Samuel Cahyawijaya∗††thanks: ∗ Equal contribution., Holy Lovenia∗, Willy
Chung∗, Rita Frieske, Zihan Liu, Pascale Fung

# Cross-Lingual Cross-Age Group Adaptation
for Low-Resource Elderly Speech Emotion Recognition

###### Abstract

Speech emotion recognition plays a crucial role in human-computer
interactions. However, most speech emotion recognition research is biased
toward English-speaking adults, which hinders its applicability to other
demographic groups in different languages and age groups. In this work, we
analyze the transferability of emotion recognition across three different
languages–English, Mandarin Chinese, and Cantonese; and 2 different age
groups–adults and the elderly. To conduct the experiment, we develop an
English-Mandarin speech emotion benchmark for adults and the elderly,
BiMotion, and a Cantonese speech emotion dataset, YueMotion. This study
concludes that different language and age groups require specific speech
features, thus making cross-lingual inference an unsuitable method. However,
cross-group data augmentation is still beneficial to regularize the model,
with linguistic distance being a significant influence on cross-lingual
transferability. We release publicly release our code at
https://github.com/HLTCHKUST/elderly_ser.

Index Terms: speech emotion recognition, cross-lingual adaptation, cross-age
adaptation, elderly, low-resource language

##  1 Introduction

Understanding human emotion is a crucial step towards better human-computer
interaction [1, 2, 3, 4]. Most studies on speech emotion recognition are
centered on young-adult people, mainly originating from English-speaking
countries [5, 6, 7, 8]. This demographic bias causes existing commercial
emotion recognition systems to inaccurately perceived emotion in the elderly
[9]. Despite the fast-growing elderly demographic in many countries [10], only
a few studies with a fairly limited amount of data work on emotion recognition
for elderly [11, 12, 13], especially from non-English-speaking countries [14].

To cope with the limited resource problem in the elderly emotion recognition,
in this work, we study the prospect of transferring emotion recognition
ability over various age groups and languages through the utilization of
multilingual pre-trained speech models, e.g., Wav2Vec 2.0 [15]. Specifically,
we aim to understand the transferability of emotion recognition ability using
only speech modality between two languages and two age groups, i.e, English-
speaking adults, English-speaking elderly, Mandarin-speaking adults, and
Mandarin-speaking elderly. To do this, we develop BiMotion, a bi-lingual bi-
age-group speech emotion recognition benchmark that covers 6 adult and elderly
speech emotion recognition datasets from English and Mandarin Chinese.
Additionally, we analyze the effect of language distance on the
transferability of emotion recognition ability from Mandarin and English using
YueMotion, a newly-constructed Cantonese speech emotion recognition dataset.

Figure 1: Elderly speech emotion data in low-resource languages is extremely
rare, which makes an in-group trained model perform poorly (a). We explore two
ways to improve it: cross-group inference (b) and cross-group augmentation
(c).

We analyze the transferability of emotion recognition ability in three ways,
i.e., 1) cross-group inference using only the source group data for training
to better understand the features distinction between each group, 2) cross-
group data augmentation to better understand the transferability across
different groups, and 3) feature-space projection to better understand the
effect of transferability across different language distance. Through a series
of experiments, we conclude that:

  * •

Very distinct speech features are needed for recognizing emotion across
different language and age groups, which makes cross-lingual inference an ill-
suited method for transferring speech emotion recognition ability.

  * •

Despite the different important speech features across groups, data
augmentation across different groups is still beneficial as it helps the model
to better regularize yielding higher evaluation performance.

  * •

Language distance plays a crucial role in cross-lingual transferability, as
more similar languages tend to share more similar speech characteristics, thus
allowing better generalization to the target language.

##  2 Related Work

###  2.1 Emotion Recognition

Various efforts have explored emotion recognition solutions for around two
decades ago through different modalities [16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. Most studies focus on methods that can estimate the subject emotion
effectively through a specific modality while some others focus on combining
multiple modalities to better estimate the subject emotion. The
transferability of emotion recognition across different cultures and language
groups has also been previously studied [26, 27, 28] showing that there are
significant differences across languages and cultures. Nevertheless, it is
only evaluated on a small-scale model that owns limited world knowledge. In
this work, we extend this analysis to pre-trained speech models and further
explore the transferability to cover an extremely low-resource group, i.e.,
the elderly in the low-resource language group.

###  2.2 Pre-trained Speech Model

Large pre-trained speech models achieve state-of-the-art performance for
various speech tasks in recent years [15, 29, 30]. These models have been
trained in an unsupervised manner on large-scale multilingual speech corpora.
The representation of these models has been shown to be effective, achieving
state-of-the-art performance in various downstream tasks including automatic
speech recognition, speaker verification, language identification, and emotion
recognition [31, 32, 33, 34, 35]. In this work, we adopt the XLSR Wav2Vec 2.0
model [15, 30], which has been pre-trained on large-scale multilingual speech
corpora. The multilingual representation learned by the model will be crucial
for evaluating the transferability of emotion recognition ability across
different languages.

##  3 Benchmark and Dataset

###  3.1 BiMotion Benchmark

Table 1: Statistics of datasets covered in BiMotion.

Dataset | Language | Age Group | #Train | #Valid | #Test
---|---|---|---|---|---
CREMA-D | English | Elderly | 150 | 42 | 300
English | Adults | 5000 | 750 | 1200
CSED | Mandarin | Elderly | 200 | 52 | 400
ElderReact | English | Elderly | 615 | 355 | 353
ESD | Mandarin | Adults | 15000 | 1000 | 1500
English | Adults | 15000 | 1000 | 1500
IEMOCAP | English | Adults | 7500 | 1039 | 1500
TESS | English | Elderly | 699 | 200 | 500
English | Adults | 700 | 201 | 500

To test the transferability of emotion recognition ability across different
languages and age groups, we develop a bilingual speech emotion recognition
benchmark namely BiMotion, that covers two languages, i.e., English and
Mandarin, and two age groups, i.e., Elderly (age $\geq$60 y.o) and Adults (age
20-59). We construct our benchmark from a collection of six publicly available
datasets, i.e., CREMA-D [11], CSED [14], ElderReact [12], ESD [36], IEMOCAP
[5], and TESS [13]. For datasets with an official split, i.e., ESD and
ElderReact, we use the provided dataset splits. For the other datasets, we
generate a new dataset split since there is no official split provided. The
statistics of the datasets in BiMotion are shown in Table 1.

###  3.2 YueMotion Dataset

To strengthen our transferability analysis, we further introduce a new speech
emotion recognition dataset covering adults in Cantonese named YueMotion. The
utterances in YueMotion are recorded by 11 speakers (4 males and 7 females)
each with a personal recording device. During the recording session, each
speaker is asked to say out loud 10 pre-defined sentences used in daily
conversations with 6 different emotions, i.e., happiness, sadness, neutral,
disgust, fear, and anger. The YueMotion dataset is used to further verify our
hypothesis regarding the transferability of emotion recognition ability. The
detailed statistics of YueMotion dataset are shown in Table 2.

Table 2: Statistics of the YueMotion dataset.

Age Group | Gender | #Train | #Valid | #Test | #All
---|---|---|---|---|---
Adults | Male | 120 | 36 | 84 | 240
Female | 210 | 63 | 147 | 420
Total Samples | 330 | 99 | 231 | 660

##  4 Methodology

We employ two methods to evaluate the transferability of emotion recognition
ability, i.e., cross-group data augmentation and cross-group inference.111To
enhance readability, we use colored underlines to mark the term cross-group
data augmentation and cross-group inference onwards. The detail of each method
is explained in the following subsection.

###  4.1 Cross-Group Inference

We define a set of language L={$l_{1}$, $l_{2}$, …, $l_{n}$} and a set of age
group A={$a_{1}$, $a_{2}$, …, $a_{m}$}. We define training and evaluation data
from a language $l_{i}$ and age group $a_{j}$ as $X^{trn}_{l_{i},a_{j}}$ and
$X^{tst}_{l_{i},a_{j}}$, respectively. To understand the transferability from
the source group to the target group, we explore a method called cross-group
inference. Following prior works on zero-shot accent/language adaptation [37,
38, 39], we explore cross-lingual and cross-age inference settings in our
experiment. Specifically, given a training dataset $X^{trn}_{l_{i},a_{j}}$, we
select three out-of-group test datasets: 1) datasets that are not in $l_{i}$
language ($X^{tst}_{\neg l_{i},a_{j}}$), 2) datasets that are not in $a_{j}$
age-group ($X^{tst}_{l_{i},\neg a_{j}}$), and 3) datasets that are not in
$l_{i}$ language and $a_{j}$ age-group $X^{tst}_{\neg l_{i},\neg a_{j}}$. We
conduct cross-group inference by training the models on a specific group
training set and evaluating them on the three out-of-group test sets. With
this method, we can analyze the effect of cross-group information on a
specific language and age group.

###  4.2 Cross-Group Data Augmentation

Prior works [40, 41] have shown that cross-domain and cross-lingual data
augmentation method effectively improves the model performance in textual and
speech modalities, especially on low-resource languages and domains. We adopt
the data augmentation technique to the speech emotion recognition task by
utilizing cross-lingual and cross-age data augmentation. Our cross-group data
augmentation injects a training dataset $X^{trn}_{l_{i},a_{j}}$ from a
language group $l_{i}$ and an age group $a_{j}$ with data from other language
and age groups producing a new augmented dataset. Given
$X^{trn}_{l_{i},a_{j}}$, there are three different data augmentation settings
possible, i.e., cross-lingual data augmentation on the same age group
resulting in $X^{trn}_{L,a_{j}}$, cross-age data augmentation on the same
language resulting in $X^{trn}_{l_{i},A}$, and cross-lingual cross-age data
augmentation resulting in $X^{trn}_{L,A}$. We then fine-tune the model on the
augmented dataset and evaluate it on $X^{tst}_{l_{i},a_{j}}$, the test set
with a language $l_{i}$ and age group $a_{j}$. Through this method, we can
analyze the influence of cross-group augmentation on a specific language and
age group.

###  4.3 Feature-Space Projection

Given a fine-tuned emotion recognition model $\theta$, we extract the speech
representation by taking the high-dimensional features before the last linear
classification layer. By using $\theta$, we extract the speech features from
data points and perform matrix decomposition [42, 43, 44, 45, 46, 47, 48] into
a 2-dimensional space with UMAP [49] for visualization. With this approach, we
can approximate the effect of augmenting cross-group data points to the
training coverage, which will determine the prediction quality of the model to
the evaluation data in a specific language $l_{i}$ and age-group $a_{j}$.

