# MRCBert: A Machine Reading Comprehension Approach for Unsupervised
Summarization

Saurabh Jain
School of Computing
National University of Singapore
<EMAIL_ADDRESS>
&Guokai Tang
School of Computing
National University of Singapore
<EMAIL_ADDRESS>
&Lim Sze Chi
School of Computing
National University of Singapore
<EMAIL_ADDRESS>

###### Abstract

When making an online purchase, it becomes important for the customer to read
the product reviews carefully and make a decision based on that. However,
reviews can be lengthy, may contain repeated, or sometimes irrelevant
information that does not help in decision making. In this paper, we introduce
MRCBert, a novel unsupervised method to generate summaries from product
reviews. We leverage Machine Reading Comprehension (Zhang et al. (2020)), i.e.
MRC, approach to extract relevant opinions and generate both rating-wise and
aspect-wise summaries from reviews. Through MRCBert 111Code:
https://github.com/saurabhhssaurabh/reviews_summarization.git we show that we
can obtain reasonable performance using existing models and transfer learning,
which can be useful for learning under limited or low resource scenarios. We
demonstrated our results on reviews of a product from the Electronics category
in the Amazon Reviews dataset. Our approach is unsupervised as it does not
require any domain-specific dataset, such as the product review dataset, for
training or fine-tuning. Instead, we have used SQuAD v1.1 dataset only to
fine-tune BERT for the MRC task. Since MRCBert does not require a task-
specific dataset, it can be easily adapted and used in other domains.

##  1 Introduction

With the rapid expansion of e-commerce platforms in recent years, online
shopping has become relatively convenient and frequent attractive offers have
made it even more popular amongst customers. One way of enabling shoppers to
make informed decisions over their purchases is through the ratings and
reviews made by previous customers. For the customer, it becomes important to
only look at key aspects relevant to specific features of the product to make
a decision quickly. However, it poses a challenge for the reader to scour
through huge amounts of reviews which can be lengthy, contain repeated
information about the product, or irrelevant information that does not give
further insight to their decision making. This makes product-based opinion
summarization in reviews important and useful for customers.

While opinion summarization has been widely done in supervised or semi-
supervised contexts, modern deep learning methods rely on large amounts of
annotated data or golden summaries that are not readily available in the
opinion-summarization domain and are expensive to produce. Hence, the
motivation of this paper is to conduct unsupervised summarization of user
opinions relevant to a product feature. More specifically, we introduce
MRCBert that uses existing open-sourced pre-trained models and demonstrate
product-based opinion summarization using the machine reading comprehension
(MRC) approach without using labeled or golden summaries.

Firstly, MRCBert performs a question answering task, in which it extracts
relevant opinions from answers through product-centric questions. Then, the
extracted opinions are input to an abstractive summarization model to produce
aspect-wise summaries. To demonstrate proof of concept, we tested our approach
on reviews from a single product using the publicly available Amazon
electronics reviews dataset.

Our contributions can be summarised as follows:

  1. 1.

A novel unsupervised method that uses Machine Reading Comprehension approach
to extract relevant opinions and generate both rating-wise and aspect-wise
summaries from reviews.

  2. 2.

We have introduced a framework to generate summaries from reviews such that it
can be easily scaled and used in any other domains. This framework does not
require any domain-specific dataset. Our model can generate both aspect-wise
and rating-wise summaries with the same pipeline, while other works have
separate approaches to generate these.

  3. 3.

Our methodology can be used in limited or low resource scenarios. We have
successfully demonstrated it with the help of a pre-trained MRC model and the
concept of transfer learning.

  4. 4.

Introduced a unique sentiment accuracy metric to perform the unsupervised
evaluation.

##  2 Related Works

Earlier works on user reviews summarization like McIlroy et al. (2015), Joung
and Kim (2021) and Phong et al. (2015) used extractive techniques for topic
modelling and keyword extraction. They are also often generically referred to
as opinion mining techniques. Other approaches also attempted to solve the
extractive opinion summarization task by finding attribute-value pairs that
are of interest using aspect-based feature mining (Bafna and Toshniwal
(2013)), or studying part-of-speech tagging to identify product features and
their corresponding opinion words with the help of a sentiment classification
module (Yang et al. (2009)).

However, these earlier works usually require golden summaries, or are rule-
based methods that may not generalize well with noisy reviews datasets like
the Amazon Reviews dataset, where there can be multiple topics unrelated to
the product, misspelled words, colloquialisms, or misused sentence syntaxes.
Later works that do unsupervised abstractive opinion summarization have pushed
boundaries. As far as we are aware, the only two earlier methods dealing with
unsupervised multi-document opinion summarization are MeanSum and Copycat.

In the work of Chu and Liu (2018), the MeanSum model consists of two main
components: (1) an auto-encoder module that learns representations for each
review and constrains the generated summaries to be in the language domain and
(2) a summarization module that learns to generate summaries that are
semantically similar to each of the input documents. The authors have done
this by training end-to-end with an autoencoder reconstruction loss to achieve
both objectives. To obtain the opinion summary of one target product, the
authors used the mean of all encoded reviews as the combined review
representation and feed that into the decoder network to produce one summary.

Unlike aspect-based summarization in the work of Liu (2012) that rewards the
diversity of opinions, Copycat aims to generate summaries that represent a
consensus i.e. dominant opinions in reviews. This is a relevant adaptation of
the problem at hand as dominant reviews are useful for quick decision-making
to get the overall feel of a product. The Copycat abstractive summarizer is
similar to MeanSum with an encoder and decoder module that takes the mean of
latent representations of reviews at inference time. However, their proposed
architecture differs by using a hierarchical variational autoencoder (VAE) to
study the latent representation of each review. Its decoder network also
directly accesses all other reviews within a product, enabling it to capture
common information across different reviews and retaining information of
stronger relevance to the product. This is done by using a pointer-generator
network that studies the distribution of words in each review and in each
product combined. The authors showed superior performance in ROUGE R1, R2, and
RL scores as compared to MeanSum. More details on ROUGE scores by Lin (2004)
are presented in Section 4.

##  3 Proposed Method

We fine-tuned BERT (Devlin et al. (2018)) and DistilBERT (Sanh et al. (2019))
models with SQuAD v1.1 (Rajpurkar et al. (2016)) dataset. Afterward, we
extracted opinions from reviews, using a machine reading comprehension
approach, which was used later to generate summaries. We have explained our
approach in detail in the following sections.

###  3.1 BERT

In recent years, BERT has been one of the most successful deep learning
architectures for NLP tasks. Although embedding layers can learn the
contextual representation of texts after training on large-scale corpora,
training a wide variety of networks for different tasks on limited datasets is
unable to learn representations effectively. Unlike ELMO (Peters et al.
(2018)) that learns additional text features by training a specific
architecture for a specific task, BERT is intended for fine-tuning approach
with a general model for all tasks. It is also desirable that an intelligent
agent should learn more from data and use minimum prior human knowledge.

Mainly there are two types of BERT architecture:

  1. 1.

BERT base: It contains 12 layers, 768 hidden dimensions, and 12 attention
heads. It has total 110M parameters.

  2. 2.

BERT large: It has 24 layers, 1024 hidden dimensions, and 24 attention heads.
It has total 340M parameters.

###  3.2 DistilBERT

DistilBERT is a faster and lighter variant of BERT with some reduction in
performance. According to the authors, it is 40% lighter and 60% faster than
BERT, while it retains 97% of its language understanding capabilities. We used
it in our work to compare its performance with BERT on review summarization
tasks when very little computation power is available.

###  3.3 Fine-tuning BERT and DistilBERT

We used SQuAD v1.1 (Rajpurkar et al. (2016)) dataset to fine-tune BERT and
DistilBERT. We used HuggingFace library222HuggingFace Library:
https://huggingface.co/ for fine-tuning. SQuAD v1.1 contains more than
$100,000$ pairs of questions and answers. We formulate fine-tuning task as
follows. Given a question $q=(q_{1},\dots,q_{m})$ and a paragraph
$p=(p_{1},\dots,p_{n})$, we formulate input as
$x=[CLS],q_{1},\dots,q_{m},[SEP],p_{1},,\dots,p_{n},[SEP]$, where [CLS] is a
dummy token and [SEP] is a separator to separate question from paragraph. Let
BERT(.) be the pre-trained BERT/DistilBERT model. We first get hidden
representation of input x by passing it to pre-trained BERT/DistilBERT model
as $h=BERT(x)\in R^{r_{h}*|x|}$, where $|x|$ is length of input sequence and
$r_{h}$ is size of hidden dimension. Then hidden representation is passed to
two separate dense layers (Huang et al. (2016)) followed by Softmax (Wu et al.
(2016)) function: $l_{1}=Softmax(W_{1}*h+b_{1})$ and
$l_{2}=Softmax(W_{2}*h+b_{2})$, where $W_{1},W_{2}\in R^{r_{h}}$ and $b1,b2\in
R$. Softmax is applied along the dimension of the sequence. Outputs $s$ and
$e$ are span indices in paragraph $p$, computed from $l_{1}$ and $l_{2}$ as:

${s=arg\hskip 5.0ptmax_{index[SEP]<s<e}(l_{1})}$ (1) ${e=arg\hskip
5.0ptmax_{index[SEP]<e<|x|}(l_{2})}$ (2)

Hence, final answer $a$ will always be a valid text span from $p_{s}$ to
$p_{e}$. We used average cross entropy (Zhang and Sabuncu (2018)) loss for
both the pointers as our training loss:

${L=\frac{-\sum(log(l_{1})OH(s))-\sum(log(l_{2})OH(e))}{2}}$ (3)

where $OH$ is representing one hot vector of ground truths $s$ and $e$.

###  3.4 Opinion Extraction

We treated the opinion extraction task as a Machine Reading Comprehension task
in which we considered a review as a paragraph and opinion on a feature as a
question. At first, we manually extracted features from product descriptions.
Since we did narrow down our scope to Computers, Tablets, and Laptops, all
products have almost similar features. Therefore, manual extraction did not
take much time. Alternatively, there are several research works available to
extract features from product descriptions. Due to a lack of resources, we
avoided using these research works. We used 2 question variants, both are
targeted to ask about a product aspect. They are of the form _"How is
[feature]?"_ and _"What is opinion on [feature]?"_. We worked with a total of
10 features. These are Display, Memory, Speaker, Sound, Processor, WiFi,
Battery, Brand, Operating System, and Camera.

We send a list of questions and a review to our fine-tuned BERT/DistilBERT
model. Every question is represented as $q=(q_{1},\dots,q_{n})$ and review as
$r=(r_{1},\dots,r_{m})$. The output of the model returns a list of indices
pairs representing answer span $s$ and $e$ for each question, where $s$
represents start index and e represents end index. Then, text phrase from the
reviews corresponding to these indices is considered as opinion on the
respective feature i.e. $o=(r_{s},\dots,r_{e}$. Then, we pass the list of
reviews and opinions to our summarization module.

