2},\,r>1\\\
n-s_{2},\,n-k=s_{1}-s_{2},\,s_{1}>s_{2},\,r<1\\\
k-s_{2},\,n-k=s_{1}-s_{2},\,s_{1}>s_{2},\,r=1\\\
k-s_{2},\,n-k=s_{1}-s_{2},\,s_{1}\leq s_{2}\end{cases}$ (A.10)

Note that the degree is well-defined, except in the case where the degree of
the leading monomial of both rational functions coincide at infinity with
positive degree. In this case, the degree of the solutions depends on the
ratio of the leading coefficients.

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