##  5 Experimental Settings

Table 3: Evaluation results of cross-lingual and cross-age groups inference on
BiMotion. Cross-all denotes cross-age cross-lingual inference. eng, zho, eld,
and adu denote English, Mandarin, elderly, and adults, respectively.

Training Data | Test Data | Avg.
---|---|---
$\bm{l_{i}}$=eng |  $\bm{l_{i}}$=eng |  $\bm{l_{i}}$=zho |  $\bm{l_{i}}$=zho
$\bm{a_{j}}$=eld |  $\bm{a_{j}}$=adu |  $\bm{a_{j}}$=eld |  $\bm{a_{j}}$=adu
Cross-all ($\bm{X^{trn}_{\neg l_{i},\neg a_{j}}}$) | 27.48 | 9.71 | 24.75 | 28.71 | 22.66
Cross-age ($\bm{X^{trn}_{l_{i},\neg a_{j}}}$) | 40.40 | 27.69 | 26.69 | 14.56 | 27.34
Cross-lingual ($\bm{X^{trn}_{\neg l_{i},a_{j}}}$) | 14.54 | 32.42 | 16.06 | 57.54 | 30.14
Baseline ($\bm{X^{trn}_{l_{i},a_{j}}}$) | 55.12 | 70.28 | 57.00 | 97.00 | 69.85

###  5.1 Baseline Models

For our experiment, we utilize the XLSR-53 Wav2Vec 2.0 [15, 30] model which is
pre-trained in 53 languages including English, Mandarin, and Cantonese.222We
use the model checkpoint from https://huggingface.co/facebook/wav2vec2-large-
xlsr-53. We compare the performance of the cross-group data augmentation and
cross-group inference settings with a simple in-group training baseline where
given a language $l_{i}$ and an age-group $a_{j}$, the model is fine-tuned
only on $X_{l_{i},a_{j}}$ and evaluated on the test set of the corresponding
language and age-group.

###  5.2 Training Settings

To evaluate the transferability of emotion recognition ability, we conduct an
extensive analysis in the BiMotion benchmark. We explore two different cross-
group transfer methods, i.e., cross-group data-augmentation (§4.2) and cross-
group inference (§4.1). Specifically, we fine-tune the pre-trained XLSR-53
Wav2Vec 2.0 model using the in-group and the cross-group augmented datasets,
i..e., $X_{l_{i},a_{j}}$, $X_{a_{j}}$, $X_{l_{i}}$, $X_{all}$, and evaluate on
all the test set covering both the in-group and the out-of-group test data,
i.e, $X^{test}_{l_{i},a_{j}}$, $X^{test}_{\neg l_{i}}$, $X^{test}_{\neg
a_{j}}$, and $X^{test}_{\neg l_{i},\neg a_{j}}$.

To further strengthen our analysis, we conduct the transferability experiment
on the YueMotion dataset. We use the Cantonese elderly and Cantonese adults
data on the YueMotion as the test set, and explore cross-group data
augmentation and cross-group inference for those test datasets. To measure the
effectiveness of the cross-group transferability on YueMotion, we utilize the
training datasets from BiMotion and compare the cross-group transfer methods
with the baseline trained only on the YueMotion training dataset.

Hyperparameters We use the same hyperparameters in all experiments. For fine-
tuning the model we use the following hyperparameter setting, i.e., AdamW
optimizer [50] with a learning rate of 5e-5, a dropout rate of 0.1, a number
of epochs of 20, and early stopping of 5 epochs.

###  5.3 Evaluation Metrics

Some speech utterances may consist of multiple emotion labels, e.g., happiness
and surprise, which makes emotion recognition a multilabel problem. By framing
the problem as multilabel, the occurrence of each emotion becomes sparse, thus
leading to an imbalanced label distribution. To consider the imbalance
distribution, we use binary F1-score to compute per-label evaluation
performance and take the weighted F1-score over different emotion labels,
which is an average of the F1 scores for each class weighted by the number of
samples in that class.

##  6 Results and Discussion

Table 4: Evaluation results of cross-lingual and cross-age groups data
augmentation on BiMotion. Cross-all denotes cross-age cross-lingual data
augmentation. eng, zho, eld, and adu denote English, Mandarin, elderly, and
adults, respectively.

Training Data | Test Data | Avg.
---|---|---
$\bm{l_{i}}$=eng |  $\bm{l_{i}}$=eng |  $\bm{l_{i}}$=zho |  $\bm{l_{i}}$=zho
$\bm{a_{j}}$=eld |  $\bm{a_{j}}$=adu |  $\bm{a_{j}}$=eld |  $\bm{a_{j}}$=adu
Cross-all ($\bm{X^{trn}_{L,A}}$) | 68.06 | 77.24 | 52.21 | 97.40 | 73.73
Cross-age ($\bm{X^{trn}_{l_{i},A}}$) | 66.96 | 75.47 | 59.93 | 97.00 | 74.84
Cross-lingual ($\bm{X^{trn}_{L,a_{j}}}$) | 54.55 | 74.13 | 54.84 | 95.82 | 57.34
Baseline ($\bm{X^{trn}_{l_{i},a_{j}}}$) | 55.12 | 70.28 | 57.00 | 97.00 | 69.85

###  6.1 Transferability via Cross-Group Inference

The results of our cross-group inference experiment are shown in Table 3.
Based on our experiments, cross-lingual, cross-age, and cross-age cross-
lingual inferences are not beneficial for improving the performance of the
target group compared to the baseline trained on the specific language and age
group, instead, they hinder the performance by a huge margin. For instance,
the best cross-age inference comes from English-Adults to English-Elderly,
however, it still hampers the performance of the in-group English-Elderly
baseline by $\sim$15% weighted F1-score. While for cross-lingual inference,
the results are much worse with more than $\sim 30\%$ lower weighted F1-score
than the corresponding in-group training baseline. This result suggests that
there are some differences in speech features used to recognize emotion
between different languages and different age groups, which makes transfer
through cross-group inference not suitable for speech emotion recognition.

Figure 2: Cross-group data augmentation on the Cantonese-Adults data. Blue,
red, olive regions represent the density plot of the training data for happy,
sad, and neutral emotions, respectively.

Figure 3: Cross-group data augmentation on the Mandarin-Elderly data. Blue,
and red regions denote the density of the training data for positive and
negative emotions, respectively.

###  6.2 Transferability via Cross-Group Data Augmentation

Despite having different characteristics, cross-group transfer through data
augmentation could provide richer and more diverse samples of emotion-induced
speech and help to avoid overfitting problems, especially in a low-resource
setting. Based on our cross-group data augmentation results in Table 4, the
English-Elderly and English-Adults groups benefit the most from the cross-
group data augmentation, gaining $\sim$12% and $\sim$5% weighted F1-score
respectively. While for the Mandarin groups, the improvement is quite small.
Mandarin-Elderly gains $\sim$3% weighted F1-score with data augmentation from
Mandarin-Elderly, while the result for the Mandarin-Adults remains the same,
which is probably due to the limited data available from the Mandarin-Elderly
group.

For cross-lingual data augmentation, the effect is apparent with the only
improvement coming from the English-Adults group, while the performance in
other groups decreases. We conjecture this is due to the huge linguistic
differences between English and Mandarin [27, 51]. Further analysis regarding
the effect of language distance is discussed on §6.3. Furthermore, a
combination of cross-lingual and cross-age data augmentation tends to improve
the performance even higher. Specifically, the performance on English-Elderly
and English-Adults increase by $\sim$13% and $\sim$17% weighted F1-score,
respectively. The performance on Mandarin-Adults group improves marginally by
$\sim$0.4% weighted F1-score. On the other hand, the performance of Mandarin-
Elderly decreases by $\sim$5% weighted F1-score compared to the in-group
training baseline. This might be caused by distributional shift due to the
large amount of augmentation from other groups with respect to the amount of
data in the Mandarin-Elderly group.

We further analyze the cross-group behavior further through feature-space
projection using the model trained on all BiMotion training data. As shown in
Figure 3, cross-age data augmentation improves the training coverage of the
model resulting in a better generalization on unseen evaluation data, despite
the speech feature differences among different age groups. This shows the
potential of leveraging cross-age data augmentation for modeling low-resource
groups such as the elderly.

###  6.3 Effect of Language Distance on Transferability

We further analyze the effect of language distance on cross-lingual data
augmentation. Specifically, we explore the effect of cross-lingual data
augmentation from Mandarin and English to Cantonese. To balance the amount of
data between English and Mandarin, we only utilize the ESD dataset [36] which
contains 15K training, 1K validation, and 1.5K utterances for each English and
Mandarin. For the Cantonese dataset, we utilize the newly constructed speech
emotion dataset, YueMotion. As shown in Table 5, the performance on the
Cantonese data increases when the model is augmented with Mandarin data and
decreases when the model is augmented with English. This follows the
linguistic similarity of languages; Mandarin and Cantonese are more similar as
both come from the same language family, i.e., Sino-Tibetan, compared to
English that comes from the Indo-European language family. Similar pattern is
also observed in the Mandarin test data when the Cantonese-Mandarin trained
model outperforms the Cantonese-Mandarin-English model by $\sim$3% weighted
F1-score.

We also analyze the cross-lingual behavior further through feature-space
projection with the Cantonese-Mandarin-English trained model. As shown in
Figure 2, the original Cantonese training data cannot cover all the test data.
When Mandarin data is added, the training coverage improved which covers more
test samples for each label. When English data is added, the training coverage
is improved, but not without shifting the training distribution, e.g., the
out-of-distribution on the sad label, causing the model to perform worse on
the Cantonese evaluation data.

Table 5: The effect of cross-lingual transferability in YueMotion.

