[affiliation=1]AnbaiJiang [affiliation=2]BingHan [affiliation=3]ZhiqiangLv
[affiliation=3]YufengDeng [affiliation=1,*]Wei-QiangZhang
[affiliation=2]XieChen [affiliation=2]YanminQian [affiliation=1,3]JiaLiu
[affiliation=1,*]PingyiFan

# AnoPatch: Towards Better Consistency in Machine Anomalous Sound Detection

###### Abstract

Large pre-trained models have demonstrated dominant performances in multiple
areas, where the consistency between pre-training and fine-tuning is the key
to success. However, few works reported satisfactory results of pre-trained
models for the machine anomalous sound detection (ASD) task. This may be
caused by the inconsistency of the pre-trained model and the inductive bias of
machine audio, resulting in inconsistency in data and architecture. Thus, we
propose AnoPatch which utilizes a ViT backbone pre-trained on AudioSet and
fine-tunes it on machine audio. It is believed that machine audio is more
related to audio datasets than speech datasets, and modeling it from patch
level suits the sparsity of machine audio. As a result, AnoPatch showcases
state-of-the-art (SOTA) performances on the DCASE 2020 ASD dataset and the
DCASE 2023 ASD dataset. We also compare multiple pre-trained models and
empirically demonstrate that better consistency yields considerable
improvement.

###### keywords:

machine audio, anomaly detection, pre-trained models, patch

##  1 Introduction

Machine anomalous sound detection (ASD) aims to predict whether the sound of a
machine is normal or anomalous when only normal sounds are provided as prior.
Recent years saw a growing need for robust ASD methods in the field of modern
manufacturing, where machine audio under normal conditions can be utilized to
detect unknown malfunctions, which has been demonstrated in some scenarios to
significantly reduce risks and boost production efficiency.††* Corresponding
authors

Like any binary classification task, the machine ASD task has to balance
between recall and precision. On one hand, to increase recall, the model is
required to overfit with the normal distribution such that any deviation from
the normal distribution yields a perceivable variation in the output. On the
other hand, to increase precision, the model is required to grasp the
universal features of normal samples and generalize well on unseen samples.
However, these requirements become quite challenging for the ASD model, since
the decision boundary can not be accurately determined when (i) only one class
is presented for training and (ii) the normal distribution is sparse.

To implicitly derive the ASD boundary, training the model by deputy tasks is a
common solution in previous literature[1, 2, 3]. However, since the normal
distribution is sparse and the training data is insufficient, training the
model from scratch tends to yield non-scalable boundaries, especially when the
machine has diverse working conditions. Thus, we opt to initialize the model
by the parameters of a large pre-trained model that has obtained common
knowledge of speech and audio by pre-training on extensive data, which enables
the model to learn scalable boundaries when fine-tuned on machine audio.

Specifically, we focus on pre-trained models that are originally intended for
audio classification and model the audio from patch level, which involves a
dual consideration of data consistency and architecture consistency. To
guarantee production efficiency, machines are operated steadily which can be
modeled as stationary processes, while speech generally possesses more
variations over time. It is believed that machine audio is more information-
sparse than speech, which is similar to the difference between vision and
language [4]. On one hand, fine-tuning audio classification models yields
better consistency between pre-training data and fine-tuning data. On the
other hand, modeling from patch level is more consistent with the inductive
bias of machine audio than modeling from frame level. Here patch level refers
to modeling the patches in audio spectrograms, while frame level refers to
modeling the time frames in waveforms or spectrograms. Recent works have
demonstrated that modeling the audio from patch level generally has superior
performances in audio classification tasks to modeling it from frame level [5,
6]. Thus, a pre-trained model with patch-level representation is adopted, and
we will empirically demonstrate its superiority for the machine ASD task.

In this paper, we propose AnoPatch, in which a ViT [7] backbone is trained to
extract patch-level representations from mel-spectrograms. The parameters of
the ViT backbone are initialized by a pre-trained audio model, and the
backbone is further fine-tuned on machine audio by classifying the metadata
associated with the machine (e.g. machine type, entity ID, speed). During
detection, all patch-level representations are merged into a general embedding
for each clip, and KNN [8] is applied to the embedding for anomaly detection.

Extensive experiments are conducted on the datasets of DCASE 2020 Task 2 [9]
and DCASE 2023 Task 2 [10], both of which are machine ASD datasets with
multiple machine types. AnoPatch achieves state-of-the-art (SOTA) performances
on both datasets without leveraging model ensembling. We also compare multiple
pre-trained speech and audio models and empirically demonstrate that
consistency is crucial when adapting pre-trained models to the machine ASD
task.

The main contributions of this paper can be summarized:

  * •

We propose AnoPatch which fine-tunes a pre-trained ViT backbone for the
machine ASD task in consideration of better consistency.

  * •

AnoPatch achieves SOTA performances on both the DCASE 2020 dataset and the
DCASE 2023 dataset.

  * •

We empirically demonstrate that pre-trained models with better consistency
tend to possess better ASD capability.

##  2 Related Work

###  2.1 Pre-trained Models for Speech and Audio Tasks

A significant amount of pre-trained models have emerged in recent years,
continuously setting up new SOTA performances on various speech and audio
tasks. Most models have a focal point on either speech tasks or audio tasks,
although some may generalize on both domains.

As for models focusing on speech tasks, Wav2Vec 2.0 [11] quantizes raw
waveforms into discrete units and models them by a transformer-based encoder.
HuBERT [12] generates pseudo labels by clustering on mel spectrograms while
adopting the network of Wav2Vec 2.0. UniSpeech [13] combines contrastive
learning with supervised learning. WavLM [14] improves HuBERT by applying
multiple data augmentations.

As for models focusing on audio tasks [15], AST [16] fine-tunes a ViT model
which is pre-trained on images for audio classification. BEATs [17]
iteratively trains a ViT backbone and an acoustic tokenizer in which the
acoustic tokenizer assigns pseudo labels for unlabeled data. Imagebind [18]
employs a ViT encoder to align audio with multiple modalities. Recently both
ATST [5] and EAT [6] utilize self-teaching techniques to unsupervisedly train
a ViT model.

It is noted that speech models tend to model speech by frames, while audio
models tend to model audio by patches.

###  2.2 Approaches for Machine ASD

Approaches for machine ASD can be roughly divided into disentangled approaches
and end-to-end approaches. Disentangled approaches [3] employ neural networks
to extract semantic features of machine audio and apply statistical anomaly
detection algorithms to these features. Commonly used detection algorithms are
KNN [8], LOF [19], and Gaussian mixture models (GMM). End-to-end models employ
neural networks to directly estimate the anomaly score. Amongst all end-to-end
models, autoencoder [9, 20] assumes that anomalous samples may have bigger
reconstruction error when trained to reconstruct normal samples, and flow
model [21] directly models the normal distribution and estimates the
likelihood.

###  2.3 Pre-trained Models for Machine ASD

Compared to the prevalent applications of pre-trained models for speech and
audio tasks, few works explored the use of pre-trained models for machine ASD,
probably because pre-trained features have not shown much superiority against
pre-defined features. Previous works [1, 22] explored the use of OpenL3 [23]
embeddings for the machine ASD task, yet none of which produced satisfactory
results. Recently Han et al. [24] explored the use of multiple pre-trained
speech models [11, 12, 13, 14], which outperform multiple systems in the DCASE
2023 challenge.

We believe such a result may result from the difference between speech and
machine audio. As pointed out in Section 1, machine audio is more stationary
and information-sparse compared to speech, which is more suitable for modeling
it from patch level. On one hand, the parameters obtained from pre-training on
speech data may not scale well on machine audio. On the other hand, since the
inductive bias is different, the architecture suitable for speech tasks may
not apply well to the machine ASD task. Therefore, we propose to fine-tune
pre-trained models that are intended for audio classification and leverage
patch-level representation.

Table 1: Comparison on the DCASE 2020 dataset between AnoPatch and previous SOTA models Models | Development set | Evaluation set | All
---|---|---|---
fan | pump | slider | ToyCar | ToyConveyor | valve | mean | fan | pump | slider | ToyCar | ToyConveyor | valve | mean | mean
No. 1 [25] | 80.65 | 83.27 | 93.41 | 92.72 | 73.28 | 94.30 | 86.27 | 89.42 | 87.69 | 93.68 | 92.04 | 82.27 | 93.51 | 89.77 | 88.02
Sub-Cluster AdaCos [1] | 82.77 | 91.81 | 98.59 | 94.01 | 76.77 | 96.63 | 90.10 | 95.42 | 92.53 | 93.54 | 93.96 | 84.94 | 97.31 | 92.95 | 91.53
MFN [3] | 83.71 | 90.82 | 98.70 | 91.97 | 71.29 | 96.49 | 88.83 | 94.72 | 92.94 | 97.58 | 94.31 | 77.54 | 94.88 | 92.00 | 90.38
STgram [2] | 91.51 | 86.85 | 98.58 | 91.06 | 69.09 | 99.04 | 89.35 | - | - | - | - | - | - | - | -
AnoPatch (Ours) | 86.46 | 93.10 | 99.20 | 96.10 | 73.20 | 97.53 | 90.93 | 95.56 | 94.34 | 99.77 | 96.00 | 83.74 | 96.26 | 94.28 | 92.58
Table 2: Comparison on the DCASE 2023 dataset between AnoPatch and previous SOTA models Models | Development set | Evaluation set | All
---|---|---|---
bearing | fan | gearbox | slider | ToyCar | ToyTrain | valve | hmean | bandsaw | grinder | shaker | ToyDrone | ToyNscale | ToyTank | Vacuum | hmean | hmean
No. 1 [26] | 64.41 | 76.27 | 74.78 | 91.83 | 51.66 | 53.17 | 85.44 | 68.11 | 60.97 | 65.18 | 63.50 | 55.71 | 84.92 | 60.72 | 92.27 | 66.97 | 67.54
No. 2 [27] | 72.39 | 62.41 | 74.41 | 87.84 | 59.10 | 58.67 | 65.53 | 67.38 | 55.47 | 64.76 | 70.98 | 52.89 | 71.90 | 70.73 | 91.48 | 66.39 | 66.88
Han et al. [24] | 57.10 | 62.76 | 67.52 | 79.11 | 63.47 | 57.35 | 67.79 | 64.31 | - | - | - | - | - | - | - | - | -
FeatEx [28] | - | - | - | - | - | - | - | 66.95 | - | - | - | - | - | - | - | 68.52 | 67.73
AnoPatch (Ours) | 70.43 | 66.65 | 58.67 | 81.88 | 58.78 | 67.16 | 53.73 | 64.24 | 69.71 | 64.06 | 80.25 | 64.49 | 85.04 | 72.60 | 92.24 | 74.23 | 68.87
Table 3: Comparison on the DCASE 2020 dataset between multiple pre-trained models Set | Domain | Models | Size | fan | pump | slider | ToyCar | ToyConveyor | valve | mean
---|---|---|---|---|---|---|---|---|---|---
dev | Speech | Wav2Vec 2.0 [11] | 316M | 79.04 | 80.16 | 89.55 | 94.04 | 70.11 | 92.49 | 84.23
HuBERT [12] | 316M | 75.81 | 84.99 | 85.13 | 92.44 | 67.91 | 92.37 | 83.11
UniSpeech [13] | 316M | 77.13 | 84.64 | 91.31 | 92.41 | 71.87 | 93.51 | 85.14
WavLM [14] | 316M | 79.37 | 85.52 | 86.96 | 92.93 | 69.23 | 94.69 | 84.78
Audio | AST [16] | 86M | 80.51 | 82.34 | 98.21 | 91.37 | 61.15 | 97.55 | 85.19
Imagebind [18] | 86M | 82.44 | 87.98 | 98.41 | 92.24 | 68.86 | 91.42 | 86.89
ATST [5] | 85M | 76.22 | 82.81 | 89.46 | 94.85 | 72.02 | 95.34 | 85.12
BEATs [17] (Adopted) | 90M | 84.01 | 92.87 | 94.72 | 96.55 | 71.42 | 99.62 | 89.87
eval | Speech | Wav2Vec 2.0 [11] | 316M | 85.67 | 81.03 | 87.86 | 93.23 | 82.29 | 93.76 | 87.31
HuBERT [12] | 316M | 82.42 | 76.76 | 95.69 | 92.30 | 78.51 | 95.61 | 86.88
UniSpeech [13] | 316M | 83.33 | 79.39 | 92.98 | 93.96 | 81.91 | 91.89 | 87.24
WavLM [14] | 316M | 84.35 | 77.72 | 94.53 | 92.40 | 77.13 | 90.16 | 86.05
Audio | AST [16] | 86M | 90.68 | 86.08 | 93.92 | 90.65 | 63.91 | 92.14 | 86.23
Imagebind [18] | 86M | 91.56 | 88.92 | 95.65 | 92.36 | 70.74 | 87.27 | 87.75
ATST [5] | 85M | 86.76 | 84.11 | 88.07 | 94.04 | 79.87 | 96.87 | 88.29
BEATs [17] (Adopted) | 90M | 93.98 | 93.83 | 98.05 | 96.95 | 84.24 | 94.53 | 93.60

