Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. *

This research was supported by Basic Science Research Program through the
National Research Foundation of Korea (NRF), funded by the Ministry of
Education, Science, Technology (No. 2019R1A2C1085113). This work was also
supported by grant No. 02-2018-043 from the Seoul National University Bundang
Hospital (SNUBH) Research Fund.

Corresponding author: Il Dong Yun (e-mail: yun@ hufs.ac.kr) and Eun Ju Chun
(humandr@snubh.org).

# Centerline Depth World for Reinforcement Learning-based Left Atrial
Appendage Orifice Localization

WALID ABDULLAH AL1  IL DONG YUN2 , AND EUN JU CHUN.3  Department of Computer
and Electronic Systems Engineering, Hankuk University of Foreign Studies,
Yongin, 17035 South Korea<EMAIL_ADDRESS> Department of
Computer and Electronic Systems Engineering, Hankuk University of Foreign
Studies, Yongin, 17035 South Korea<EMAIL_ADDRESS> Department of
Radiology, Seoul National University Bundang Hospital, Seongnam, 13620 South
Korea<EMAIL_ADDRESS>
###### Abstract

Left atrial appendage (LAA) closure (LAAC) is a minimally invasive implant-
based method to prevent cardiovascular stroke in patients with non-valvular
atrial fibrillation. Assessing the LAA orifice in preoperative CT angiography
plays a crucial role in choosing an appropriate LAAC implant size and a proper
C-arm angulation. However, accurate orifice localization is hard because of
the high anatomic variation of LAA, and unclear position and orientation of
the orifice in available CT views. Deep localization models also yield high
error in localizing the orifice in CT image because of the tiny structure of
orifice compared to the vastness of CT image. In this paper, we propose a
centerline depth-based reinforcement learning (RL) world for effective orifice
localization in a small search space. In our scheme, an RL agent observes the
centerline-to-surface distance and navigates through the LAA centerline to
localize the orifice. Thus, the search space is significantly reduced
facilitating improved localization. The proposed formulation could result in
high localization accuracy comparing to the expert-annotations in $98$ CT
images. Moreover, the localization process takes about $8$ seconds which is
$18$ times more efficient than the existing method. Therefore, this can be a
useful aid to physicians during the preprocedural planning of LAAC.

###### Index Terms:

appendage closure, appendage occlusion, left atrial appendage, centerline
depth, orifice detection, orifice localization, reinforcement learning

=-15pt

##  I Introduction

Figure 1: The proposed centerline depth world for LAA orifice localization.
(Left) centerline from LAA to LA, and the orifice (red region) are shown in
the cropped input volume. (Right) RL world for the corresponding centerline.
Red square mark indicates the localization agent position $i$. Observing the
local centerline depth state $s$, the agent moves forward or backward along
the centerline to converge at the orifice position $p$ indicated by the red
vertical line. Agent decides the optimal moving action $a$ based on its policy
$\pi(\boldsymbol{s},a)$.

Atrial fibrillation (AF) characterized by rapid and irregular atrial beating
is one of the most common heart abnormalities [1]. According to the report by
Zoni-Berisso et al. [2], AF affects $2$–$3$% of the population, with a high
increase comparing to $0.3$–$1$% in $2005$ [3]. One of the most prevalent
sites of thrombus formation leading to AF-associating cardiovascular stroke is
the left atrial appendage (LAA) [4]. To treat high-risk AF-patients, a
minimally invasive implant-based strategy called LAA closure (LAAC) is widely
used [5]. In this strategy, a closure device is implanted at the appendage
orifice to prevent the thrombus from entering into the bloodstream. For
preoperative interventional planning, LAA assessment deserves duteous
attention where the major concern to the physicians during the assessment is
the detection of the orifice. Accurate identification of the orifice plays a
significant role because it contributes to choosing the occlusion device of an
appropriate size and obtaining a proper C-arm angulation for intervention.

Left atrial appendage can be viewed as an anatomical projection from the main
left atrial (LA) chamber, and the LAA orifice (or, ostium) is the narrow
opening of the appendage to the atrium (Fig. 1). Preoperative planning of LAAC
utilizes CT angiography to acquire 3D measurements during the orifice
assessment. However, localizing the appendage orifice is a challenging task
due to significant variation in shape, size, and orientation of the appendage.
Moreover, conventional CT views do not allow a clear understanding of the
appendage anatomy. In addition, the appendage and the left atrium share a
common intensity with no separation wall against each other. Consequently,
accurate orifice localization becomes laborious and time-consuming for the
physicians. Therefore, a computational method for appendage orifice
localization can be useful to the physicians by reducing their effort and
speeding up the preoperative planning.

With the available LAA orifice assessment works mainly limited to manual
approaches [6, 7, 8], the major research-focus is on the LAA segmentation
where no indication is provided for the orifice. Prior appendage segmentation
works are generally semi-automatic requiring a manual volume-of-interest (VOI)
annotation. In most of these works, the primary goal is to segment the left
atrium, where appendage is included in the segmentation result without any
distinction [9, 10]. Zheng et al. [11, 12] suggests segmenting the left atrium
by separate shape-constrained segmentation of different parts of the LA
including the appendage. Later, the gap between the main LA chamber and the
appendage is filled by projecting the proximal ring of appendage towards the
main chamber. Because the orifice usually resides in this gap, such smooth
projection may risk oversimplifying the irregular orifice shape.

There are a number of works treating LAA segmentation as the major objective.
Grasland-Mongarin et al. [13] also presented shape-constrained deformable
models to segment the cardiac chambers, where LAA was extracted using mesh
inflation. However, model-based approaches face problem in coping with the
varying LAA structure. Model-free approaches usually have two steps: (i) axial
slice-wise 2D segmentation proposals (using parametric max-flow [14] or fully
convolutional networks [15]), and (ii) 3D refinement (using 3D conditional
random field [15]). Such model-free approaches could cope better with the
varied anatomy of LAA. Nonetheless, all these segmentation approaches include
a part of the left atrium in their segmented results without presenting a
clear separation for the orifice.

Leventić et al. [16] separated the appendage from the left atrium in their
region growing-based segmentation. The separation plane is obtained from three
additional points manually marked in the segmented volume, supposedly
representing the LAA orifice. Recently, they extended their work to compute
the orifice location following a semi-automatic approach requiring a threshold
to obtain initial mask and a seed-point in LAA [17]. To the best of our
knowledge, this is the only existing computational method dealing with orifice
localization.

The orifice detection approach of Leventić et al. [17] is based on scanning
the cross-sectional area along the centerline. Because cross-sections inside
the LA continues to be larger after the orifice, this method identifies the
largest continuous rise in the cross-sectional areas. The orifice location is
detected at the closest local minima before such rise. The optimal cross-
section for each point in the centerline is exhaustively determined by taking
the minimum area from multiple cross-sectional planes tilting the plane by up
to 40 degrees. Overall, the whole process become computationally expensive
taking about 2.5 minutes [17].

Most importantly, using such a fixed rule for localizing the orifice may not
be a robust solution considering the highly varied anatomy of the appendage.
The unavoidable noise and artifacts in CT can also become potential threats
against such localization rule. False local minima may arise posing threat to
identifying both the largest uninterrupted rise inside LA and the closest
local minima before.

Machine learning approaches showed improvements in several medical image
analysis tasks under such high variation [18, 19]. With the advent of deep
learning, several end-to-end localization models have been proposed. Such deep
models can predict location of keypoints or landmarks of interest directly
from image. Convolutional neural network (CNN)-based cascaded regression
network [20, 21] is a widely used architecture to regress the direct location.
For anatomical landmark localization in medical images, spatially regressing
Gaussian heatmaps [22, 23] around the target location showed fruitful results.
However, training such end-to-end models to localize the orifice from raw
image is difficult because of the tiny orifice structure compared to the vast
search space of 3D CT image.

In this work, we aim at ensuring robust LAA orifice localization under the
high anatomical variation. Instead of setting a fixed rule [17] for detecting
the orifice, we suggest a learning-based approach to utilize the rich context
in expert-annotated data. In our approach, a reinforcement learning (RL) agent
is trained to navigate through LAA centerline to localize the orifice. For
effective learning, we propose a centerline depth-based RL environment, which
has a small search space to enable easier learning, yet showing strong
potential for discriminating the orifice location. Due to the training
difficulty using raw image, we equip the agent with the centerline-to-surface
distance information for simpler but better representation about the
anatomical progression. Fig. 1 illustrates the proposed RL world for orifice
localization. For centerline seed initialization, the proposed semi-automatic
method requires a simple click roughly at the LAA tip, which is clearly
visible in the orthogonal CT views.

For improved evaluation, we use a dataset significantly larger than the
dataset used in the previous orifice localization work [17]. Our dataset also
includes real patients who have undergone the LAAC procedure. Besides
comparing the orifice center location similar to the previous study [17], we
also compare the orifice plane and area against the expert orifice annotations
for a fruitful evaluation of the detected orifice. We show that the proposed
localization method yields significantly lower detection error compared with
the rule-based [17] and the other learning-based solutions. Taking only $8$
seconds, the proposed orifice localization method can be a useful guide to the
physicians during preoperative planning of LAAC.

##  II Methodology

To accelerate the proprocedural assessment of LAA orifice in CT angiography,
we suggest a robust orifice localization approach formulating the localization
process as a reward-maximization problem of a RL agent. Using the LAA
centerline as the search space for localization, we propose a simple RL world
called the centerline depth world. The localization agent in such a world
interacts by moving one step at a time along the centerline, to eventually
converge near the orifice position after a number of steps.

Centerline depth refers to the distance of centerline from the nearest
surface-point. Usually, anatomy of the left atrial appendage suggests a
gradually increased cross-sectional area from the LAA orifice to the LA unlike
the area from LAA tip to the orifice [17]. Therefore, the alteration of
centerline depth from the LAA to LA provides powerful information for the
agent to localize the orifice. Localizing the tiny orifice in the vast CT
image using an end-to-end deep localization model [20, 22] is difficult.
Instead, the centerline can serve as a small but useful search space to track
the anatomy. Such centerline-based search strategy is also used in the
previous orifice localization study [17] utilizing the above anatomical
progression towards the atrium. The previous study computed the minimal cross-
sectional area at each centerline point to search for the closest local minima
before the largest uninterrupted increase. However, the desired cross-
sectional areas along the centerline is highly noisy and computationally
expensive. The depth computation, on the other hand, is highly efficient and
has relatively low noise, while providing an equivalent representation (shown
later in Fig. 4).

In our RL world, the agent observes the local depth-map and navigates through
the centerline to finally reach the orifice. We first describe the procedure
of computing the centerline from a given CT image, then move to presenting the
proposed centerline depth world for orifice detection.

###  II-A LAA Centerline Extraction

This process requires a manually annotated initial seed roughly at the LAA
tip. Marking this initial seed is convenient and fast because the process is
not sensitive to the seed position as long as it resides inside the LAA tip
before the LAA neck. From the initial seed, we obtain the LAA centerline in
three steps. Firstly, we binarize the CT image extracting the organs-of-
interest (OOI) from the background. Secondly, we perform the Euclidean
distance transform from the binarized image. Finally, we grow the centerline
from LAA to LA using the resultant distance map.

Let us denote the initial LAA seed by $\boldsymbol{x}_{s}$. The OOI extraction
should be extensive in nature so that the resultant region includes a
significant part of the left atrium beyond the appendage orifice. Otherwise,
the following detection using the axis-to-surface distance can be problematic
due to insufficient data related to the progression after the orifice.
Existing LAA segmentation techniques could be used for appendage extraction.
However, as discussed in the previous section, the model-based approaches
among those has the problem of coping with structural variation [15]. The
supervised slice-wise mask prediction with 3D CRF-based refinement comes with
a large computational cost [15, 14]. Supervised approaches also requires
laboriously marking the ground truth for 3D segmentation mask for all the
training volumes.