Training Data | Test Data
---|---
$\bm{l_{i}}$=yue |  $\bm{l_{i}}$=eng |  $\bm{l_{i}}$=zho
Cross-lingual (yue+eng+zho) | 47.59 | 92.61 | 93.75
Cross-lingual (yue+eng) | 43.59 | 92.51 | -
Cross-lingual (yue+zho) | 51.60 | - | 97.00
Baseline (yue) | 45.33 | - | -

##  7 Conclusion

We investigate the transferability of speech emotion recognition ability to
achieve a better generalization for the elderly in low-resource languages. We
construct an English-Mandarin speech emotion benchmark for adults and the
elderly, namely BiMotion, from six publicly available datasets. We also
construct, YueMotion, a low-resource speech emotion dataset for adults in
Cantonese to analyze the effect of language distance in cross-lingual data
augmentation. Based on the experiments, we conclude: 1) significantly distinct
speech features are necessary to recognize emotion across different language
and age groups, 2) although the speech features may vary across different
groups, cross-group data augmentation is still beneficial to better generalize
the model, and 3) language distance substantially affects the effectiveness of
cross-lingual transferability. Our analysis lays the groundwork for developing
more effective speech emotion recognition models for low-resource groups,
e.g., the elderly in low-resource languages.

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$b_{-}$ and $b_{+}$ under the erasing map $I$, it follows that
$B(\tilde{b}_{-})\cdot B(\tilde{b}_{+})=1$, so $(\tilde{b}_{-},\tilde{b}_{+})$
gives a cyclic coordinate function for the 4-dimensional punctured Milnor
fiber. ∎

We mention here one potential application of the notions of coordinate
functions and cyclic coordinate functions, namely proving the non-formality of
the $L_{\infty}$-structures on the cochain complexes $\mathit{SC}^{\ast}(M)$
and $\mathit{SC}^{\ast}_{S^{1}}(M)$.

######  Definition 61.

An $L_{\infty}$-algebra $\mathcal{A}$ is called formal if it has a minimal
model $H^{\ast}(\mathcal{A})$ with $\ell_{k}=0$ for $k\geq 3$. It is called
intrinsically formal if for any $L_{\infty}$-structure on
$H^{\ast}(\mathcal{A})$, we necessarily have $\ell_{k}=0$ for $k\geq 3$.

Let $M_{d}$ be the complement of $d\geq n+2$ generic hyperplanes in
$\mathbb{CP}^{n}$, then it admits a symplectic embedding into
$\mathbb{C}^{n}$. A generalization of our argument should lead to the identity
(1.21), with the string homology $\mathbb{H}_{\ast}^{S^{1}}$ replaced with the
$S^{1}$-equivariant symplectic cohomology $\mathit{SH}_{S^{1}}^{\ast}(M_{d})$.
On the other hand, it follows from [39], Theorem 2.3 that $M_{d}$ does not
admit a cyclic quasi-dilation, so in particular it does not admit a cyclic
coordinate function. It follows from (1.21) that there must be some $k\geq 3$
such that $\tilde{\ell}_{k}\neq 0$ on $\mathit{SH}_{S^{1}}^{\ast}(M_{d})$. In
particular, this shows that the $L_{\infty}$-algebra
$\mathit{SH}_{S^{1}}^{\ast}(M_{d})$ cannot be intrinsically formal. Since it
is reasonable to expect that the $L_{\infty}$-structure
$(\tilde{\ell}_{k})_{k\geq 1}$ is homotopy equivalent to the geometric
$L_{\infty}$-structure on $\mathit{SC}_{S^{1}}^{\ast}(M_{d})$ constructed by
counting holomorphic curves (cf. [46]), we conjecture that
$\mathit{SC}_{S^{1}}^{\ast}(M_{d})$ isn’t formal as an $L_{\infty}$-algebra.
Since $\mathbb{C}^{n}$ has vanishing symplectic cohomology, similar argument
can be carried out in the non-equivariant case to show the non-intrinsic
formality (and conjecturally non-formality) of the $L_{\infty}$-structure
$(\ell_{k})_{k\geq 1}$ on $\mathit{SC}^{\ast}(M_{d})$. The non-existence of a
coordinate function follows from similar argument as in the proof of [39],
Theorem 2.3, since $M_{d}$ cannot be uniruled by cylinders. Note that these
non-formality results fits with the geometric intuition that the affine
variety $M_{d}$ is uniruled by $d$-punctured spheres.

###  5.2 Compact symplectic manifolds

We briefly discuss in this subsection how to combine the original idea of
Fukaya [17] and the new ingredients in this paper to obtain restrictions on
Lagrangian embedding in certain compact symplectic manifolds.

In [17], Fukaya applied similar ideas as described in the introduction to the
compact symplectic manifold $\mathbb{CP}^{n}$ and proved the following:

######  Theorem 62 ([17], Theorem 14.1).

Let $L\subset\mathbb{CP}^{n}$ be a closed Lagrangian submanifold, which is a
$K(\pi,1)$ space and $\mathit{Spin}$. Then there exists
$\bar{a}\in\pi_{2}(\mathbb{CP}^{n},L)$ such that $L$ bounds a holomorphic disc
in the class $\bar{a}$ with Maslov index 2. Moreover,
$a=\partial\bar{a}\in\pi_{1}(L)$ is non-zero, and its centralizer
$Z_{a}\subset\pi_{1}(L)$ has finite index.

######  Remark 63.

When $X=Y\times Z$, where $Y$ is a closed Kähler manifold with a subcritical
polarization in the sense of Biran-Cieliebak [6], and $Z$ is chosen so that
$X$ is monotone, Theorem 62 is proved by Damian [14] for monotone Lagrangian
submanifolds $L\subset X$ which are $K(\pi,1)$ spaces.

Theorem 62 is parallel to Corollary 8 in this paper. In particular, it implies
that a Lagrangian $L\subset\mathbb{CP}^{n}$ cannot be a hyperbolic manifold.
This is a special case of the Viterbo-Eliashberg theorem [15], which states
that if $L\subset X$ is a closed Lagrangian submanifold in a uniruled smooth
algebraic variety $X$, then $L$ cannot be hyperbolic.

It is therefore desirable to have generalizations of Theorem 62 for closed
Lagrangian submanifolds $L$ in a more general class of uniruled smooth
algebraic varieties $X$. Inspired by the work of Biran-Cieliebak [6], we
introduce the following notion.

######  Definition 64.

Let $X$ be a closed Kähler manifold. If there is a complex hypersurface
$\Sigma\subset X$ such that the complement $X\setminus\Sigma$ is a Weinstein
manifold which admits a cyclic dilation. Then we shall call the couple
$(X,\Sigma)$ a polarization with finite first Gutt-Hutchings capacity.

In view of Remark 63, it is natural to expect the following.

######  Conjecture 65.

Let $X$ be a Kähler manifold which admits a polarization $(X,\Sigma)$ with
finite first Gutt-Hutchings capacity. Then the conclusion of Theorem 62 holds
for any closed Lagrangian submanifold $L\subset X$ which is a $K(\pi,1)$ space
and Spin.

It is not clear how to prove Conjecture 65 in general. Here let us take a look
at a special case, which is inspired by the works of Abouzaid-Smith [4] and
Ganatra-Pomerleano [27]. Let $X$ be a smooth projective variety, and
$\Sigma\subset X$ is a normal crossing divisor, with smooth components
$\Sigma_{1},\cdots,\Sigma_{m}$. Assume that

$K_{X}^{-1}\cong\mathcal{O}(\sum_{i=1}^{m}a_{i}\Sigma_{i}).$ (5.9)

Let $I\subset\\{1,\cdots,m\\}$ be a subset, we introduce the notations
$\Sigma_{I}:=\bigcap_{i\in I}\Sigma_{i}$,
$\Sigma_{I}^{\circ}:=\Sigma_{I}\setminus\bigcup_{j\notin I}\Sigma_{j}$.
Locally near $\Sigma_{I}\subset X$, there is an $|I|$-fold intersection
$U_{I}$ of the tubular neighborhoods $U_{i}$ of each $\Sigma_{i}$, and the
iterated projection $U_{I}\rightarrow\Sigma_{I}$ is a symplectic fibration
with structure group $U(1)^{|I|}$, with fibers a product of standard
symplectic 2-discs. Denote by $S_{I}\rightarrow\Sigma_{I}$ the associated
$T^{|I|}$-bundle over $\Sigma_{I}$, and by $S_{I}^{\circ}$ its restriction to
$\Sigma_{I}^{\circ}$.

Let $J_{X}$ be an almost complex structure on $X$ which is compatible with the
symplectic form $\omega_{X}$, and let $L\subset X$ be a closed Lagrangian
submanifold which is disjoint from $\Sigma$. For a $J_{X}$-holomorphic map
$u:(D,\partial D)\rightarrow(X,L)$ with $u(0)\in\Sigma_{I}^{\circ}$, one can
choose a local complex coordinate $z$ near the origin so that the normal
component $u_{i}$ of $u$ has an expansion

$u_{i}(z)=a_{i}z^{v_{i}}+O(|z|^{v_{i}+1}),$ (5.10)

where $v_{i}\in\mathbb{N}$ and the coefficient $a_{i}\neq 0$ is the
$v_{i}$-jet normal to $\Sigma_{i}$ modulo higher order terms, which takes
value in $(T_{0}^{\ast}D)^{\otimes v_{i}}\otimes N_{u(0)}\Sigma_{i}$, where
$N\Sigma_{i}$ is the normal bundle of $\Sigma_{i}\subset X$. Now fix an
asymptotic marker $\ell$ at $0\in D$, it specifies via $a_{i}$ a non-zero
vector in $N_{u(0)}\Sigma_{i}\setminus\\{0\\}$. Denote its image in the real
projectivization $\mathbb{P}N_{u(0)}\Sigma_{i}$ by $[a_{i}]$. Taking the
direct sum over $i\in I$ gives a point $\left[\bigoplus_{i\in I}a_{i}\right]$
in the fiber $S_{I,u(0)}$ of the torus bundle $S_{I}\rightarrow D_{I}$. Let
$\mathbf{v}=(v_{1},\cdots,v_{m})\in\mathbb{Z}_{\geq 0}^{m}$, where $v_{i}=0$
for $i\notin I$, the enhanced evaluation map of $u$ at $0$ is defined as

$\mathit{Ev}_{0}^{\mathbf{v}}(u):=\left(u(0),\left[\bigoplus_{i\in
I}a_{i}\right]\right),$ (5.11)

which keeps track of both the incidence condition and the tangency condition
of the holomorphic curve $u$ at $0$ with the divisors $\Sigma_{i}$, $i\in I$.