##  3 Proposed Method

Figure 1: Architecture of AnoPatch

###  3.1 Backbone

Figure 1 illustrates the overall architecture of AnoPatch, in which a ViT
backbone is adopted to extract patch-level representations of machine audio.
The parameters of the ViT backbone are initialized from BEATs [17] in
consideration of better consistency between pre-training and fine-tuning,
since BEATs is pre-trained on AudioSet for audio classification purposes. The
input of AnoPatch is a mel-spectrogram which is then split into 16 $\times$ 16
patches. The ViT backbone models each patch as a single token and outputs an
embedding for each patch. An attentive statistics pooling layer from ECAPA-
TDNN [29] is employed to merge patch embeddings into an utterance embedding
$u_{i}$. Finally, the utterance embedding $u_{i}$ is linearly mapped to a low-
dimensional embedding $x_{i}$ which is further utilized for anomaly detection.

###  3.2 Fine-Tuning

AnoPatch is fine-tuned on machine audio by classifying the metadata associated
with the machine, in which each unique combination of the metadata is
considered as a new label. Since the classification task is rather easy,
ArcFace loss [30] is adopted to reinforce the task, which can be formulated as
follows:

$L=-\frac{1}{N}\sum\limits_{i=1}^{N}\log\frac{e^{s\cos(\theta_{y_{i}}+m)}}{e^{s\cos(\theta_{y_{i}}+m)}+\sum\limits_{j=1,j{\neq}y_{i}}^{c}e^{s\cos\theta_{j}}}$
(1)

where $y_{i}$ is the label of sample $i$, and $s$ and $m$ are two
hyperparameters. $\theta_{j}$ is the angle between the embedding of sample $i$
and the registered embedding of the j-th class, which is the j-th column of
the weight $W$ of the classification head:

$\theta_{j}=\arccos\left(\frac{W_{j}^{T}x_{i}}{{\lVert
W_{j}\rVert}_{2}{\cdot}{\lVert x_{i}\rVert}_{2}}\right)$ (2)

where $x_{i}$ is the final output of the backbone, $T$ represents the
transpose operation, and $\lVert{\cdot}\rVert$ denotes the L2 norm distance.

To further strengthen the fine-tuning task, SpecAug [31] is adopted before
splitting the spectrogram into patches, which is consistent with BEATs. It is
noted that time warping is removed since it may be associated with anomalies.

###  3.3 Anomaly Detection

KNN [8] is adopted as the anomaly detection backend. Embeddings of normal
samples form a memory bank, and the cosine distance between a query embedding
and its closest neighbor is utilized as the anomaly score ($k=1$). For the
DCASE 2020 dataset, KNN is individually applied to clips of the same machine
type and machine ID. For the DCASE 2023 dataset, KNN is individually applied
to each machine type, and we also adopt a soft scoring mechanism [10] where
two KNN detectors are initialized by source samples and target samples
respectively, and the minimum score given by two detectors is selected.

##  4 Experiment

###  4.1 Datasets

Experiments are conducted on the datasets of DCASE 2020 Task 2 [9] and DCASE
2023 Task 2 [10], which feature 7 and 14 machine types respectively. The DCASE
2020 dataset can be divided into a development set and an evaluation set, both
of which contain a training subset with only normal audio clips, and a test
subset with normal and anomalous audio clips. Machine type and machine ID are
provided as metadata, where machine ID corresponds to different entities of a
type of machine. The DCASE 2023 dataset shares a similar structure but
introduces domain shift to reflect diverse working conditions. That is, for
the training subset of each machine type, 990 clips are from the source
domain, while only 10 clips are from the target domain. Machine types and
attributes of working conditions are provided as metadata.

Performances are evaluated by the area under the receiver operating
characteristic (ROC) curve (AUC) and the partial-AUC (pAUC) following the
challenge rules [9, 10]. For the DCASE 2020 dataset, we report the arithmetic
mean of all AUC and pAUC for each machine type, and an arithmetic mean of the
whole set. For the DCASE 2023 dataset, we report the harmonic mean of all AUC
and pAUC for each machine type, and a harmonic mean of the whole set.

###  4.2 Baseline Systems

Multiple SOTA models are adopted as baselines. For the DCASE 2020 dataset, we
adopt the best system of the challenge [25] and three single models [1, 3, 2]
as baselines. Since MobileFaceNet (MFN) [32] is frequently adopted in previous
literature [2, 3] with competitive performances, we implement it on the DCASE
2020 dataset as a decent baseline. For the DCASE 2023 dataset, we adopt the
top two systems of the challenge and two single models [24, 28] as baselines.
It is noted that most top-tier systems of both challenges are ensemble models,
while AnoPatch is a single model.

###  4.3 Implementation Details

Table 4: Detailed hyperparameter settings Items | DCASE 2020 | DCASE 2023
---|---|---
Number of classes | 41 | 167
SpecAug-freq | 40 | 80
SpecAug-time | 80 | 80
Total steps | 10k
Batch size | 32
Gradient accumulation | 8
Optimizer | AdamW [33]
Learning rate | 1e-4
Warmup steps | 960

The ViT backbone consists of 12 layers with 90M parameters in total, which is
initialized from BEATs (iteration 3 of pre-training). Identical with BEATs,
the audio waveform is converted to a mel-spectrogram with a windows size of
25ms, a hop size of 10ms, and 128 Mel bins, and then the network transforms
the whole spectrogram into a 128-dim embedding. Each unique combination of
machine type and machine ID is regarded as a unique class for the DCASE 2020
dataset, while each unique combination of machine type and attributes is
regarded as a unique class for the DCASE 2023 dataset. AnoPatch is trained by
data of all machine types, and all parameters are learnable. Experiments are
conducted separately on two datasets on an NVIDIA RTX A6000 GPU. Detailed
hyperparameter settings are listed in Table 4.

###  4.4 Experiment Results

Table 1 compares AnoPatch with previous SOTA models on the DCASE 2020 dataset.
AnoPatch achieves the best performance with a mean of 92.58 on both sets,
setting up a new milestone of 94.28 on the evaluation set.

Table 2 compares AnoPatch with previous SOTA models on the DCASE 2023 dataset.
As expected, AnoPatch outperforms all baseline models with a mean of 68.87 on
both sets, also setting up a new milestone of 74.23 on the evaluation set.

###  4.5 Comparison between Pre-trained Models

Multiple large pre-trained models for speech and audio tasks are compared on
the machine ASD task. Four speech models are employed, namely Wav2Vec 2.0
[11], HuBERT [12], UniSpeech [13], and WavLM [14], all of which are pre-
trained on speech datasets with a network suitable for grasping the dense
semantics in speech, inheriting data inconsistency and architecture
inconsistency. Four audio models are employed, namely AST [16], Imagebind
[18], ATST [5], and BEATs [17], all of which are trained on audio datasets and
model the audio from patch level. It is noted that AST is initialized from a
ViT model pre-trained on images, thus inheriting data inconsistency. All
models are fine-tuned by the same labels and ArcFace loss. Four speech models
are implemented following [24], while weighted sum, status augmentation, and
transformer pooling are removed for fair comparison. The fine-tuning processes
of three other audio models are mainly the same as BEATs. If the audio model
has incorporated a special token that aggregates the features over time (e.g.
[cls] token), this token is adopted as the utterance embedding $u_{i}$. If
not, an attentive statistics pooling layer is appended to the model to
aggregate the features. It is noted that SpecAug is also removed for a fair
comparison.

Table 3 compares the performances of these pre-trained models on the DCASE
2020 dataset. As for the development set, the best performance of speech
models is 85.14, while the worst performance of audio models is 85.19. As for
the evaluation set, all audio models except AST outperform the best speech
models, which can be explained by the data inconsistency of AST. The
improvement even goes up to 4.73 and 6.29 on two subsets. Thus, it is believed
that pre-trained audio models leveraging patch-level representation shall have
superior performances for the machine ASD task.

##  5 Conclusion

In this paper, we focused on adapting large pre-trained models to the machine
ASD task. It was believed that the inconsistency between pre-training and
fine-tuning deprecates the performances of previous attempts, which was
empirically proved by the comparison in Table 3. Therefore, we proposed
AnoPatch which contains a ViT backbone initialized by BEATs. As a result,
AnoPatch demonstrates SOTA results on the DCASE 2020 dataset and the DCASE
2023 dataset.