###  3.5 Summarization Model

As Copycat has claimed superior performance over MeanSum as an abstractive
opinion summarization model, we experimented with pre-trained Copycat as a
summarization module for our extracted opinions.333Copycat:
https://github.com/abrazinskas/Copycat-abstractive-opinion-summarizer. For
more robust testing of MRCBert, we further experiment using a summarization
model that is not tailored for opinion summarization. The objective is to
investigate if MRCBert can perform with summarization models that are used for
out-of-domain datasets, therefore further testing its scalability to other
problem domains. As the unsupervised summarization models found thus far are
both summarising opinions from reviews, the next resort is to use a supervised
summarization model for this investigation. We used the DistilBART-12-6-cnn
Summarizer available on HuggingFace. 444DistilBART Summarizer API:
https://huggingface.co/sshleifer/distilbart-cnn-12-6.

Copycat has been introduced in Section 2. In more detail, Copycat trains a
Variational AutoEncoder to learn two latent representations: $z_{i}$ the
latent representation of a review $r_{i}$ and a product latent representation
$c$. During summary generation, Copycat infers the mean representation of a
product given all $N$ reviews of the product (Eq. 4), and then infer the mean
representation of the review (Eq. 5). To generate a summarizing review, the
conditional language model is used to sample a review as in Eq. 6.
$p_{\theta}(r|z^{*},r_{1:N})$ in Eq. 6 is done with the help of a pointer-
generator network that maintains two-word distributions: one that assigns
probabilities to words being generated using a fixed vocabulary, and another
captures probabilities of words to be copied directly from other reviews.

${c^{*}=\mathbb{E}_{c\sim q_{\phi}(c|r_{1:N})}[c]}$ (4)

${z^{*}=\mathbb{E}_{z\sim p_{\theta}(z|c^{*})}[z]}$ (5)

${r^{*}=\arg\max p_{\theta}(r|z^{*},r_{1:N})[z]}$ (6)

The DistilBART-12-6-cnn has been trained on CNN/Dailymail (See et al. (2017))
and Extreme Summarization (XSum) datasets (Narayan et al. (2018)). These
datasets contain document-summary pairs from news articles of various topics
and are a different summarization problem domain from MRCBert, where the
latter focuses on opinion summarization. Therefore, the DistilBART-12-6-cnn is
suitable for our use case of investigating the scalability of MRCBert using
pre-trained models from other problem domains.

DistilBART-12-6-cnn originates as one of the applications of the Bart model
introduced in the work of Lewis et al. (2019). It uses a standard seq2seq
architecture with a bidirectional encoder and a left-to-right autoregressive
decoder and can be fine-tuned for multiple downstream tasks such as the
summarization task applied in MRCBert.

Besides using Copycat and DistilBART-12-6-cnn as the chosen summarization
models to compare with, MRCBert uses a combination of the two. MRCBert uses
the generated reviews by Copycat, which captures majority sentiment and
handles misspelled words, colloquialisms, or misused sentence syntaxes with
the idea that majority reviews are written in proper sentences. Then using the
generated reviews, MRCBert taps on DistilBART-12-6-cnn to summarize the
generated reviews. Section 6 covers a comparison of the 2 summarization models
when used separately and MRCBert, which uses a fusion of the two.

###  3.6 MRCBert Summarization Framework

We put together the opinion extraction module using MRC and the summarization
module to generate rating-wise and aspect-wise summaries into a novel
framework called MRCBert (Figure 1).

Figure 1: The proposed MRCBert summarization framework.

##  4 Evaluation Metrics

To evaluate our generated summaries, we use the following set of metrics to
evaluate their effectiveness. Given a set of summaries that consists of tens
of thousands of summaries generated by our pipeline, evaluating them all
manually would be labor-intensive if not impossible. Therefore we will use
some metrics where each of these metrics can automatically evaluate some
aspects of the summaries.

###  4.1 Sentiment Accuracy Metric

We trained a sentiment prediction model using BERT Devlin et al. (2018) as a
preprocessing layer and as an encoding layer, we attached an output dense
layer of size (None,6) that predicts the following: given a text that contains
either review or summary, output the sentiment ranked from 0 to 5, which
corresponds to the sentiment scale, for product reviews, the scale corresponds
to ratings that user gave when they submitted their review text. We make the
weights of both BERT layers trainable so that they can capture review-related
linguistic knowledge during training. In total the trainable parameters of our
model is 81,135,366 parameters. We trained the model on a subset of Amazon
Reviews Dataset’s Electronic Category from the work of McAuley et al. (2015)
that consists of 300,000 reviews for 20 epochs using an initial learning rate
of $5\times 10^{-5}$, batch size of 128 and ADAMW Loshchilov and Hutter (2019)
as optimizer. The results of our trained model are shown in table 1

Table 1:  Sentiment model result on test set | Categorical Accuracy | Categorical Loss
---|---|---
Sentiment Prediction Model | 0.746 | 0.674

To evaluate our approach, we score our model using the difference between
predicted sentiment and an average rating of the reviews being summarized
using equation 7. We denote the score as $S_{sentiment}$.

${S_{sentiment}=1-\frac{1}{N}\sum_{i=1}^{N}\left[\log_{6}\left(\left|\frac{1}{k}\sum_{j=1}^{k}R(x_{j})-S(s_{i})\right|+1\right)\right]}$
(7)

Where $S(s_{i})$ is the sentiment score computed from the $i^{th}$ summary,
$R(x_{j})$ is the rating given $j^{th}$ review, $k$ is the total number of
reviews being used to generate summary and $N$ is the total number of
summaries, $log_{6}$ is a notation trick we use to make the score fall between
$(0,1)$.

###  4.2 ROUGE Scores

Not only that we want to make sure the sentiment in our summary and the
average sentiment in the reviews that the summary is generated from is
similar. We also want to ensure that our summary contains similar information
that is found in the set of reviews that the summary comes from. For example,
if the reviews all mentioned screen size, we want to ensure that our summary
also captures some summarized information about screen size. For this purpose,
we use ROUGE-1 score to measure the overlap of unigram (each word) between the
reviews and summary, and ROUGE-2 score to measure the overlap of bigram (two
words) between the review and reference summary.

To evaluate our approach, we score our model using the unigram and bigram
difference between generated summary and the reviews being summarized using
equation 8. We denote the score as $S_{Rouge}$.

${S_{Rouge}=\frac{1}{N}\sum_{i=1}^{N}\left[\frac{1}{k}\sum_{j=1}^{k}ROUGE_{i}(s_{i},x_{j}))\right]}$
(8)

Where $k$ is the total number of reviews being used to generate summary, $N$
is the total number of summaries and $ROUGE_{i}(s_{i},x_{j})$ is the ROUGE-I
score generated between $j^{th}$ review and $i^{th}$ summary.

##  5 Experimental Setup

###  5.1 Dataset

We used SQuAD v1.1 (Rajpurkar et al. (2016)) and Amazon Reviews Data(2018) (Ni
et al. (2019)) in our work. We used SQuAD to fine tune BERT (Devlin et al.
(2018)) and DistilBERT (Sanh et al. (2019)) architectures. SQuAD dataset
contains more than 100,000 pairs of question-answers extracted from more than
500 Wikipedia articles. For the summarization task, we have used the Amazon
Reviews dataset. This dataset contains 233.1 million reviews collected in the
range of May 1996 to October 2018. These reviews belong to products from 29
categories. Due to lack of resources, we focused only on the Electronics
category. This category contains reviews of electronic gadgets and their
accessories. Again, we did narrow down our scope and focused on reviews
belonging to Computers, Tablets, and Laptops. In total, there were 131256
reviews in these categories.

###  5.2 Data Preparation

For fine-tuning BERT and DistilBERT, we have used the same SQuAD dataset as
provided by the authors without data pre-processing. However, when applied to
the Amazon Reviews dataset for the opinion extraction task, data pre-
processing was done.

As an extension to our analysis of MRCBert’s performance, we tested MRCBert
with and without our data pre-processing module. This is not including those
pre-processing steps of the pre-trained models. When conducted without pre-
processing, raw data from Amazon reviews were used as inputs to extract
opinions. When conducted with pre-processing, we removed stop words, symbols,
and numbers, and did lemmatization.

Although reviews have been provided as JSON objects, few reviews contain HTML
tags and overlapped values of two or more keys. These issues created problems
in parsing JSON objects. Therefore, we have removed such reviews in our work
and dropped duplicated reviews (i.e. exact match in reviews and ratings).
Finally, we have in total 4886 reviews from a single product.

###  5.3 Experimental Details

We executed our experiments on Tesla T4 16GB GPU machine with CUDA 11.2
installed. The BERT model for fine-tuning has 768 hidden sizes, 12 attention
heads, and 12 hidden layers. We fine-tuned BERT for 3 epochs, it took around
3.85 hours to fine-tune. It had in total 87714 samples and per second 1.581
samples were processed. Maximum CPU memory usage was 0.17GB and maximum GPU
memory usage was 9GB. We achieved an F1 score of 96.67 on the validation
dataset and an F1 score of 88.60 on the test dataset.

The DistilBERT model has hidden size 768, 12 attention heads, and 6 hidden
layers. It was fine-tuned for 3 epochs and it took 4.16 hours to fine-tune. It
had total 87714 samples and per second 1.464 samples were processed. Maximum
CPU memory usage was 0.17GB and maximum GPU memory usage was 4.57GB. We
achieved an F1 score of 85.67 on the test dataset.

One limitation of the Copycat is that it currently supports only review
summarization for up to 8 reviews. Hence, when applied to our dataset
comprising of 4886 reviews, Copycat outputs multiple summaries, each summary
corresponds to a group of up to 8 input reviews. As we are looking to have a
single summary either product-wise (1 summary), ratings-wise (5 summaries per
rating on the scale of 5), and aspects-wise (10 summaries for 10 aspects), we
need a methodology to post-process and condense the multiple summaries into a
single paragraph.

To do so, we split the multiple summaries into multiple sentences. Each
sentence will have its associated sentence embedding (Reimers and Gurevych
(2019)). All sentence embeddings then go through agglomerative clustering with
a distance threshold of 1.5 to keep the length of the post-processed summary
moderate. We then sample the maximum length sentence in each cluster and join
them to form a single summary. The maximum length sentence was chosen instead
of the minimum length, as we find that the minimum length sentence could at
times capture little or no information.

This post-processing is not needed when using the DistilBART-12-6-cnn
Summarizer which can take in variable-length inputs. Hence, we joined all
input sentences or extracted opinions to form a single document input to
DistilBART-12-6-cnn Summarizer and it outputs a single summary.

##  6 Results

###  6.1 Without additional pre-processing

This section presents the evaluation results when no additional data pre-
processing was conducted. Note that this is excluding the data pre-processing
modules in the pre-trained models that were used.

####  6.1.1 Rating-wise Reviews

We evaluated the rating-wise summaries with and without the opinion extraction
module to see how well the generated summaries reflect the input review
ratings through the use of sentiment accuracy scores. When the summarization
model used is Copycat, $S_{sentiment}$ is higher on average when given the
actual reviews (without opinion extraction) as compared to using extracted
opinions. However, when the summarization model used is DistilBART-12-6-cnn,
$S_{sentiment}$ on average is the same. Hence, there is no clear distinction
in Table 2 that shows whether sentiment scores are better with or without
opinion extraction.

A characteristic of rating-wise summaries that could have led to inconclusive
results in Table 2 may be the presence of mixed sentiments in the reviews. For
example, there is a review rated 2 out of 5 but also has some positive
experience. An excerpt of this review says: _"Though the iPad is a fun toy to
have around, the apps for it are pretty fun, and it has internet browsing
capabilities, it only has about 80% of the utility of using a computer."._ As
a result, the generated summary by DistilBART-12-6-cnn with opinions extracted
has captured some positive sentiment: _"The camera takes lousy, grainy photos.
(…) The iPad is a fun toy to have around."_.