For $\bar{a}\in H_{2}(X,L)$ and
$\mathbf{v}_{l}=(v_{l,1},\cdots,v_{l,m})\in\mathbb{Z}_{\geq 0}^{m}$ supported
on $I_{l}\subset\\{1,\cdots,m\\}$, let

${}_{l}\mathcal{R}_{k+1}(L,\mathbf{v}_{l},\bar{a})$ (5.12)

be the moduli space of pairs

$\left((D;z_{0},\cdots,z_{k},p_{1},\cdots,p_{l};\ell),u\right),$ (5.13)

where $z_{0},\cdots,z_{k}\in\partial D$, the auxiliary marked points
$p_{1},\cdots,p_{l}$ satisfy (4.24), and the asymptotic marker $\ell$ at $0\in
D$ points towards $p_{l}$. $u:(D,\partial D)\rightarrow(X,L)$ is a
$J_{X}$-holomorphic map such that $u(0)\cdot\Sigma_{i}=v_{l,i}$ for $i\in
I_{l}$, and $[u]=\bar{a}$. The enhanced evaluation map gives rise to a map

$\mathit{Ev}_{0}^{\mathbf{v}_{l}}:{}_{l}\mathcal{R}_{k+1}(L,\mathbf{v}_{l},\bar{a})\rightarrow
S_{I_{l}}^{\circ}.$ (5.14)

For any locally finite chain $\alpha_{l}\in
C_{\ast}^{\mathit{BM}}(S_{I_{l}}^{\circ};\mathbb{K})$, we form the fiber
product

${}_{l}\mathcal{R}_{k+1}(L,\mathbf{v}_{l},\alpha_{l},\bar{a}):={}_{l}\mathcal{R}_{k+1}(L,\mathbf{v}_{l},\bar{a})\times_{\mathit{Ev}_{0}^{\mathbf{v}_{l}}}\alpha_{l}.$
(5.15)

The compactification
${}_{l}\overline{\mathcal{R}}_{k+1}(L,\mathbf{v}_{l},\alpha_{l},\bar{a})$ is
an admissible K-space, with its codimension 1 boundary covered by the
following strata:

$\bigsqcup_{\begin{subarray}{c}k_{1}+k_{2}=k+1\\\ 1\leq i\leq k_{1}\\\
\bar{a}_{1}+\bar{a}_{2}=\bar{a}\end{subarray}}\left({}_{l}\overline{\mathcal{R}}_{k_{1}+1}(L,\mathbf{v}_{l},\alpha_{l},\bar{a}_{1})\textrm{
}{{}_{i}\times_{0}}\textrm{
}\overline{\mathcal{R}}_{k_{2}+1}(L,\bar{a}_{2})\sqcup\overline{\mathcal{R}}_{k_{1}+1}(L,\bar{a}_{1})\textrm{
}{{}_{i}\times_{0}}\textrm{
}{}_{l}\overline{\mathcal{R}}_{k_{2}+1}(L,\mathbf{v}_{l},\alpha_{l},\bar{a}_{2})\right),$
(5.16)
${}_{j}\overline{\mathcal{P}}(\mathbf{v}_{l},\alpha_{l},x_{j})\times{}_{l-j}\overline{\mathcal{R}}^{1}_{k+1}(x_{j},L,\bar{a}),0\leq
j\leq l,$ (5.17)
${}_{l-1}\overline{\mathcal{R}}^{S^{1}}_{k+1}(L,\mathbf{v}_{l},\alpha_{l},\bar{a}),$
(5.18)
${}_{l}^{j,j+1}\overline{\mathcal{R}}^{S^{1}}_{k+1}(L,\mathbf{v}_{l},\alpha_{l},\bar{a}),1\leq
j\leq l-1,$ (5.19)

where ${}_{j}\mathcal{P}(\mathbf{v}_{l},\alpha_{l},x_{j})$ are moduli spaces
of (parametrized) logarithmic PSS (Piunikhin-Salamon-Schwarz) maps studied in
[27], with $x_{j}$ being an orbit of the vector field $X_{H_{t}}$ of some
admissible Hamiltonian function (defined using the radial coordinate on the
collar neighborhood of the divisor complement $X\setminus\Sigma$). When $j=0$,
this is defined as the fiber product

$\mathcal{P}(\mathbf{v}_{l},\alpha_{l},x_{j}):=\mathcal{P}(\mathbf{v}_{l},x_{j})\times_{\mathit{Ev}_{0}^{\mathbf{v}_{l}}}\alpha_{l},$
(5.20)

where $\mathcal{P}(\mathbf{v}_{l},x_{j})$ is the moduli space of maps
$u:\mathbb{C}\rightarrow X$ satisfying the Floer equation $(du-
X_{H_{t}}\otimes dt)^{0,1}=0$, with the obvious asymptotic and tangency
conditions. In general, ${}_{j}\mathcal{P}(\mathbf{v}_{l},\alpha_{l},x_{j})$
is defined as the space of the same maps $u:\mathbb{C}\rightarrow X$, but from
a varying family of domains obtained by equipping $\mathbb{C}$ with the
auxiliary marked points $p_{l-j+1},\cdots,p_{l}$, which are strictly radially
ordered in the sense that

$0<|p_{l}|<\cdots<|p_{l-j+1}|.$ (5.21)

As before, the boundary strata (5.16) come from disc bubbles along the
Lagrangian boundary $L$, (5.17) arise from breakings of planes at the origin
$0\in D$ when $|p_{l-j+1}|\rightarrow 0$, (5.18) corresponds to the locus
where $|p_{1}|=\frac{1}{2}$, and (5.19) are the loci defined by
$|p_{j}|=|p_{j+1}|$.

A count of rigid elements in the moduli spaces
${}_{j}\overline{\mathcal{P}}(\mathbf{v},\alpha,x)$ for varying $\alpha\in
C_{\ast}^{\mathit{BM}}(S_{I}^{\circ};\mathbb{K})$ and
$\mathbf{v}\in\mathbb{Z}_{\geq 0}^{m}$ supported on $I\subset\\{1,\cdots,m\\}$
defines a map

$\mathit{PSS}^{\mathit{log}}_{j}:C_{\mathit{log}}^{\ast+2j}(X,\Sigma)\rightarrow\mathit{SC}^{\ast}(M)$
(5.22)

on the logarithmic cochain complex

$C_{\mathit{log}}^{\ast}(X,\Sigma):=\bigoplus_{I\subset\\{1,\cdots,m\\}}t^{\mathbf{v}}C_{2n-|I|-\ast}^{\mathit{BM}}(S_{I}^{\circ};\mathbb{K}),$
(5.23)

When $l=0$, the positive energy part of this map is a chain map on admissible
cochains $C_{\mathit{log}}^{\ast}(X,\Sigma)_{\mathit{ad}}\subset
C_{\mathit{log}}^{\ast}(X,\Sigma)$ (cf. [27], Definition 3.20), which induces
on the cohomology level the logarithmic PSS map
$H_{\mathit{log}}^{\ast}(X,\Sigma)_{\mathit{ad}}\rightarrow\mathit{SH}^{\ast}_{+}(M)$
studied in [27].

Now suppose that there exist a family of locally finite chains
$(\alpha_{l})_{l\geq 0}$ and vectors $(\mathbf{v}_{l})_{l\geq 0}$ such that
$\sum_{l=0}^{\infty}\mathit{PSS}_{l}^{\mathit{log}}(\alpha_{l}t^{\mathbf{v}_{l}})$
hits the identity $e_{M}\in\mathit{SC}^{0}(M)$, and
$\mathit{PSS}_{j}^{\mathit{log}}(\alpha_{l}t^{\mathbf{v}_{l}})=0$ for $j<l$,
then by pushing forward the virtual fundamental chains of the moduli spaces in
(5.16)-(5.19), similar arguments as in Section 4.5 would imply the identity
(1.21), and therefore confirm Conjecture 65 for Lagrangian submanifolds
$L\subset X$ which are disjoint from $\Sigma$.

When $X\setminus\Sigma$ admits a dilation
$b\in\mathit{SH}^{1}(X\setminus\Sigma)$, and the dilation comes from an
admissible cochain $\alpha t^{\mathbf{v}}$, i.e.
$\mathit{PSS}^{\mathit{log}}(\alpha t^{\mathbf{v}},\mathfrak{b})=b$, where
$\mathfrak{b}\in C^{\ast}(X\setminus\Sigma;\mathbb{K})$ is a bounding cochain
for $\alpha t^{\mathbf{v}}$, it follows from [27], Lemmas 4.29 and 4.30 that
there exists an $\tilde{\alpha}\in
C_{\ast}^{\mathit{BM}}(S_{I}^{\circ};\mathbb{K})$ such that
$\mathit{PSS}^{\mathit{log}}(\tilde{\alpha}t^{\mathbf{v}})$ hits the identity
$e_{M}\in\mathit{SC}^{0}(M)$. In particular, this confirms Conjecture 65 for
the pairings $(X,\Sigma)$ studied in [27], Sections 6.1 and 6.4, where a
dilation can be constructed via the logarithmic PSS map.

##  Appendix A Kuranishi structures and virtual fundamental chains

For reader’s convenience, we collect in this appendix some basic notions and
useful facts in the theory of Kuranishi structures. Our references here are
[19, 20, 21, 22] and [32]. Let $X$ be a separable, metrizable topological
space.

###  A.1 Basic notions

Instead of writing down all the definitions, we will only recall here the most
basic notions and provide the reference for their variations and
generalizations, so the reader may use this section as a dictionary for the
notions in Kuranishi theory used in the main content of our paper. Following
[32], we shall define Kuranish charts using manifolds instead of orbifolds.
This is sufficient for our purposes because sphere bubbles do not appear for
Lagrangian submanifolds in Liouville manifolds.

######  Definition 66.

A Kuranishi chart of $X$ is a quadruple $\mathcal{U}=(U,\mathcal{E},s,\psi)$
such that

  * (i)

$U$ is a smooth manifold,

  * (ii)

$\mathcal{E}$ is a smooth vector bundle on $U$,

  * (iii)

$s$ is a smooth section of $\mathcal{E}$,

  * (iv)

$\psi:s^{-1}(0)\rightarrow X$ is a homeomorphism onto an open subset of $X$.

We call $U$ a Kuranishi neighborhood, $\mathcal{E}$ an obstruction bundle, $s$
a Kuranishi map, and $\psi$ a parametrization. $\dim\mathcal{U}:=\dim
U-\mathrm{rank}\;E$ is called the dimension of $\mathcal{U}$. An orientation
of $\mathcal{U}$ is a pair of orientations of $U$ and $E$.