In contrast, we adopt a model-free 3D geodesic distance based unsupervised
approach. The LA and LAA has an even and easily distinguishable intensity
distribution compared with the background (Fig. 2). Therefore, we can simply
obtain the desired LAA and LA regions based on the geodesic distance from the
LAA seed. Obtaining the desired region directly from 3D geodesic distance
instead of 2D slice-wise prediction enables faster computation. Geodesic
distance-based extraction also allows the seed to grow where it finds similar
intensity without any constraint. Therefore, the resultant region can cover
the entire appendage following the intensity including the anatomical
irregularities. This advantage is also discussed in [15]. The major problem of
the this approach is reported to be the inclusion of the adjacent left
superior pulmonary vein (LSPV) [15] because small leakage may occasionally
exist from LAA to LSPV. However, this does not affect our centerline depth-
based localization. This is because depth is calculated as the centerline
distance from the nearest surface (or background) and LAA surface is closer
than the LSPV surface.

Figure 2: Intensity distribution of a CT image (in Hounsfield unit).
Intensity-based geodesic distance thresholding can extract the LA and LAA
regions.

First, we obtain a fixed-size VOI relative to the seed location, so that it
sufficiently encloses the appendage and a part of the LA. This may also
include the left superior pulmonary vein (LSPV). Inside the VOI, we compute
the geodesic distance for all the voxels with respect to the localized seed
$\boldsymbol{x}_{s}$. Finally, we can extract the extended LAA by thresholding
the distance.

Initially, we grow the seed by a radius of $5$ voxels to obtain a small seed
region $\Omega$. For the reference voxel-set in $\Omega$, the geodesic
distance at any position $\boldsymbol{x}$ indicates the distance of the
shortest geodesic path, which is defined as follows:

$\begin{split}\mathcal{D}(\boldsymbol{x};\Omega)&=\min_{\boldsymbol{y}\in\Omega}{\min_{\zeta\in\mathcal{P}_{\boldsymbol{x},\boldsymbol{y}}}\sum_{\boldsymbol{p}\in\zeta}\sqrt{\big{(}|\nabla_{\zeta}\boldsymbol{p}|^{2}+\alpha|\nabla_{\zeta}I(\boldsymbol{p})|^{2}\big{)}}}\end{split}$
(1)

where $\zeta$ iterates over all paths from $\boldsymbol{x}$ to
$\boldsymbol{y}$, $\mathcal{P}_{\boldsymbol{x},\boldsymbol{y}}$, and
$\nabla_{\zeta}$ indicates the gradient following a path $\zeta$. $I$ denotes
the intensity. $\alpha$ sets the contribution of image gradient over spatial
distance. In our experiment, we simply use $\alpha=1$. Such geodesic distance
is also used by Criminisi et al. [24] for segmentation. As opposed to the
usual segmentation labeling decided by the comparative closeness to a
foreground or background seed, a distance threshold parameter $\lambda$
decides the segmentation in our case of a single foreground seed. In our
experiment, a fixed threshold value $\lambda=0.3$ is used for thresholding the
normalized geodesic distance. As we showed in Fig. 2, the intensity
distribution of our desired region is apparently different from the
background. Therefore, the threshold value does not have to be precise. We
obtained the above threshold value by visual assessment through a trial-and-
error process.

We compute the geodesic distance following the raster scan update scheme used
by [24]. Two sets of forward and backward updates are performed. In the
forward update, we visit and update each voxel using upper-lower slice, top-
bottom, left-right scanning. Backward scan follows the opposite direction. The
current voxel is updated using the geodesic distance information of its
neighbours that have been visited in the current scan.

After obtaining the binarized CT image with foreground including the LA, LAA,
etc., we compute the Euclidean distance transform [25] to have the distance of
each voxel from its nearest background. Note that, this is equivalent to the
voxel-wise distance to the nearest surface of the extracted region. Let us
denote this distance map by $\hat{D}$ . We call this the depth of a voxel with
respect to the surface.

We grow the centerline by tracking the maximum depth voxels from the initial
LAA seed $\boldsymbol{x}_{s}$ to the LA. At each step $i$, we decide the next
centerline voxel $\boldsymbol{x}_{i+1}$ by finding the local maxima among the
unvisited neighboring voxels.Thus, we represent the centerline by all of the
sequentially tracked voxels as: $\mathcal{M}=(\boldsymbol{x}_{i})_{i=1}^{T}$.
Here, $T$ is the fixed length of the centerline. In our experiment, $T=300$
was sufficient to grow the desired centerline.

The resultant centerline $\mathcal{M}$ serves as the search space for the
orifice detection. Anatomy of the left atrial appendage suggests a gradually
increased cross-sectional area from the LAA orifice to the LA unlike the area
from LAA tip to the orifice. Instead of computing the cross-sectional area
about the computed axial points in $\mathcal{M}$, we use the axis-to-surface
distance (i.e., the distance of the axial points to the nearest surface)
because it renders equivalent information. Moreover, the axis-to-surface
distance about an axial point $\boldsymbol{x}_{i}\in\mathcal{M}$ can be
obtained directly from the earlier Euclidean distance transform map
$\hat{\mathcal{D}}(\boldsymbol{x}_{i})$ without any further computation.

###  II-B Centerline Depth World

The proposed RL world is defined based on the computed centerline
$\mathcal{M}$ and the depth along the centerline voxels. We denote the
centerline depth by a vector $\boldsymbol{\hat{d}}=(\hat{d}_{i})_{i=1}^{T}$
where $\hat{d}_{i}=\hat{\mathcal{D}}(\boldsymbol{x}_{i})$ is the depth of the
$i$-th voxel in the centerline. Fig. 1 illustrates the centerline depth world.
Note that, the agent position in this world suggests the index to a centerline
voxel in the actual CT image. We define the target orifice position $p$ in
this world to be the index to the closest centerline voxel comparing with the
expert annotated orifice center. Initiating from any position index $i$, the
goal of the RL agent is to localize this orifice voxel index $p$ in the
centerline.

The previous study applied a fixed rule to track the closest local minima
before the largest uninterrupted cross-sectional increase (supposedly
occurring in the LA after the orifice). This can be sensitive to the highly
varying LAA anatomy. For example, there are some cases where there are no
local minima at the LAA neck because LAA neck is larger than the tip.
Additionally, false local minima may arise due to the high noise in CT.

To handle such issues, we train an agent to optimize its policy for the
expert-annotated data containing varied LAA images. The proposed world is a
simple 1D world, yet capable of providing discriminative knowledge for the
orifice localization agent. The agent initialized at any position index $i$ in
the centerline, observes the local centerline depth, and decides an action to
move forward or backward along the centerline. Thus taking an episode of steps
along the centerline, it eventually converges to the target $p$-th position in
the centerline.

Relating the key elements of RL formulation, state $\boldsymbol{s}$ can be
referred to as the local centerline depth map centered at the agent position.
Therefore, for any position $i$, the state
$\boldsymbol{s}_{i}=(d_{j})_{j=i-k}^{i+k}$ for a state-length of $2k+1$.
Observing this state, the agent decides to move forward or backward along the
centerline to reach the orifice in the future. Therefore, the action-space can
be denoted as: $A=\\{forward,backward\\}$. The policy $\pi(\boldsymbol{s},a)$
represents the agent-behavior model that outputs the optimal action
probabilities $P(a);a\in A$ for a given state $\boldsymbol{s}$. We denote
agent-transition by $(i,a,i^{\prime})$ indicating a transition from position
$i$ to $i^{\prime}$ on the centerline by action $a$.

For training the policy, a discrete reward signal $R$ is fed for each
transition, to encourage moves that draws the agent closer to the target or
discourage otherwise. We also encourage a reward of convergence when the agent
is at a small distance away from the target. Similar rewarding scheme is also
showed to be effective in previous RL-based anatomical landmark localization
using full image space [26]. For a transition $(i,a,i^{\prime})$, we present
our reward function as follows:

$R(i,a,i^{\prime})=\begin{cases}+2,&\text{ if }|p-i|\leq\tau\\\ +1,&\text{ if
}|p-i^{\prime}|<|p-i|\\\ -1,&\text{ otherwise}\end{cases}$ (2)

In deep reinforcement learning, policy $\pi$ is represented by a deep neural
network. Similar to the widely used policy networks for localization in 2D and
3D image space [27, 26], our policy model also begins with a stack of
trainable convolutional layers to extract discriminative features from the
input state. These features are then fed to a fully connected multi-layer
perceptron model with two output nodes for giving probabilities the two
possible actions. Appendix A describes the policy network description in
detail.

The goal of training is to optimize the policy model to maximize the episodic
reward. Policy is usually optimized by stochastic gradient ascent over a
number of epochs. At each epoch, the agent first runs several episodes using
its current policy and gather experiences, where an experience
$(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r)$ consists of the state
$\boldsymbol{s}$, chosen action $a$, next state $\boldsymbol{s}^{\prime}$, and
the reward $r$. Sampling mini-batches from these experiences, the policy is
then updated by taking gradient steps towards maximizing the expected reward.

The gradients for the above-described policy-update are estimated following
the policy gradient theorem [28], which uses the log-likelihood of actions in
its objective function. Despite the wide use of this objective function, it
causes destructively large policy updates and is not robust [29, 30]. We use
the recently proposed proximal policy optimization (PPO) [29], which renders
more robust and stable policy update. While in policy gradient theorem, log-
probability is used to trace the impact of actions, PPO uses the ratio of the
current policy to the old, i.e.,
$\rho(\pi)=\pi(\boldsymbol{s},a)/\pi_{\text{old}}(\boldsymbol{s},a)$.
Furthermore, this impact is clipped between $[1-\delta,1+\delta]$ in the
objective function to prevent large policy changes. The clipped objective
function is as follows:

$\begin{split}L(\pi)=\mathbb{E}_{(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r)}&\big{[}\min\big{(}\rho(\pi)A(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r),\\\
&\text{
clip}(\rho(\pi),1-\delta,1+\delta)A(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r)\big{)}\big{]}\end{split}$
(3)

Here, $A(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r)$ indicates the advantage
of the current policy for state $s$, over the previous state-value
$V(\boldsymbol{s})$. State-value function $V(\boldsymbol{s})$ gives the
discounted cumulative reward achievable through state $\boldsymbol{s}$ in the
long-term. The discounted cumulative reward of the current policy for a
transition $(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r)$ is estimated as:
$r+\gamma V(\boldsymbol{s}^{\prime})$, the discounted state-value of the next
state added to the current reward. Here, $0<\gamma<1$ is the discount factor.
Now, the advantage is computed by taking the increase of this cumulative
reward from the value of state $\boldsymbol{s}$ :

$A(\boldsymbol{s},a,\boldsymbol{s}^{\prime},r)=r+\gamma
V(\boldsymbol{s}^{\prime})-V(\boldsymbol{s})$ (4)

The state-value function $V$ is also represented by a deep neural network.
Parameters of this value network is also updated at each epoch during
training.

During testing with the trained policy $\pi$, the agent is initialized at a
random position in the centerline depth world. Using the policy, it takes
optimal actions to subsequently update its position until it moves back and
forth at the orifice position. Setting the position of convergence in the
centerline as the center of the orifice, we initialize a perpendicular plane
as the orifice plane. Later, we refine this plane by tilting the plane upto
$20$ degrees about each axis and finding the plane with minimum cross-section.

Using the centerline depth input, localizing the orifice in the centerline can
also be done by other learning algorithms. For example, a 1D deep regressor
can be used to directly estimate the orifice location in the centerline. Also,
a spatial regressor could also be used to regress heatmaps around the orifice
location. Though the search space is significantly reduced using centerline
depth, the input dimension is still higher having a length equal to the
centerline length. Collecting sufficient CT images to fit a deep model for
this can be challenging. The RL-formulation, on the other hand, extracts
states from the original centerline depth. These states are used as the
training samples for policy training, which are much more in number gathered
by agent-exploration.