On the other hand, the rating 2 generated summary by DistilBART-12-6-cnn
without opinion extraction captures only negative opinions. It says, _"The
iPad is extremely slow for a 32 gb (…) The software is outdated and the home
bottom is broken."_. As a result of mixed sentiments present in reviews, the
$S_{sentiment}$ scores for rating 2 summaries using DistilBART-12-6-cnn with
opinion generated is poorer compared to without. We hypothesize likely a
similar trend for other ratings.

Table 2: Sentiment accuracy scores for rating-wise reviews with and without opinion extraction, using Copycat and DistilBART-12-6-cnn summarization models | Copycat | DistilBART-12-6-cnn
---|---|---
Rating |  $S_{sentiment}$ without opinion extraction |  $S_{sentiment}$ with opinion extraction |  $S_{sentiment}$ without opinion extraction |  $S_{sentiment}$ with opinion extraction
1 |  1.000 |  1.000 |  1.000 |  1.000
2 |  0.226 |  0.226 |  1.000 |  0.613
3 |  0.613 |  0.387 |  0.387 |  0.387
4 |  1.000 |  1.000 |  0.613 |  0.613
5 |  1.000 |  0.613 |  0.613 |  1.000

####  6.1.2 Aspect-wise Reviews

Results in Figure 2 show that MRCBert captures sentiment better than Copycat
and DistilBART-12-6-cnn separately. The generated summaries also used relevant
words from the original reviews as seen from the high precision. While the
Recall is low, the low occurrence of 1-grams from the input reviews present in
the generated summaries is intuitive as the input reviews are usually lengthy
and may contain irrelevant information or incorrect use of words.

Figure 2: Sentiment accuracy and ROUGE-1 precision, recall and F1 scores for
aspect-wise summaries, comparing MRCBert against Copycat and
DistilBART-12-6-cnn. A1-A10 indicate the 10 aspects elaborated in Section 3.4.

###  6.2 Overall Results

In table 3, we show the sentiment score, ROUGE-1 and ROUGE-2 score for
Copycat, DistilBART-12-6-cnn and our model- MRCBert, these results are
obtained by averaging each model’s performance.

Table 3: Overall results with pre-processing |  | ROUGE-1 | ROUGE-2
---|---|---|---
| $S_{sentiment}$ | F1 | Precision | Recall | F1 | Precision | Recall
Copycat | 0.675 | 0.479 | 0.992 | 0.331 | 0.132 | 0.868 | 0.074
DistilBART-12-6-cnn | 0.687 | 0.526 | 0.990 | 0.378 | 0.148 | 0.717 | 0.087
MRCBert | 0.798 | 0.533 | 0.980 | 0.386 | 0.162 | 0.630 | 0.099

###  6.3 Qualitative Analysis

In this section, we present some generated summaries for qualitative analysis.
A summary about the display of an iPad after giving using MRCBert: _"Product
arrived on time and in good condition, works great, easy to use, good
graphics. Screen is a bit darker than I expected it to be but it works great.
Screen protector is not clear on the screen, but screen resolution is not as
clear as the screen resolution. Sim card is a good size, but not compatible
with sim card."_.

While the summary generated is linguistically coherent and talks about the
display of the iPad as designed, the last sentence about the sim card is not
logical. We have yet to find a better evaluation metric or methodology that
can capture logical fallacies of our generated summaries, and research that
can flag logical fallacies in language modeling will be beneficial to consider
as future work.

We also looked into rating-wise reviews to look for contrasting opinions and
as expected, the rating 5 summary captured positive sentiment, unlike the
rating 1 summary. An example rating 1 summary is as follows: _The screen is
black. Damaged without proper cable. It freezes all the time. It can’t even
open Safari. Not worth the purchase. I’m returning it.. out garageband app. It
was a fraud. Cannot figure out how to switch screens cannot operate it at
all._

Meanwhile, an example rating 5 summary: _Product is awesome, excellent seller,
great product. For his birthday and is very happy with its operation. iPad
looks new, works well and came in great condition to be used. Packed safely,
worked like new. Fast shipping! Great product - works perfect. Arrived about a
week early & I am VERY glad I ordered it._

##  7 Future Works

We mentioned earlier we would like to look into a summary generation that
enables both linguistic and logical coherence as future work. Besides that, as
we have demonstrated MRCBert for only electronics would like to extend it to
include a larger range of products with varying attributes. Currently, our
components are fragmented as we need to process BERT/DistilBERT and summary
generation modules separately. The vision next is to adopt end-to-end
modeling. Apart from these, the current work does not differentiate between
genuine and fake reviews and generates summaries from all the reviews
uniformly. Looking at generating summaries from only genuine reviews can be a
potential future research topic. One possible idea is to weigh summaries
according to some credibility score of reviewers. In our evaluation of the
summaries, we used ROUGE-1, ROUGE-2, sentiment analysis, and qualitative
analysis, however the set of metrics that we used is by no mean the most
complete set of metrics that evaluate summaries to full extent, as for future
work, a better set of metrics can be developed to evaluate summaries, this
will guide researchers in developing summarization models in future. Lastly,
our work thus far studies only mono-lingual reviews, but we can also look to
summarize multi-lingual reviews.

## References

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##  8 Appendix

Figure 3: ROUGE-2 scores for aspect-wise summaries, comparing MRCBert, Copycat, and DistilBART-12-6-cnn. Table 4: Question Variants Feature |  Variant 1 |  Variant 2
---|---|---
Display |  How is display? |  What is opinion on display?
Memory |  How is memory? |  What is opinion on memory?
Speaker |  How is speaker? |  What is opinion on speaker?
Sound |  How is sound? |  What is opinion on sound?
Processor |  How is processor? |  What is opinion on processor?
WiFi |  How is WiFi? |  What is opinion on WiFi?
Battery |  How is battery? |  What is opinion on battery?
Brand |  How is brand? |  What is opinion on brand?
Operating System |  How is operating system? |  What is opinion on operating system?
Camera |  How is camera? |  What is opinion on camera?
Table 5: Sample summaries generated from MRCBert using opinion extraction Feature |  Summary
---|---
Display |  Product arrived on time and in good condition. Gps unit works great, but it does not work very well on my ipad. Keyboard is slow, but not the best of all of the features I have. Best gps device I have ever owned. Great value for the money you pay for.
Wireless Connection |  Sound quality of this computer is excellent, but I wish it had. works great for the price I would recommend it to anyone who want to. product arrived on time and in good condition. Screen protector is a great value for the price .. however, I can’t. was not able to return the screen protector, but it did not work.
Battery |  This is an excellent product for the price I paid for it and the price. The display is nice, but not as good as the original one I have. The sound quality of this tablet is not bad, but not worth the money. This is a great little device to use with use to be able to clean up and clean up.
Operating System |  The sound quality of these speakers are great for the price you can’t go wrong. The ipad touch screen is great but the screen resolution is not clear enough to be <UNK>. The speakers on the ipad stand up to the touch screen but it can be not compatible with the iPad touch screen.
Processor |  This is a great tablet for the price you can’t go wrong it will be the best. The screen is a bit darker than I thought it would be but otherwise it. works well for what I need it to do, but have not been able to connect to my <UNK>page spread.
Brand |  The screen is a great touch screen protector but the screen doesn’t stay lit up. The ipad is not compatible with the ipad 2. The screen protector arrived on time and in good condition and the price was excellent. I would recommend this to anyone looking for a good price and a good deal of easy to use.
Memory |  I have used this product for several months now and it has been holding up well. The cable is a great value for the price .. I have not had any problems. The only thing I dislike about is not to be <UNK>is not the best ipad.
Sound |  The sensitivity of the sensitivity is not very what else can be said to say about a sensitivity machine. The product arrived on time and in good condition as advertised. The best tablet for the price you can’t go wrong <UNK>the best ipad I have ever owned.
Camera |  The screen protector is a great addition to the touch screen, but battery life is a bit long but it does not take up much space on. The product arrived in a timely manner and packaged well packed with no damage to. the product arrived on time and in good condition.. I can’t wait to use it on my ipad 2 or 3 times a month.
Quality of Speaker |  Reef is a great product for the price you can’t go wrong with the <UNK>. it does what it says it will do, however, I wish it had a little. good power cord for the price I would recommend it to anyone looking for a good case cover for the price. this is the best ipad I have ever owned.