To globalize the notion of a Kuranishi chart, we need the following two
definitions.

######  Definition 67.

Let $\mathcal{U}_{i}=(U_{i},\mathcal{E}_{i},s_{i},\psi_{i})$, $i=1,2$ be
Kuranishi charts of $X$. An embedding of Kuranishi charts
$\Phi:\mathcal{U}_{1}\rightarrow\mathcal{U}_{2}$ is a pair
$\Phi=(\phi,\hat{\phi})$ such that

  * (i)

$\phi:U_{1}\rightarrow U_{2}$ is an embedding of smooth manifolds,

  * (ii)

$\hat{\phi}:\mathcal{E}_{1}\rightarrow\mathcal{E}_{2}$ is an embedding of
smooth vector bundles over $\phi$,

  * (iii)

$\hat{\phi}\circ s_{1}=s_{2}\circ\phi$,

  * (iv)

$\psi_{2}\circ\phi=\psi_{1}$ on $s_{1}^{-1}(0)$,

  * (v)

for any $x\in s_{1}^{-1}(0)$, the covariant derivative

$D_{\phi(x)}s_{2}:\frac{T_{\phi(x)}U_{2}}{(D_{x}\phi)(T_{x}U_{1})}\rightarrow\frac{(\mathcal{E}_{2})_{\phi(x)}}{\hat{\phi}((\mathcal{E}_{1})_{x})}$
(A.1)

is an isomorphism.

######  Definition 68.

Let $\mathcal{U}_{i}=(U_{i},\mathcal{E}_{i},s_{i},\psi_{i})$, $i=1,2$ be
Kuranishi charts of $X$. A coordinate change in weak sense (resp. strong
sense) from $\mathcal{U}_{1}$ to $\mathcal{U}_{2}$ is a triple
$\Phi_{21}=(U_{21},\phi_{21},\hat{\phi}_{21})$ satisfying the first two (resp.
all three) of the requirements below:

  * (i)

$U_{21}$ is an open subset of $U_{1}$;

  * (ii)

$(\phi_{21},\hat{\phi}_{21})$ is an embedding of Kuranishi charts
$\mathcal{U}_{1}|_{U_{21}}\rightarrow\mathcal{U}_{2}$;

  * (iii)

$\psi_{1}\left(s_{1}^{-1}(0)\cap
U_{21}\right)=\mathrm{im}\psi_{1}\cap\mathrm{im}\psi_{2}$.

######  Definition 69.

A Kuranishi structure $\widehat{\mathcal{U}}$ of $X$ of dimension $d$ consists
of

  * •

a Kuranishi chart of dimension $d$
$\mathcal{U}_{p}=(U_{p},\mathcal{E}_{p},s_{p},\psi_{p})$ at $p$ for every
$p\in X$,

  * •

a coordinate change in the weak sense
$\Phi_{pq}=(U_{pq},\phi_{pq},\hat{\phi}_{pq}):\mathcal{U}_{q}\rightarrow\mathcal{U}_{p}$
for every $p\in X$ and $q\in\mathrm{im}\psi_{p}$ so that
$\Phi_{pp}=(U_{p},\mathit{id},\mathit{id})$,

such that

  * (i)

$o_{q}\in U_{pq}$ for every $q\in\mathrm{im}\psi_{p}$,

  * (ii)

for every $p\in X$, $q\in\mathrm{im}\psi_{p}$ and
$r\in\psi_{q}\left(s^{-1}(0)\cap U_{pq}\right)$, there holds
$\Phi_{pr}|_{U_{pqr}}=\Phi_{pq}\circ\Phi_{qr}|_{U_{pqr}}$, where
$U_{pqr}:=\phi_{qr}^{-1}(U_{pq})\cap U_{pr}$.

The pair $(X,\widehat{\mathcal{U}})$ is called a K-space of dimension $d$. We
say the Kuranishi structure $\widehat{\mathcal{U}}$ is oriented if each
Kuranishi chart $\mathcal{U}_{p}$ is oriented and $\phi_{pq}$ preserves
orientations for every $p\in X$ and $q\in\mathrm{im}\psi_{p}$.

The analogy of the notion of a manifold with corners in the theory of
Kuranishi structures is called an admissible K-space, see [22], Chapters 17
and 25 for details.

Just as one can attach collar neighborhoods to manifolds with boundary, the
same can be done for admissible K-spaces, see [22], Chapter 17 for the theory
of collared Kuranishi structures. In particular, for $\tau>0$, the notion of a
$\tau$-outer collaring of an admissible K-space $X$ used in Section 4.5 is
defined in [22], Definition 17.29.

######  Definition 70.

Let $\widehat{\mathcal{U}}$ be a Kuranishi structure of $X$, and $Y$ a
topological space.

  * (i)

A strongly continuous map $\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow Y$
assigns a continuous map $f_{p}:U_{p}\rightarrow Y$ for each $p\in X$ such
that $f_{p}\circ\phi_{pq}=f_{q}$ holds on $U_{pq}$.

  * (ii)

When $Y$ is a smooth manifold, we say that $\hat{f}$ is strongly smooth if
each $f_{p}$ is smooth.

The notion of a strongly smooth map generalizes to that of an admissible map
for admissible K-spaces.

In order to define the pushforward of virtual fundamental chains of K-spaces
via strongly continuous and submersive maps, we need to equip the K-spaces
with CF-perturbations. Roughly speaking, these are triples
$\mathcal{S}_{x}=\left(W_{x},\omega_{x},\\{\mathfrak{s}_{x}^{\varepsilon}\\}\right)$
for every point $x\in V_{x}\subset U_{p}$ satisfying certain conditions, where
$\mathfrak{s}_{x}^{\varepsilon}$ is a family of sections of $\mathcal{E}_{p}$
over the manifold chart $V_{x}$ of $U_{p}$, parametrized by an open
neighborhood $W_{x}$ of $0$ in some finite-dimensional vector space, and
$\omega_{x}$ is a top degree differential form on $W_{x}$. See [22],
Definition 7.4 for details.

######  Definition 71.

Let $(X,\widehat{\mathcal{U}})$ be a K-space and $N$ a manifold with corners.

  * (i)

A strongly continuous map $\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow N$ is
said to be a corner stratified smooth map if $f_{p}:U_{p}\rightarrow N$ is a
corner stratified smooth map ([22], Definition 26.1) for any $p\in X$.

  * (ii)

A corner stratified smooth map $\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow
N$ is a corner stratified weak submersion if $f_{p}:U_{p}\rightarrow N$ is a
corner stratified submersion for any $p\in X$.

  * (iii)

Let $\widehat{\mathcal{S}}$ be a CF-perturbation of $X$. We say that a corner
stratified smooth map $\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow N$ is a
corner stratified strong submersion with respect to $\widehat{\mathcal{S}}$ if
the following holds. Let $p\in X$ and $(U_{p},\mathcal{E}_{p},s_{p},\psi_{p})$
is a Kuranishi chart at $p$. Let $(V_{x},\phi_{x})$ be a manifold chart of
$U_{p}$ and
$\left(W_{x},\omega_{x},\\{\mathfrak{s}_{x}^{\varepsilon}\\}\right)$ be a
representative of $\widehat{\mathcal{S}}$ in the chart $V_{x}$, then

$f\circ\psi_{p}\circ\phi_{x}\circ\mathit{pr}_{V}:(\mathfrak{s}_{x}^{\varepsilon})^{-1}(0)\rightarrow
N$ (A.2)

is a corner stratified submersion, where $\mathit{pr}_{V}$ is the projection
$V_{x}\times W_{x}\rightarrow V_{x}$.

To achieve the compatibility of CF-perturbations, we need the notion of a
thickening of Kuranishi structures, see [22], Definition 5.3.

###  A.2 $C^{0}$-approximation

We recall Irie’s $C^{0}$-approximation lemma ([32], Sections 7.5 and 9) and
its consequences.

######  Definition 72.

Let $(X,\widehat{\mathcal{U}})$ be a K-space, $(Y,d_{Y})$ be a metric space,
and $\hat{f},\hat{g}:(X,\widehat{\mathcal{U}})\rightarrow Y$ be strongly
smooth maps. For any $\varepsilon>0$, we say that $\hat{f}$ and $\hat{g}$ are
$\varepsilon$-close, if

$d_{Y}\left(f_{p}(x),g_{p}(x)\right)<\varepsilon$ (A.3)

for every $p\in X$ and $x\in U_{p}$.

######  Theorem 73 ($C^{0}$-approximation lemma).

Let $(X,\widehat{\mathcal{U}})$ be a K-space and
$\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow\mathcal{L}_{k+1}^{\mathit{con}}$
be a strongly continuous map such that
$\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow
L$ is strongly smooth for every $0\leq j\leq k$. Let $Z\subset X$ be a closed
subset and
$\hat{g}:(Z,\widehat{\mathcal{U}}|_{Z})\rightarrow\mathcal{L}_{k+1}$ is a
strongly smooth map such that

  * •

$\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{g}=\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}|_{Z}$
for every $0\leq j\leq k$,

  * •

$\hat{g}$ is $\varepsilon$-close to $\hat{f}|_{Z}$ with respect to
$d_{\mathcal{L}_{k+1}}$.

If $\varepsilon<\rho_{L}$, where $\rho_{L}$ is the constant fixed in Section
4.4, there exists an open substructure $\widehat{\mathcal{U}}_{0}$ of
$\widehat{\mathcal{U}}$, and a strongly smooth map
$\hat{g}^{\prime}:(X,\widehat{\mathcal{U}}_{0})\rightarrow\mathcal{L}_{k+1}$
such that

  * •

$\hat{g}^{\prime}$ is $\varepsilon$-close to
$\hat{f}|_{\widehat{\mathcal{U}}_{0}}$,

  * •

$\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{g}^{\prime}=\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}|_{\widehat{\mathcal{U}}_{0}}$
for every $0\leq j\leq k$,

  * •

$\hat{g}^{\prime}=\hat{g}$ on $\widehat{\mathcal{U}}_{0}|_{Z}$.

The $C^{0}$-approximation lemma, combined with [22], Proposition 17.81, yields
the following.