##  6 Acknowledgements

This work was supported by the National Key Research and Development Program
of China (Grant NO.2021YFA1000500(4)) and the National Natural Science
Foundation of China under Grant No. 62276153.

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Momentum Sections in Hamiltonian Mechanics and Sigma Models

Momentum Sections in Hamiltonian Mechanics
and Sigma Models

Noriaki IKEDA

N. Ikeda

Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga
525-8577, Japan<EMAIL_ADDRESS>
Received May 24, 2019, in final form September 29, 2019; Published online
October 03, 2019

We show a constrained Hamiltonian system and a gauged sigma model have a
structure of a momentum section and a Hamiltonian Lie algebroid theory
recently introduced by Blohmann and Weinstein. We propose a generalization of
a momentum section on a pre-multisymplectic manifold by considering gauged
sigma models on higher-dimensional manifolds.

symplectic geometry; Lie algebroid; Hamiltonian mechanics; nonlinear sigma
model

53D20; 70H33; 70S05

##  1 Introduction

Recently, relations of physical systems with a Lie algebroid structure and its
generalizations have been found and analyzed in many contexts. For instance, a
Lie algebroid [23] and a generalization such as a Courant algebroid appear in
topological sigma models [18], T-duality [9], quantizations, etc.

Blohmann and Weinstein [2] have proposed a generalization of a momentum map
and a Hamiltonian $G$-space on a Lie algebra (a Lie group) to Lie algebroid
setting, based on analysis of the general relativity [1]. It is called a
momentum section and a Hamiltonian Lie algebroid. This structure is also
regarded as reinterpretation of compatibility conditions of geometric
quantities such as a metric $g$ and a closed differential form $H$ with a Lie
algebroid structure, which was analyzed by Kotov and Strobl [21].

In this paper, we reinterpret geometric structures of physical theories as a
momentum section theory, and discuss how momentum sections naturally appear in
physical theories. Moreover, from this analysis, we will find a proper
definition of a momentum section on a pre-multisymplectic manifold.

We analyze a constrained Hamiltonian mechanics system with a Lie algebroid
structure discussed in the paper [19], and a two-dimensional gauged sigma
models [17] with a $2$-form B-field and one-dimensional boundary. In a
constrained Hamiltonian mechanics system, we consider a Hamiltonian and
constraint functions inhomogeneous with respect to the order of momenta. Then,
a zero-th order term in constraints is essentially a momentum section. In a
two-dimensional gauged sigma model, a pre-symplectic form is a B-field, and a
one-dimensional boundary term is a momentum section. Two examples are very
natural physical systems, thus, we can conclude that a momentum section is an
important geometric structure in physical theories.

Recently, a two-dimensional gauged sigma model with a $2$-form B-field with
three-dimensional Wess–Zumino term [17] is analyzed related to T-duality in
string theory [5, 6, 10, 11, 12, 13]. For such an application, it is
interesting to generalize a momentum section in a pre-multisymplectic
manifold. The string theory sigma model with an NS $3$-form flux $H$ is the
pre-$2$-symplectic case in our theory.

In this paper, we consider an $n$-dimensional gauged sigma model with
$(n+1)$-dimensional Wess–Zumino term. The Wess–Zumino term is constructed from
a closed $(n+1)$-form $H$, which defines a pre-$n$-plectic structure on a
target manifold $M$. For gauging, we introduce a vector bundle $E$ over $M$, a
connection $A$ on a world volume $\Sigma$ and a Lie algebroid connection
$\Gamma$ on a target vector bundle $E$. Consistency conditions of gauging give
geometric conditions on a series of extra geometric quantities
$\eta^{(k)}\in\Omega^{k}\big{(}M,\wedge^{n-k}E^{*}\big{)}$, $k=0,\dots,n-1$.
From this analysis, we propose a definition of a momentum section on a pre-
multisymplectic manifold. This definition comes from a natural physical
example, a gauged sigma model. We see that our definition of a momentum
section on a pre-multisymplectic manifold is a generalization of a momentum
map on a multisymplectic manifold [8, 14].

This paper is organized as follows. In Section 2, we explain definitions of a
momentum section and a Hamiltonian Lie algebroid. In Section 3, we show a
constrained Hamiltonian system has a momentum section. In Section 4, we
discuss a two-dimensional gauged sigma model with boundary and show a boundary
term gives a momentum section. In Section 5, we consider gauging conditions of
an $n$-dimensional gauged sigma model with a WZ term and propose a
generalization of a momentum section on a pre-multisymplectic manifold.
Section 6 is devoted to discussion and outlook.

##  2 Momentum section and Hamiltonian Lie algebroid

In this section, we review a momentum section and a Hamiltonian Lie algebroid
introduced in [2].

###  2.1 Lie algebroid

A Lie algebroid is a unified structure of a Lie algebra, a Lie algebra action
and vector fields on a manifold.

######  Definition 2.1.

Let $E$ be a vector bundle over a smooth manifold $M$. A Lie algebroid
$(E,\rho,[-,-])$ is a vector bundle $E$ with a bundle map $\rho\colon
E\rightarrow TM$ and a Lie bracket
$[-,-]\colon\Gamma(E)\allowbreak\times\Gamma(E)\rightarrow\Gamma(E)$
satisfying the Leibniz rule,

$\displaystyle[e_{1},fe_{2}]=f[e_{1},e_{2}]+\rho(e_{1})f\cdot e_{2},$

where $e_{i}\in\Gamma(E)$ and $f\in C^{\infty}(M)$.

A bundle map $\rho$ is called an anchor map.

######  Example 2.2.

Let a manifold $M$ be one point $M=\\{pt\\}$. Then a Lie algebroid is a Lie
algebra $\mathfrak{g}$.

######  Example 2.3.

If a vector bundle $E$ is a tangent bundle $TM$ and $\rho=\mathrm{id}$, then a
bracket $[-,-]$ is a normal Lie bracket of vector fields and
$(TM,\mathrm{id},[-,-])$ is a Lie algebroid.

######  Example 2.4.

Let $\mathfrak{g}$ be a Lie algebra and assume an infinitesimal action of
$\mathfrak{g}$ on a manifold $M$. The infinitesimal action $\mathfrak{g}\times
M\rightarrow M$ determines a map $\rho\colon M\times\mathfrak{g}\rightarrow
TM$. The consistency of a Lie bracket requires a Lie algebroid structure on
$(E=M\times\mathfrak{g},\rho,[-,-])$. This Lie algebroid is called an action
Lie algebroid.

######  Example 2.5.

An important nontrivial Lie algebroid is a Lie algebroid induced from a
Poisson structure. A bivector field $\pi\in\Gamma\big{(}{\wedge}^{2}TM\big{)}$
is called a Poisson structure if $[\pi,\pi]_{S}=0$, where $[-,-]_{S}$ is a
Schouten bracket on $\Gamma(\wedge^{\bullet}TM)$.

Let $(M,\pi)$ be a Poisson manifold. Then, we can define a bundle map,
$\pi^{\sharp}\colon T^{*}M\rightarrow TM$ by
$\pi^{\sharp}(\alpha)(\beta)=\pi(\alpha,\beta)$ for all
$\beta\in\Omega^{1}(M)$. A Lie bracket on $\Omega^{1}(M)$ is defined by the so
called Koszul bracket,

$\displaystyle[\alpha,\beta]_{\pi}=L_{\pi^{\sharp}(\alpha)}\beta-
L_{\pi^{\sharp}(\beta)}\alpha-{\rm d}(\pi(\alpha,\beta)),$

where $\alpha,\beta\in\Omega^{1}(M)$. Then,
$\big{(}T^{*}M,\pi^{\sharp},[-,-]_{\pi}\big{)}$ is a Lie algebroid.

One can refer to many other examples, for instance, in [23].

###  2.2 Lie algebroid differential

We consider a space of exterior products of sections,
$\Gamma(\wedge^{\bullet}E^{*})$ on a Lie algebroid $E$. Its element is called
an $E$-differential form. We can define a Lie algebroid differential
${}^{E}{\rm
d}\colon\Gamma\big{(}{\wedge}^{m}E^{*}\big{)}\rightarrow\Gamma\big{(}{\wedge}^{m+1}E^{*}\big{)}$
such that $\big{(}{}^{E}{\rm d}\big{)}^{2}=0$.

######  Definition 2.6.

A Lie algebroid differential ${}^{E}{\rm
d}\colon\Gamma\big{(}{\wedge}^{m}E^{*}\big{)}\rightarrow\Gamma\big{(}{\wedge}^{m+1}E^{*}\big{)}$
is defined by

$\displaystyle{}^{E}{\rm
d}\alpha(e_{1},\dots,e_{m+1})=\sum_{i=1}^{m+1}(-1)^{i-1}\rho(e_{i})\alpha(e_{1},\dots,\check{e_{i}},\dots,e_{m+1})$
$\displaystyle\hphantom{{}^{E}{\rm
d}\alpha(e_{1},\dots,e_{m+1})=}{}+\sum_{i,j}(-1)^{i+j}\alpha([e_{i},e_{j}],e_{1},\dots,\check{e_{i}},\dots,\check{e_{j}},\dots,e_{m+1}),$

where $\alpha\in\Gamma\big{(}{\wedge}^{m}E^{*}\big{)}$ and
$e_{i}\in\Gamma(E)$.

It is useful to describe Lie algebroids by means of $\mathbb{Z}$-graded
geometry [26]. A graded manifold ${\cal M}$ with local coordinates $x^{i}$,
$i=1,\dots,\dim M$, and $q^{a}$, $a=1,\dots,\operatorname{rank}E$, of degree
zero and one, respectively, are denoted by ${\cal M}=E[1]$ for some rank $r$
vector bundle $E$, where the degree one basis $q^{a}$ is identified by a
section in $E^{*}$, i.e., we identify
$C^{\infty}(E[1])\simeq\Gamma(\wedge^{\bullet}E)$. A product for homogeneous
elements $f,g\in C^{\infty}({\cal M})$ has a property, $fg=(-1)^{|f||g|}gf$,
where $|f|$ is degree of $f$. Especially, $q^{a}q^{b}=-q^{b}q^{a}$. We
introduce a derivation of degree $-1$, $\frac{\partial}{\partial q^{a}}$
satisfying $\frac{\partial}{\partial q^{a}}q^{b}=\delta^{b}_{a}$. Here, a
derivation is a linear operator on a space of functions satisfying the Leibniz
rule.