Table 6: All results from Copycat, DistilBART-12-6-cnn and MRCBert using opinion extraction and without |  |  |  | ROUGE-1 | ROUGE-2
---|---|---|---|---|---
Type |  OE |  model |  S |  F1 |  P |  R |  F1 |  P |  R
all-reviews |  no |  Copycat |  0.707 |  0.718 |  0.801 |  0.698 |  0.409 |  0.564 |  0.398
rating1 |  no |  Copycat |  1.000 |  0.728 |  0.780 |  0.727 |  0.461 |  0.654 |  0.421
rating2 |  no |  Copycat |  0.226 |  0.000 |  0.000 |  0.000 |  0.000 |  0.000 |  0.000
rating3 |  no |  Copycat |  0.613 |  0.740 |  0.751 |  0.774 |  0.451 |  0.500 |  0.500
rating4 |  no |  Copycat |  1.000 |  0.687 |  0.805 |  0.646 |  0.416 |  0.621 |  0.391
rating5 |  no |  Copycat |  1.000 |  0.693 |  0.818 |  0.643 |  0.404 |  0.681 |  0.340
how_is_display |  yes |  Copycat |  0.716 |  0.409 |  0.872 |  0.282 |  0.098 |  0.722 |  0.054
how_is_wireless_connection |  yes |  Copycat |  0.685 |  0.430 |  0.863 |  0.303 |  0.103 |  0.701 |  0.058
how_is_battery |  yes |  Copycat |  0.869 |  0.399 |  0.869 |  0.273 |  0.090 |  0.724 |  0.050
how_is_operating_system |  yes |  Copycat |  0.693 |  0.406 |  0.875 |  0.278 |  0.101 |  0.707 |  0.056
how_is_processor |  yes |  Copycat |  0.698 |  0.373 |  0.862 |  0.250 |  0.082 |  0.700 |  0.045
how_is_brand |  yes |  Copycat |  0.754 |  0.426 |  0.917 |  0.290 |  0.106 |  0.768 |  0.059
how_is_memory |  yes |  Copycat |  0.700 |  0.440 |  0.880 |  0.310 |  0.115 |  0.735 |  0.065
how_is_sound |  yes |  Copycat |  0.711 |  0.372 |  0.867 |  0.250 |  0.087 |  0.632 |  0.048
how_is_camera |  yes |  Copycat |  0.839 |  0.385 |  0.874 |  0.260 |  0.100 |  0.716 |  0.056
how_is_quality_of_speaker |  yes |  Copycat |  0.819 |  0.456 |  0.905 |  0.320 |  0.112 |  0.761 |  0.063
rating1 |  yes |  Copycat |  1.000 |  0.433 |  0.871 |  0.303 |  0.096 |  0.761 |  0.053
rating2 |  yes |  Copycat |  0.226 |  0.426 |  0.905 |  0.292 |  0.102 |  0.778 |  0.056
rating3 |  yes |  Copycat |  0.387 |  0.422 |  0.903 |  0.290 |  0.098 |  0.799 |  0.054
rating4 |  yes |  Copycat |  1.000 |  0.405 |  0.893 |  0.274 |  0.101 |  0.772 |  0.056
rating5 |  yes |  Copycat |  0.613 |  0.420 |  0.883 |  0.289 |  0.103 |  0.773 |  0.057
all-reviews |  no |  DistilBART-12-6-cnn |  0.707 |  0.665 |  0.886 |  0.565 |  0.381 |  0.580 |  0.343
rating1 |  no |  DistilBART-12-6-cnn |  1.000 |  0.782 |  0.823 |  0.795 |  0.440 |  0.571 |  0.429
rating2 |  no |  DistilBART-12-6-cnn |  1.000 |  0.745 |  0.804 |  0.740 |  0.453 |  0.523 |  0.495
rating3 |  no |  DistilBART-12-6-cnn |  0.387 |  0.710 |  0.849 |  0.639 |  0.455 |  0.546 |  0.464
rating4 |  no |  DistilBART-12-6-cnn |  0.613 |  0.690 |  0.879 |  0.610 |  0.402 |  0.661 |  0.354
rating5 |  no |  DistilBART-12-6-cnn |  0.613 |  0.719 |  0.872 |  0.653 |  0.395 |  0.631 |  0.344
how_is_display |  yes |  DistilBART-12-6-cnn |  0.181 |  0.445 |  0.924 |  0.311 |  0.130 |  0.645 |  0.077
how_is_wireless_connection |  yes |  DistilBART-12-6-cnn |  0.685 |  0.420 |  0.943 |  0.286 |  0.111 |  0.673 |  0.063
how_is_battery |  yes |  DistilBART-12-6-cnn |  0.869 |  0.407 |  0.909 |  0.277 |  0.103 |  0.576 |  0.060
how_is_operating_system |  yes |  DistilBART-12-6-cnn |  0.693 |  0.430 |  0.925 |  0.296 |  0.119 |  0.627 |  0.069
how_is_processor |  yes |  DistilBART-12-6-cnn |  0.861 |  0.416 |  0.919 |  0.285 |  0.106 |  0.595 |  0.061
how_is_brand |  yes |  DistilBART-12-6-cnn |  0.754 |  0.439 |  0.931 |  0.301 |  0.136 |  0.579 |  0.081
how_is_memory |  yes |  DistilBART-12-6-cnn |  0.700 |  0.417 |  0.942 |  0.282 |  0.121 |  0.706 |  0.070
how_is_sound |  yes |  DistilBART-12-6-cnn |  0.183 |  0.363 |  0.941 |  0.237 |  0.095 |  0.595 |  0.055
how_is_camera |  yes |  DistilBART-12-6-cnn |  0.181 |  0.440 |  0.939 |  0.305 |  0.121 |  0.623 |  0.071
how_is_quality_of_speaker |  yes |  DistilBART-12-6-cnn |  0.819 |  0.402 |  0.947 |  0.266 |  0.117 |  0.696 |  0.067
rating1 |  yes |  DistilBART-12-6-cnn |  1.000 |  0.459 |  0.872 |  0.329 |  0.129 |  0.516 |  0.077
rating2 |  yes |  DistilBART-12-6-cnn |  0.613 |  0.479 |  0.920 |  0.342 |  0.135 |  0.606 |  0.080
rating3 |  yes |  DistilBART-12-6-cnn |  0.387 |  0.391 |  0.950 |  0.258 |  0.112 |  0.680 |  0.064
rating4 |  yes |  DistilBART-12-6-cnn |  0.613 |  0.436 |  0.908 |  0.301 |  0.124 |  0.637 |  0.072
rating5 |  yes |  DistilBART-12-6-cnn |  1.000 |  0.430 |  0.936 |  0.293 |  0.128 |  0.686 |  0.074
how_is_display |  yes |  MRCBert |  0.839 |  0.471 |  0.890 |  0.342 |  0.128 |  0.496 |  0.078
how_is_wireless_connection |  yes |  MRCBert |  0.685 |  0.460 |  0.896 |  0.330 |  0.125 |  0.497 |  0.076
how_is_battery |  yes |  MRCBert |  0.869 |  0.393 |  0.875 |  0.267 |  0.109 |  0.569 |  0.064
how_is_operating_system |  yes |  MRCBert |  0.693 |  0.429 |  0.834 |  0.306 |  0.132 |  0.534 |  0.080
how_is_processor |  yes |  MRCBert |  0.861 |  0.446 |  0.829 |  0.325 |  0.106 |  0.525 |  0.063
how_is_brand |  yes |  MRCBert |  0.794 |  0.415 |  0.930 |  0.279 |  0.124 |  0.550 |  0.073
how_is_memory |  yes |  MRCBert |  0.859 |  0.448 |  0.877 |  0.318 |  0.134 |  0.467 |  0.082
how_is_sound |  yes |  MRCBert |  0.845 |  0.456 |  0.845 |  0.334 |  0.117 |  0.481 |  0.072
how_is_camera |  yes |  MRCBert |  0.839 |  0.431 |  0.861 |  0.305 |  0.112 |  0.496 |  0.066
how_is_quality_of_speaker |  yes |  MRCBert |  0.819 |  0.485 |  0.900 |  0.350 |  0.139 |  0.502 |  0.085


#  The $\alpha$-induction of Superconformal Nets

ZIYUN XU
Graduate School of Mathematical Sciences
The University of Tokyo, Komaba, Tokyo, 153-8914, Japan
e-mail<EMAIL_ADDRESS>
###### Abstract

The successful application of $\alpha$-induction to the conformal field theory
has given rise to many amazing results. For example, the classification of
conformal nets in the discrete series. In this paper, however, we study
$\alpha$-induction of graded local conformal nets. The main difference lies
between locality in the conformal net cases and graded locality in the graded
local conformal net cases. We show that the inclusions of graded local
conformal nets give braided subfactors. We check that the $\alpha$-induction
is still valid in terms of graded local conformal nets. Moreover, the simple
current extension originally defined in terms of conformal nets still works
for graded local conformal nets. Consequently, compared to [chklx], we give a
shorter proof of classification of $N=2$ superconformal nets in the discrete
series according to Gannon’s list [gannonlist].

##  1 Introduction

The subfactor theory initiated by Jones [jones] has opened many surprising
connections of operator algebras to various branches of mathematics and
physics, such as $3$-dimensional topology, quantum groups and statistical
mechanics. The conformal net [fg], [wass] is a central aspect of subfactor
theory concerning two dimensional conformal field theory. Methods of subfactor
theory have turned out to be powerful in the operator algebraic approach to
conformal field theory. $\alpha$-induction was first introduced in [lr] and
studied in [xu], with a more general setting studied in [be1], [be2], [be3].
In [bek], $\alpha$-induction was defined and studied in a fully general
setting. By applying $\alpha$-induction to certain subfactors [kl], they gave
a classification of conformal nets with central charge less than one and two
dimensional full conformal nets [kl2].
A superconformal net is an operator algebraic approach to two dimensional
superconformal field theory. The $N=1$ superconformal nets were first
introduced and studied in [ckl] and $N=2$ superconformal nets were introduced
and studied in [chklx]. In [chklx] they studied the $\alpha$-induction of the
bose subnet of a given $N=2$ superconformal net. By analyzing fermionization
of the inclusion of bose subnets, they gave a classification of $N=2$
superconformal nets in the discrete series.
In this paper, we directly study the $\alpha$-induction of a given graded
local conformal net and give a shorter classification of $N=2$ superconformal
nets in the discrete series. We show that the general idea of [chklx] is still
valid for the $\alpha$-induction of graded local conformal nets and their
classification process can be simplified. More precisely, the general idea of
classifying $N=2$ superconformal nets in the discrete series is similar to
classifying conformal nets in the discrete series. Given an $N=2$
superconformal net $\mathcal{A}$ in the discrete series, we have a net of
subfactors $\mathcal{A}_{c}(I)\subset\mathcal{A}(I)$, where $\mathcal{A}_{c}$
is the $N=2$ super-Virasoro net. From one of those subfactors, we can get a
modular invariant (i.e., a matrix $Z_{\lambda,\mu}$) by $\alpha$-induction. On
the other hand, given a modular invariant, we can construct the dual canonical
endomorphism as $\theta=\bigoplus Z_{0,\lambda}\lambda$. Then we get a
subfactor associated with the dual canonical endomorphism. We show that the
map $\mathcal{A}\to Z$ is a bijection.

##  2 Preliminaries

###  2.1 Braided subfactors and $\alpha$-induction

Let $N,M$ be infinite factors. We denote by Mor$(N,M)$ the set of unital
$*$-homomorphisms from $N$ to $M$. The statistical dimension of
$\lambda\in$Mor$(N,M)$ is defined as
$d_{\lambda}\coloneqq[M:\lambda(N)]^{\frac{1}{2}}$, where $[M:\lambda(N)]$ is
the minimal index [jones], [kosaki]. A morphism $\lambda\in$Mor$(N,M)$ is
called irreducible if $\lambda(N)\subset M$ is irreducible, i.e.
$\lambda(N)^{\prime}\cap M=\mathbb{C}$. Two morphisms
$\lambda,\lambda^{\prime}\in$Mor$(N,M)$ are called unitary equivalent if there
is a unitary $u\in M$ such that $\lambda^{\prime}=u^{*}\lambda u$. The unitary
equivalence class $[\lambda]$ of a morphism $\lambda\in$Mor$(N,M)$ is called
an $M$-$N$ sector. For sectors, we have a notion of direct sums, products and
conjugates(cf. and the references therein for more details). For
$\lambda,\mu\in$Mor$(N,M)$ we denote Hom$(\lambda,\mu)=\\{m\in
M:m\lambda(n)=\mu(n)m,n\in N\\}$
and$\langle\lambda,\mu\rangle=\operatorname{dim}\operatorname{Hom}(\lambda,\mu)$.
Let $N$ be a type III factor equipped with a system $\Delta\subset$Mor$(N,N)$
of endomorphisms in the sense of ([bek], definition 2.1). This means morphisms
in $\Delta$ are irreducible and have finite statistical dimensions and they
are different as sectors. In addition, they form a closed fusion algebra. Then
$\Sigma(\Delta)\subset$Mor$(N,N)$ denotes the set of morphisms which decompose
as sectors into finite sums of elements in $\Delta$. We assume that $\Delta$
is braided in the sense of [bek] and we extend the braiding to
$\Sigma(\Delta)$. We then consider a braided subfactor $N\subset M$, i.e. they
are both type III factors and that the dual canonical endomorphism sector
$[\theta]$ decomposes in a finite sum of sectors of morphisms in $\Delta$,
i.e. $\theta\in\Sigma(\Delta)$. Here $\theta=\bar{\iota}\iota$ with
$\iota:N\to M$ being the injection homomorphism and $\bar{\iota}\in$Mor$(M,N)$
being a conjuagte morphism. Note that this forces the statistical dimension of
$\theta$ and thus the index of the subfactor $N\subset M$ to be finite. Then
we can define $\alpha$-induction by

$\alpha^{\pm}_{\lambda}=\bar{\iota}^{-1}\circ\operatorname{Ad}(\epsilon(\lambda,\theta)^{\pm})\circ\lambda\circ\bar{\iota}$

for $\lambda\in\Sigma(\Delta)$. Then $\alpha^{\pm}_{\lambda}\in$Mor$(M,M)$ and
satisfy $\alpha^{\pm}_{\lambda}|_{N}=\lambda$. The ambichiral system
${}_{M}\mathcal{X}^{0}_{M}\subset$Mor$(M,M)$ is the subset corresponding to
subsectors of $\alpha^{+}_{\lambda}$ and $\alpha^{-}_{\mu}$ when $\lambda$ and
$\mu$ vary in $\Delta$. By theorem 4.2 in [bek2], there is a non-degenerating
braiding on the ambichiral structure ${}_{M}\mathcal{X}^{0}_{M}$ satisfying
braided fusion equations in the sense of corollary $2.8$ in [be1].