######  Theorem 74 ([32], Theorem 7.33).

Suppose we are given the following data

  * •

$k\in\mathbb{Z}_{\geq 0}$, $\tau\in(0,1)$ and $\varepsilon\in(0,\rho_{L})$;

  * •

a $\tau$-collared K-space $(X,\widehat{\mathcal{U}})$;

  * •

a $\tau$-collared strongly continuous map
$\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow\mathcal{L}_{k+1}^{\mathit{con}}$
such that
$\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow
L$ is a $\tau$-collared admissible map for every $0\leq j\leq k$, and
$\mathit{ev}_{0}^{\mathcal{L}}\circ\hat{f}$ is a stratified weak submersion;

  * •

for every $l\in\mathbb{N}$, a $\tau$-collared Kuranishi structure
$\widehat{\mathcal{U}}_{l}^{+}$ on $\widehat{S}_{l}(X)$ which is a thickening
of $\widehat{\mathcal{U}}|_{\widehat{S}_{l}(X)}$;

  * •

for $l_{1},l_{2}\in\mathbb{N}$, an $\frac{(l_{1}+l_{2})!}{l_{1}!l_{2}!}$-fold
covering of $\tau$-collared K-spaces

$\widehat{S}_{l_{1}}\left(\widehat{S}_{l_{2}}(X),\widehat{\mathcal{U}}_{l_{2}}^{+}\right)\rightarrow\left(\widehat{S}_{l_{1}+l_{2}}(X),\widehat{\mathcal{U}}_{l_{1}+l_{2}}^{+}\right)$
(A.4)

such that the following diagrams commute for $l_{1},l_{2},l_{3}\in\mathbb{N}$:

${\widehat{S}_{l_{1}}\left(\widehat{S}_{l_{2}}\left(\widehat{S}_{l_{3}}(X),\widehat{\mathcal{U}}_{l_{3}}^{+}\right)\right)}$${\widehat{S}_{l_{1}+l_{2}}\left(\widehat{S}_{l_{3}}(X),\widehat{\mathcal{U}}_{l_{3}}^{+}\right)}$${\widehat{S}_{l_{1}}\left(\widehat{S}_{l_{2}+l_{3}}(X),\widehat{\mathcal{U}}_{l_{2}+l_{3}}^{+}\right)}$${\left(\widehat{S}_{l_{1}+l_{2}+l_{3}}(X),\widehat{\mathcal{U}}_{l_{1}+l_{2}+l_{3}}^{+}\right)}$
(A.5)
${\widehat{S}_{l_{1}}\left(\widehat{S}_{l_{2}}(X,\widehat{\mathcal{U}})\right)}$${\widehat{S}_{l_{1}}\left(\widehat{S}_{l_{2}}(X),\widehat{\mathcal{U}}_{l_{2}}^{+}\right)}$${\widehat{S}_{l_{1}+l_{2}}(X,\widehat{\mathcal{U}})}$${\left(\widehat{S}_{l_{1}+l_{2}}(X),\widehat{\mathcal{U}}_{l_{1}+l_{2}}^{+}\right)}$
(A.6)

  * •

a $\tau$-collared CF-perturbation $\widehat{\mathcal{S}}_{l}^{+}$ of
$\left(\widehat{S}_{l}(X),\widehat{\mathcal{U}}_{l}^{+}\right)$ for every
$l\in\mathbb{N}$, such that the pullback of
$\widehat{\mathcal{S}}_{l_{1}+l_{2}}^{+}$ by (A.4) coincides with the
restriction of $\widehat{\mathcal{S}}_{l_{2}}^{+}$ for every
$l_{1},l_{2}\in\mathbb{N}$;

  * •

a $\tau$-collared admissible map
$\hat{f}_{l}^{+}:\left(\widehat{S}_{l}(X),\widehat{\mathcal{U}}_{l}^{+}\right)\rightarrow\mathcal{L}_{k+1}$
for every $l\in\mathbb{N}$ such that

    * –

the pullback of $\hat{f}_{l_{1}+l_{2}}^{+}$ by (A.4) coincides with the
restriction of $\hat{f}_{l_{2}}^{+}$ for every $l_{1},l_{2}\in\mathbb{N}$,

    * –

for every $0\leq j\leq k$,
$\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}_{l}^{+}:\left(\widehat{S}_{l}(X),\widehat{\mathcal{U}}_{l}^{+}\right)\rightarrow
L$ coincides with the restriction of
$\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}$ to $\widehat{S}_{l}(X)$ via the
KK-embedding
$\widehat{\mathcal{U}}|_{\widehat{S}_{l}(X)}\rightarrow\widehat{\mathcal{U}}_{l}^{+}$,

    * –

$\mathit{ev}_{0}^{\mathcal{L}}\circ\hat{f}_{l}^{+}:\left(\widehat{S}_{l}(X),\widehat{\mathcal{U}}_{l}^{+}\right)\rightarrow
L$ is a stratified strong submersion with respect to
$\widehat{\mathcal{S}}_{l}^{+}$,

    * –

$\hat{f}_{l}^{+}$ is $\varepsilon$-close to $\hat{f}|_{\widehat{S}_{l}(X)}$.

Then, for any $\tau^{\prime}\in(0,\tau)$, there exist the following data.

  * (i)

A $\tau^{\prime}$-collared Kuranishi structure $\widehat{\mathcal{U}}^{+}$ on
$X$, which is a thickening of $\widehat{\mathcal{U}}$.

  * (ii)

An isomorphism of $\tau^{\prime}$-collared Kuranishi structures
$\widehat{\mathcal{U}}^{+}|_{\widehat{S}_{l}(X)}\cong\widehat{\mathcal{U}}^{+}_{l}$
for every $l\in\mathbb{N}$.

  * (iii)

A $\tau^{\prime}$-collared CF-perturbation $\widehat{\mathcal{S}}^{+}$ of
$(X,\widehat{\mathcal{U}}^{+})$ such that
$\widehat{\mathcal{S}}^{+}|_{\widehat{S}_{l}(X)}$ coincides with
$\widehat{\mathcal{S}}^{+}_{l}$ via the isomorphism of K-spaces
$\widehat{\mathcal{U}}^{+}|_{\widehat{S}_{l}(X)}\cong\widehat{\mathcal{U}}^{+}_{l}$.

  * (iv)

A $\tau^{\prime}$-collared admissible map
$\hat{f}^{+}:(X,\widehat{\mathcal{U}}^{+})\rightarrow\mathcal{L}_{k+1}$ such
that

    * –

$\hat{f}^{+}$ is $\varepsilon$-close to $\hat{f}$;

    * –

for every $0\leq j\leq k$, $\mathit{ev}_{j}^{\mathcal{L}}\circ\hat{f}^{+}$
coincides with $\mathit{ev}_{j}\circ\hat{f}$ with respect to the KK-embedding
$\widehat{\mathcal{U}}\rightarrow\widehat{\mathcal{U}}^{+}$;

    * –

$\mathit{ev}_{0}\circ\hat{f}^{+}:(X,\widehat{\mathcal{U}}^{+})\rightarrow L$
is a stratified strong submersion with respect to $\widehat{\mathcal{S}}^{+}$.

######  Remark 75.

There is a version of Theorem 74 for $\tau$-collared K-spaces
$(X,\widehat{\mathcal{U}})$, and a $\tau$-collared strong continuous map

$\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow[a,b]^{\boxplus\tau}\times\mathcal{L}_{k+1}^{\mathit{con}},$
(A.7)

where $a<b$ are real numbers and $[a,b]^{\boxplus\tau}:=[a-\tau,b+\tau]$.

######  Proposition 76.

Let $X$ be the $\tau$-outer collaring of one of the moduli spaces
$\overline{\mathcal{R}}_{k+1}(L,\bar{a};P)$,
$\overline{\mathcal{R}}_{k+2,\vartheta}(L,\bar{a};P)$,
${}_{l}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\bar{a}};P)$,
${}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^{1}}(x,L,\ring{\bar{a}};P)$ or
${}_{l}^{j,j+1}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\bar{a}};P)$, where
the number $\tau$ is chosen as in Remark 47, then the thickening
$\widehat{\mathcal{U}}^{+}$ of the Kuranishi structure and the CF-perturbation
$\widehat{\mathcal{S}}^{+}$ that we obtained on $X$ in Theorem 74 can be taken
to be cyclically invariant.

###### Proof.

According to Fukaya [18], Sections 3 and 5, for these moduli spaces we can
start in Theorem 74 with the Kuranishi structures
$\widehat{\mathcal{U}}_{l}^{+}$ and CF-perturbations
$\widehat{\mathcal{S}}_{l}^{+}$ that are invariant under the cyclic
permutations of the boundary marked points $z_{0},\cdots,z_{k}$. In fact, as
we have seen in the proof of Theorem 44, we can first apply Theorem 74 to the
boundaries and corners of the moduli spaces defined using domains without
boundary marked points, which gives us a thickening
$\widehat{\mathcal{U}}^{+}$ of the original Kuranishi structure
$\widehat{\mathcal{U}}$. The pullback of $\widehat{\mathcal{U}}^{+}$ via the
forgetful map $f_{k+1}$ (cf. (4.95)) which erases all the boundary marked
points then equips $X$ with a cyclic invariant $\frac{1}{2}$-collared
Kuranishi structure, extending the cyclic invariant Kuranishi structures on
its boundaries and corners, which are obtained by pullbacks as well. Similar
considerations apply to CF-perturbations. ∎

###  A.3 Integration along the fiber

We recall the integration along the fiber associated to a strongly smooth map,
and the compatibility properties satisfied by the de Rham chains defined in
this way. The main references here are [22], Chapter 7 and [32], Section 7.1.

######  Theorem 77 ([32], Theorem 7.2).

Let $(X,\widehat{\mathcal{U}})$ be a compact, oriented K-space of dimension
$d$ without boundary, equipped with a strongly smooth map
$\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow\mathcal{L}_{k+1}$, a
differential form $\hat{\omega}$, and a CF-perturbation
$\widehat{\mathcal{S}}=(\widehat{\mathcal{S}}^{\varepsilon})_{0<\varepsilon\leq
1}$. We assume that $\widehat{\mathcal{S}}$ is transversal to $0$, and
$\mathit{ev}_{0}\circ\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow L$ is
strongly submersive with respect to $\widehat{\mathcal{S}}$. Then one can
define a de Rham chain

$\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\in
C_{d-|\hat{\omega}|}^{\mathit{dR}}(\mathcal{L}_{k+1})$ (A.8)

for sufficiently small $\varepsilon$, so that Stokes’ formula and fiber
product formula hold.