The most general degree plus one vector field on ${\cal M}$ has the form

$\displaystyle Q=\rho_{a}^{i}(x)q^{a}\frac{\partial}{\partial
x^{i}}-\frac{1}{2}C^{c}_{ab}(x)q^{a}q^{b}\frac{\partial}{\partial q^{c}},$

where $\rho_{a}^{i}(x)$ and $C^{c}_{ab}(x)$ are local functions of $x$. Since
$q^{a}$ is of degree $1$, $C^{c}_{ab}=-C^{c}_{ba}$.

Let $e_{a}$ be a local basis in $E$ dual to the basis corresponding to the
coordinates $q^{a}$. Two functions $\rho$ and $C$ in $Q$ define a bundle map
$\rho\colon E\rightarrow TM$ and a bilinear bracket on $\Gamma(E)$ by means of
$\rho(e_{a}):=\rho_{a}^{i}\partial_{i}$ and $[e_{a},e_{b}]:=C^{c}_{ab}e_{c}$.
Then, one can prove that these satisfy the definition of a Lie algebroid, iff

$\displaystyle Q^{2}=0.$

Identifying functions on
$C^{\infty}(E[1])\simeq\Gamma(\wedge^{\bullet}E^{*})$, $Q$ corresponds to a
Lie algebroid differential ${}^{E}{\rm d}$. In the remaining of the paper, we
identify $C^{\infty}(E[1])\simeq\Gamma(\wedge^{\bullet}E^{*})$, and $Q$ to
${}^{E}{\rm d}$.

###  2.3 Momentum section

In this section, a momentum section on a Lie algebroid $E$ is defined [2]. For
definition, we suppose a pre-symplectic form $B\in\Omega^{2}(M)$ on a base
manifold $M$, i.e., a closed $2$-form which is not necessarily nondegenerate.
A Lie algebroid $(E,\rho,[-,-])$ is one over a pre-symplectic manifold
$(M,B)$.

We introduce a connection (a linear connection) on $E$, i.e., a covariant
derivative $D\colon\Gamma(E)\rightarrow\Gamma(E\otimes T^{*}M)$, satisfying
$D(fe)=fDe+{\rm d}f\otimes e$ for a section $e\in\Gamma(E)$ and a function
$f\in C^{\infty}(M)$. A connection is extended to
$\Gamma(M,\wedge^{\bullet}T^{*}M\otimes E)$ as a degree $1$ operator.

In order to define a momentum section, we consider an $E^{*}$-valued 1-form
$\gamma\in\Omega^{1}(M,E^{*})$ defined by

$\displaystyle\langle\gamma(v),e\rangle=-B(v,\rho(e)),$

where $e\in\Gamma(E)$ and $v\in\mathfrak{X}(M)$ is a vector field. Here
$\langle-,-\rangle$ is a natural pairing of $E$ and $E^{*}$. We introduce the
following three conditions for a Lie algebroid $E$ on a pre-symplectic
manifold $(M,B)$.

  1. (H1)

$E$ is _presymplectically anchored with respect to $D$_ if

$\displaystyle D\gamma=0,$ (2.1)

where $D$ is a dual connection on $E^{*}$ defined by

$\displaystyle{\rm d}\langle\mu,e\rangle=\langle
D\mu,e\rangle+\langle\mu,De\rangle,$

for all sections $\mu\in\Gamma(E^{*})$ and $e\in\Gamma(E)$. The dual
connection extends to a degree 1 operator on $\Omega^{k}(M,E^{*})$.

  2. (H2)

A section $\mu\in\Gamma(E^{*})$ is a _$D$ -momentum section_ if

$\displaystyle D\mu=\gamma.$ (2.2)

  3. (H3)

A $D$-momentum section $\mu$ is _bracket-compatible_ if

$\displaystyle{}^{E}{\rm
d}\mu(e_{1},e_{2})=-\langle\gamma(\rho(e_{1})),e_{2}\rangle,$ (2.3)

for all sections $e_{1},e_{2}\in\Gamma(E)$. We note these conditions have
already appeared in [21] as compatibility conditions of geometric quantities
as a metric and a closed differential form with a Lie algebroid structure. The
equation (H1) is the same as equation (6) and (H2) is equation (7) in [21].

A Hamiltonian Lie algebroid is defined as follows.

######  Definition 2.7.

A Lie algebroid $E$ with a connection $D$ and a section $\mu\in\Gamma(E^{*})$
is called weakly Hamiltonian if (H1) and (H2) are satisfied. If the condition
is satisfied on a neighborhood of every point in $M$, it is called locally
weakly Hamiltonian.

######  Definition 2.8.

A Lie algebroid $E$ with a connection $D$ and a section $\mu\in\Gamma(E^{*})$
is called is called Hamiltonian if (H1), (H2) and (H3) are satisfied. If the
condition is satisfied on a neighborhood of every point in $M$, it is called
locally Hamiltonian.

A bracket-compatible $D$-momentum section, i.e., conditions (H2) and (H3) are
sufficient in our examples in later section. Thus, in this paper, we mainly
consider a bracket compatible momentum section $\mu\in\Gamma(E^{*})$
satisfying (H2) and (H3), and do not necessarily require that $E$ is
presymplectically anchored, i.e., (H1). In this case, $D^{2}\mu$ is not
necessarily zero.

###  2.4 Lie algebra case: momentum map

A momentum section is a generalization of a momentum map on a symplectic
manifold with a Lie group action. The definition of a momentum section (H1),
(H2) and (H3) reduces to the definition of a momentum map if a Lie algebroid
$E$ is an action Lie algebroid.

Suppose $B$ is nondegenerate, i.e., $B$ is a symplectic form. Consider an
action Lie algebroid on $E=M\times\mathfrak{g}$. It means that an
infinitesimal Lie algebra action is given by a bundle map
$\rho\colon\mathfrak{g}\times M\rightarrow TM$, such that

$\displaystyle[\rho(e_{1}),\rho(e_{2})]=\rho([e_{1},e_{2}]).$

The bracket in left hand side is a Lie bracket of vector fields. In this case,
we can take a zero connection, $D={\rm d}$. Then, three axioms of a momentum
section reduce to the following equations.

  1. (H1)

$\displaystyle{\rm d}\gamma(e)={\rm d}(\iota_{\rho(e)}B)={\cal
L}_{\rho(e)}B=0.$ (2.4)

This means that $\rho(e)$ is a symplectic vector field.

  2. (H2)

A section $\mu\in\Gamma(M\times\mathfrak{g}^{*})$ is regarded as a map
$\mu\colon M\rightarrow\mathfrak{g}^{*}$. Equation (2.2) is that a map $\mu$
is a Hamiltonian for the vector field $\rho(e)$,

$\displaystyle{\rm d}\mu(e)=\iota_{\rho(e)}B.$ (2.5)

Equation (2.5) leads equation (2.4).

  3. (H3)

${\rm d}\mu=\gamma$, i.e., ${\rm d}\mu=-B(\rho,-)$.

Equation (2.3) is equivalent to

$\displaystyle\operatorname{ad}^{*}_{e_{1}}\mu(e_{2})=\mu([e_{1},e_{2}]).$
(2.6)

for $e_{1},e_{2}\in\mathfrak{g}$. This means that $\mu$ is
$\mathfrak{g}$-equivariant.

Independent conditions are (2.5) and (2.6), which are the definition of an
infinitesimally equivariant momentum map.

Many examples of momentum sections which are not momentum maps have been
discussed in [2]. One can refer to more examples.

##  3 Constrained Hamiltonian system

We discuss examples of physical systems which have momentum sections and
Hamiltonian Lie algebroid structures. In this section, we consider a
constrained Hamiltonian mechanics system in $1+0$ dimension analyzed in [19].

Let $(N=T^{*}M,\omega_{\rm can})$ be a symplectic manifold over a smooth
manifold $M$, where $\omega_{\rm can}$ is a canonical symplectic form on $N$.
We take Darboux coordinates $\big{(}x^{i},p_{i}\big{)}$ such that $\omega_{\rm
can}={\rm d}x^{i}\wedge{\rm d}p_{i}$. On this symplectic manifold, we consider
a dynamical system. Assume a Hamiltonian $H\in C^{\infty}(N)$, and $r$
constraint functions $\Phi_{a}=\Phi_{a}(x,p)$, satisfying the following
compatibility condition:

There exist local matrix functions $\lambda_{a}^{b}=\lambda_{a}^{b}(x,p)$ such
that

$\displaystyle\\{H,\Phi_{a}\\}=\lambda^{b}_{a}\Phi_{b},$ (3.1)

where $\\{-,-\\}$ is the Poisson bracket induced by the symplectic form
$\omega_{\rm can}$. This ensures that $H$ is preserved by the Hamiltonian flow
of the constraints on the constraint surface $C:=\\{\Phi_{a}=0\\}$.

Moreover, suppose constraint functions are of the first class, i.e., they
satisfy

$\displaystyle\\{\Phi_{a},\Phi_{b}\\}=C^{c}_{ab}\Phi_{c},$ (3.2)

for some functions $C^{c}_{ab}=C^{c}_{ab}(x,p)$ on $N$.

We assume that constraints $\Phi_{a}$, $a=1,\dots,r$, are _irreducible_ ,
i.e., $\varphi_{C}^{*}({\rm d}\Phi_{1}\wedge\dots\wedge{\rm d}\Phi_{r})$ is
everywhere non-zero, where $\varphi_{C}\colon C\to N$ is the canonical
embedding map of the constraint surface into the original phase space.
Moreover, two sets of irreducible constraints $\Phi_{a}$, $a=1,\dots,r$, and
$\tilde{\Phi}_{a}$, $a=1,\dots,r$, are _equivalent_ if there exist local
matrix functions $M_{a}^{b}=M_{a}^{b}(x,p)$ such that

$\displaystyle\tilde{\Phi}_{a}=M^{b}_{a}\Phi_{b}$ (3.3)

holds true and the matrix $\big{(}M_{a}^{b}\big{)}_{a,b=1}^{r}$ is invertible
when restricted to $C$.

We take setting of the paper [19]. We require the canonical symplectic form
$\omega_{\rm can}={\rm d}x^{i}\wedge{\rm d}p_{i}$. Then, there is a natural
grading of functions with respect to the monomial degree in the momenta
$p_{i}$. The space of order $i$ or less than $i$ functions is denoted by
$C^{\infty}_{\leq i}(T^{*}M)$.