###  2.2 $Q$-systems and classification

Let $M$ be an infinite factor. A $Q$-system [longo] is a triple of an
endomorphism of $M$ and isometries
$v\in\operatorname{Hom}(\operatorname{id},\gamma),w\in\operatorname{Hom}(\gamma,\gamma^{2})$
satisfying the following identities:

$\displaystyle v^{*}w=\gamma(v^{*})w\in\mathbb{R}_{>0},\gamma(w)w=w^{2}.$

If $N\subset M$ is a finite index subfactor, the associated canonical
endomorphism gives rise to a $Q$-system. Conversely, any $Q$-system determines
an extension $N\subset M$ such that $\gamma$ is the canonical endomorphism:
$N$ is given by $N=\\{x\in M:wx=\gamma(x)w\\}$.

##  3 Graded Local Conformal Nets

The concept of graded local conformal nets is a generalization of the notion
of local conformal nets and has been explicitly introduced and studied in
[ckl].
Let $S^{1}=\\{z\in\mathbb{C}:|z|=1\\}$ be the unit circle. Let
$\operatorname{Diff}(S^{1})$ be the infinite dimensional real Lie group of
orientation-preserving smooth diffeomorphisms of $S^{1}$ and denote by
$\operatorname{Diff}(S^{1})^{(n)},n\in\mathbb{N}\cup\\{\infty\\}$ the
corresponding $n$-cover. In particular,
$\operatorname{Diff}(S^{1})^{(\infty)}$ is the universal covering group of
$\operatorname{Diff}(S^{1})$. By identifying
$\operatorname{PSL}(2,\mathbb{R})$ with the group of Möbius transformations on
$S^{1}$ we can consider it as a three dimensional Lie subgroup of
$\operatorname{Diff}(S^{1})$. We denote by
$\operatorname{PSL}(2,\mathbb{R})^{(n)}\subset\operatorname{Diff}(S^{1})^{(n)},n\in\mathbb{N}\cup\\{\infty\\}$
the corresponding $n$-cover so that
$\operatorname{PSL}(2,\mathbb{R})^{(\infty)}$ is the universal covering group
of $\operatorname{PSL}(2,\mathbb{R})$. We denote by
$\dot{g}\in\operatorname{Diff}(S^{1})$ the image of
$g\in\operatorname{Diff}(S^{1})^{(\infty)}$ under the covering map.
Let $\mathcal{I}$ denote the set of all connected non-dense open intervals in
$S^{1}$. For $I\in\mathcal{I}$, $I^{\prime}$ denotes the interior of
$S^{1}\backslash I$. Given $I\in\mathcal{I}$, the subgroup
$\operatorname{Diff}(S^{1})_{I}$of diffeomorphisms localized in $I$ is defined
as the stabilizer of $I$ in $\operatorname{Diff}(S^{1})$, namely the subgroup
of $\operatorname{Diff}(S^{1})$ whose elements are the diffeomorphisms acting
trivially on $I^{\prime}$.

######  Definition 1.

A graded local conformal net $\mathcal{A}$ on $S^{1}$ is a map
$I\mapsto\mathcal{A}(I)$ from the set of intervals(connected, non dense)
$\mathcal{I}$ to the set of von Neumann algebras acting on a common infinite
dimensional separable Hilbert space $\mathcal{H}$ which satisfies the
following properties:

  1. 1.

Isotony. $\mathcal{A}(I)\subset\mathcal{A}(J)$ if $I,J\in\mathcal{I}$ and
$I\subset J$.

  2. 2.

Möbius covariance. There is a strongly continuous unitary representation $U$
of $\operatorname{PSL}(2,\mathbb{R})^{(\infty)}$ such that

$U_{g}\mathcal{A}(I)U^{*}_{g}=\mathcal{A}(\dot{g}I),g\in\operatorname{PSL}(2,\mathbb{R})^{(\infty)},I\in\mathcal{I}.$

  3. 3.

Positive energy. The conformal Hamiltonian $L_{0}$ is positive.

  4. 4.

Existence and uniqueness of the vacuum. There exists a $U$-invariant vector
$\Omega\in\mathcal{H}$ which is unique up to a phase and cyclic for
$\bigvee_{I\in\mathcal{I}}\mathcal{A}(I)$, the von Neumann algebra generated
by the algebras $\mathcal{A}(I),I\in\mathcal{I}$.

  5. 5.

Graded locality. There exists a self adjoint unitary(the grading unitary)
$\Gamma$ on $\mathcal{H}$ satisfying
$\Gamma\mathcal{A}(I)\Gamma=\mathcal{A}(I)$ for all $I\in\mathcal{I}$ and
$\Gamma\Omega=\Omega$ and such that $\mathcal{A}(I^{\prime})\subset
Z\mathcal{A}(I)^{\prime}Z^{*},\ I\in\mathcal{I}$, where
$Z\coloneqq\frac{1-i\Gamma}{1-\Gamma}$.

  6. 6.

Diffeomorphism covariance. There is a strongly continuous projective unitary
representation of $\operatorname{Diff}(S^{1})^{(\infty)}$, denoted again by
$U$, extending the unitary representation of
$\operatorname{PSL}(2,\mathbb{R})^{(\infty)}$ and such that

$U_{g}\mathcal{A}(I)U^{*}_{g}=\mathcal{A}(\dot{g}I),\
g\in\operatorname{Diff}(S^{1})^{(\infty)},\ I\in\mathcal{I},$

and such that

$U_{g}xU^{*}_{g}=x,x\in\mathcal{A}(I^{\prime}),\
g\in\operatorname{Diff}(S^{1})^{(\infty)}_{I},\ I\in\mathcal{I}.$

A local conformal net is a graded local conformal net with trivial gauge
unitary $\Gamma=1$. The bose subnet of a graded local conformal net
$\mathcal{A}$ is defined as the fixed point subnet $\mathcal{A}^{\gamma}$ with
$\gamma=\operatorname{Ad}\Gamma$. The restriction of $\mathcal{A}^{\gamma}$ to
the bose subspace of $\mathcal{H}$ is a local conformal net with respect to
the restriction to this subspace of the projective representation $U$ of
$\operatorname{Diff}(S^{1})^{(\infty)}$.

Some consequences [ckl],[fg], [gl], [cw] of the preceding definition are:

  1. 1.

Reeh-Schlieder Property. $\Omega$ is cyclic and separating for every
$\mathcal{A}(I),I\in\mathcal{I}$.

  2. 2.

Bisognano-Wichmann Property. Let $I\in\mathcal{I}$ and let $\Delta_{I},J_{I}$
be the modular operator and the modular conjugation of
$(\mathcal{A}(I),\Omega)$. Then we have

$U(\delta_{I}(-2\pi t))=\Delta_{I}^{it},t\in\mathbb{R}.$

Moreover, the unitary representation
$U:\operatorname{PSL}(2,\mathbb{R})^{(\infty)}\to B(\mathcal{H})$ extends to
an (anti-)unitary representation of
$\operatorname{PSL}(2,\mathbb{R})\rtimes\mathbb{Z}/2$ determined by

$U(r_{I})=ZJ_{I},$

acting covariantly on $\mathcal{A}$. Here $(\delta_{I}(t))_{t\in\mathbb{R}}$
is the one-parameter dilation subgroup of $\operatorname{PSL}(2,\mathbb{R})$
with respect to $I$ and $r_{I}$ the reflection of the interval onto the
complement $I^{\prime}$.

  3. 3.

Graded Haag Duality. $\mathcal{A}(I^{\prime})=Z\mathcal{A}(I)^{\prime}Z^{*}$,
for $I\in\mathcal{I}$.

  4. 4.

Outer regularity.
$\mathcal{A}(I_{0})=\bigcap_{I\in\mathcal{I},I\supset\bar{I_{0}}}\mathcal{A}(I),I_{0}\in\mathcal{I}$.

  5. 5.

Additivity. If $I=\bigcup_{\alpha}I_{\alpha}$ with
$I,I_{\alpha}\in\mathcal{I}$, then
$\mathcal{A}(I)=\bigvee_{\alpha}\mathcal{A}(I_{\alpha})$.

  6. 6.

Factoriality. $\mathcal{A}(I)$ is a type III1-factor for $I\in\mathcal{I}$.

  7. 7.

Irreducibility. $\bigvee_{I\in\mathcal{I}}\mathcal{A}(I)=B(\mathcal{H})$.

  8. 8.

Vacuum Spin-Statistics theorem. $e^{i2\pi L_{0}}=\Gamma$.

  9. 9.

Uniqueness of Covariance. For fixed $\Omega$, the strongly continuous
projective representation $U$ of $\operatorname{Diff}(S^{1})^{(\infty)}$
making the net covariant is unique.

##  4 Representations of Graded Local Conformal Nets

In this section, we assume that $\mathcal{A}$ is a graded local conformal net,
although certain notions and results are valid in the Möbius covariance case.

######  Definition 2.

A Doplicher-Haag-Roberts (DHR) representation $\lambda$ of $\mathcal{A}$ is a
map $I\to\lambda_{I}$ that associates to an interval $I\in\mathcal{I}$ a
normal representation $\lambda_{I}$ of $\mathcal{A}(I)$ on a fixed Hilbert
space $\mathcal{H}_{\lambda}$ such that

$\lambda_{\tilde{I}}|_{\mathcal{A}(I)}=\lambda_{I},\ I\subset\tilde{I},$

for $\tilde{I}\in\mathcal{I}$.

We say that a representation $\lambda$ on $\mathcal{H}_{\lambda}$ is
diffeomorphism covariant if there exists a projective unitary representation
$U_{\lambda}$ of $\operatorname{Diff}(S^{1})^{\infty}$ on
$\mathcal{H}_{\lambda}$ such that

$\lambda_{\dot{g}I}(U_{g}xU^{*}_{g})=U_{\lambda}(g)\lambda_{I}(x)U^{*}_{\lambda}(g),\
x\in\mathcal{A}(I),\ g\in\operatorname{Diff}(S^{1})^{\infty}.$

Here $\dot{g}$ denotes the image of $g$ in $\operatorname{Diff}(S^{1})$. A
Möbius covariant representation is analogously defined.

######  Definition 3.

A representation $\lambda$ on $\mathcal{H}_{\lambda}$ is graded if there
exists a self adjoint unitary $\Gamma_{\lambda}$ on $\mathcal{H}_{\lambda}$
such that

$\lambda_{I}(\Gamma x\Gamma)=\Gamma_{\lambda}\lambda_{I}(x)\Gamma_{\lambda},\
x\in\mathcal{A}(I),$

for $I\in\mathcal{I}$.