The Stokes’ formula and the fiber product formula in Theorem 77 are stated as
follows.

######  Theorem 78 (Stokes’ formula).

For sufficiently small $\varepsilon>0$, there holds

$\partial\left(\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\right)=(-1)^{|\hat{\omega}|+1}\hat{f}_{\ast}(X,\widehat{\mathcal{U}},d\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon}).$
(A.9)

Suppose for $i=1,2$, we are given the following data:

  * •

a compact oriented K-space $(X_{i},\widehat{\mathcal{U}}_{i})$ of dimension
$d_{i}$;

  * •

a strongly smooth map
$\hat{f}_{i}:(X_{i},\widehat{\mathcal{U}}_{i})\rightarrow\mathcal{L}_{k_{i}+1}$;

  * •

a differential form $\hat{\omega}_{i}$ on $(X_{i},\widehat{\mathcal{U}}_{i})$;

  * •

a CF-perturbation $\widehat{\mathcal{S}}_{i}$ on
$(X_{i},\widehat{\mathcal{U}}_{i})$ such that
$\mathit{ev}_{0}\circ\hat{f}_{i}:(X_{i},\widehat{\mathcal{U}}_{i})\rightarrow
L$ is strongly submersive with respect to $\widehat{\mathcal{S}}_{i}$.

For each $1\leq j\leq k_{1}$, we can take fiber product of K-spaces and define

$(X_{12},\widehat{\mathcal{U}}_{12}):=(X_{1},\widehat{\mathcal{U}}_{1})\textrm{
}{{}_{\mathit{ev}_{j}\circ\hat{f}_{1}}\times_{\mathit{ev}_{0}\circ\hat{f}_{2}}}\textrm{
}(X_{2},\widehat{\mathcal{U}}_{2}).$ (A.10)

On can also define fiber product of CF-perturbations
$\widehat{\mathcal{S}_{12}}:=\widehat{\mathcal{S}}_{1}\times\widehat{\mathcal{S}}_{2}$
on $(X_{12},\widehat{\mathcal{U}}_{12})$ ([22], §10.2). Finally, define a
differential form $\hat{\omega}_{12}$ on $(X_{12},\widehat{\mathcal{U}}_{12})$
by

$\hat{\omega}_{12}:=(-1)^{(d-|\hat{\omega}|_{1}-n)|\hat{\omega}_{2}|}\hat{\omega}_{1}\times\hat{\omega}_{2},$
(A.11)

and a strongly smooth map
$\hat{f}_{12}:(X_{12},\widehat{\mathcal{U}}_{12})\rightarrow\mathcal{L}_{k_{1}+k_{2}}$
by

$(f_{12})_{p_{1},p_{2}}(x_{1},x_{2}):=\mathit{con}_{j}\left((f_{1})_{p_{1}}(x_{1}),(f_{2})_{p_{2}}(x_{2})\right),$
(A.12)

where $x_{1}\in U_{p_{1}}$, $x_{2}\in U_{p_{2}}$, and $\mathit{ev}_{j}\circ
f_{p_{1}}(x_{1})=\mathit{ev}_{0}\circ f_{p_{2}}(x_{2})$.

######  Theorem 79 (Fiber product formula).

We have

$(\hat{f}_{12})_{\ast}(X_{12},\widehat{\mathcal{U}}_{12},\hat{\omega}_{12},\widehat{\mathcal{S}}_{12}^{\varepsilon})=(\hat{f}_{1})_{\ast}(X_{1},\widehat{\mathcal{U}}_{1},\hat{\omega}_{1},\widehat{\mathcal{S}}_{1}^{\varepsilon})\circ_{j}(\hat{f}_{2})_{\ast}(X_{2},\widehat{\mathcal{U}}_{2},\hat{\omega}_{2},\widehat{\mathcal{S}}_{2}^{\varepsilon})$
(A.13)

for sufficiently small $\varepsilon>0$.

######  Remark 80.

There are versions of Theorem 77 for admissible K-spaces
$(X,\widehat{\mathcal{U}})$, in which case the strongly smooth map $\hat{f}$
is replaced with an admissible map, and the strong submersion
$\mathit{ev}_{0}\circ\hat{f}$ is replaced with a stratified strong submersion,
and the Stokes’ formula now takes the form

$\partial\left(\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\right)=(-1)^{|\hat{\omega}|}(\hat{f}|_{\partial
X})_{\ast}(\partial X,\widehat{\mathcal{U}}|_{\partial
X},\hat{\omega}|_{\partial X},\widehat{\mathcal{S}}|_{\partial
X})+(-1)^{|\hat{\omega}|+1}\hat{f}_{\ast}(X,\widehat{\mathcal{U}},d\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon}),$
(A.14)

where $\partial X$ is the normalized boundary of $X$, which is itself a
K-space.

As another variation, we can consider admissible maps
$\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow[a,b]\times\mathcal{L}_{k+1}$, in
which case $\partial X=\partial_{h}X\sqcup\partial_{v}X$ is decomposed as
horizontal and vertical boundaries, where
$\partial_{h}X=f^{-1}\left(\\{a,b\\}\times\mathcal{L}_{k+1}\right)$. Let
$\partial_{-}X=f^{-1}(\\{a\\}\times\mathcal{L}_{k+1})$ and
$\partial_{+}X=f^{-1}(\\{b\\}\times\mathcal{L}_{k+1})$. In this case, we
require

$\left(\mathit{pr}_{[a,b]}\circ\hat{f},\mathit{ev}_{0}\circ\mathit{pr}_{\mathcal{L}_{k+1}}\circ\hat{f}\right):(X,\widehat{\mathcal{U}})\rightarrow[a,b]\times
L$ (A.15)

to be a corner-stratified strong submersion, and the pushforward defines a
relative de Rham chain

$\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\in\overline{C}_{d-|\hat{\omega}|-1}^{\mathit{dR}}(\mathcal{L}_{k+1})$
(A.16)

for sufficiently small $\varepsilon>0$, which satisfies

$e_{\pm}\left(\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\right)=(\hat{f}|_{\partial_{\pm}X})_{\ast}\left(\partial_{\pm}X,\widehat{\mathcal{U}}|_{\partial_{\pm}X},\hat{\omega}|_{\partial_{\pm}X},\widehat{\mathcal{S}}^{\varepsilon}|_{\partial_{\pm}X}\right)\in
C_{d-|\hat{\omega}|-1}^{\mathit{dR}}(\mathcal{L}_{k+1}).$ (A.17)

The Stokes’ formula in this case is

$\begin{split}\partial\left(\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\right)&=(-1)^{|\hat{\omega}|}(\hat{f}|_{\partial_{h}X})_{\ast}\left(\partial
X,\widehat{\mathcal{U}}|_{\partial_{h}X},\hat{\omega}|_{\partial_{h}X},\widehat{\mathcal{S}}|_{\partial_{h}X}\right)\\\
&+(-1)^{|\hat{\omega}|+1}\hat{f}_{\ast}(X,\widehat{\mathcal{U}},d\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon}).\end{split}$
(A.18)

For the purposes of this paper (cf. Remark 45), we need an additional property
of the de Rham chains defined by integration along the fiber. For any $a\in
H_{1}(L;\mathbb{Z})$, and $x$ a 1-periodic orbit of the Hamiltonian vector
field $X_{H_{t}}$, let $(X,\widehat{\mathcal{U}})$ be one of the
$\frac{1}{2}$-collared K-spaces
$\overline{\overline{\mathcal{R}}}_{k+1}(L,\bar{a};P)$,
$\overline{\overline{\mathcal{R}}}_{k+1,\vartheta}(L,\bar{a};P)$,
${}_{l}\overline{\overline{\mathcal{R}}}_{k+1}^{1}(x,L,\ring{\bar{a}};P)$,
${}_{l-1}\overline{\overline{\mathcal{R}}}_{k+1}^{S^{1}}(x,L,\ring{\bar{a}};P)$
or
${}_{l}^{j,j+1}\overline{\overline{\mathcal{R}}}_{k+1}^{1}(x,L,\ring{\bar{a}};P)$
considered in Section 4.5. Cyclic permutations of the labels of the boundary
marked points $z_{0},\cdots,z_{k}$ defines a $\mathbb{Z}_{k+1}$-action on $X$,
whose generator is a map $\kappa:X\rightarrow X$. It follows from Theorem 44,
(vii) and Proposition 76 that $X$ can be equipped with a Kuranishi structure
$\widehat{\mathcal{U}}$ and a CF-perturbation
$\widehat{\mathcal{S}}^{\varepsilon}$ that is cyclic invariant, therefore the
map $\kappa$ extends to a map

$\kappa:(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\rightarrow(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon}),$
(A.19)

if the differential form $\hat{\omega}$ is cyclic invariant. The following
theorem is a straightforward consequence of the cyclic invariance of these
data. For simplicity, we assume below that $P=\\{m\\}$. The statement for the
$P=[m,m+1]$ case is similar, simply replace $R_{k}$ with $\overline{R}_{k}$.

######  Theorem 81 (Cyclic permutation formula).

Let the quadruple
$(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})$
be as above, and assume that $\varepsilon>0$ is sufficiently small. Then for
$1\leq i\leq k$,

$\hat{f}_{\ast}\left(\kappa^{i}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\right)=(-1)^{ki}(R_{k})^{i}_{\ast}\left(\hat{f}_{\ast}(X,\widehat{\mathcal{U}},\hat{\omega},\widehat{\mathcal{S}}^{\varepsilon})\right).$
(A.20)

Here, the admissible map $\hat{f}:(X,\widehat{\mathcal{U}})\rightarrow
P\times\mathcal{L}_{k+1}$ is given by one of the maps (4.149), (4.150),
(4.151), (4.152) or (4.154).

###### Proof.

We only need to justify the signs. Note that cyclic permutation of the
boundary marked points from $z_{0},z_{1},\cdots,z_{k}$ to
$z_{k},z_{0}\cdots,z_{k-1}$ changes the orientations of the abstract moduli
spaces from $-dz_{1}\wedge\cdots\wedge dz_{k}$ to $dz_{2}\wedge\cdots\wedge
dz_{k}\wedge dz_{1}$ (with $z_{0}$ fixed at $1$, and neglecting the choices of
orientations for the interior marked points $p_{1},\cdots,p_{l}$, which all
agree. cf. [26], Remark 46). Thus a sign $(-1)^{k}$ needs to be introduced for
each such permutation. ∎

##  Appendix B Orientations of moduli spaces

In this appendix, we discuss the orientations of the moduli spaces considered
in this paper and compute the signs $\varepsilon_{1},\cdots,\varepsilon_{22}$
in Theorem 44, (iv). As before, we assume $M$ is a Liouville manifold with
$c_{1}(M)=0$ and $L\subset M$ is a Lagrangian submanifold which is equipped
with some fixed choice of a $\mathit{Spin}$ structure relative to the
$\mathbb{Z}_{2}$-gerbe $\alpha$.