As a typical example which appears in physical applications, we consider the
case of $\Phi_{a}\in C^{\infty}_{\leq 1}(T^{*}M)$ and $H\in C^{\infty}_{\leq
2}(T^{*}M)$. These imply

$\displaystyle\Phi_{a}=\rho_{a}^{i}(x)p_{i}+\alpha_{a}(x),$ (3.4)

and

$\displaystyle H=\frac{1}{2}g^{ij}(x)p_{i}p_{j}+\beta^{i}(x)p_{i}+V(x).$ (3.5)

Here $\rho_{a}^{i}(x)$, $\alpha_{a}(x)$, $g^{ij}(x)$, $\beta^{i}(x)$ and
$V(x)$ are local functions of $x$.

We show that this Hamiltonian mechanics system has a momentum section and a
Hamiltonian Lie algebroid structure.

###  3.1 Lie algebroid structure on constraints

First, we see equation (3.2) with (3.4). As explained in [19], this equation
requires an (anchored almost) Lie algebroid structure. Counting an order of
$p_{i}$ in the equivalence condition (3.3), matrix functions $M^{a}_{b}$ are
functions of $x$. Then, a global structure is a rank $r$ vector bundle $E$
over $M$ with transition functions $(M^{a}_{b})_{a,b=1}^{r}$.

The Poisson bracket reduces the order by one or less than one since
$\\{p_{i},x^{j}\\}=\delta_{i}^{j}$ and $\\{p_{i},p_{j}\\}=0$. Thus, the
equality (3.2) implies $C^{c}_{ab}\in C^{\infty}_{0}(T^{*}M)\cong
C^{\infty}(M)$, which is uniquely determined due to the irreducibility
condition. The 1st order of $p$ of equation (3.2) takes the form,
$[\rho_{a},\rho_{b}]^{i}=C^{c}_{ab}(x)\rho_{c}^{i}$, i.e., globally,

$\displaystyle[\rho(e_{1}),\rho(e_{2})]=\rho([e_{1},e_{2}]),$ (3.6)

for $e_{1},e_{2}\in\Gamma(E)$.

Next we apply (3.2) to the Jacobi identity
$\\{\\{\Phi_{a},\Phi_{b}\\},\Phi_{c}\\}+\operatorname{cycl}(abc)=0$. The 1st
order of $p_{i}$ gives

$\displaystyle\big{(}C_{ab}^{e}C_{ce}^{d}+\partial_{j}C_{ab}^{d}\rho_{c}^{j}+\operatorname{cycl}(abc)\big{)}{\rho^{i}_{d}}=0.$
(3.7)

From the irreducibility condition on the constraints and the above identity,
we may deduce

$\displaystyle
C_{ab}^{e}C_{ce}^{d}+\rho_{a}^{j}\partial_{j}C_{bc}^{d}+\operatorname{cycl}(abc)=\sigma^{d}_{abc},$

for some functions $\sigma_{abc}^{d}$ skewsymmetric in the lower indices and
$\sigma^{d}_{abc}\rho^{i}_{d}=0$. If the anchor map $\rho$ is assumed
injective, we have $\sigma^{d}_{abc}=0$ and

$\displaystyle
C_{ab}^{e}C_{ce}^{d}+\rho_{a}^{j}\partial_{j}C_{bc}^{d}+\operatorname{cycl}(abc)=0.$
(3.8)

It is now straightforward to verify that equations (3.6) and (3.8) yield Lie
algebroid axioms, where the anchor map $\rho\colon E\rightarrow TM$ is defined
by $\rho(e_{a})=\rho^{i}_{a}(x)\partial_{i}$ and the Lie bracket is defined by
$[e_{a},e_{b}]=C_{ab}^{c}(x)e_{c}$ for a basis $e_{a}$ of the fiber of $E$. We
remark that the equivalence (3.3) takes care of the equivalence of the two
sides to not depend on the choice of a chosen frame.

A vector bundle with a bundle map $\rho\colon E\rightarrow TM$ and a bilinear
bracket $[-,-]$ is an anchored almost Lie algebroid $(E,\rho,[-,-])$ if $\rho$
and $[-,-]$ satisfy

$\displaystyle[\rho(e_{1}),\rho(e_{2})]=\rho([e_{1},e_{2}]),$
$\displaystyle[e_{1},fe_{2}]=f[e_{1},e_{2}]+\rho(e_{1})f\cdot e_{2}.$

If $\rho$ is not injective, we can take an anchored almost Lie algebroid since
it satisfies equation (3.7). We do not discuss these cases in this paper.

We can take a more general algebroid satisfying
$\sigma^{d}_{abc}\rho^{i}_{d}=0$ such as a Courant algebroid. We leave such
cases to other analysis.

The second term $\alpha_{a}$ in $\Phi_{a}$ is considered as components of an
$E$-1-form, $\alpha=\alpha_{a}(x)e^{a}\in\Gamma(E^{*})$, where $e^{a}$ is a
basis on $E^{*}$. The Poisson bracket (3.2) is equivalent to the condition on
$\alpha$,

$\displaystyle{}^{E}{\rm d}\alpha=0.$ (3.9)

On the other hand, $\alpha$ is determined by (3.4) only up to additions of the
form $\alpha_{a}\mapsto\alpha_{a}+\rho_{a}^{i}(x)\partial_{i}f(x)$, for a
function $f$ on $M$, which does not modify the symplectic form. Since such
additions to $\alpha$ are the ${}^{E}{\rm d}$-exact ones, we see that 0th
order deformations of $p$ in first class constraints (3.4) are parametrized by
the Q-cohomology of the Lie algebroid at degree one,

$\displaystyle[\alpha]\in H_{Q}^{1}(E[1]).$

Equation (3.2) and injective assumption for $\rho$ gives a Lie algebroid
structure on $E$ and equation (3.9).

###  3.2 Hamiltonian, metric and connection

In this section, we explain geometric structures induced from the Hamiltonian
(3.5) and the Poisson bracket (3.1) discussed in [19]. Suppose that in (3.5)
the symmetric matrix $g^{ij}$ has an inverse. Then a symmetric tensor $g^{ij}$
corresponds to an inverse of a metric $g$ on $M$.
$\beta:=\beta^{i}\partial_{i}$ is a vector field and $V(x)$ is a global
potential function on $M$. Counting order of $p$ in equation (3.1),
$\lambda_{a}^{b}$ is a 1st order function of $p$, thus it is assumed that
$\lambda_{a}^{b}=g^{ij}(x)\Gamma_{aj}^{b}(x)p_{i}+\tau_{a}^{b}(x)$. From
consistency of equation (3.1) with transition functions $M^{a}_{b}$ given by
equivalence of $\Phi_{a}$, we obtain

$\displaystyle\Gamma^{\prime}=M\Gamma M^{-1}+{\rm d}MM^{-1},$
$\displaystyle\tau^{\prime}=M\tau M^{-1}+\big{(}\beta,{\rm d}MM^{-1}\big{)},$
(3.10)

where $\Gamma_{a}^{b}=\Gamma_{aj}^{b}{\rm d}x^{j}$. $\Gamma_{a}^{b}$
transforms as a connection 1-form on $E$. $\tau_{a}^{b}$ is a local matrix
transforming in equation (3.10).111An interesting $\tau$ is
$\tau=(\Gamma,\beta)$, which case is analyzed later in this section.

We can absorb the term linear in the momenta in the Hamiltonian,
$\beta^{i}\mapsto 0$, at the expense of redefining the potential $V$ and the
$E$-1-forms $\alpha$ and simultaneously twisting the symplectic form
$\omega_{\rm can}$ by a magnetic field $B={\rm d}A\in\Omega^{2}(M)$ as

$\displaystyle\omega=\omega_{\rm can}+B,$

where $A_{i}=g_{ij}\beta^{j}$ and $A=A_{i}(x){\rm d}x^{i}$. The globally
defined $2$-form $B={\rm d}A$ is obviously regarded as a pre-symplectic form
since ${\rm d}B=0$.

By the above redefinition, constraints and the Hamiltonian become

$\displaystyle\Phi_{a}^{\prime}=\rho_{a}^{i}(x)p_{i}+\alpha^{\prime}_{a}(x),$
$\displaystyle H=\frac{1}{2}g^{ij}(x)p_{i}p_{j}+V^{\prime}(x).$

Here, $\alpha^{\prime}$ is an E-1-form defined by
$\langle\alpha^{\prime},e\rangle=\langle\alpha,e\rangle-\iota_{\rho(e)}A$ for
all $e\in\Gamma(E)$, and $V^{\prime}$ is defined by
$V^{\prime}(x)=V(x)-\frac{1}{2}g(\beta,\beta)$. Equations (3.2) and (3.1)
change but are similar equations,

$\displaystyle\\{\Phi^{\prime}_{a},\Phi^{\prime}_{b}\\}=C_{ab}^{c}\Phi^{\prime}_{c},$
(3.11) $\displaystyle\\{H,\Phi^{\prime}_{a}\\}=\lambda_{a}^{\prime
b}\Phi^{\prime}_{b},$ (3.12)

where $\tau^{\prime}=\tau-g^{-1}(\Gamma,A)$ and
$\lambda^{\prime}=\lambda-g^{-1}(\Gamma,A)=g^{-1}(\Gamma,p)+\tau^{\prime}$.

After the above redefinition, we show that geometric structure described by
equations (3.11) and (3.12) have a structure of a momentum section.

The 1st order term of $p$ in equation (3.11) gives the same conditions as
(3.2), i.e., (3.11) requires a Lie algebroid structure on the vector bundle
$E$ with the same anchor map $\rho$ and Lie bracket $[-,-]$ before the
redefinition. In the 0th order term of $p$ in equation (3.11), the affine
constraints $\alpha$ changes to

$\displaystyle{}^{E}{\rm d}\alpha^{\prime}=-\rho^{*}(B),$ (3.13)

since the new symplectic form $\omega$ gives the Poisson bracket
$\\{p_{i},p_{j}\\}=B_{ij}$. Here $\rho^{*}$ is the induced map of the anchor
to $\Omega^{\bullet}(M)$, mapping ordinary differential forms to
$E$-differential forms. In particular,
$\rho^{*}(B)=\frac{1}{2}B_{ij}\rho_{a}^{i}\rho_{b}^{j}q^{a}q^{b}\in\Gamma\big{(}{\wedge}^{2}E^{*}\big{)}$.
Equation (3.13) is the same as equation (2.3) in the condition (H3) by
identifying $\mu=\alpha^{\prime}$.

Let us analyze equation (3.12). As already pointed, the transformation
property of $\Gamma^{a}_{bi}$ under the transition function $M^{a}_{b}$ shows
$\Gamma^{a}_{bi}$ is a connection 1-form, thus this defines a Lie algebroid
connection $D\colon\Gamma(E)\rightarrow\Gamma(E\otimes T^{*}M)$. $D$ and
$\rho$ can be combined to define an $E$-connection
${}^{E}\nabla\colon\Gamma(TM)\rightarrow\Gamma(TM\otimes E^{*})$ on $TM$:

$\displaystyle{}^{E}\nabla_{\\!e}v:={\cal L}_{\rho(e)}v+\rho(D_{v}e),$ (3.14)

where $v\in\mathfrak{X}(M)$ and $e\in\Gamma(E)$. An $E$-connection is extended
to a tensor product space of $TM$ and $T^{*}M$.