##  III Results and Discussion

Figure 3: Orifice localization error of different methods. Previously attempted rule-based method showed a high error due to the sensitivity to the varied LAA anatomy in the CT images. End-to-end method using CNN also had high localization error, indicating the localization difficulty in full image space. Centerline depth-based methods showed relatively low error because of the reduced search space and simpler input feature. The proposed RL-based method could further lower the error by using its explorative learning with small states. TABLE I: Orifice assessment error compared with expert annotation. | Method | Center difference (mm) | Orientation difference (∘) | Area difference ($\text{mm}^{2}$)
---|---|---|---|---
| Mean $\pm$ SD | Median | Mean $\pm$ SD | Median | Mean $\pm$ SD | Median
Overall | CD:RL | $\boldsymbol{2.547\pm 1.931}$ | $\boldsymbol{2.067}$ | $\boldsymbol{9.895\pm 6.236}$ | $\boldsymbol{9.982}$ | $\boldsymbol{72.399\pm 74.365}$ | $\boldsymbol{44.165}$
CD:heatmap regress | $3.806\pm 4.207$ | $2.582$ | $14.573\pm 8.799$ | $12.354$ | $75.472\pm 71.646$ | $54.029$
CD:location regress | $4.113\pm 3.597$ | $3.485$ | $14.396\pm 8.959$ | $13.581$ | $102.676\pm 128.085$ | $54.821$
CNN location regress | $6.231\pm 5.464$ | $4.851$ | $15.198\pm 10.375$ | $12.194$ | $142.168\pm 163.259$ | $102.160$
Rule-based | $8.904\pm 8.058$ | $5.900$ | $18.213\pm 12.208$ | $15.192$ | $355.463\pm 534.821$ | $162.037$
LAAO Patients | CD:RL | $\boldsymbol{2.166\pm 1.113}$ | $\boldsymbol{1.994}$ | $\boldsymbol{9.519\pm 6.899}$ | $\boldsymbol{8.133}$ | $\boldsymbol{84.745\pm 70.192}$ | $\boldsymbol{48.563}$
CD:heatmap regress | $4.075\pm 6.193$ | $2.183$ | $15.282\pm 11.762$ | $12.605$ | $121.863\pm 186.677$ | $50.682$
CD:location regress | $3.539\pm 1.744$ | $3.409$ | $10.887\pm 8.076$ | $10.703$ | $127.723\pm 148.023$ | $56.594$
CNN location regress | $6.709\pm 4.329$ | $6.349$ | $17.542\pm 13.335$ | $16.699$ | $304.937\pm 308.462$ | $192.025$
Rule-based | $7.102\pm 8.894$ | $5.322$ | $14.423\pm 17.069$ | $9.584$ | $324.109\pm 446.462$ | $168.534$

We evaluated the proposed orifice localization method using CT image of $98$
cases. $7$ cases among these are from actual LAAC patient who underwent
through the appendage occlusion procedure using the WATCHMAN implant [31]. We
randomly split the dataset to obtain $80\%$ of the data for training the
policy. The remaining $20\%$ of the data is used as the test set for
evaluating our localization model. During the split, the $7$ actual cases are
kept in the test set. All the CT images are obtained from a local clinical
site. Expert annotations for the orifice are also obtained from the same site.
The pixel spacing of the volumes ranges from about $0.3$ to $0.5$ mm, where
the inter-slice spacing ranges from $0.45$ to $0.50$ mm. The patients were
about $67\pm 8$ years old, about $36\%$ of them being female.

Because the previous orifice localization work [17] used a small dataset (of
$17$ cases), we re-implemented their method using our data for fair
comparison. In the result table and figures, we label this method as _rule-
based_. To show the advantage of the centerline-based search, we also compare
the end-to-end deep localization models based on CNN to predict orifice
location directly from CT image. We implemented both the direct location
estimation network [20] and the heatmap regression network [23]. However, they
had slight difference in overall performance. Therefore, we only mention the
results of direct location regression and identify this as _CNN location
regress_ across tables and figures.

We also implemented 1D versions of such models for the centerline-based search
space, where the regression network directly predicts orifice location from
the centerline depth input. This is useful for evaluating the necessity of
using a RL agent to localize instead of direct regression from the centerline
depth. All the centerline depth-based learning methods including ours are
labeled with the prefix _CD_ indicating centerline depth.

We plot the localization error distribution of different methods in Fig. 3.
Here, the error represents the Euclidean distance of the localized orifice
center from the expert annotated center location. For comprehensive
evaluation, we also compare the orientation and area of the orifice in Table
I. Orientation error is presented as the angular difference between the normal
vectors of the detected and expert-identified orifice. In Table I, we also
present the results for the actual LAAO patients separately.

Figure 4: Visualization of orifice location detected by different methods.
(Top) red, blue, green and magenta regions in the volumetric render indicate
the resultant orifice by the expert, proposed method, centerline depth-based
heatmap regression and previous rule-based approach, respectively. The black
line is the centerline from LAA to LA. (Bottom) the resultant orifice
locations are marked in the centerline depth map using the same colors. The
proposed method could robustly localize the orifice near the expert-annotated
location. The previous rule-based technique may fall in false local minima
outside the neck (middle and right). RL-based method with its simpler model
and small input states also resulted in better localization than the heatmap
regression approach in cases where the enlargement starts much before the
orifice.

The centerline depth-based localization methods could give significantly lower
error than the end-to-end deep localization method. This shows the difficulty
in localizing the small orifice using raw image with full search space.
Limiting the search space to the centerline and using centerline depth input
contributed to comparatively easier learning because it provides rich
relatable information for tracking the anatomy. The previously proposed rule-
based method [17] showed relatively larger error. The irregular LAA structure
affected the identification of the desired local minima before the largest
uninterrupted cross-sectional increase.

The proposed RL-based centerline search gave the lowest error with a reduced
error variance. Having an average localization error of $2.547\pm 1.931$ mm,
the proposed method showed significant improvement over the other methods
($p$-value $<0.02$). Centerline-based direct regression models used the small
training data directly obtained from the given CT images. On the other hand,
the proposed method utilizes the agent to explore thousands of sequential
states for policy-training at each epoch. Moreover, the underlying state-
distribution is also discovered by the agent as it attempts to perform
different localization episodes through sequential state-transition. Thus,
some states helpful for localization process are given more importance.
Whereas, sample class-weight tuning is a crucial step in heatmap-regression
based localization.

In Fig. 4, we illustrate the localized orifice position for different cases.
The first case shows an example where all the methods showed low localization
error. In the second case, LAA neck is gradually enlarged from the tip.
Therefore, the rule-based method gave incorrect localization assigning the
local minima at the tip as the orifice. In the last case, there are multiple
local minima within the enlargement from the orifice to the LA. As a result,
the rule-based method falsely identified the orifice among such minima. Though
the centerline-based deep localization model localized the orifice at a small
distance in the previous two cases, it showed a high error for the last case.
Because the tip is smaller than the neck, visually the anatomical enlargement
starts earlier from the tip, such model proposed the orifice near the tip.
Whereas, the actual orifice location falls within the enlargement, which is
correctly localized by the proposed method.

Figure 5: Model sensitivity to seed initialization. Detected orifice locations
for different seed-points had a low variance. (Left) volumetric render of the
seed-points inside cropped input volume. (Middle) Localization errors for
different seed points. (Right) Seed points shown in the axial view. Figure 6:
Model dependency on state-length. Too small states fail to provide sufficient
information to guide the agent decision. Increasing the state-size improves
the performance. Too big states reduce state-variation making the state-
discrimination harder.

To show the clinical impact of our method, we compared our localization
results with the inter-observer variance. To do so, we obtained annotations
for partial data from a second expert observer and compare these with the
original annotations. The inter-observer difference of the orifice center was
$2.82\pm 1.42$ mm, which is larger than the difference of the proposed method.
The whole localization process after manual seed initialization takes only
about $8$ seconds, which is $18$ times faster than the previous approach [17].
Use of the linear centerline depth computation instead of exhaustive scanning
for the minimum cross-sectional area contributed to such efficiency. Thus, the
proposed method can be potentially useful for the physicians during the LAAO
planning.

###  III-A Dependency on state-length

State-based sequential modeling is the key reason to enable a simpler decision
model with small but numerous training states. Therefore, state-length becomes
an important parameter. We observed the localization performance of the
proposed model by varying the state-size from $31$ to $71$. We plot the
results in Fig. 6. Smaller states provide little information for the agent to
decide optimal moves. Therefore, increasing the state-length allows the agent
to utilize more information to effectively choose the action. However, after a
certain state-length, the accuracy begins to degrade. Such bigger states
reduces state-variation losing the local focus. Therefore, it becomes
difficult for the agent to discriminate the states. In our experiment, a
state-size of $51$ was optimal.

###  III-B Sensitivity to seed initialization

Because our semi-automatic method requires manual seed initialization for
growing the LAA centerline, analyzing the sensitivity to such initialization
is necessary. Identifying the seed-point is rather simple because this does
not have to be a precise location. Rather this can be anywhere in the LAA tip
before the LAA neck. Though orifice identification is difficult in orthogonal
CT views, LAA tip can be easily distinguished. Fig. 5(right) shows various LAA
tip seed-points in axial view.

In Fig. 5, we plot the localization performance for different seed-points from
all over the LAA before the orifice. Resultant positions for different seed-
points had a low variance of $0.302$. Therefore, manual annotation can be done
with enough relaxation. From the annotated seed, we grow the centerline by
tracking the local maxima in the depth map. Regardless of the initial
position, the first few steps allows to reach the actual centerline (maximum
depth nearby) crossing the highest gradient path in the depth map. Afterwards,
the tracking follows along the desired centerline through the orifice upto the
LA. Therefore, all such seed points eventually end up being grown into the
actual centerline, causing similar localization results.

##  IV Conclusion

To ensure robust LAA orifice localization, we proposed a reinforcement
learning-based approach. We formulated a centerline depth world where a
localization agent performs a search for the orifice on the LAA centerline
observing local centerline depth as small states. The proposed approach showed
improved robustness compared with the previously rule-based approach. It also
showed significant improvement over the end-to-end CNN-based localization, by
using a reduced search space and simpler input. Moreover, comparing deep
learning-based methods using the same reduced search space, we observed that
the proposed RL formulation enabled more effective learning by its sequential
exploration of small training states. Therefore, the proposed localization
approach can be a useful aid to the physicians during the LAA occlusion
planing.

##  Appendix A Policy Network and RL hyperparameters

The initial part of the policy network has three 1D convolutional units to
output high-level feature from the input state. Each unit has two
convolutional layers with kernel-size of $3$, using rectified linear unit
(ReLU) activation. At the end of each unit, there is a 1D max-pooling layer
having a pool-size of $2$ and stride of $2$ to reduce the spatial dimension.
The latter part of the network consists of three fully connected (FC) layers.
The final FC layer has two output nodes and uses softmax-gating to produce
probabilities for the actions in our binary action space.

During the training, an $\epsilon$-greedy policy is used for episodic state-
exploration, where the agent greedily chooses an action based on the policy
with probability $\epsilon$ and it chooses a random action with probability
$1-\epsilon$. In our experiment, $\epsilon=0.7$ is used. The discount factor
$\gamma$ in (4) is set to $0.9$. Both these parameter-values are widely used
in different RL studies [27, 26]. The optimal learning rate was chosen to be
$1\mathrm{e}{-5}$, after trial-and-error process using different learning
rates ranging from $0.1$ to $1\mathrm{e}{-7}$.

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# 3-form Yang-Mills based on 2-crossed modules

Danhua Song<EMAIL_ADDRESS>Ke Wu<EMAIL_ADDRESS>Jie Yang
<EMAIL_ADDRESS>School of Mathematical Sciences, Capital Normal University,
Beijing 100048, China

###### Abstract

In this paper, we study the higher Yang-Mills theory in the framework of
higher gauge theory. It was shown that the 2-form electromagnetism can be
generalized to the 2-form Yang-Mills theory with the group $U(1)$ replaced by
a crossed module of Lie groups . To extend this theory to even higher
structure, we develop a 3-form Yang-Mills theory with a 2-crossed module of
Lie groups. First, we give an explicit construction of non-degenerate
symmetric $G$-invariant forms on the 2-crossed module of Lie algebras. Then,
we derive the 3-Bianchi-Identities for 3-curvatures. Finally, we create a
3-form Yang-Mills action and obtain the corresponding field equations.

crossed module of groups, 2-crossed module of groups, 3-connection, 3-Bianchi-
Identities, 3-form Yang-Mills

In modern formal theoretical physics a certain idea is very productive: the
higher-dimensional extended objects are thought to be the basic constituents
of matter and mediators of fundamental interactions. At present, higher gauge
theory Baez.2010  seems to be the geometrically most promising technique to
describe the dynamics of the higher-dimensional extended objects where gauge
fields and field strengths are higher degree forms. Among them, the 2-gauge
theory Baez2005HigherGT ; Bar ; JBUS  and the 3-gauge theory
Faria_Martins_2011 ; Saemann:2013pca ; doi:10.1063/1.4870640 ; Fiorenza:2012mr
have been studied deeply, whose higher algebraic structures and higher
geometrical structures can be found in, e.g. TRMV ; JBUS ; U ; ACJ ; FH . In
this article, we are concerned about the non-abelian higher gauge fields
arising in a great deal of physical contexts, such as six dimensional
superconformal field theory Saemann:2013pca , quantum gravity TRMV , string
theory USJ  and M-theory HSG ; SP , and so on.