The orientation of the moduli space $\overline{\mathcal{R}}_{k+1}(L,\beta)$
has been fixed in [32], Section 7.2.3. We shall orient
$\overline{\mathcal{R}}_{k+1,\vartheta}(L,\beta)$ in the same way as
$\overline{\mathcal{R}}_{k+1}(L,\beta)$. More precisely, let
$\mathcal{R}(L,\beta)$ be the moduli space of pseudoholomorphic discs without
boundary marked points, which carries its canonical orientation defined in
[19], Section 8.1. $\overline{\mathcal{R}}_{k+1,\vartheta}(L,\beta)$ is
oriented so that the natural isomorphism

$T\left(\overline{\mathcal{R}}_{k+1,\vartheta}(L,\beta)\right)\oplus
T\left(\mathit{Aut}(D)\right)\cong
T\left(\overline{\mathcal{R}}(L,\beta)\right)\oplus T(\partial D)^{\oplus k}$
(B.1)

is orientation-preserving, where $\partial D$ is equipped with the
anticlockwise orientation, and $\mathit{Aut}(D)$ is oriented via the
diffeomorphism $\mathit{Aut}(D)\cong(\partial D)^{3}$. It follows from the
sign computation in the case of $\overline{\mathcal{R}}_{k}(L,\beta)$ that

$\varepsilon_{1}=(k_{1}-i)(k_{2}-1)+n+k_{1}-1$ (B.2)

in (4.78).

To compute the signs
$\varepsilon_{2},\varepsilon_{3},\varepsilon_{4,0},\cdots,\varepsilon_{4,l},\varepsilon_{5},\varepsilon_{6}$
in (4.79), we first pick orientations for the abstract moduli spaces. On a
representative of the moduli spaces ${}_{l}\mathcal{R}_{k+1}^{1}$ and
${}_{l-1}\mathcal{R}_{k+1}^{S^{1}}$ which fixes $z_{0}=1$ and $\zeta=0$, we
pick the volume forms

$-r_{1}\cdots r_{l}dz_{1}\wedge\cdots\wedge dz_{k}\wedge dr_{l}\wedge
d\theta_{l}\wedge\cdots\wedge dr_{1}\wedge d\theta_{1}$ (B.3)

and

$r_{1}\cdots r_{l}dz_{1}\wedge\cdots\wedge dz_{k}\wedge d\theta_{1}\wedge
dr_{l}\wedge d\theta_{l}\wedge\cdots\wedge dr_{2}\wedge d\theta_{2}$ (B.4)

for ${}_{l}\mathcal{R}_{k+1}^{1}$ and ${}_{l-1}\mathcal{R}_{k+1}^{S^{1}}$,
respectively, where $(r_{j},\theta_{j})$ is the polar coordinates for the
auxiliary marked point $p_{j}$. For the moduli spaces
${}_{l}\mathcal{R}_{k+1,\tau_{i}}^{1}$, we shall equip them with orientations
which are compatible with the auxiliary-rescaling map (4.36). More precisely,
for a representative with $z_{0}=1$ and $\zeta=0$, we pick the volume form

$r_{1}\cdots r_{l}dz_{1}\wedge\cdots\wedge dz_{k}\wedge dz_{f}\wedge
dr_{l}\wedge d\theta_{l}\wedge\cdots\wedge dr_{1}\wedge d\theta_{1}.$ (B.5)

The space of $l$-point angle decorated cylinders admits a canonical complex
orientation, and we trivialize the $\mathbb{R}$-action on it by choosing
$\partial_{s}$ as the vector field inducing the action. The quotient gives the
moduli space ${}_{l}\mathcal{M}$. Using this convention, we have an
isomorphism

$\langle\partial_{s}\rangle\otimes\lambda^{\mathit{top}}\left(T{}_{l}\overline{\mathcal{M}}(x,y)\right)\cong
o_{x}\otimes o_{y}^{-1},$ (B.6)

where $o_{x}$ and $o_{y}$ are orientation lines associated to the Hamiltonian
orbits $x$ and $y$, respectively, and $\lambda^{\mathit{top}}$ stands for the
top degree exterior power. As in the definition of the Floer differential,
when counting rigid elements of ${}_{l}\overline{\mathcal{M}}(x,y)$ in the
definition of the operations $\delta_{l}$, we twist by the sign $(-1)^{|y|}$.

The moduli space ${}_{l}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\beta})$ is
oriented by combining the ideas of [5], Appendix A and [19], Chapter 8. More
precisely, let $D_{u}$ be the linearization of the perturbed Cauchy-Riemann
operator, then we have an isomorphism of line bundles

$\det(D_{u})\otimes
o_{x}\otimes\kappa_{x}^{\alpha}\cong\lambda^{\mathit{top}}(\mathit{TL}),$
(B.7)

where $\kappa_{x}^{\alpha}$ is the background line determined by our choice of
the relative $\mathit{Spin}$ structure on $L$, the orbit $x$, and the
background class $[\alpha]\in H^{2}(M;\mathbb{Z}_{2})$ (cf. [5], Definition
3.1). Combining (B.6) and (B.7), it follows that on a boundary stratum of the
form
${}_{j}\overline{\mathcal{M}}(x,y_{j})\times{}_{l-j}\overline{\mathcal{R}}(y_{j},L,\ring{\beta})$,
we have a natural isomorphism

$\lambda^{\mathit{top}}\left(T{}_{l-j}\overline{\mathcal{R}}(y_{j},L,\ring{\beta})\right)\otimes\langle\partial_{s}\rangle\otimes\lambda^{\mathit{top}}\left(T{}_{j}\overline{\mathcal{M}}(x,y_{j})\right)\cong\lambda^{\mathit{top}}(\mathit{TL})\otimes(o_{x}\otimes\kappa_{x}^{\alpha})^{-1}.$
(B.8)

Comparing the orientation of
${}_{j}\overline{\mathcal{M}}(x,y_{j})\times{}_{l-j}\overline{\mathcal{R}}(y_{j},L,\ring{\beta})\subset\partial{}_{l}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\beta})$
with (B.8), we obtain

$\varepsilon_{4,j}=n+|y_{j}|.$ (B.9)

For the boundary strata
${}_{l}^{j,j+1}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\beta})$ and
${}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^{1}}(x,L,\ring{\beta})$, we may
orient them so that the natural maps

${}_{l}^{j,j+1}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\beta})\rightarrow
S^{1}\times{}_{l-1}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\beta})$ (B.10)

and

${}_{l-1}\overline{\mathcal{R}}_{k+1}^{S^{1}}(x,L,\ring{\beta})\rightarrow
S^{1}\times{}_{l-1}\overline{\mathcal{R}}_{k+1}^{1}(x,L,\ring{\beta})$ (B.11)

induced from the forgetful maps (4.27) and (4.28) are oriented
diffeomorphisms. This provides us with an inductive way to orient the moduli
spaces so that

$\varepsilon_{5}=\varepsilon_{6}=0.$ (B.12)

It remains to determine the signs $\varepsilon_{2}$ and $\varepsilon_{3}$. For
this purpose, we introduce the moduli spaces
${}_{l}\widetilde{\mathcal{R}}^{1}(x,L,\ring{\beta})$ of pairs
$\left((S,p_{1},\cdots,p_{l};\ell),u\right)$, where $u:S\rightarrow M$
satisfies (4.45) and $[u]=\ring{\beta}$, but now we do not quotient by
$\mathit{Aut}(S)\cong S^{1}$. Similarly, define
$\widetilde{\mathcal{R}}(L,\beta)$ to be the space of pseudoholomorphic maps
$u:(D,\partial D)\rightarrow(M,L)$ with $[u]=\beta$, but without modding out
$\mathit{Aut}(D)\cong\mathit{PSL}(2,\mathbb{R})$. Consider the gluing map

$\mathit{gl}:{}_{l}\widetilde{\mathcal{R}}^{1}(x,L,\ring{\beta}_{1})\textrm{
}{{}_{1}\times_{-1}}\textrm{
}\widetilde{\mathcal{R}}(L,\beta_{2})\rightarrow{}_{l}\widetilde{\mathcal{R}}^{1}(x,L,\ring{\beta}),\textrm{
}\ring{\beta}_{1}+\beta_{2}=\ring{\beta},$ (B.13)

where the notation ${}_{1}\times_{-1}$ means the fiber product on the left-
hand side is taken with respect to the evaluation maps at $1\in\partial S$ on
the first component, and at $-1\in\partial D$ on the second component. The
argument of [19], Lemma 8.3.10 applies to our case and implies the following:

######  Lemma 82.

The map $\mathit{gl}$ is orientation-preserving.

Using this lemma, the signs $\varepsilon_{2}$ and $\varepsilon_{3}$ can be
computed in the same way as [19], Section 8.3. We have

$\varepsilon_{2}=(k_{1}-i)(k_{2}-1)+n+k\textrm{ and
}\varepsilon_{3}=(k_{1}-i)(k_{2}-1)+n+1.$ (B.14)

The remaining signs
$\varepsilon_{7},\cdots,\varepsilon_{13},\varepsilon_{14,0},\cdots,\varepsilon_{14,l},\varepsilon_{15},\varepsilon_{16}$
are determined from the signs
$\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4,0},\cdots,\varepsilon_{4,l},\varepsilon_{5},\varepsilon_{6}$
using the formulae

$\partial([m,m+1]\times X)=\\{m+1\\}\times X\sqcup(-1)\\{m\\}\times
X\sqcup(-1)[m,m+1]\times\partial X,$ (B.15)
$[m,m+1]\times(X\times_{L}Y)=([m,m+1]\times X)\times_{[m,m+1]\times
L}([m,m+1]\times Y),$ (B.16)

where $X$ and $Y$ are admissible K-spaces.

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Department of Mathematics, Columbia University, NY 10027, United States

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