Equation (3.12) then gives three conditions by considering it to 2nd, 1st, and
0th order in the momenta. To 2nd order, we obtain the geometrical
compatibility equation,

$\displaystyle{}^{E}\nabla g=0,$

on the metric $g$.

To 1st order, we get another condition on the system of constraints, It
relates the exterior covariant derivative of $\alpha^{\prime}$ induced by $D$,
$D\alpha^{\prime}\in\Gamma(E^{*}\otimes T^{*}M)$, to the anchor map $\rho$,
now regarded as a section of $E^{*}\otimes TM$:

$\displaystyle D\alpha^{\prime}=\gamma+(\tau^{\prime t}\otimes
g_{\flat})\rho,$ (3.15)

where $\gamma\in\Omega^{1}(M,E^{*})$ is a 1-form taking a value on $E^{*}$
appeared in the definition of a momentum section, $\tau^{\prime t}\colon
E^{*}\to E^{*}$, the transposed of $\tau^{\prime}$, and $g_{\flat}\colon TM\to
T^{*}M$, $v\mapsto\iota_{v}g$, as maps on the corresponding sections. To 0th
order one finds that the potential $V^{\prime}$ has to satisfy

$\displaystyle{}^{E}{\rm d}V^{\prime}=\tau^{\prime}(\alpha^{\prime}).$

If $\tau^{\prime}=0$, equation (3.15) becomes

$\displaystyle D\alpha^{\prime}=\gamma,$

which is the condition (H2), i.e., equation (2.2), since
$\mu=\alpha^{\prime}$. The condition $\tau^{\prime}=0$ is $\tau=g(\Gamma,A)$.
The remaining condition of a momentum section is the condition (H1), i.e.,
equation (2.1), $D\gamma=0$. In general, this constrained Hamiltonian system
does not satisfy (2.1).222Using an $E$-connection (3.14) and its extension to
$\Omega^{2}(M)$, equation (2.1) is equivalent to ${}^{E}\nabla B=0$, where
$\gamma_{ia}=-B_{ij}\rho^{j}_{a}$ and
${}^{E}\nabla_{a}B_{ij}=\rho_{a}^{k}\partial_{k}B_{ij}+\partial_{i}\rho_{a}^{k}B_{kj}+\partial_{j}\rho_{a}^{k}B_{ik}+\rho^{k}_{b}\Gamma_{aj}^{b}B_{ik}+\rho^{k}_{b}\Gamma_{ai}^{b}B_{kj}=0$.

Therefore, we obtain the following result:

######  Theorem 3.1.

We consider the constraint Hamiltonian system satisfying equations (3.1) and
(3.2) with constraints (3.4) and a Hamiltonian (3.5). Then, $B={\rm
d}(g(\beta,-))$ is a pre-symplectic form. If $\rho$ is injective and
$\tau^{\prime}=\tau-g(\Gamma,A)=\tau-(\Gamma,\beta)=0$,
$\alpha^{\prime}=\alpha-\iota_{\rho}A$ is a bracket compatible $D$-momentum
section on a Lie algebroid $E$ with respect to a connection $D$ defined by a
connection $1$-form $\Gamma^{b}_{a}$. Moreover, if
$D\gamma=D(-\iota_{\rho}B)=0$, it is pre-symplectically anchored.

In $\tau^{\prime}\neq 0$ case, this constrained Hamiltonian system gives a
generalization of a momentum section. It is interesting to see this
generalization in a future work.

##  4 Two-dimensional sigma model with boundary

In this section, we analyze a momentum section and a Hamiltonian Lie algebroid
structure in a two-dimensional sigma model. If a base manifold is in two
dimensions and with boundary, a momentum section naturally appears.

Let $\Sigma$ be a two-dimensional manifold and $M$ be a $d$-dimensional target
manifold. $X\colon\Sigma\rightarrow M$ is a smooth map from $\Sigma$ to $M$.
We start at the following sigma model action with a $2$-form B-field,

$\displaystyle S=\frac{1}{2}\int_{\Sigma}g_{ij}(X){\rm d}X^{i}\wedge*{\rm
d}X^{j}+b_{ij}(X){\rm d}X^{i}\wedge{\rm d}X^{j},$ (4.1)

where $g$ is a metric and $b\in\Omega^{2}(M)$ is a closed $2$-form on $M$.
$g_{ij}(X)$ and $b_{ij}(X)$ are their pullbacks to $\Sigma$. This action is
invariant under diffeomorphisms on a worldsheet $\Sigma$ and on a target space
$M$.

We analyze a general condition that the action $S$ is invariant under other
symmetries on $M$. In a general setting, an element of a vector space $V$, or
more generally, a section of the vector bundle $E$ on $M$, $e\in\Gamma(E)$
acts on $M$ as an infinitesimal transformation generated by a vector field. A
transformation is determined by defining a bundle map to a tangent bundle,
$\rho\colon E\rightarrow TM$. Suppose that $\rho$ define an infinitesimal
gauge transformation of $X$ as

$\displaystyle\delta X^{i}=\rho(\epsilon)^{i}=\rho^{i}_{a}(X)\epsilon^{a},$
(4.2)

where $i=1,2,\dots,d$ are indices of local coordinates on $M$,
$\epsilon\in\Gamma(X^{*}E)$ is a parameter (a gauge parameter), and
$\rho(e_{a})=\rho^{i}_{a}(X)\partial_{i}$ by taking a basis of $E$, $e_{a}$.

By straight computations, the action (4.1) is in invariant under the
transformation (4.2), iff

$\displaystyle{\cal L}_{\rho(e_{a})}g=0,$ (4.3) $\displaystyle{\cal
L}_{\rho(e_{a})}b={\rm d}\beta_{a},$ (4.4)
$\displaystyle[\rho(e_{a}),\rho(e_{b})]=\rho([e_{a},e_{b}]),$ (4.5)

where ${\cal L}$ is a Lie derivative and $\beta_{a}\in\Omega^{1}(M,E^{*})$ is
a 1-form taking a value on $E^{*}$. A vector field $\rho(e_{a})$ satisfying
equation (4.3) is called a Killing vector field. From equation (4.5), a vector
bundle is an anchored almost Lie algebroid.

In this paper, $E$ is a Lie algebroid. In this case, the action $S$ is
invariant if equations (4.3) and (4.4) are satisfied.

###  4.1 Gauged sigma model

We can generalize the above theories by gauging the action (4.1). ‘Gauging’ is
a deformation of the action using a connection 1-form
$A\in\Omega^{1}(\Sigma,X^{*}E)$.

A pullback of a basis of a 1-form on $M$, ${\rm d}X^{i}$, is ‘gauged’ using a
covariant derivative with respect to a connection $A$ as

$\displaystyle F^{i}=DX^{i}={\rm d}X^{i}-\rho^{i}_{a}(X)A^{a}.$

We can assume $A^{a}$ has a genuine infinitesimal gauge transformation,

$\displaystyle\delta A^{a}={\rm d}\epsilon^{a}+[A,\epsilon]^{a}={\rm
d}\epsilon^{a}+C_{bc}^{a}A^{b}\epsilon^{c},$

however, $C_{bc}^{a}=C_{bc}^{a}(X)$ is not necessarily constant but a local
function on $M$. We consider a target space covariant version of the gauge
transformation by introducing (a pullback of) a connection on $M$,
$\Gamma_{bi}^{a}(X)$:333We can consider a more general ansatz of a gauge
transformation as $\delta A^{a}=D\epsilon^{a}+[A,\epsilon]^{a}={\rm
d}\epsilon^{a}+C_{bc}^{a}A^{b}\epsilon^{c}+\Delta A^{a}$, where $\Delta A^{a}$
is a 1-form taking a value on a pullback of $E$, which is linear with respect
to the infinitesimal parameter $\epsilon^{a}$ [11, 12, 13].

$\displaystyle\delta A^{a}={\rm
d}\epsilon^{a}+C_{bc}^{a}(X)A^{b}\epsilon^{c}+\Gamma_{bi}^{a}(X)\epsilon^{b}DX^{i},$

where the gauge transformation is covariant under the target space
diffeomorphism. In summary, we choose gauge transformations,

$\displaystyle\delta X^{i}=\rho^{i}_{a}(X)\epsilon^{a},$ (4.6)
$\displaystyle\delta A^{a}={\rm
d}\epsilon^{a}+C_{bc}^{a}(X)A^{b}\epsilon^{c}+\Gamma_{bi}^{a}(X)\epsilon^{b}DX^{i}.$
(4.7)

The action (4.1) is generalized to a gauged sigma model action by ‘gauging’
the symmetry to infinitesimal transformations (4.6) and (4.7). Since the
manifold $\Sigma$ has boundary, we take the following ansatz for a gauged
sigma model action:

$\displaystyle
S=\frac{1}{2}\int_{\Sigma}g_{ij}(X)DX^{i}\wedge*DX^{j}+b_{ij}(X){\rm
d}X^{i}\wedge{\rm d}X^{j}+\int_{\partial\Sigma}\eta_{i}(X){\rm
d}X^{i}+\mu_{a}(X)A^{a},$ (4.8)

where the last two terms are the most general possible boundary terms with
some arbitrary local functions $\eta_{i}(X)$ and $\mu_{a}(X)$.
$\eta_{i}(X)dX^{i}$ is a pullback of a 1-form on a target space $M$ and
$\mu_{a}(X)$ is a pullback of an element $\Gamma(E^{*})$ on a target space
$M$. Requiring (4.8) is invariant under gauge transformations (4.6) and (4.7),
we obtain geometric conditions for a metric $g$, a $2$-form $b$ and $\rho$ and
a bracket $[-,-]$. We obtain the following conditions for the metric, $\rho$
and a bracket,

$\displaystyle{\cal L}_{\rho(e_{a})}g=\Gamma_{a}^{b}\vee\iota_{\rho(e_{b})}g,$
(4.9) $\displaystyle[\rho(e_{a}),\rho(e_{b})]=\rho([e_{a},e_{b}]),$ (4.10)

where $\vee$ is a symmetric product of 1-forms. Equation (4.9) is equivalent
to ${}^{E}\nabla g=0$. Though equation (4.10) is satisfied if $(E,\rho,[-,-])$
is an anchored almost Lie algebroid, we suppose $(E,\rho,[-,-])$ is a true Lie
algebroid now.