The electromagnetic theory is the simplest abelian gauge theory where the
gauge field is described by the connection 1-form $A$ on a $U(1)$ bundle. When
the charged particles are extended to $1$-dimensional charged strings, one can
get the $2$-form electrodynamics HP  where the $2$-form gauge field can be
described in terms of the $2$-form $B$ known as the Kalb-Ramond field
Kalb:1974yc . The $2$-form gauge field $B$ is described by a connection on a
$U(1)$ gerbe as the categorized version of a $U(1)$ bundle YZ . Furthermore,
there exists a generalization known as $p$-form electrodynamics, where the
$p$-form gauge fields play the role of gauge fields 10.1007/BF01889624 ; EE ;
YTMM . When the group $U(1)$ is replaced by a nonabelian counterpart, one can
obtain the Yang-Mills theory Mathai_2006 ; pittphilsci10008 . Therefore, it
raises a natural question of whether one can generalize Yang-Mills theory to a
kind of higher-form Yang-Mills theory. The main contribution of this work is
to develop a general method for 2-form, 3-form Yang-Mills theories.

In the related work 2002hep.th….6130B , Baez generalized the Yang-Mills theory
to the 2-form Yang-Mills theory based on the $2$-form electrodynamics. The
gauge group is replaced by a crossed module of groups
$\left(H,G;\alpha,\vartriangleright\right)$ Beaz ; Crans ; Brown , where
$\alpha:H\longrightarrow G$ is a Lie group homomorphism and
$\vartriangleright$ is a smooth action of the Lie group $G$ on the Lie group
$H$ by automorphisms. A simple example of a Lie crossed module is $G=H=U(N)$
with $\alpha$ the identity map and the $\vartriangleright$ the adjoint action.
An associated differential crossed module
$\left(\mathcal{h},\mathcal{g};\tilde{\alpha},\widetilde{\vartriangleright}\right)$
can be constructed, where
$\tilde{\alpha}:\mathcal{h}\longrightarrow\mathcal{g}$ is a Lie algebra map
and $\widetilde{\vartriangleright}$ is a left action of the Lie algebra
$\mathcal{g}$ of $G$ on the Lie algebra $\mathcal{h}$ of $H$ by derivations.
Obviously, the differential version of the above example is
$\mathcal{g}=\mathcal{h}=\mathcal{u}(N)$ with $\widetilde{\vartriangleright}$
being the adjoint action and $\tilde{\alpha}$ being the identity map. We
perform the crossed module of groups using squares of the form

$g_{1}$$g_{2}$$h$

where $g_{1}$, $g_{2}\in G$, $h\in H$ and $\alpha(h)=g_{2}g^{-1}_{1}$. Thus,
we can calculate the crossed module with squares given by horizontally

$g_{1}$$g_{2}$$h_{1}$$g_{3}$$h_{2}$=$g_{1}$$g_{3}$$h_{1}\circ
h_{2}$$h_{1}\circ h_{2}:=h_{2}h_{1}$

and vertically

$g_{1}$$g_{2}$$h_{1}$$g_{3}$$g_{4}$$h_{2}$=$g_{1}g_{3}$$g_{2}g_{4}$$h_{1}\star
h_{2}$$h_{1}\star h_{2}:=h_{1}(g_{1}\vartriangleright h_{2})$

Here, horizontal composition of squares is denoted by a circle “$\circ$”, and
vertical composition is denoted by a star “$\star$”. Squares admit horizontal
and vertical inverses, defined by

$g_{2}$$g_{1}$$h^{-h}$$g^{-1}_{2}$$g^{-1}_{1}$$h^{-v}$

where $h^{-h}=h^{-1}$ and $h^{-v}=g_{1}^{-1}\vartriangleright h^{-1}$. In
addition, the vertical multiplication is the group multiplication of the group
$H$, but the horizontal one is not. There are some slightly different
descriptions, see Baez.2010 ; HP ; MORTON2020103548  for more details. Based
on this algebraic structure, Baez constructed the 2-form Yang-Mills action and
obtained the corresponding field equations. More information about the 2-form
Yang-Mills theory can be found in 2002hep.th….6130B . In the special case when
$H$ is trivial, the 2-form Yang-Mills equations reduce to the ordinary Yang-
Mills equations with gauge group $G$ pittphilsci10008 . When $G$ is trivial
and $H=U(1)$, the 2-form Yang-Mills equations reduce to those of 2-form
electromagnetism HP .

In another recent work Gastel , Gastel gave two definitions of 2-Yang-Mills
and 3-Yang-Mills theory which are linear if the good gauges are chosen.
However, what we focus on here is different from his ideas. In this paper, we
extend the 2-form Yang-Mills theory of Baez’s work to a 3-form Yang-Mills
theory. Consequently, we need to introduce a new algebraic structure, known as
2-crossed module of groups, which is described by three groups TRMV ;
Faria_Martins_2011 ; doi:10.1063/1.4870640 ; TRMV1 ; YHMNRY ; TRMV3 ;
Kamps20022groupoidEI ; Mutlu1998FREENESSCF ; Porter10thecrossed . We denote a
2-crossed module of Lie groups as
$(L,H,G;\beta,\alpha,\vartriangleright,\bm{\left\\{,\right\\}})$ consisting of
three Lie groups $L$, $H$ and $G$, and two group homomorphisms $\beta$ and
$\alpha$

$L\stackrel{{\scriptstyle\beta}}{{\longrightarrow}}H\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}G,$

and the smooth left action $\vartriangleright$ by automorphisms of $G$ on $L$
and $H$, and on itself by conjugation, i.e.

$g\vartriangleright(e_{1}e_{2})=(g\vartriangleright e_{1})(g\vartriangleright
e_{2}),\ \ \ \ \ \ \ \ (g_{1}g_{2})\vartriangleright
e=g_{1}\vartriangleright(g_{2}\vartriangleright e),$ (1)

for any $g,g_{1},g_{2}\in G,e,e_{1},e_{2}\in H$ or $L$, and a $G$-equivariant
smooth function $\bm{\left\\{,\right\\}}:H\times H\longrightarrow L$ called
the Peiffer lifting, such that

$g\vartriangleright\bm{\left\\{}h_{1},h_{2}\bm{}=\bm{\left\\{}g\vartriangleright
h_{1},g\vartriangleright h_{2}\bm{},$ (2)

for any $g\in G$ and $h_{1},h_{2}\in H$ and they satisfy certain relations .
There is a left action of $H$ on $L$ by automorphisms
$\vartriangleright^{\prime}$ which is defined by

$h\vartriangleright^{\prime}l=l\bm{\left\\{}\beta(l)^{-1},h\bm{},\ \ \ \ \ \ \
\forall l\in L,h\in H.$ (3)

This together with the homomorphism $\beta:L\longrightarrow H$ determines a
crossed module $(L,H;\beta,\vartriangleright^{\prime})$.

There is a differential 2-crossed module, also called a 2-crossed module of
Lie algebras, corresponding to the 2-crossed module of Lie groups
$(L,H,G;\beta,\alpha,\vartriangleright,\bm{\left\\{,\right\\}})$. This
definition appeared in AAAA ; E . We mark it with
$(\mathcal{l},\mathcal{h},\mathcal{g};\tilde{\beta},\tilde{\alpha},\widetilde{\vartriangleright},\left\\{,\right\\})$
consisting of three Lie algebras $\mathcal{l}$, $\mathcal{h}$ and
$\mathcal{g}$ respectively corresponding to the three Lie groups $L$, $H$ and
$G$, and two Lie algebra maps $\tilde{\beta}$ and $\tilde{\alpha}$

$\mathcal{l}\stackrel{{\scriptstyle\tilde{\beta}}}{{\longrightarrow}}\mathcal{h}\stackrel{{\scriptstyle\tilde{\alpha}}}{{\longrightarrow}}\mathcal{g},$

and left action $\widetilde{\vartriangleright}$ of $\mathcal{g}$ on
$\mathcal{l},\mathcal{h}$ and $\mathcal{g}$ by automorphisms, and a
$\mathcal{g}$-equivariant bilinear map
$\left\\{,\right\\}:\mathcal{h}\times\mathcal{h}\longrightarrow\mathcal{l}$
called the Peiffer lifting, such that

$X\widetilde{\vartriangleright}\left\\{Y_{1},Y_{2}\right\\}=\left\\{X\widetilde{\vartriangleright}Y_{1},Y_{2}\right\\}+\left\\{Y_{1},X\widetilde{\vartriangleright}Y_{2}\right\\},\
\ \ \ \ \ \ \forall X\in\mathcal{g},Y_{1},Y_{2}\in\mathcal{h},$ (4)

and they satisfy appropriate conditions Faria_Martins_2011 . Analogously to
the 2-crossed module of Lie groups case, there is a left action
$\widetilde{\vartriangleright}^{\prime}$ of $\mathcal{h}$ on $\mathcal{l}$
which is defined by

$Y\widetilde{\vartriangleright}^{\prime}Z=-\left\\{\tilde{\beta}(Z),Y\right\\},\
\ \ \ \ \ \ \forall Y\in\mathcal{h},Z\in\mathcal{l}.$ (5)

This together with the homomorphism
$\tilde{\beta}:\mathcal{l}\longrightarrow\mathcal{h}$ defines a differential
crossed module
$(\mathcal{l},\mathcal{h};\tilde{\beta},\widetilde{\vartriangleright}^{\prime})$.

In consideration of the crossed module of groups
$(L,H;\beta,\vartriangleright^{\prime})$ and its calculations with squares as
shown above, we perform the 2-crossed module of groups using cubes of the form

$l$$g_{1}$$g_{2}$$g_{3}$$g_{4}$$h_{1}$$h_{2}$

where $g_{1}$, $g_{2}$, $g_{3}$, $g_{4}\in G$, $h_{1}$, $h_{2}\in H$, $l\in L$
and $\beta(l)=h_{2}h^{-1}_{1}$. There are three types of compositions between
cubes given by FIG. 1 where $h_{1}\circ h_{3}=h_{3}h_{1}$, $h_{2}\circ
h_{4}=h_{4}h_{2}$, $l\star
l^{\prime}=l(h_{1}\vartriangleright^{\prime}l^{\prime})$, and FIG. 2 where
$l\circ l^{\prime}=l^{\prime}l$, and FIG. 3 where $h_{1}\star
h_{3}=h_{1}(g_{1}\vartriangleright h_{3})$, $h_{2}\star
h_{4}=h_{2}(g_{3}\vartriangleright h_{4})$, $l\star
l^{\prime}=l(h_{1}\vartriangleright^{\prime}l^{\prime})$. Cubes admit
horizontal and vertical inverses, defined by FIG. 6 where
$h^{-h}_{1}=h^{-1}_{1}$, $h^{-h}_{2}=h^{-1}_{2}$,
$l^{-v}=h^{-1}_{1}\vartriangleright^{\prime}l^{-1}$, and FIG. 6 where
$l^{-h}=l^{-1}$, and FIG. 6 where $h_{1}^{-v}=g_{1}^{-1}\vartriangleright
h_{1}^{-1}$, $h_{2}^{-v}=g_{3}^{-1}\vartriangleright h_{2}^{-1}$,
$l^{-v}=h_{1}^{-1}\vartriangleright^{\prime}l^{-1}$. Here, we restrict
ourselves to the particular 2-crossed module of groups in which
$(H,G;\alpha,\vartriangleright)$ is also a crossed module.