We shall say a representation $\lambda$ on $\mathcal{H}_{\lambda}$ is graded
if there exists a selfadjoint unitary $\Gamma_{\lambda}$ on
$\mathcal{H}_{\lambda}$ such that

$\lambda_{I}(\Gamma x\Gamma)=\Gamma_{\lambda}\lambda_{I}(x)\Gamma_{\lambda},\
x\in\mathcal{A}(I),$

for $I\in\mathcal{I}$. The following results are due to proposition $12$ in
[ckl], :

######  Proposition 1.

Let $\lambda$ be an irreducible representation of $\mathcal{A}$. Then
$\lambda$ is graded and diffeomorphism covariant.

######  Proposition 2.

Let $\pi$ be a representation of the graded local conformal net, and suppose
the Hilbert spaces $\mathcal{H}$ and $\mathcal{H}_{\pi}$ to be separable.
Given an interval $I\in\mathcal{I}$, there exists a localized endomorphism
$\rho$ of $\mathcal{A}$ unitarily equivalent to $\pi$,
$\pi=\operatorname{Ad}U\circ\rho$. Moreover, the unitary $U$ satisfies
$U^{*}\Gamma_{\pi}=\Gamma U^{*}$.

Fix an interval $I\in\mathcal{I}$. Denote by
$\operatorname{Rep}^{I}(\mathcal{A})$ the category of graded endomorphisms of
$\mathcal{A}$ localized in $I$, whose object set $\Delta_{I}(\mathcal{A})$
consists of endomorphisms of $\mathcal{A}$ localized in $I$ and morphisms
between objects are intertwiners, i.e.
$\operatorname{Hom}_{\mathcal{A}(I)}(\lambda,\mu)=\\{x\in
Z\mathcal{A}(I)Z^{*}:x\lambda(a)=\mu(a)x,a\in\mathcal{A}(I)\\}$. Denote by
$\langle\lambda,\mu\rangle_{\mathcal{A}(I)}$ the dimension of
$\operatorname{Hom}_{\mathcal{A}(I)}(\lambda,\mu)$. Denote by
$\operatorname{DHR}(\mathcal{A})$ the category of DHR representations of
$\mathcal{A}$, whose objects are DHR representations and morphisms are
intertwiners between representations. By the above proposition, we get the
equivalence
$\operatorname{DHR}(\mathcal{A})\cong\operatorname{Rep}^{I}(\mathcal{A})$.

######  Definition 4.

A conformal net $\\{M(I)\\}_{I\in\mathcal{I}}$ is completely rational if its
representation category $\operatorname{Rep}^{I}(M)$ is a modular tensor
category.

######  Proposition 3.

Let $\mathcal{A}$ be a graded local conformal net whose bose subnet
$\mathcal{A}_{b}$ is completely rational. Then
$\operatorname{Rep}^{I}(\mathcal{A})$ is a braided tensor category.

###### Proof.

Consider the subfactor $\mathcal{A}_{b}(I)\subset\mathcal{A}(I)$, where
$\mathcal{A}_{b}$ is the Bose subnet of $\mathcal{A}$. For any
$\lambda\in\Delta_{I}(\mathcal{A})$, its restriction
$\lambda_{b}\coloneqq\lambda|_{\mathcal{A}_{b}}$ gives a DHR representation of
the Bose subnet. By proposition 26 in [ckl], we have
$\lambda=\alpha^{+}_{\lambda_{b}}=\alpha^{-}_{\lambda_{b}}$, so $\lambda$ is
ambichiral in the sense of [bek2], section $2$. By [bek2] section $2$, there
is a unitary braiding structure satisfying braided fusion equations on the
ambichiral system which gives the braiding structure on
$\operatorname{Rep}^{I}(\mathcal{A})$. ∎

######  Remark 1.

The braided tensor category structure on vertex operator (super)algebras is
known and studied in section $2$ of [ckm]. In our case, the structure
morphisms of the $Q$-system are even morphisms satisfying identities in
definition $2.25$ of [ckm]. However, in this paper, we mainly focus on the
graded locality and its influence on counting multiplicities of the dual
canonical endomorphism hence we do not study the supercateogry structure in
detail.

######  Remark 2.

In [cgh], to find a graded local extension $\mathcal{A}\subset\mathcal{B}$ for
a given graded local conformal net $\mathcal{A}$, they first fix a $Q$-system
$\Theta_{b}$ in $\operatorname{Rep}(\mathcal{A}_{b})$. $\Theta_{b}$ determines
the inclusion $\mathcal{A}_{b}\subset\mathcal{B}_{b}$. Then they search for
another $Q$-system $\Theta_{f}$ which is a $\Theta_{b}$-module in
$\operatorname{Rep}(\mathcal{A}_{b})$ with twist $-1$. $\Theta_{f}$
corresponds to the extension $\mathcal{B}_{b}\subset\mathcal{B}$. The expected
extension $\mathcal{A}\subset\mathcal{B}$ is given by the $Q$-system
$\Theta_{b}\oplus\Theta_{f}$, which is the graded conformal net analogous to
the superalgebra object in the sense of definition $2.25$ of [ckm]. In
addition, as in our case, we directly carry out $\alpha$-induction on
$\mathcal{A}_{b}\subset\mathcal{B}$. Therefore, our approach and [cgh] are two
different points of view of the same $Q$-system.

##  5 Simple Current Extensions of Graded Local Conformal Nets

In this section, we study simple current extensions of graded local conformal
nets. Let $(\mathcal{A},\mathcal{H}_{0},\Omega_{0},\Gamma_{0})$ be a graded
local net on $\mathcal{H}_{0}$. Assume $\sigma$ is an order $n$ automorphism
in DHR($\mathcal{A}$). We also extend $\sigma(\Gamma_{0})=\Gamma_{0}$. Define
a representation $\pi:\mathcal{A}\to B(\mathcal{H})$ by:

$\pi(a)\xi\coloneqq\oplus_{p=0}^{n-1}\sigma^{p}(a)\xi_{p}\ \text{and}\
\pi(\Gamma_{0})\xi\coloneqq\oplus_{p=0}^{n-1}\Gamma_{0}\xi_{p},$ (1)

where $\mathcal{H}=\mathcal{H}_{0}^{\oplus n}$ and
$\xi=\oplus_{p=0}^{n-1}\xi_{p}$. Denote
$\mathcal{B}(I)\coloneqq\pi(\mathcal{A}(I))$,
$\Omega_{p}\coloneqq\delta_{p,0}\Omega_{0}$,
$\Gamma\coloneqq\oplus_{p=0}^{n-1}\Gamma_{0}$,
$Z\coloneqq\frac{1-i\Gamma}{1-i}$. Then
$\Gamma\pi(x)\Gamma=\pi(\Gamma_{0}x\Gamma_{0})$. Assume
$\sigma\in\Delta(I_{0})$. For any interval $I$, we can take unitary $u_{I}$
such that $[u_{I},\Gamma_{0}]=0$ and $\text{Ad}(u_{I})\circ\sigma$ localized
in $I$. Define $f_{u_{I}}\in B(\mathcal{H})$, by

$(f_{u_{I}}\xi)_{p}\coloneqq\sigma^{p-1}(u^{*}_{I})\xi_{p-1},\
p\in\mathbb{Z}_{n}.$ (2)

It is easy to check

$(f_{u_{I}}^{*}\xi)_{p}=\sigma^{p}(u_{I})\xi_{p+1}.$

Then, we have $f_{u_{I}}^{*}\pi(a)f_{u_{I}}=\pi\circ\sigma_{I}(a)$ for any
$a\in\mathcal{A}$. Note that $f_{u_{I}}\in\mathcal{B}(I^{\prime})^{\prime}$
since $\mathcal{B}(I)$ may not be Haag dual. Now we define the extended net
$\mathcal{C}$ on $\mathcal{H}$ as

$\mathcal{C}(I)\coloneqq\langle\mathcal{B}(I),Z^{*}f_{u_{I}}Z\rangle^{\prime\prime}.$

Note that $f_{u_{I}}=f_{\operatorname{id}}\pi(u_{I}^{*})$, where
$f_{\operatorname{id}}$ is defined by replacing $u_{I}$ with identity in (2).
The definition of the extended net is independent of the choice of
$f_{u_{I}}$. Indeed, if $\text{Ad}(\hat{u_{I}})\circ\sigma\in\Delta(I)$, for
another $\hat{u_{I}}$, we have
$u_{I}\hat{u_{I}}^{*}\in\mathcal{A}(I^{\prime})^{\prime}=Z_{0}\mathcal{A}(I)Z_{0}^{*}$.
Then $Z^{*}f_{u_{I}}Z$ and $Z^{*}f_{\hat{u_{I}}}Z$ only differ by an element
of $\mathcal{B}(I)$ because $\pi(Z_{0}aZ_{0}^{*})=Z\pi(A)Z^{*}$. In addition,
we have:

$\Gamma f_{u_{I}}\Gamma=\Gamma
f_{\operatorname{id}}\pi(u_{I}^{*})\Gamma=f_{\operatorname{id}}\pi(\Gamma_{0}u_{I}^{*}\Gamma_{0})=f_{u_{I}}.$
(3)

where we use the fact that
$[\Gamma,f_{\operatorname{id}}]=[\Gamma_{0},u_{I}]=0$. Therefore
$[Z,f_{u_{I}}]=0$ and we can rewrite
$\mathcal{C}(I)=\langle\mathcal{B}(I),f_{u_{I}}\rangle^{\prime\prime}$.

######  Lemma 1.

In the above setting, we have,
$f_{\operatorname{id}}\pi(u_{I}^{*})=\pi(\sigma_{0}^{-1}(u_{I}))f_{\operatorname{id}}$.

###### Proof.

Take $\xi\in\mathcal{H}$. Then
$(f_{\operatorname{id}}\pi(u_{I}^{*}))\xi)_{p}=(\pi(u_{I}^{*}\xi)_{p-1}=\sigma^{p-1}(u_{I})\xi_{p-1}$.
On the other hand,
$(\pi(\sigma_{0}^{-1}(u_{I}))f_{\operatorname{id}}\xi)_{p}=\sigma^{p-1}(u_{I})(f_{\operatorname{id}}\xi)_{p}=\sigma^{p-1}(u_{I})\xi_{p-1}$.
∎

######  Lemma 2 ([be2] lemma $3.6$).

In the above setting, we have,
$f_{u_{I_{2}}}f_{u_{I_{1}}}=\epsilon^{\pm}(\sigma,\sigma)f_{u_{I_{1}}}f_{u_{I_{2}}}$,
if $I_{1}\cap I_{2}=\emptyset$.

###### Proof.

First notice that $f_{u_{I}}=f_{\operatorname{id}}\pi(u_{I}^{*})$ for
$I\in\mathcal{I}$.