Next we analyze a condition for a $2$-form B-field $b$. Using ${\rm d}b=0$,
the gauge transformation for $S_{b}=\frac{1}{2}\int_{\Sigma}b_{ij}(X){\rm
d}X^{i}\wedge{\rm d}X^{j}$ is

$\displaystyle\delta S_{b}=\int_{\Sigma}{\cal
L}_{\rho(\epsilon)}b=\int_{\Sigma}{\rm
d}\iota_{\rho(\epsilon)}b=\int_{\partial\Sigma}\iota_{\rho(\epsilon)}b.$

Thus, requirement of gauge invariance of the total action $\delta S=0$ gives
the conditions including quantities of boundary terms. In local coordinates,
straight computations give three equations,

$\displaystyle\mu_{a}=-\eta_{i}\rho^{i}_{a},$ (4.11)
$\displaystyle\rho^{j}_{a}b_{ji}+\rho^{j}_{a}\partial_{j}\eta_{i}+\eta_{j}\partial_{i}\rho^{j}_{a}+\Gamma^{b}_{ai}\mu_{b}=0,$
(4.12)
$\displaystyle\rho^{i}_{a}\partial_{i}\mu_{b}-C^{c}_{ab}\mu_{c}-\rho^{i}_{b}\Gamma^{c}_{ai}\mu_{c}=0.$
(4.13)

The first condition (4.11) is $\mu(e)=-\iota_{\rho(e)}\eta$ for
$e\in\Gamma(E)$, the second and third conditions (4.12) and (4.13) are
equivalent to (H2) and (H3), where we identify $B=b+{\rm d}\eta$. Thus, we
obtain the following result.

######  Theorem 4.1.

We consider a gauged sigma model with boundary, (4.8). Let
$\mu(e)=-\iota_{\rho(e)}\eta\in\Gamma(E^{*})$ and $B=b+{\rm
d}\eta\in\Omega^{2}(M)$. Let $D$ be a connection defined by $\Gamma_{a}^{b}$.
Then, $\mu$ is a bracket compatible $D$-momentum section with respect to the
connection $D$ with a pre-symplectic form $B$. If $B$ satisfies (H1), it is
pre-symplectically anchored.

Finally, we comment a gauge algebra generated by gauge transformations (4.6)
and (4.7). Gauge transformations must consist of a closed algebra at least on
an orbit of equations of motion. Closure conditions of a gauge algebra
generated $[\delta_{1},\delta_{2}]\sim\delta_{3}$ by equations (4.6) and (4.7)
impose extra equations for $X^{i}$ and $A^{a}$, which are topological
conditions.

This gives topological condition on a external gauge field $A^{a}$ and
geometry of $M$. One can refer to [3] and [27] for more analysis related
T-duality. See also [16].

##  5 Momentum section on pre-multisymplectic manifold

In this section, we propose a generalization of a momentum section to a pre-
multisymplectic manifold. Our strategy is to generalize a gauged sigma model
in Section 4. We generalize a $2$-form B-field $b$ to a higher $(n+1)$-form
$h$ and a two-dimensional manifold $\Sigma$ to a higher-dimensional manifold.
We naturally obtain a generalization of a momentum section from consistency of
these gauged sigma models.

###  5.1 Gauged sigma model in $\boldsymbol{n}$ dimensions with Wess–Zumino
term

We introduce a pre-$n$-plectic manifold.

######  Definition 5.1.

A pre-$n$-plectic manifold is $(M,h)$, where $M$ is a smooth manifold and $h$
is a closed $(n+1)$-form on $M$.

A pre-$n$-plectic manifold is also called a pre-multisympletic manifold for
$n\geq 2$. A pre-$n$-plectic manifold is called an $n$-plectic manifold if $h$
is nondegenerate, i.e., if $\iota_{v}h=0$ for a vector field
$v\in\mathfrak{X}(M)$ is equivalent to $X=0$.

We also introduce a metric $g$ on $M$. Moreover, by introducing an
$(n+1)$-dimensional manifold $\Xi$ with boundary $\Sigma=\partial\Xi$, and a
map $X\colon\Sigma\rightarrow M$, we consider a nonlinear sigma model with a
Wess–Zumino term. Compatibility conditions of $h$ and $g$ are determined from
gauge invariance of the following sigma model action with a Wess–Zumino term:

$\displaystyle S=\int_{\Sigma}\frac{1}{2}g_{ij}(X){\rm d}X^{i}\wedge*{\rm
d}X^{j}+\int_{\Xi}\frac{1}{(n+1)!}h_{i_{1}\dots i_{n+1}}(X){\rm
d}X^{i_{1}}\wedge\dots\wedge{\rm d}X^{i_{n+1}},$ (5.1)

where $g(X)$ is a pullback of a metric $g$ and
$h(X)=\frac{1}{(n+1)!}h_{i_{1}\dots i_{n+1}}(X){\rm
d}X^{i_{1}}\wedge\dots\wedge{\rm d}X^{i_{n+1}}$ in the second term is a
pullback of a closed $(n+1)$-form $h$ on $M$. The $n=2$ case is most important
since this is string sigma model with an NS-flux $3$-form $H$.

Invariance conditions of $S$ under the transformation (4.2) of $X$ as in
Section 4 gives a similar condition,

$\displaystyle{\cal L}_{\rho(e_{a})}g=0,$ $\displaystyle{\cal
L}_{\rho(e_{a})}h={\rm d}\beta_{a},$
$\displaystyle[\rho(e_{a}),\rho(e_{b})]=\rho([e_{a},e_{b}]),$ (5.2)

where $\beta$ is an $n$-form taking a value on $E^{*}$. Equation (5.2) require
an anchored almost Lie algebroid structure on a target vector bundle $E$.

We consider gauging of an $n$-dimensional sigma model (5.1) by introducing a
connection $A\in\Omega^{1}(\Sigma,X^{*}E)$ and gauge transformations (4.6) and
(4.7). Here, we consider the case that a vector bundle $E$ is a Lie algebroid
for (5.2) again. We take a Hull–Spence type ansatz [17] for a gauged action,
but in our case a gauge structure is not a Lie algebra but a Lie algebroid.
The ansatz is

$\displaystyle S=S_{g}+S_{h}+S_{\eta},$ (5.3)

where

$\displaystyle S_{g}=\int_{\Sigma}\frac{1}{2}g_{ij}DX^{i}\wedge*DX^{j},$
$\displaystyle S_{h}=\int_{\Xi}\frac{1}{(n+1)!}h_{i_{1}\dots i_{n+1}}(X){\rm
d}X^{i_{1}}\wedge\dots\wedge{\rm d}X^{i_{n+1}},$ $\displaystyle
S_{\eta}=\int_{\Sigma}\sum_{k=0}^{n}\frac{1}{k!(n-k)!}\eta^{(k)}_{i_{1}\dots
i_{k}a_{k+1}\dots a_{n}}(X){\rm d}X^{i_{1}}\wedge\dots\wedge{\rm
d}X^{i_{k}}\wedge A^{a_{k+1}}\wedge\dots\wedge A^{a_{n}},$

where $\eta^{(k)}$ is a pullback of a $k$-form on $M$ taking a value on
$\wedge^{n{-}k}\\!E^{*}$, i.e.,
$\eta^{(k)}\\!\in\\!X^{*}\Omega^{k}(M,\wedge^{n{-}k}\\!E^{*})$.

We require gauge invariance of the above gauged action under the gauge
transformations (4.6) and (4.7), which are the same ones as in two-dimensional
case Section 4. Requirement of gauge invariance imposes conditions for
pullbacks of coefficient functions $g\in\Gamma\big{(}S^{2}T^{*}M\big{)}$,
$h\in\Omega^{n+1}(M)$ and
$\eta^{(k)}\in\Omega^{k}\big{(}M,\wedge^{n-k}E^{*}\big{)}$. These identities
gives geometric identities of a metric $g$, $H$ and $\eta^{(k)}$ on the vector
bundle $E$ on $M$ before pullbacks.444We use the same notation for geometric
quantities on $M$ and their pullbacks. We propose that this structure gives a
momentum section in a pre-$n$-plectic manifold.

From concrete computations, the condition of $g$ is

$\displaystyle{\cal L}_{\rho(e_{a})}g=\Gamma_{a}^{b}\vee\iota_{\rho(e_{b})}g,$

as in the case of the two-dimensional sigma model. For $h$ and $\eta^{(k)}$ on
$M$, we obtain the following conditions on $M$:555Note that we obtain
identities on $h\in\Omega^{n+1}(M)$ and
$\eta^{(k)}\in\Omega^{k}\big{(}M,\wedge^{n-k}E^{*}\big{)}$ from conditions for
their pullbacks in the gauged sigma model (5.3). two algebraic conditions,

$\displaystyle\eta^{(k-1)}(e_{k},\dots,e_{n})=(-1)^{k}\iota_{\rho(e_{k})}\eta^{(k)}(e_{k+1},\dots,e_{n})+\operatorname{cycl}(e_{k},\dots,e_{n}),$
(5.4)
$\displaystyle\iota_{\rho(e_{k})}\eta^{(k)}(e_{k+1},\dots,e_{k+m},\dots,e_{n})+\iota_{\rho(e_{k+m})}\eta^{(k)}(e_{k+1},\dots,e_{k},\dots,e_{n})=0,$
$\displaystyle\qquad k=1,\dots,n-1,\quad m=1,\dots,n-k,$ (5.5)

and three differential equations,

$\displaystyle D\eta^{(n-1)}(e)=\iota_{\rho(e)}\widetilde{h},\qquad k=n,$
(5.6) $\displaystyle{\cal
L}_{\rho}(e)\eta^{(k)}(e_{k+1},\dots,e_{n})+\sum_{i=1}^{n-k}(-1)^{i}\eta^{(k)}([e,e_{k+i}],e_{k+1},\dots,\check{e}_{k+i},\dots,e_{n})$
$\displaystyle\qquad{}+\sum_{i=1}^{n-k}(-1)^{i}\langle\Gamma,\rho(e)\rangle\wedge\eta^{(k)}(e_{k+1},\dots,e_{n})$
$\displaystyle\qquad{}-\sum_{i=1}^{n-k}(-1)^{i}\Gamma(e)\wedge\iota_{\rho(e_{k+i})}\eta^{(k)}(e_{k+1},\dots,\check{e}_{k+i},\dots,e_{n})$
$\displaystyle\qquad{}+\sum_{i=1}^{n-k}(-1)^{i}\langle\iota_{\rho(e_{k+i})}\Gamma(e)\stackrel{{\scriptstyle\wedge}}{{,}}\eta^{(k)}(e_{k+1},\dots,\check{e}_{k+i},\dots,e_{n})\rangle=0,\qquad
k=1,\dots,n-1,\\!\\!\\!$ (5.7) $\displaystyle{\cal
L}_{\rho}(e)\eta^{(0)}(e_{1},\dots,e_{n})+\sum_{i=1}^{n}(-1)^{i}\eta^{(0)}([e,e_{k+i}],e_{k+1},\dots,\check{e}_{k+i},\dots,e_{n})$
$\displaystyle\qquad{}+\sum_{i=1}^{n}(-1)^{i}\langle\iota_{\rho(e_{i})}\Gamma(e)\stackrel{{\scriptstyle\wedge}}{{,}}\eta^{(0)}(e_{1},\dots,\check{e}_{i},\dots,e_{n})\rangle=0,\qquad
k=0,$ (5.8)

where $\widetilde{h}=h+{\rm d}\eta^{(n)}$, $e,e_{i}\in\Gamma(E)$,
$i=k,\dots,n$, $\Gamma$ is a connection 1-form on $E$, and $\langle-,-\rangle$
is a natural pairing of $E^{*}$ and $E$. Notation
$\stackrel{{\scriptstyle\wedge}}{{,}}$ means both a wedge product on
$\Omega^{k}(M)$ and a pairing of $E$ and $E^{*}$. Note that $\delta
S_{h}=\int_{\Xi}{\cal
L}_{\rho(\epsilon)}h=\int_{\Sigma}\iota_{\rho(\epsilon)}h$ since ${\rm d}h=0$.
For $k=n-1$, equation (5.7) is also written as