$l$$l^{\prime}$$g_{1}$$g_{2}$$g_{3}$$g_{4}$$g_{5}$$g_{6}$$h_{1}$$h_{2}$$h_{3}$$h_{4}$:=$l\star
l^{\prime}$$g_{1}$$g_{5}$$g_{3}$$g_{6}$$h_{1}\circ h_{3}$$h_{2}\circ h_{4}$
Figure 1: The first horizontal composition
$l$$g_{1}$$g_{2}$$g_{3}$$g_{4}$$h_{1}$$h_{2}$$g_{5}$$g_{6}$$h_{3}$$l^{\prime}$:=$g_{1}$$g_{2}$$g_{5}$$g_{6}$$h_{1}$$h_{3}$$l\circ
l^{\prime}$ Figure 2: The second horizontal composition
$l$$g_{1}$$g_{2}$$g_{3}$$g_{4}$$h_{1}$$h_{2}$$l^{\prime}$$g_{5}$$g_{6}$$g_{7}$$g_{8}$$h_{3}$$h_{4}$:=$l\star
l^{\prime}$$g_{1}g_{5}$$g_{2}g_{6}$$g_{3}g_{7}$$g_{4}g_{8}$$h_{1}\star
h_{3}$$h_{2}\star h_{4}$ Figure 3: The vertical composition

$l^{-v}$$g_{2}$$g_{1}$$g_{4}$$g_{3}$$h^{-h}_{1}$$h^{-h}_{2}$ Figure 4: The
first horizontal inverse

$l^{-h}$$g_{3}$$g_{4}$$g_{1}$$g_{2}$$h_{1}$$h_{2}$ Figure 5: The second
horizontal inverse

$l^{-v}$$g_{1}^{-1}$$g_{2}^{-1}$$g_{3}^{-1}$$g_{4}^{-1}$$h_{1}^{-v}$$h_{2}^{-v}$
Figure 6: The vertical inverse

In order to construct the 3-form Yang-Mills action, we introduce some kind of
$G$-invariant bilinear forms on the 2-crossed module of Lie algebras. More
precisely speaking, we generalize the $G$-invariant bilinear forms on the a
crossed module of Lie algebras Martins:2010ry ; Radenkovi__2019 . Firstly, we
generalize the mixed relations of $(H,G;\alpha,\vartriangleright)$ and
$(\mathcal{h},\mathcal{g};\tilde{\alpha},\widetilde{\vartriangleright})$ to
the mixed relations of
$(L,H,G;\beta,\alpha,\vartriangleright,\bm{\left\\{,\right\\}})$ and
$(\mathcal{l},\mathcal{h},\mathcal{g};\tilde{\beta},\tilde{\alpha},\widetilde{\vartriangleright},\left\\{,\right\\})$.
When $(H,G;\alpha,\vartriangleright)$ is a crossed module of groups, there are
two mixed relations Martins:2010ry  induced by the action via
$\vartriangleright$ on $H$ and also denoted by $\vartriangleright$

$\displaystyle\tilde{\alpha}(g\vartriangleright Y)=g\tilde{\alpha}(Y)g^{-1},\
\ \ \ \ \ \ \forall g\in G,Y\in\mathcal{h},$ (6)
$\displaystyle\alpha(h)\vartriangleright Y=hYh^{-1},\ \ \ \ \ \ \ \forall h\in
H,Y\in\mathcal{h}.$ (7)

Besides, $(L,H;\beta,\vartriangleright^{\prime})$ is also a crossed module of
groups, i.e. satisfying

$\displaystyle\beta(h\vartriangleright^{\prime}l)=h\beta(l)h^{-1},\ \ \ \ \ \
\ \forall h\in H,l\in L,$ (8)
$\displaystyle\beta(l)\vartriangleright^{\prime}l^{\prime}=ll^{\prime}l^{-1},\
\ \ \ \ \ \ \forall l,l^{\prime}\in L.$ (9)

Thus there are also the mixed relations, corresponding to the crossed module
of Lie algebras
$(\mathcal{l},\mathcal{h};\tilde{\beta},\widetilde{\vartriangleright}^{\prime})$,
induced by the action via $\vartriangleright^{\prime}$ on $L$ and also denoted
by $\vartriangleright^{\prime}$

$\displaystyle\tilde{\beta}(h\vartriangleright^{\prime}Z)=h\tilde{\beta}(Z)h^{-1},\
\ \ \ \ \ \ \forall h\in H,Z\in\mathcal{l},$ (10)
$\displaystyle\beta(l)\vartriangleright^{\prime}Z=lZl^{-1},\ \ \ \ \ \ \
\forall l\in L,Z\in\mathcal{l}.$ (11)

A symmetric non-degenerate $G$-invariant form in
$(\mathcal{l},\mathcal{h},\mathcal{g};\tilde{\beta},\tilde{\alpha},\widetilde{\vartriangleright},\left\\{,\right\\})$
is given by a triple of non-degenerate symmetric bilinear forms
$\langle,\rangle_{\mathcal{g}}$ in $\mathcal{g}$ ,
$\langle,\rangle_{\mathcal{h}}$ in $\mathcal{h}$ and
$\langle,\rangle_{\mathcal{l}}$ in $\mathcal{l}$ such that

  1. 1.

$\langle,\rangle_{\mathcal{g}}$ is $G$-invariant, i.e.

$\langle gXg^{-1},gX^{\prime}g^{-1}\rangle_{\mathcal{g}}=\langle
X,X^{\prime}\rangle_{\mathcal{g}},\ \ \ \forall g\in
G,X,X^{\prime}\in\mathcal{g};$

  2. 2.

$\langle,\rangle_{\mathcal{h}}$ is $G$-invariant, i.e.

$\langle g\vartriangleright Y,g\vartriangleright
Y^{\prime}\rangle_{\mathcal{h}}=\langle Y,Y^{\prime}\rangle_{\mathcal{h}},\ \
\ \ \ \ \ \forall g\in G,Y,Y^{\prime}\in\mathcal{h};$

  3. 3.

$\langle,\rangle_{\mathcal{l}}$ is $G$-invariant, i.e.

$\langle g\vartriangleright Z,g\vartriangleright
Z^{\prime}\rangle_{\mathcal{l}}=\langle Z,Z^{\prime}\rangle_{\mathcal{l}},\ \
\ \ \ \ \ \forall g\in G,Z,Z^{\prime}\in\mathcal{l}.$

Note that $\langle,\rangle_{\mathcal{h}}$ is necessarily $H$-invariant. Since

$\langle
hYh^{-1},hY^{\prime}h^{-1}\rangle_{\mathcal{h}}=\langle\alpha(h)\vartriangleright
Y,\alpha(h)\vartriangleright Y^{\prime}\rangle_{\mathcal{h}}=\langle
Y,Y^{\prime}\rangle_{\mathcal{h}},\ \ \ \ \forall h\in
H,Y,Y^{\prime}\in\mathcal{h},$

where we have used the mixed relation (7). In the case when the Peiffer
lifting or the map $\beta$ is trivial consequently,
$\langle,\rangle_{\mathcal{l}}$ is $H$-invariant, i.e.

$\langle
h\vartriangleright^{\prime}Z,h\vartriangleright^{\prime}Z^{\prime}\rangle_{\mathcal{l}}=\langle
Z,Z^{\prime}\rangle_{\mathcal{l}},\ \ \ \ \forall h\in
H,Z,Z^{\prime}\in\mathcal{l}.$

Besides, $\langle,\rangle_{\mathcal{l}}$ is necessarily $L$-invariant based on
the $H$-invariance of $\langle,\rangle_{\mathcal{l}}$, i.e.

$\langle
lZl^{-1},lZ^{\prime}l^{-1}\rangle_{\mathcal{l}}=\langle\beta(l)\vartriangleright^{\prime}Z,\beta(l)\vartriangleright^{\prime}Z^{\prime}\rangle_{\mathcal{l}}=\langle
Z,Z^{\prime}\rangle_{\mathcal{l}},\ \ \ \ \forall l\in
L,Z,Z^{\prime}\in\mathcal{l},$

where the mixed relation (11) is used. See Radenkovi__2019  for details .

There are no compatibility conditions among the bilinear forms
$\langle,\rangle_{\mathcal{l}}$, $\langle,\rangle_{\mathcal{h}}$ and
$\langle,\rangle_{\mathcal{g}}$. From the well-known fact that any
representation of $G$ can be made unitary if $G$ is a compact group, one can
get the following Lemma.

###### Lemma 1

Let $(L,H,G;\beta,\alpha,\vartriangleright,\bm{\left\\{,\right\\}})$ be a
2-crossed module of Lie groups with the group $G$ being compact in the real
case, or having a compact real form in the complex case. Then one can
construct $G$-invariant symmetric non-degenerate bilinear forms
$\langle,\rangle_{\mathcal{g}}$, $\langle,\rangle_{\mathcal{h}}$ and
$\langle,\rangle_{\mathcal{l}}$ in the associated differential 2-crossed
module
$(\mathcal{l},\mathcal{h},\mathcal{g};\tilde{\beta},\tilde{\alpha},\widetilde{\vartriangleright},\left\\{,\right\\})$.
Furthermore these forms can be chosen to be positive definite.

These invariance conditions imply that:

$\displaystyle\langle[X,X^{\prime}],X^{\prime\prime}\rangle_{\mathcal{g}}=-\langle
X^{\prime},[X,X^{\prime\prime}]\rangle_{\mathcal{g}},$ (12)
$\displaystyle\langle[Y,Y^{\prime}],Y^{\prime\prime}\rangle_{\mathcal{h}}=-\langle
Y^{\prime},[Y,Y^{\prime\prime}]\rangle_{\mathcal{h}},$ (13)
$\displaystyle\langle[Z,Z^{\prime}],Z^{\prime\prime}\rangle_{\mathcal{l}}=-\langle
Z^{\prime},[Z,Z^{\prime\prime}]\rangle_{\mathcal{l}}.$ (14)

One can define two bilinear antisymmetric maps
$\sigma:\mathcal{h}\times\mathcal{h}\longrightarrow\mathcal{g}$ by the rule:

$\displaystyle\langle\sigma(Y,Y^{\prime}),X\rangle_{\mathcal{g}}=-\langle
Y,X\widetilde{\vartriangleright}Y^{\prime}\rangle_{\mathcal{h}},\ \ \ \ \ \ \
\forall X\in\mathcal{g},Y,Y^{\prime}\in\mathcal{h},$ (15)

and $\kappa:\mathcal{l}\times\mathcal{l}\longrightarrow\mathcal{g}$ by the
rule:

$\displaystyle\langle\kappa(Z,Z^{\prime}),X\rangle_{\mathcal{g}}=-\langle
Z,X\widetilde{\vartriangleright}Z^{\prime}\rangle_{\mathcal{l}},\ \ \ \ \ \ \
\forall X\in\mathcal{g},Z,Z^{\prime}\in\mathcal{l}.$ (16)

Due to $\sigma(Y^{\prime},Y)=-\sigma(Y,Y^{\prime})$ and
$\kappa(Z^{\prime},Z)=-\kappa(Z,Z^{\prime})$, then one has

$\displaystyle\langle
Y,X\widetilde{\vartriangleright}Y^{\prime}\rangle_{\mathcal{h}}=-\langle
Y^{\prime},X\widetilde{\vartriangleright}Y\rangle_{\mathcal{h}}=-\langle
X\widetilde{\vartriangleright}Y,Y^{\prime}\rangle_{\mathcal{h}},$ (17)
$\displaystyle\langle
Z,X\widetilde{\vartriangleright}Z^{\prime}\rangle_{\mathcal{l}}=-\langle
Z^{\prime},X\widetilde{\vartriangleright}Z\rangle_{\mathcal{l}}=-\langle
X\widetilde{\vartriangleright}Z,Z^{\prime}\rangle_{\mathcal{l}}.$ (18)

Further, one needs to define two bilinear maps
$\eta_{1}:\mathcal{l}\times\mathcal{h}\longrightarrow\mathcal{h}$ and
$\eta_{2}:\mathcal{l}\times\mathcal{h}\longrightarrow\mathcal{h}$ by the rule:

$\displaystyle\langle\left\\{Y,Y^{\prime}\right\\},Z\rangle_{\mathcal{l}}=-\langle
Y,\eta_{1}(Z,Y^{\prime})\rangle_{\mathcal{h}}=-\langle
Y^{\prime},\eta_{2}(Z,Y)\rangle_{\mathcal{h}},$ (19)

for each $Y$, $Y^{\prime}\in\mathcal{h}$, and $Z\in\mathcal{l}$. See
Radenkovi__2019  for more information about these maps. To obtain the 2-form
Yang-Mills equations, one defines a map
$\alpha^{*}:\mathcal{g}\longrightarrow\mathcal{h}$ in 2002hep.th….6130B  by
the rule:

$\displaystyle\langle
Y,\alpha^{*}(X)\rangle_{\mathcal{h}}=\langle\tilde{\alpha}(Y),X\rangle_{\mathcal{g}},\
\ \ \ \ \ \ \forall X\in\mathcal{g},Y\in\mathcal{h}.$ (20)

Here, to obtain the 3-form Yang-Mills equations, we have to first define a map
$\beta^{*}:\mathcal{h}\longrightarrow\mathcal{l}$ by the rule:

$\displaystyle\langle
Z,\beta^{*}(Y)\rangle_{\mathcal{l}}=\langle\tilde{\beta}(Z),Y\rangle_{\mathcal{h}},\
\ \ \ \ \ \ \forall Y\in\mathcal{h},Z\in\mathcal{l}.$ (21)

The maps $\sigma$, $\kappa$, $\eta_{1}$, $\eta_{2}$, $\alpha^{*}$ and
$\beta^{*}$ are very closely linked with our approach to the construction of
the 3-form Yang-Mills.