$\displaystyle f_{u_{I_{2}}}f_{u_{I_{1}}}$
$\displaystyle=f_{\operatorname{id}}\pi(u_{I_{2}}^{*})f_{\operatorname{id}}\pi(u_{I_{1}}^{*})$
$\displaystyle=\pi(\sigma^{-1}(u^{*}_{I_{2}})\sigma^{-2}(u^{*}_{I_{1}}))f^{2}_{\operatorname{id}}$
$\displaystyle=\pi(\sigma^{-1}(u^{*}_{I_{2}})\sigma^{-2}(u^{*}_{I_{1}})\sigma^{-2}(u_{I_{2}})\sigma^{-1}(u_{I_{1}}))f_{\operatorname{id}}\pi(u_{I_{1}})^{*}f_{\operatorname{id}}\pi(u_{I_{2}})^{*}$
$\displaystyle=\sigma^{-2}\circ\pi(\sigma(u^{*}_{I_{2}})u^{*}_{I_{1}}u_{I_{2}}\sigma(u_{I_{1}}))f_{u_{I_{1}}f_{u_{I_{2}}}}$
$\displaystyle=\sigma^{-2}(\epsilon^{\pm}(\sigma,\sigma))f_{u_{I_{1}}f_{u_{I_{2}}}}$

where we repeatedly use lemma 1. ∎

######  Proposition 4.

The extension net $\\{\mathcal{C}(I)\\}_{I\in\mathcal{I}}$ is graded local if
and only if $\epsilon(\sigma,\sigma)=1$.

###### Proof.

Since $\mathcal{A}(I)$ is relatively graded local in $\mathcal{B}(I)$, it only
suffices to check

$f_{u_{I_{2}}}f_{u_{I_{1}}}=f_{u_{I_{1}}}f_{u_{I_{2}}},$ (4)

which is guaranteed by the above lemma. ∎

To sum up the preceding argument, we have the following result:

######  Theorem 1.

The net $\\{\mathcal{C}(I)\\}_{I\in\mathcal{I}}$ is a graded local conformal
net.

##  6 From Inclusion of Nets to Modular Invariants

Let $\mathcal{A}\subset\mathcal{B}$ be an inclusion of graded local conformal
nets on $(\mathcal{H},\Omega)$ with the same gauge unitary $\Gamma$. Let
$\gamma:\mathcal{B}(I)\to\mathcal{A}(I)$ be the canonical endomorphism and
$\theta=\gamma|_{\mathcal{A}(I)}$ be the dual canonical endomorphism. Then
$\mathcal{B}(I)=\mathcal{A}(I)v$, where
$v\in\operatorname{Hom}_{\mathcal{B}(I)}(\operatorname{id},\gamma)$ and
$w\in\operatorname{Hom}_{\mathcal{A}(I)}(\operatorname{id},\theta)$ are
isometries such that $w$ induces the conditional expectation
$E(b)=w^{*}\gamma(b)w,b\in\mathcal{B}(I)$ and
$w^{*}v=|\mathcal{B}:\mathcal{A}|^{\frac{1}{2}}=w^{*}\gamma(v)$. Here,
$|\mathcal{B}:\mathcal{A}|$ is the index of the subfactor
$\mathcal{A}(I)\subset\mathcal{B}(I)$. The triple $(\theta,v,w)$ as above with
some compatible conditions defines a $Q$-system.

######  Proposition 5 ([be1] corollary $2.8$).

(Braided fusion equation)For $\lambda,\mu,\nu,\rho\in\Delta_{\mathcal{A}}(I)$,
and $f\in\text{Hom}_{\mathcal{A}(I)}(\lambda\circ\mu,\nu)$ we have:

$\displaystyle\rho(f)\epsilon^{\pm}(\lambda,\rho)\lambda(\epsilon^{\pm}(\mu,\rho))$
$\displaystyle=\epsilon^{\pm}(\nu,\rho)f.$ $\displaystyle
f\lambda(\epsilon^{\pm}(\rho,\mu))\epsilon^{\pm}(\rho,\lambda)$
$\displaystyle=\epsilon^{\pm}(\rho,\nu)\rho(f).$

In the following, we set $\epsilon(\lambda,\mu)=\epsilon^{+}(\lambda,\mu)$.

######  Lemma 3 ([be1] lemma $3.1$).

For $\lambda\in\Delta_{I}(\mathcal{A})$, we have

$\operatorname{Ad}(\epsilon(\lambda,\theta))\circ\lambda\circ\gamma(v)=\theta(\epsilon(\lambda,\theta)^{*})\gamma(v).$

######  Proposition 6.

Take $\theta,\gamma,v$ as above. We have

$\epsilon(\theta,\theta)v^{2}=\epsilon(\theta,\theta)^{*}v^{2}=v^{2}.$
$\epsilon(\theta,\theta)\gamma(v)=\epsilon(\theta,\theta)^{*}\gamma(v)=\gamma(v).$

###### Proof.

Take $I_{+}$ another interval disjoint with $I$. Take $u\in\mathcal{A}$ which
intertwines $\gamma$ and $\gamma_{+}$. Particularly, $uv$ intertwines $\theta$
and $\theta_{+}$. Moreover, $uv$ intertwines $\text{id}_{\mathcal{B}}$ and
$\theta_{+}$ which implies $uv\in\mathcal{B}(I^{\prime}_{+})^{\prime}$ and
$[uv,v]=0$. $uvv=vuv=\theta(u)v^{2}$ with
$\epsilon(\theta,\theta)=u^{*}\theta(u)$, we get
$\epsilon(\theta,\theta)v^{2}=u^{*}\theta(u)v^{2}=v^{2}$. By the fact that
$v^{2}=\gamma(v)v$, we have
$\epsilon(\theta,\theta)\gamma(v)vv^{*}=\gamma(v)vv^{*}$, taking conditional
expectation on both sides, we get
$\epsilon(\theta,\theta)\gamma(v)=\gamma(v)$. ∎

######  Lemma 4.

Let $t\in Z\mathcal{B}(I)Z^{*}$ such that $t\lambda(a)=\mu(a)t$ for any
$a\in\mathcal{A}(I)$, then
$t\in\text{Hom}_{\mathcal{B}(I)}(\alpha^{+}_{\lambda},\alpha^{+}_{\mu})$.

###### Proof.

Let $s=\gamma(t)$. Then clearly
$s\in\operatorname{Hom}_{\mathcal{A}(I)}(\theta\circ\lambda,\theta\circ\mu)$.
By proposition 5, we have

$\epsilon(\theta\circ\mu,\theta)s=\theta(s)\epsilon(\theta,\theta)\theta(\epsilon(\lambda,\theta)).$

Since
$\epsilon(\theta\circ\mu,\theta)=\epsilon(\theta,\theta)\theta(\epsilon(\lambda,\theta)$,
we have

$s\theta(\epsilon(\lambda,\theta)^{*})=\theta(\epsilon(\mu,\theta)^{*})\epsilon(\theta,\theta)^{*}\theta(s)\epsilon(\theta,\theta).$

By repeatedly using proposition 6 and lemma 3 we have,

$\displaystyle
s\operatorname{Ad}(\epsilon(\lambda,\theta))\circ\lambda\circ\gamma(v)$
$\displaystyle=s\theta(\epsilon(\lambda,\theta)^{*})\gamma(v)$
$\displaystyle=\theta(\epsilon(\mu,\theta)^{*})\epsilon(\theta,\theta)^{*}\theta(s)\epsilon(\theta,\theta)\gamma(v)$
$\displaystyle=\theta(\epsilon(\mu,\theta)^{*})\epsilon(\theta,\theta)^{*}\theta(s)\gamma(v)$
$\displaystyle=\theta(\epsilon(\mu,\theta)^{*})\epsilon(\theta,\theta)^{*}\gamma(v)s$
$\displaystyle=\theta(\epsilon(\mu,\theta)^{*})\gamma(v)s$
$\displaystyle=\operatorname{Ad}(\epsilon(\mu,\theta))\circ\mu\circ\gamma(v)s.$

Taking $\gamma^{-1}$ on both sides of the above identity, we obtain
$t\alpha^{+}_{\lambda}(v)=\alpha^{+}_{\mu}(v)t$. ∎

######  Lemma 5 ([be1] lemma $3.8$).

For $a\in\mathcal{A}(I)$, $av=0$ implies $a=0$. For $b\in\mathcal{B}(I)$,
$w^{*}\gamma(b)=0$ implies $b=0$.

######  Theorem 2.

For $\lambda,\mu\in\Delta_{\mathcal{A}}(I)$,
$\langle\alpha^{+}_{\lambda},\alpha^{+}_{\mu}\rangle_{\mathcal{B}(I)}=\langle\theta\circ\lambda,\mu\rangle_{\mathcal{A}(I)}$.

###### Proof.

Let $x\in\operatorname{Hom}_{\mathcal{B}(I)}(\alpha_{\lambda},\alpha_{\mu})$.
Then, for $a\in\mathcal{A}(I)$,

$w^{*}\gamma(x)\theta\circ\lambda(a)=w^{*}\gamma(x\lambda(a))=w^{*}\gamma(\mu(a)x)=w^{*}\theta\circ\mu(a)x=\mu(a)w^{*}\gamma(x).$

Here, we use the intertwining property of $x$ and $w$. The above identity
means
$w^{*}\gamma(x)\in\operatorname{Hom}_{\mathcal{A}(I)}(\theta\circ\lambda,\mu)$
and the map $x\to w^{*}\gamma(x)$ is injective by lemma 5.
On the other hand, let
$y\in\operatorname{Hom}_{\mathcal{A}(I)}(\theta\circ\lambda,\mu)$. Then, for
$a\in\mathcal{A}(I)$,

$yv\lambda(a)=y\theta\circ\lambda(a)v=\mu(a)yv.$

Here, we use the intertwining property of $v$ and $y$. By lemma 4, we have
$yv\in\operatorname{Hom}_{\mathcal{B}(I)}(\alpha_{\lambda},\alpha_{\mu})$ and
by lemma 5, the map $y\to yv$ is injective. ∎

######  Corollary 1.

In the above setting, define the modular invariant matrix
$Z_{\lambda,\mu}=\langle\alpha^{+}_{\lambda},\alpha^{-}_{\mu}\rangle_{\mathcal{B}(I)}$,
where $\lambda,\mu\in\Delta_{I}(\mathcal{A})$. Then, we have
$Z_{0,\mu}=\langle\theta,\mu\rangle_{\mathcal{A}(I)}$ for any
$\mu\in\Delta_{I}(\mathcal{A})$, where $0$ denotes the identity(vacuum
representation).