$\displaystyle{}^{E}{\rm
d}\eta^{(n-1)}(e_{1},e_{2})-D\eta^{(n-2)}(e_{1},e_{2})=0.$

In $n=1$, equations (5.4)–(5.8) reduce to conditions of a momentum section
(H2) and (H3) by setting $\mu=\eta^{(0)}$, $\gamma=\eta^{(1)}$ and
$B=\widetilde{h}$. In $n=2$, equations (5.4)–(5.8) give gauging conditions of
target geometry in [11].

It is natural to impose the following condition corresponding to the condition
(H1),

$\displaystyle D\iota_{\rho}\widetilde{h}=0.$ (5.9)

However, this condition is not needed for gauge invariance of a gauged sigma
model. As a result, we need not impose this condition on the definition of a
momentum section.

Finally, we obtain the following definition of a multimomentum section on a
pre-mutlisymplectic manifold.

######  Definition 5.2.

Let $(M,\widetilde{h})$ be a pre-$n$-plectic manifold, where $\widetilde{h}$
is a closed $(n+1)$-form, and $(E,\rho,[-,-])$ be a Lie algebroid over $M$. We
define the following three conditions corresponding to (H1), (H2) and (H3).

(HM1)

$E$ is a _pre- $n$-plectically anchored with respect to $D$_ if

$\displaystyle D\gamma=0,$

where $\gamma=\iota_{\rho}\widetilde{h}\in\Omega^{n}(M,E^{*})$.

(HM2)

$\eta^{(n-1)}\in\Omega^{n-1}(M,E^{*})$ is a _$D$ -multimomentum
$(D$-momentum$)$ section_ if it satisfies equation (5.6),

$\displaystyle D\eta^{(n-1)}(e)=\iota_{\rho(e)}\widetilde{h}.$

(HM3)

We define a descent set of multimomentum sections
$\big{(}\eta^{(k)}\big{)}_{k=0}^{n-2}$ by equations (5.4) and (5.5), where
$\eta^{(k)}\in\Omega^{k}\big{(}M,\wedge^{n-k}E^{*}\big{)}$. A
$D$-multimomentum section and its descents
$\big{(}\eta^{(k)}\big{)}_{k=0}^{n-1}$ are _bracket-compatible_ if (5.7) and
(5.8) are satisfied.

Under this definition, we have the same definition of a weakly Hamiltonian Lie
algebroid, Definition 2.7, and a Hamiltonian Lie algebroid, Definition 2.8,
but a momentum section is a set of multimomentum sections $\eta^{(k)}$ on a
pre-multisymplectic manifold $\big{(}M,\widetilde{h}\big{)}$. A Hamiltonian
Lie algebroid on a pre-multisymplectic manifold is defined as follows.

######  Definition 5.3.

A Lie algebroid $E$ with a connection $D$ and a section
$\eta^{(n-1)}\in\Omega^{n-1}(M,E^{*})$ is called weakly Hamiltonian if (HM1)
and (HM2) are satisfied. If the condition is satisfied on a neighborhood of
every point in $M$, it is called locally weakly Hamiltonian.

######  Definition 5.4.

A Lie algebroid $E$ with a connection $D$ and a section
$\eta^{(k)}\in\Omega^{k}\big{(}M,\wedge^{n-k}E^{*}\big{)}$, $k=0,\dots,n-1$ is
called Hamiltonian if (HM1), (HM2) and (HM3) are satisfied. If the condition
is satisfied on a neighborhood of every point in $M$, it is called locally
Hamiltonian.

We summarize a geometric structure of a gauge sigma model with a $(n+1)$-form
flux $h$ using the terminology of multimomentum sections.

######  Theorem 5.5.

We consider an $n$-dimensional gauged sigma model with WZ term (5.3). Then,
$\eta^{(k)}\in\Omega^{k}\big{(}M,\wedge^{n-k}E^{*}\big{)}$, $k=0,\dots,n-1$,
are a bracket compatible $D$-multimomentum section and descents with a
pre-$n$-plectic form $\widetilde{h}=h+{\rm d}\eta^{(n)}$. If $\widetilde{h}$
satisfies (HM1), it is pre-$n$-plectically anchored.

###  5.2 Momentum map on multisymplectic manifold: Lie algebra case

Let a Lie algebroid be an action Lie algebroid $E=M\times\mathfrak{g}$. Then,
we can take a trivial connection ${\rm d}=D$, and a momentum section on a
pre-$n$-plectic manifold reduces to a (multi)momentum map on a pre-symplectic
manifold.

Conditions (5.4)–(5.8) reduce to

$\displaystyle\eta^{(k-1)}(e_{k},\dots,e_{n})=(-1)^{k}\operatorname{ad}^{*}_{e_{k}}\eta^{(k)}(e_{k+1},\dots,e_{n})+\operatorname{cycl}(e_{k},\dots,e_{n}),$
$\displaystyle\operatorname{ad}^{*}_{e_{k}}\eta^{(k)}(e_{k+m},\dots,e_{k+m},\dots,e_{n})+\operatorname{ad}^{*}_{e_{k+1}}\eta^{(k)}(e_{k+1},\dots,e_{k},\dots,e_{n})=0,$
$\displaystyle\qquad k=1,\dots,n-1,\quad m=1,\dots,n-k,$ $\displaystyle{\rm
d}\eta^{(n-1)}=\iota_{\rho_{a}}\widetilde{h},\qquad k=n,$ (5.10)
$\displaystyle{\rm
d}\eta^{(k-1)}(e,e_{k+1},\dots,e_{n})=\operatorname{ad}^{*}_{e}\eta^{(k)}(e_{k+1},\dots,e_{n})$
$\displaystyle\qquad{}-\sum_{i=k}^{n}(-1)^{i-1}\eta^{(k)}([e,e_{i}],e_{k+1},\dots,\check{e}_{i},\dots,e_{n}),\qquad
k=1,\dots,n-1,$
$\displaystyle\operatorname{ad}^{*}_{e}\eta^{(0)}(e_{1},\dots,e_{n})=\sum_{i=1}^{n}(-1)^{i-1}\eta^{(0)}([e,e_{i}],e_{1},\dots,\check{e}_{i},\dots,e_{n}),\qquad
k=0.$

A pre-$n$-plectically anchored condition equation (5.9) is trivially satisfied
from equation (5.10),

$\displaystyle{\rm d}\iota_{\rho}\widetilde{h}=0.$

This condition already appeared in [21].

The above conditions are a direct generalization of a momentum map
(multimomentum map) on a multisymplectic manifold with a Lie group action [8,
14] by setting $\eta^{(k)}=0$ for $k=0,\dots,n-2$. In this case,
$\eta^{(n-1)}$ is a multimomentum map.

##  6 Discussion and outlook

We have shown that a simple constrained Hamiltonian mechanics and a two-
dimensional gauged sigma model with boundary have a momentum section and a
Hamiltonian Lie algebroid structure. By generalizing a gauged sigma model to a
higher-dimensional gauged sigma model with WZ term, we have proposed a theory
of a multimomentum section on a pre-multisymplectic manifold.

It is important to compare other generalizations of a moment map theory to a
multisymplectic manifold such as Madsen–Swann’s multimoment map on the $n$-th
Lie kernel [24, 25], a homotopy moment map [7], and a weak moment map [15].

Though we proposed a momentum section on a pre-multisymplectic manifold (5.4)
and (5.8) from consistency conditions of a higher-dimensional gauged nonlinear
sigma model, their geometrical structures should be analyzed more. These
structure are described by a Lie algebroid differential ${}^{E}d$ and a
covariant derivative $D$.

In all examples in our paper, the pre-symplectically anchored condition (H1)
is not necessary for consistency of structures. Conditions (H2) and (H3) are
essential for physical applications. More examples are needed for deeper
understanding of a momentum section theory.

We have assumed an anchor map $\rho$ is injective in this paper. However we
should relax this condition. If an anchor map $\rho$ is not necessarily
injective, we can consider more general algebroid such as a Courant algebroid
[22], a Lie 3-algebroid [20], and higher algebroids, as a symmetry of a gauged
sigma model. This direction is related to a Lie group action on a Courant
algebroid and the reduction [4]. These generalizations are left for future
analysis.

We considered an infinitesimal version, i.e., an action of a Lie algebroid on
a pre-(multi) symplectic manifold. A globalization to a Lie groupoid
corresponding to a generalization of a Lie group action is a next problem.

A next step of physical systems in this paper is quantization. One possible
quantization is an equivariant localization using the Duistermaat–Heckman
formula, which has been already discussed in [2]. In this paper, we have
obtained more concrete physical models for applications of the localization.
For this purpose, the condition (H1) looks like essential since we need an
equivariant differential such that $D^{2}=0$.

Since a momentum section and a Hamiltonian Lie algebroid structure is a
natural structure on a gauged sigma model, we can hope to obtain new physical
results from analysis of a Hamiltonian Lie algebroid.

### Acknowledgments

The author would like to thank Yuji Hirota, Kohei Miura, Satoshi Watamura and
Alan Weinstein for useful comments. He thanks the referees for their careful
reading of the manuscript and especially for their helpful comments.

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