In order to calculate more efficiently, we introduce the component notation.
Given a Lie algebra $\mathcal{g}$, there is a vector space
$\Lambda^{k}(M,\mathcal{g})$ of $\mathcal{g}$-valued differential $k$-forms on
the manifold $M$. For
$A=\sum\limits_{a}A^{a}X_{a}\in\Lambda^{k_{1}}(M,\mathcal{g})$,
$A^{\prime}=\sum\limits_{b}A^{\prime b}X_{b}\in\Lambda^{k_{2}}(M,\mathcal{g})$
for some scalar differential $k$-forms $A^{a}$, $A^{\prime b}$ and elements
$X_{a},X_{b}\in\mathcal{g}$, define

$A\wedge A^{\prime}:=\sum\limits_{a,b}A^{a}\wedge A^{\prime b}X_{a}X_{b}\
,\quad A\wedge^{\left[,\right]}A^{\prime}:=\sum\limits_{a,b}A^{a}\wedge
A^{\prime b}\left[X_{a},X_{b}\right],\quad dA=\sum\limits_{a}dA^{a}X_{a},$

then there is an identity

$A\wedge^{\left[,\right]}A^{\prime}=A\wedge
A^{\prime}-(-1)^{k_{1}k_{2}}A^{\prime}\wedge A.$

In the above, we can also choose $\mathcal{g}$ to be $\mathcal{h}$ or
$\mathcal{l}$. For
$B=\sum\limits_{b}B^{b}Y_{b}\in\Lambda^{k_{1}}(M,\mathcal{h})$,
$B^{\prime}=\sum\limits_{b}B^{\prime
b}Y_{b}\in\Lambda^{k_{2}}(M,\mathcal{h})$, $\forall$ $Y_{a}$,
$Y_{b}\in\mathcal{h}$, define

$A\wedge^{\widetilde{\vartriangleright}}B:=\sum\limits_{a,b}A^{a}\wedge
B^{b}X_{a}\widetilde{\vartriangleright}Y_{b}\ ,\quad
B\wedge^{\left\\{,\right\\}}B^{\prime}:=\sum\limits_{a,b}B^{a}\wedge B^{\prime
b}\left\\{Y_{a},Y_{b}\right\\},\ \
\tilde{\alpha}(B):=\sum\limits_{a}B^{a}\tilde{\alpha}(Y_{a}).$

For $C=\sum\limits_{a}C^{a}Z_{a}\in\Lambda^{k_{1}}(U,\mathcal{l})$,
$C^{\prime}=\sum\limits_{b}C^{\prime
b}Z_{b}\in\Lambda^{k_{2}}(U,\mathcal{l})$, define

$A\wedge^{\widetilde{\vartriangleright}}C:=\sum\limits_{a,b}A^{a}\wedge
C^{b}X_{a}\widetilde{\vartriangleright}Z_{b}\ ,\ \
B\wedge^{\widetilde{\vartriangleright}^{\prime}}C:=\sum\limits_{a,b}B^{a}\wedge
C^{b}Y_{a}\widetilde{\vartriangleright}^{\prime}Z_{b}\ ,\ \
\tilde{\beta}(C):=\sum\limits_{a}C^{a}\tilde{\beta}(Z_{a}),$

where
$Y_{a}\widetilde{\vartriangleright}^{\prime}Z_{b}=-\left\\{\tilde{\beta}(Z_{b}),Y_{a}\right\\}$
by using (5).

Furthermore, we have non-degenerate $G$-invariant forms in
$\Lambda^{k}(M,\mathcal{g})$, $\Lambda^{k}(M,\mathcal{h})$ and
$\Lambda^{k}(M,\mathcal{l})$ induced by $\langle,\rangle_{\mathcal{g}}$,
$\langle,\rangle_{\mathcal{h}}$ and $\langle,\rangle_{\mathcal{l}}$ and we
denote them by $\langle,\rangle$. Then we have

$\langle A,A^{\prime}\rangle:=\sum\limits_{a,b}A^{a}\wedge A^{\prime b}\langle
X_{a},X_{b}\rangle_{\mathcal{g}},\quad\langle
B,B^{\prime}\rangle:=\sum\limits_{a,b}B^{a}\wedge B^{\prime b}\langle
Y_{a},Y_{b}\rangle_{\mathcal{h}},\quad\langle
C,C^{\prime}\rangle:=\sum\limits_{a,b}C^{a}\wedge C^{\prime b}\langle
Z_{a},Z_{b}\rangle_{\mathcal{l}}.$

There are identities

$\langle A,A^{\prime}\rangle=(-1)^{k_{1}k_{2}}\langle
A^{\prime},A\rangle,\qquad\langle B,B^{\prime}\rangle=(-1)^{k_{1}k_{2}}\langle
B^{\prime},B\rangle,\qquad\langle C,C^{\prime}\rangle=(-1)^{k_{1}k_{2}}\langle
C^{\prime},C\rangle,$

using the symmetry of $\langle,\rangle_{\mathcal{g}}$,
$\langle,\rangle_{\mathcal{h}}$ and $\langle,\rangle_{\mathcal{l}}$.

There is an important proposition in doi:10.1063/1.4870640 , and we use it to
calculate the 3-form Yang-Mills equations.

###### Proposition 1

  1. 1.

For $A\in\Lambda^{k}(M,\mathcal{g})$,
$A^{\prime}\in\Lambda^{k^{\prime}}(M,\mathcal{g})$ and
$C\in\Lambda^{*}(M,\mathcal{l})$,

$\tilde{\beta}(A\wedge^{\widetilde{\vartriangleright}}C)=A\wedge^{\widetilde{\vartriangleright}}\tilde{\beta}(C),$
(22) $A\wedge^{\widetilde{\vartriangleright}}A^{\prime}=A\wedge
A^{\prime}+(-1)^{kk^{\prime}+1}A^{\prime}\wedge A.$ (23)

  2. 2.

For $A\in\Lambda^{k}(M,\mathcal{g})$,
$B_{1}\in\Lambda^{t_{1}}(M,\mathcal{h})$,
$B_{2}\in\Lambda^{t_{2}}(M,\mathcal{h})$ and $W\in\Lambda^{*}(M,\mathcal{w})$
$(\mathcal{w}=\mathcal{g},\mathcal{h},\mathcal{l})$,

$d(A\wedge^{\widetilde{\vartriangleright}}W)=dA\wedge^{\widetilde{\vartriangleright}}W+(-1)^{k}A\wedge^{\widetilde{\vartriangleright}}dW,$
(24)
$d(B_{1}\wedge^{\left\\{,\right\\}}B_{2})=dB_{1}\wedge^{\left\\{,\right\\}}B_{2}+(-1)^{t_{1}}B_{1}\wedge^{\left\\{,\right\\}}dB_{2},$
(25)
$A\wedge^{\widetilde{\vartriangleright}}(B_{1}\wedge^{\left\\{,\right\\}}B_{2})=(A\wedge^{\widetilde{\vartriangleright}}B_{1})\wedge^{\left\\{,\right\\}}B_{2}+(-1)^{kt_{1}}B_{1}\wedge^{\left\\{,\right\\}}(A\wedge^{\widetilde{\vartriangleright}}B_{2}),$
(26)

The following propositions provide identities of non-degenerate symmetric
$G$-invariant forms in a Lie 2-crossed module.

###### Proposition 2

For $A_{1}\in\Lambda^{k_{1}}(M,\mathcal{g})$,
$A_{2}\in\Lambda^{k_{2}}(M,\mathcal{g})$,
$A_{3}\in\Lambda^{k_{3}}(M,\mathcal{g})$,
$B_{1}\in\Lambda^{t_{1}}(M,\mathcal{h})$,
$B_{2}\in\Lambda^{t_{2}}(M,\mathcal{h})$,
$B_{3}\in\Lambda^{t_{3}}(M,\mathcal{h})$,
$C_{1}\in\Lambda^{q_{1}}(M,\mathcal{l})$,
$C_{2}\in\Lambda^{q_{2}}(M,\mathcal{l})$, and
$C_{3}\in\Lambda^{q_{3}}(M,\mathcal{l})$ we have

$\displaystyle\langle
A_{1}\wedge^{[,]}A_{2},A_{3}\rangle=(-1)^{k_{1}k_{2}+1}\langle
A_{2},A_{1}\wedge^{[,]}A_{3}\rangle,$ (27) $\displaystyle\langle
B_{1}\wedge^{[,]}B_{2},B_{3}\rangle=(-1)^{t_{1}t_{2}+1}\langle
B_{2},B_{1}\wedge^{[,]}B_{3}\rangle,$ (28) $\displaystyle\langle
C_{1}\wedge^{[,]}C_{2},C_{3}\rangle=(-1)^{q_{1}q_{2}+1}\langle
C_{2},C_{1}\wedge^{[,]}C_{3}\rangle.$ (29)

Proof:We can get these identities easily by using (12), (13) and (14).

$\square$

###### Proposition 3

For $A\in\Lambda^{k}(M,\mathcal{g})$,
$B_{1}\in\Lambda^{t_{1}}(M,\mathcal{h})$,
$B_{2}\in\Lambda^{t_{2}}(M,\mathcal{h})$,
$C_{1}\in\Lambda^{q_{1}}(M,\mathcal{l})$ and
$C_{2}\in\Lambda^{q_{2}}(M,\mathcal{l})$, we have

$\displaystyle\langle
B_{1},A\wedge^{\widetilde{\vartriangleright}}B_{2}\rangle=(-1)^{t_{2}(k+t_{1})+kt_{1}+1}\langle
B_{2},A\wedge^{\widetilde{\vartriangleright}}B_{1}\rangle=(-1)^{kt_{1}+1}\langle
A\wedge^{\widetilde{\vartriangleright}}B_{1},B_{2}\rangle,$ (30)
$\displaystyle\langle
C_{1},A\wedge^{\widetilde{\vartriangleright}}C_{2}\rangle=(-1)^{q_{2}(k+q_{1})+kq_{1}+1}\langle
C_{2},A\wedge^{\widetilde{\vartriangleright}}C_{1}\rangle=(-1)^{kq_{1}+1}\langle
A\wedge^{\widetilde{\vartriangleright}}C_{1},C_{2}\rangle.$ (31)

Proof:We can get those identities easily by using (17) and (18).