##  7 $N=2$ Super-Virasoro Nets and Their Representations

######  Definition 5.

For any $t\in\mathbb{R}$, the $N=2$ super-Virasoro algebra
$\operatorname{SVir}^{N=2,t}$ is the infinite dimensional Lie superalgebra
generated by linearly independent even elements $L_{n},J_{n}$ and odd elements
$G_{r}^{\pm}$ where $n\in\mathbb{Z}$, $r,s\in\frac{1}{2}\mp t+\mathbb{Z}$,
together with an even central element $c$ and with commutation relations:

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n,0},$
$\displaystyle[L_{m},J_{n}]$ $\displaystyle=-nJ_{n+m},$
$\displaystyle[J_{m},J_{n}]$ $\displaystyle=\frac{c}{3}m\delta_{m+n,0},$
$\displaystyle[L_{n},G_{r}^{\pm}]$
$\displaystyle=(\frac{n}{2}-r)G^{\pm}_{n+r},$
$\displaystyle[J_{n},G_{s}^{\pm}]$
$\displaystyle=\pm\frac{1}{2}G^{\pm}_{n+s},$
$\displaystyle[G_{r}^{+},G_{s}^{+}]$ $\displaystyle=[G_{r}^{-},G_{s}^{-}]=0,$
$\displaystyle[G_{r}^{+},G_{s}^{-}]$
$\displaystyle=2L_{r+s}+(r-s)J_{r+s}+\frac{c}{3}(r^{2}-\frac{1}{4})\delta_{r+s,0}.$

The Neveu-Schwarz(NS) $N=2$ super-Virasoro algebra is the super-Virasoro
algebra with $t=0$, while the Ramond(R) super-Virasoro algebra is the one with
$t=\frac{1}{2}$.
We are interested in irreducible unitary representations. Here, we mainly
consider Neveu-Schwarz super-Virasoro algebra since it has the vacuum
representation. In this case, the algebra of zero modes is abelian and the
irreducibility implies that the lowest energy space is one dimensional and it
is spanned by a single unit vector $\Omega_{c,h,q}$ such that
$L_{0}\Omega_{c,h,q}=h\Omega_{c,h,q}$ and
$J_{0}\Omega_{c,h,q}=q\Omega_{c,h,q}$. The real numbers $c,h,q$ completely
determine the representation up to unitary equivalence.

######  Theorem 3 ([urep1], [urep2], [urep3]).

For any irreducible unitary representation of the Neveu-Schwarz $N=2$ super-
Virasoro algebra the corresponding values of $c,h,q$ satisfy one of the
following conditions:

  1. 1.

NS1 $c\geq 3$ and $2h-2nq+(\frac{c}{3}-1)(n^{2}-\frac{1}{4})\geq 0$ for all
$n\in\frac{1}{2}+\mathbb{Z}$.

  2. 2.

NS2 $c\geq 3$ and $2h-2hq+(\frac{c}{3}-1)(n^{2}-\frac{1}{4})=0$,
$2h-2(n+\operatorname{sgn}(n))q+(\frac{c}{3}-1)[(n+\operatorname{sgn}(n))^{2}-\frac{1}{4}]<0$
for some $n\in\mathbb{Z}+\frac{1}{2}$ and
$2(\frac{c}{3}-1)h-q^{2}+\frac{c}{3}\geq 0$.

  3. 3.

$c=\frac{3n}{n+2},h=\frac{l(l+2)-m^{2}}{4(n+2)},q=\frac{-m}{n+2}$, where
$n,l,m\in\mathbb{Z}$ satisfy $n\geq 0,0\leq l\leq n,l+m\in 2\mathbb{Z}$ and
$|m|\leq l$.

Conditions NS1, NS2, NS3, are also sufficient, namely if values $c,h,q$
satisfy one of them then there exists a corresponding irreducible unitary
representation. In particular, all the values in the discrete series of
representations (NS3) with $c=\frac{3n}{n+2}$ are realized by the coset
construction for the inclusion $U(1)_{2n+4}\subset
SU(2)_{n}\otimes\operatorname{CAR}^{\otimes 2}$ for every nonnegative integer
$n$. Here $\operatorname{CAR}^{\otimes 2}$ denotes the theory generated by two
real chiral free Fermi fields.

In fact, associating to the vacuum representation with central charge $c$ of
the Neveu-Schwarz $N=2$ super-Virasoro algebra for every allowed $c$, we can
define a graded local conformal net, which leads to the definition of $N=2$
superconformal net. This is defined in [chklx] definition $3.5$ by smearing
certain fields operators.

######  Definition 6.

The $N=2$ super-Virasoro net with central charge $c$ is defined as follows:

$\mathcal{A}_{c}(I)=\\{e^{iJ(f)},e^{iL(f)},e^{iG^{k}(f)}:f\in
C_{c}^{\infty}(I,\mathbb{R}),k=1,2\\}^{\prime\prime},I\in\mathcal{I}.$

Here,

$G^{1}_{r}\coloneqq\frac{G^{+}_{r}+G^{-}_{r}}{\sqrt{2}},\
G^{2}_{r}\coloneqq-i\frac{G^{+}_{r}-G^{-}_{r}}{\sqrt{2}},\
r\in\frac{1}{2}+\mathbb{Z}.$

######  Definition 7.

An $N=2$ superconformal net is a graded local conformal net $\mathcal{A}$ with
central charge $c$ containing the $N=2$ super-Virasoro net $\mathcal{A}_{c}$
as a covariant subnet and such that the corresponding representations of
$\operatorname{Diff}(S^{1})^{(\infty)}$ agree.

For any allowed central charge $c=\frac{3n}{n+2}$ in the discrete series,
$\mathcal{A}_{c}(I)=\mathcal{A}_{U(1)_{2n+4}}(S^{1})^{{}^{\prime}}\bigcap(\mathcal{A}_{SU(2)_{n}}(I)\otimes\mathcal{A}_{\text{CAR}^{\otimes
2}}(I))$, by theorem $5.10$ in [chklx]. DHR representations are labeled by
$(l,m)$ satisfying $l=0,1,2,...,n$,
$m=0,1,2,...,2n+3\in\mathbb{Z}/(2n+4)\mathbb{Z}$, with $l-m\in 2\mathbb{Z}$
with the identification $(l,m)=(n-l,m+n+2)$.
Fusion rules:

$(l_{1},m_{1})(l_{2},m_{2})=\bigoplus_{|l_{1}-l_{2}|\leq
l\leq\text{min}\\{l_{1}+l_{2},2n-l_{1}-l_{2}\\},l+l_{1}+l_{2}\in
2\mathbb{Z}}(l,m_{1}+m_{2}).$

The restriction of $(l,m)$ to the Bose subnet is given by
$(l,m,0)\oplus(l,m,2)$,
$(l,m)=\alpha_{(l,m,0)}$. Statistics phases and dimensions:

$\omega_{(l,m)}=\exp{(\frac{l(l+2)-m^{2}}{4(n+2)}2\pi i)},\
d_{(l,m)}=\frac{\sin\frac{(l+1)\pi}{n+2}}{\sin\frac{\pi}{n+2}}.$

##  8 Classification of $N=2$ Superconformal Nets

According to Gannon’s list [gannonlist], extensions of $N=2$ super-Virasoro
nets split into two parts: simple current extensions of some subgroups of the
maximal cyclic group and exceptional ones. Firstly, we discuss the set
consisting of DHR representations with dimension $1$ and statistics $1$. Note
that this set may not be a group in general.

  1. 1.

Assume $n$ is odd. Then $G=\mathbb{Z}_{2(k+2)}$, with the generator
$\sigma=(n,1)$, and relations $(n,1)^{2N}=(0,2N),\ (n,1)^{2N+1}=(n,2N+1)$,
$N\in\mathbb{N}$.

  2. 2.

Assume $n$ is even. With relations $(n,0)^{2}=(0,0),(n,0)(0,2N)=(n,2N)$, we
have $G\cong\mathbb{Z}_{n+2}\times\mathbb{Z}_{2}$, through the group
homomorphism:
$(n,2N)\mapsto(0,2N)\times(n,0),(0,2N)\mapsto(0,2N)\times(0,0)$. The generator
for $\mathbb{Z}_{n+2}$ is $(0,2)$ and the generator for $\mathbb{Z}_{2}$ is
$(n,0)$.

Therefore, if we want to take a simple current extension for some subgroup
$H$, this $H$ must be a subgroup of one of the above groups. Furthermore, we
have to make sure the generating representations have statistics phase $1$.

  1. (A)

When $k$ is odd, take the smallest positive integer $p$ such that
$\omega_{(k,1)^{p}}=1$. The maximal cyclic group is
$G=\langle(k,1)^{p}\rangle$, which is the cyclic group generated by
$(k,1)^{p}$.

  2. (B)

When $k\equiv 0\pmod{4}$, $\omega_{(k,0)}=1$. Let $p$ be the smallest positive
integer such that $\omega_{(0,2)^{p}}=1$. The maximal cyclic group is
$G=\langle(0,2)^{p}\rangle\times\langle(k,0)\rangle$.

  3. (C)

When $k\equiv 2\pmod{4}$, $\omega_{(k,0)}=-1$, which implies that
$\omega_{(0,\frac{M}{2})}\in\\{-1,i,-i\\}$. Let $M$ be the smallest positive
even integer such that $\omega_{(0,M)}=1$. We have
$\omega_{(0,\frac{M}{2})}=-1$, $\omega_{(k,\frac{M}{2})}=1$. Indeed, by
assumption, $\frac{M^{2}}{8(k+2)}$ and $\frac{k}{2}$ are odd then

$\omega_{(k,\frac{M}{2})}=\exp(\frac{k}{2}\pi i-\frac{M^{2}}{8(k+2)}\pi i)=1.$
(5)

Therefore, the maximal cyclic group $G$ is

$\\{(0,0),(k,\frac{M}{2}),(0,M),\cdots\\}=\langle(k,\frac{M}{2})\rangle\cong
H\times\mathbb{Z}_{2}.$

If $\omega_{(0,\frac{M}{2})}=\pm i$, we have $\omega_{(k,\frac{M}{2})}=\pm i$.
Consequently, the maximal cyclic group is $H=\langle(0,M)\rangle$.

Finally, for exceptional cases, we have:

  1. (a)

$(E_{6})$ for $k=10$, $\theta=(0,0)\oplus(6,0)$. This is a dual canonical
endomorphism because it arises from a conformal embedding $SU(2)_{10}\subset
SO(5)_{1}$ as in [kl]. The $Q$-system structure on $\theta$ is unique up to
unitary equivalence.

  2. (b)

$(E_{6})$ for $k=10$, $\theta=(0,0)\oplus(0,12)$, This is a dual canonical
endomorphism because it arises from a conformal embedding $U(1)_{12}\subset
U(1)_{3}$. The DHR representation $(0,12)$ has dimension $1$ and statistics
phase $1$, so it is realized as a crossed product by $\mathbb{Z}/2\mathbb{Z}$
and hence the $Q$-system structure is unique up to unitary equivalence.

  3. (c)

$(E_{6})$ for $k=10$, $\theta=(0,0)\oplus(6,0)\oplus(0,12)\oplus(6,12)$. This
is the combination of the above two extensions. Namely, we first consider the
extension in (a) and make a crossed product extension by
$\mathbb{Z}/2\mathbb{Z}$ as in (b). For the above two reasons, the $Q$-system
structure of $\theta$ is unique up to unitary equivalence.

  4. (d)

$(E_{8})$ For $k=28$, $\theta=(0,0)\oplus(10,0)\oplus(18,0)\oplus(28,0)$. This
is a dual canonical endomorphism because it arises from a conformal embedding
$SU(2)_{28}\subset(G_{2})_{1}$. As in (a), the $Q$-system structure of
$\theta$ is unique up to unitary equivalence.

######  Theorem 4.

The complete list of $N=2$ superconformal nets with $c<3$ in the discrete
series is listed as follows:

  1. 1.

A simple current extension arising from a subgroup of the maximal cyclic group
appearing in the above (A), (B) and (C).

  2. 2.

The exceptional cases related to $E_{6}$ and $E_{8}$ as in the above (a), (b),
(c) and (d).

## Acknowledgment

The author would thank Yasuyuki Kawahigashi for bringing about this question
and for his constant support and advice. The author would thank Sebastiano
Carpi and Roberto Longo for reading the draft and for their suggestions.

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