$\square$

We can define a bilinear map
$\overline{\sigma}:\Lambda^{t_{1}}(M,\mathcal{h})\times\Lambda^{t_{2}}(M,\mathcal{h})\longrightarrow\Lambda^{t_{1}+t_{2}}(M,\mathcal{h})$
by

$\overline{\sigma}(B_{1},B_{2}):=\sum\limits_{a,b}B_{1}^{a}\wedge
B_{2}^{b}\sigma(Y_{a},Y_{b}),$

for $B_{1}=\sum\limits_{a}B_{1}^{a}Y_{a}\in\Lambda^{t_{1}}(M,\mathcal{h})$ ,
$B_{2}=\sum\limits_{b}B_{2}^{b}Y_{b}\in\Lambda^{t_{2}}(M,\mathcal{h})$. Since
$\sigma$ is antisymmetric, we have

$\overline{\sigma}(B_{1},B_{2})=(-1)^{t_{1}t_{2}+1}\overline{\sigma}(B_{2},B_{1}).$

Let $A=\sum\limits_{c}A^{c}X_{c}\in\Lambda^{k}(M,\mathcal{g})$, then

$\displaystyle\langle\overline{\sigma}(B_{1},B_{2}),A\rangle$
$\displaystyle=\sum\limits_{a,b,c}B_{1}^{a}\wedge B_{2}^{b}\wedge
A^{c}\langle\sigma(Y_{a},Y_{b}),Y_{c}\rangle_{\mathcal{g}}=-\sum\limits_{a,b,c}B_{1}^{a}\wedge
B_{2}^{b}\wedge A^{c}\langle
Y_{a},X_{c}\widetilde{\vartriangleright}Y_{b}\rangle_{\mathcal{g}}$
$\displaystyle=(-1)^{kt_{2}+1}\sum\limits_{a,b,c}B_{1}^{a}\wedge A^{c}\wedge
B_{2}^{b}\langle
Y_{a},X_{c}\widetilde{\vartriangleright}Y_{b}\rangle_{\mathcal{g}}$
$\displaystyle=(-1)^{kt_{2}+1}\langle
B_{1},A\wedge^{\widetilde{\vartriangleright}}B_{2}\rangle,$

by using (15), i.e.

$\displaystyle\langle\overline{\sigma}(B_{1},B_{2}),A\rangle=(-1)^{kt_{2}+1}\langle
B_{1},A\wedge^{\widetilde{\vartriangleright}}B_{2}\rangle,$ (32)

and the following identity holds

$\displaystyle\langle
A,\overline{\sigma}(B_{1},B_{2})\rangle=(-1)^{t_{1}t_{2}+1}\langle
A\wedge^{\widetilde{\vartriangleright}}B_{2},B_{1}\rangle.$ (33)

Similarly, we can define a bilinear map
$\overline{\kappa}:\Lambda^{q_{1}}(M,\mathcal{l})\times\Lambda^{q_{2}}(M,\mathcal{l})\longrightarrow\Lambda^{q_{1}+q_{2}}(M,\mathcal{l})$
by

$\overline{\kappa}(C_{1},C_{2}):=\sum\limits_{a,b}C_{1}^{a}\wedge
C_{2}^{b}\kappa(Z_{a},Z_{b}),$

for $C_{1}=\sum\limits_{a}C_{1}^{a}Z_{a}\in\Lambda^{q_{1}}(M,\mathcal{l})$ and
$C_{2}=\sum\limits_{b}C_{2}^{b}Z_{b}\in\Lambda^{q_{2}}(M,\mathcal{l})$. Since
$\kappa$ is antisymmetric, we have

$\overline{\kappa}(C_{1},C_{2})=(-1)^{q_{1}q_{2}+1}\overline{\kappa}(C_{2},C_{1}).$

Let

$A=\sum\limits_{c}A^{c}X_{c}\in\Lambda^{k}(M,\mathcal{g}),$

then

$\displaystyle\langle\overline{\kappa}(C_{1},C_{2}),A\rangle$
$\displaystyle=\sum\limits_{a,b,c}C_{1}^{a}\wedge C_{2}^{b}\wedge
A^{c}\langle\overline{\kappa}(Z_{a},Z_{b}),Z_{c}\rangle_{\mathcal{g}}=-\sum\limits_{a,b,c}C_{1}^{a}\wedge
C_{2}^{b}\wedge A^{c}\langle
Z_{a},X_{c}\widetilde{\vartriangleright}Z_{b}\rangle_{\mathcal{g}}$
$\displaystyle=(-1)^{kq_{2}+1}\sum\limits_{a,b,c}C_{1}^{a}\wedge A^{c}\wedge
C_{2}^{b}\langle
Z_{a},X_{c}\widetilde{\vartriangleright}Z_{b}\rangle_{\mathcal{g}}$
$\displaystyle=(-1)^{kq_{2}+1}\langle
C_{1},A\wedge^{\widetilde{\vartriangleright}}C_{2}\rangle.$

by using (16), i.e.

$\displaystyle\langle\overline{\kappa}(C_{1},C_{2}),A\rangle=(-1)^{kq_{2}+1}\langle
C_{1},A\wedge^{\widetilde{\vartriangleright}}C_{2}\rangle,$ (34)

and there is an identity

$\displaystyle\langle
A,\overline{\kappa}(C_{1},C_{2})\rangle=(-1)^{q_{1}q_{2}+1}\langle
A\wedge^{\widetilde{\vartriangleright}}C_{2},C_{1}\rangle.$ (35)

Finally, we define bilinear maps
$\overline{\eta_{i}}:\Lambda^{q}(M,\mathcal{l})\times\Lambda^{t}(M,\mathcal{h})\longrightarrow\Lambda^{q+t}(M,\mathcal{h})$
by

$\overline{\eta_{i}}(C,B):=\sum\limits_{a,b}C^{b}\wedge
B^{a}\eta_{i}(Z_{b},Y_{a}),\ \ \ \ i=1,2$

for $B=\sum\limits_{a}B^{a}Y_{a}\in\Lambda^{t}(M,\mathcal{h})$,
$C=\sum\limits_{b}C^{b}Z_{b}\in\Lambda^{q}(M,\mathcal{l})$. There are two
identities

$\displaystyle\langle
B_{1}\wedge^{\left\\{,\right\\}}B_{2},C\rangle=(-1)^{t_{2}q+1}\langle
B_{1},\overline{\eta_{1}}(C,B_{2})\rangle=(-1)^{t_{1}(t_{2}+q)+1}\langle
B_{2},\overline{\eta_{2}}(C,B_{1})\rangle,$ (36)

by using (19).

For $A\in\Lambda^{k}(M,\mathcal{g})$, $B\in\Lambda^{t}(M,\mathcal{h})$ and
$C\in\Lambda^{q}(M,\mathcal{l})$, we have

$\displaystyle\langle
B,\alpha^{*}(A)\rangle=\langle\tilde{\alpha}(B),A\rangle,$ (37)
$\displaystyle\langle C,\beta^{*}(B)\rangle=\langle\tilde{\beta}(C),B\rangle,$
(38)

being induced by 20 and 21.

Up to present, we have established the algebraic structures of 3-form Yang-
Mills theory. Next, we derive the 3-Bianchi-Identities on account of the above
properties for Lie algebra valued differential forms. The 3-connection, on
$d$-dimensional spacetime manifold $M$ ($d\geq 4$), is described by a 1-form
$A$ valued in the Lie algebra $\mathcal{g}$, a 2-form $B$ valued in the Lie
algebra $\mathcal{h}$, and a 3-form $C$ valued in the Lie algebra
$\mathcal{l}$. We refer the interested readers to doi:10.1063/1.4870640  for
more details. The corresponding 3-curvature in this case is given by

$\Omega_{1}:=dA+A\wedge
A,\qquad\Omega_{2}:=dB+A\wedge^{\widetilde{\vartriangleright}}B,\qquad\Omega_{3}:=dC+A\wedge^{\widetilde{\vartriangleright}}C+B\wedge^{\left\\{,\right\\}}B.$

There are a fake 1-curvature and a fake 2-curvature defined as

$F_{1}:=dA+A\wedge A-\tilde{\alpha}(B),\qquad
F_{2}:=dB+A\wedge^{\widetilde{\vartriangleright}}B-\tilde{\beta}(C).$

We call the 3-connection $(A,B,C)$ fake 1-flat, if the fake 1-curvature
vanishes, i.e. $dA+A\wedge A=\tilde{\alpha}(B)$. Similarly, we call the
3-connection $(A,B,C)$ fake 2-flat, if the fake 2-curvature vanishes, i.e.
$dB+A\wedge^{\widetilde{\vartriangleright}}B=\tilde{\beta}(C)$. If the
3-curvature 4-form vanishes, i.e. $\Omega_{3}=0$, the 3-connection $(A,B,C)$
will be called 3-flat. When $(A,B,C)$ is both fake 1-flat and fake 2-flat, we
say that the 3-connection $(A,B,C)$ is fake-flat. Otherwise, the 3-connection
$(A,B,C)$ is not fake-flat.

###### Theorem 1

If $(A,B,C)$ is any 3-connection on $M$, then its fake 3-curvature
$(\Omega_{1},F_{1},F_{2})$ satisfies the 3-Bianchi-Identities:

$dF_{1}+A\wedge^{[,]}F_{1}=-\tilde{\alpha}(F_{2}+\tilde{\beta}(C)),$ (39)
$dF_{2}+A\wedge^{\widetilde{\vartriangleright}}F_{2}=(F_{1}+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}B-\tilde{\beta}(\Omega_{3}-B\wedge^{\left\\{,\right\\}}B),$
(40)
$d\Omega_{3}+A\wedge^{\widetilde{\vartriangleright}}\Omega_{3}=(F_{1}+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}C+(F_{2}+\tilde{\beta}(C))\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}(F_{2}+\tilde{\beta}(C)).$
(41)

Proof:

$\displaystyle dF_{1}$ $\displaystyle=d(dA+A\wedge
A-\tilde{\alpha}(B))=dA\wedge A-A\wedge dA-d(\tilde{\alpha}(B))$
$\displaystyle=(F_{1}-A\wedge A+\tilde{\alpha}(B))\wedge
A-A\wedge(F_{1}-A\wedge A+\tilde{\alpha}(B))-\tilde{\alpha}(dB)$
$\displaystyle=F_{1}\wedge A+\tilde{\alpha}(B)\wedge A-A\wedge
F_{1}-A\wedge\tilde{\alpha}(B)-\tilde{\alpha}(dB)$
$\displaystyle=-A\wedge^{[,]}F_{1}-A\wedge^{[,]}\tilde{\alpha}(B)-\tilde{\alpha}(dB)$
$\displaystyle=-A\wedge^{[,]}F_{1}-\tilde{\alpha}(A\wedge^{\widetilde{\vartriangleright}}B+dB)$
$\displaystyle=-A\wedge^{[,]}F_{1}-\tilde{\alpha}(F_{2}+\tilde{\beta}(C)),$

by using
$\alpha(A\wedge^{\widetilde{\vartriangleright}}B)=A\wedge^{\left[,\right]}\tilde{\alpha}(B)$,
and

$\displaystyle dF_{2}$
$\displaystyle=d(dB+A\wedge^{\widetilde{\vartriangleright}}B-\tilde{\beta}(C))=dA\wedge^{\widetilde{\vartriangleright}}B-A\wedge^{\widetilde{\vartriangleright}}dB-d(\tilde{\beta}(C))$
$\displaystyle=(F_{1}-A\wedge
A+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}B-A\wedge^{\widetilde{\vartriangleright}}(F_{2}-A\wedge^{\widetilde{\vartriangleright}}B+\tilde{\beta}(C))-\tilde{\beta}(dC)$
$\displaystyle=F_{1}\wedge^{\widetilde{\vartriangleright}}B-(A\wedge
A)\wedge^{\widetilde{\vartriangleright}}B+\tilde{\alpha}(B)\wedge^{\widetilde{\vartriangleright}}B-A\wedge^{\widetilde{\vartriangleright}}F_{2}+A\wedge^{\widetilde{\vartriangleright}}(A\wedge^{\widetilde{\vartriangleright}}B)-A\wedge^{\widetilde{\vartriangleright}}\tilde{\beta}(C)-\tilde{\beta}(dC)$
$\displaystyle=(F_{1}+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}B-\tilde{\beta}(A\wedge^{\widetilde{\vartriangleright}}C+dC)-A\wedge^{\widetilde{\vartriangleright}}F_{2}$
$\displaystyle=(F_{1}+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}B-\tilde{\beta}(\Omega_{3}-B\wedge^{\left\\{,\right\\}}B)-A\wedge^{\widetilde{\vartriangleright}}F_{2},$

by using (22) and ${\widetilde{\vartriangleright}}$ is homomorphism, and

$\displaystyle d\Omega_{3}$
$\displaystyle=d(dC+A\wedge^{\widetilde{\vartriangleright}}C+B\wedge^{\left\\{,\right\\}}B)=dA\wedge^{\widetilde{\vartriangleright}}C-A\wedge^{\widetilde{\vartriangleright}}dC+d(B\wedge^{\left\\{,\right\\}}B)$
$\displaystyle=dA\wedge^{\widetilde{\vartriangleright}}C-A\wedge^{\widetilde{\vartriangleright}}(\Omega_{3}-A\wedge^{\widetilde{\vartriangleright}}C-B\wedge^{\left\\{,\right\\}}B)+dB\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}dB$
$\displaystyle=dA\wedge^{\widetilde{\vartriangleright}}C-A\wedge^{\widetilde{\vartriangleright}}\Omega_{3}+(A\wedge
A)\wedge^{\widetilde{\vartriangleright}}C+A\wedge^{\widetilde{\vartriangleright}}(B\wedge^{\left\\{,\right\\}}B)+dB\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}dB$
$\displaystyle=(F_{1}+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}C-A\wedge^{\widetilde{\vartriangleright}}\Omega_{3}+(A\wedge^{\widetilde{\vartriangleright}}B)\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}(A\wedge^{\widetilde{\vartriangleright}}B)+dB\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}dB$
$\displaystyle=(F_{1}+\tilde{\alpha}(B))\wedge^{\widetilde{\vartriangleright}}C-A\wedge^{\widetilde{\vartriangleright}}\Omega_{3}+(F_{2}+\tilde{\beta}(C))\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}(F_{2}+\tilde{\beta}(C)),$

by using (24),(25),(26) and ${\widetilde{\vartriangleright}}$ is homomorphism.

$\square$

When the 3-connection $(A,B,C)$ is fake-flat, the the 3-Bianchi-Identities
become

$d\Omega_{1}+A\wedge^{[,]}\Omega_{1}=0,$ (42)
$d\Omega_{2}+A\wedge^{\widetilde{\vartriangleright}}\Omega_{2}=\Omega_{1}\wedge^{\widetilde{\vartriangleright}}B,$
(43)
$d\Omega_{3}+A\wedge^{\widetilde{\vartriangleright}}\Omega_{3}=\Omega_{1}\wedge^{\widetilde{\vartriangleright}}C+\Omega_{2}\wedge^{\left\\{,\right\\}}B+B\wedge^{\left\\{,\right\\}}\Omega_{2}.$
(44)

Generalizing the Yang-Mills action and the 2-form Yang-Mills action in
2002hep.th….6130B , we write down the following 3-form Yang-Mills action as a
function of the 3-connection (A, B, C) in 3-form Yang-Mills gauge theory:

$S=\int_{M}\langle F_{1},\ast F_{1}\rangle+\langle F_{2},\ast
F_{2}\rangle+\langle\Omega_{3},\ast\Omega_{3}\rangle.$

One can obtain the field equations by setting the variation of the action to
zero:

$\displaystyle\delta S=2\int_{M}\langle\delta F_{1},\ast
F_{1}\rangle+\langle\delta F_{2},\ast
F_{2}\rangle+\langle\delta\Omega_{3},\ast\Omega_{3}\rangle=0.$

The first section is as follows:

$\displaystyle\langle\delta F_{1},\ast F_{1}\rangle$
$\displaystyle=\langle\delta(dA+A\wedge A-\tilde{\alpha}(B)),\ast
F_{1}\rangle$ $\displaystyle=\langle d\delta A,\ast F_{1}\rangle+\langle
A\wedge^{[,]}\delta A,\ast F_{1}\rangle-\langle\tilde{\alpha}(\delta B),\ast
F_{1}\rangle$ $\displaystyle=\langle\delta A,d\ast F_{1}\rangle+\langle\delta
A,A\wedge^{[,]}\ast F_{1}\rangle-\langle\delta B,\alpha^{*}(\ast
F_{1})\rangle$ $\displaystyle=\langle\delta A,d\ast F_{1}+A\wedge^{[,]}\ast
F_{1}\rangle-\langle\delta B,\alpha^{*}(\ast F_{1})\rangle,$

by using $\delta(A\wedge A)=A\wedge^{[,]}\delta A$, (27) and (37). The second
section is as follows :

$\displaystyle\langle\delta F_{2},\ast F_{2}\rangle$
$\displaystyle=\langle\delta(dB+A\wedge^{\widetilde{\vartriangleright}}B-\tilde{\beta}(C)),\ast
F_{2}\rangle$ $\displaystyle=\langle d\delta B,\ast F_{2}\rangle+\langle\delta
A\wedge^{\widetilde{\vartriangleright}}B,\ast F_{2}\rangle+\langle
A\wedge^{\widetilde{\vartriangleright}}\delta B,\ast
F_{2}\rangle-\langle\tilde{\beta}(\delta C),\ast F_{2}\rangle$
$\displaystyle=-\langle\delta B,d\ast F_{2}\rangle-\langle\delta
A,\overline{\sigma}(\ast F_{2},B)\rangle-\langle\delta
B,A\wedge^{\widetilde{\vartriangleright}}\ast F_{2}\rangle-\langle\delta
C,\beta^{*}(\ast F_{2})\rangle$ $\displaystyle=-\langle\delta
A,\overline{\sigma}(\ast F_{2},B)\rangle-\langle\delta B,d\ast
F_{2}+A\wedge^{\widetilde{\vartriangleright}}\ast F_{2}\rangle-\langle\delta
C,\beta^{*}(\ast F_{2})\rangle,$

by using $\delta(A\wedge^{{\widetilde{\vartriangleright}}}B)=\delta
A\wedge^{\widetilde{\vartriangleright}}B+A\wedge^{\widetilde{\vartriangleright}}\delta
B$, (33), (30) and (38). The third section is as follows :

$\displaystyle\langle\delta\Omega_{3},\ast\Omega_{3}\rangle$
$\displaystyle=\langle\delta(dC+A\wedge^{\widetilde{\vartriangleright}}C+B\wedge^{\left\\{,\right\\}}B),\ast\Omega_{3}\rangle$
$\displaystyle=\langle d(\delta C),\ast\Omega_{3}\rangle+\langle\delta
A\wedge^{\widetilde{\vartriangleright}}C,\ast\Omega_{3}\rangle$
$\displaystyle\qquad+\langle A\wedge^{\widetilde{\vartriangleright}}\delta
C,\ast\Omega_{3}\rangle+\langle\delta
B\wedge^{\left\\{,\right\\}}B,\ast\Omega_{3}\rangle+\langle
B\wedge^{\left\\{,\right\\}}\delta B,\ast\Omega_{3}\rangle$
$\displaystyle=\langle\delta C,d\ast\Omega_{3}\rangle+(-1)^{d-1}\langle\delta
A,\overline{\kappa}(\ast\Omega_{3},C)\rangle$
$\displaystyle\qquad+\langle\delta
C,A\wedge^{\widetilde{\vartriangleright}}\ast\Omega_{3}\rangle-\langle\delta
B,\overline{\eta_{1}}(\ast\Omega_{3},B)\rangle-\langle\delta
B,\overline{\eta_{2}}(\ast\Omega_{3},B)\rangle$
$\displaystyle=(-1)^{d-1}\langle\delta
A,\overline{\kappa}(\ast\Omega_{3},C)\rangle-\langle\delta
B,\overline{\eta_{1}}(\ast\Omega_{3},B)+\overline{\eta_{2}}(\ast\Omega_{3},B)\rangle$
$\displaystyle\qquad+\langle\delta
C,d\ast\Omega_{3}+A\wedge^{\widetilde{\vartriangleright}}\ast\Omega_{3}\rangle,$

by using (35),(31) and (36). Thus we have

$\displaystyle\delta S=$ $\displaystyle 2\int_{M}\langle\delta A,d\ast
F_{1}+A\wedge^{[,]}\ast
F_{1}+(-1)^{d-1}\overline{\kappa}(\ast\Omega_{3},C)-\overline{\sigma}(\ast
F_{2},B)\rangle$ $\displaystyle\ \ -\langle\delta B,d\ast
F_{2}+A\wedge^{\widetilde{\vartriangleright}}\ast
F_{2}+\overline{\eta_{1}}(\ast\Omega_{3},B)+\overline{\eta_{2}}(\ast\Omega_{3},B)+\alpha^{*}(\ast
F_{1})\rangle$ $\displaystyle\ \ +\langle\delta
C,d\ast\Omega_{3}+A\wedge^{\widetilde{\vartriangleright}}\ast\Omega_{3}-\beta^{*}(\ast
F_{2})\rangle.$

We see that the variation of the action vanishes for $\delta A$, $\delta B$
and $\delta C$ if only if the following field equations hold:

$\displaystyle d\ast F_{1}+A\wedge^{[,]}\ast F_{1}=\overline{\sigma}(\ast
F_{2},B)+(-1)^{d}\overline{\kappa}(\ast\Omega_{3},C),$ (45) $\displaystyle
d\ast F_{2}+A\wedge^{\widetilde{\vartriangleright}}\ast
F_{2}=-\overline{\eta_{1}}(\ast\Omega_{3},B)-\overline{\eta_{2}}(\ast\Omega_{3},B)-\alpha^{*}(\ast
F_{1}),$ (46) $\displaystyle
d\ast\Omega_{3}+A\wedge^{\widetilde{\vartriangleright}}\ast\Omega_{3}=\beta^{*}(\ast
F_{2}).$ (47)

And when the 3-connection $(A,B,C)$ is fake-flat, the field equations become

$\displaystyle
d\ast\Omega_{1}+A\wedge^{[,]}\ast\Omega_{1}=\overline{\sigma}(\ast\Omega_{2},B)+(-1)^{d}\overline{\kappa}(\ast\Omega_{3},C),$
(48) $\displaystyle
d\ast\Omega_{2}+A\wedge^{\widetilde{\vartriangleright}}\ast\Omega_{2}=-\overline{\eta_{1}}(\ast\Omega_{3},B)-\overline{\eta_{2}}(\ast\Omega_{3},B),$
(49) $\displaystyle
d\ast\Omega_{3}+A\wedge^{\widetilde{\vartriangleright}}\ast\Omega_{3}=0.$ (50)

Though one may wonder about 4-gauge theory, to the best of our knowledge, it
has not been defined yet. The notion of a 3-crossed module, which should be
the foundation of 4-gauge theory, has been developed in AKU ; KOO . Ideally,
in higher gauge theory, the Yang-Mills theory may be generalized to a kind of
”$n$-form Yang-Mills theory” in accordance with the chosen $n$-group structure
$(n>3)$ Roberts2007TheIA . There is no doubt that one may encounter a lot of
difficulties.

To summarize, the relationships between the related models are concluded in
FIG. 7.

3-YM (L,H,G)2-YM (H,G)3-Elec (U(1))2-Elec (U(1))(1-)YM (G)(1-)Elec (U(1))$L,H$
: trivial$G=U(1)$3-YM : 3-form Yang-Mills 2-YM : 2-form Yang-Mills (1-)YM :
(1-form)Yang-Mills 3-Elec : 3-form Electromagnetism 2-Elec : 2-form
Electromagnetism (1-)Elec : (1-form) Electromagnetism $L$ :
trivial$(H,G)$$L,H$ : trivial$G$$H,G$ : trivial$L=U(1)$$L,G$ :
trivial$H=U(1)$$G$ : trivial$H=U(1)$$H$ : trivial$G=U(1)$$G=U(1)$$H$ :
trivial$G$ Figure 7: The relationships

###### Acknowledgements.

We are grateful to Kai Lou and Fuhao Zhang for reading the manuscript and
giving their comments. This work is supported by the National Natural Science
Foundation of China (Nos.11871350, NSFC no. 11971322).

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