11institutetext: LIACS, Leiden University, the Netherlands
11email<EMAIL_ADDRESS>22institutetext: LIACS, Leiden University,
the Netherlands
22email<EMAIL_ADDRESS>33institutetext: LIACS, Leiden
University, the Netherlands
33email<EMAIL_ADDRESS>
# Transfer Learning and Curriculum Learning
in Sokoban

Zhao Yang 11 Mike Preuss 22 Aske Plaat 33

###### Abstract

Transfer learning can speed up training in machine learning, and is regularly
used in classification tasks. It reuses prior knowledge from other tasks to
pre-train networks for new tasks. In reinforcement learning, learning actions
for a behavior policy that can be applied to new environments is still a
challenge, especially for tasks that involve much planning. Sokoban is a
challenging puzzle game. It has been used widely as a benchmark in planning-
based reinforcement learning. In this paper, we show how prior knowledge
improves learning in Sokoban tasks. We find that reusing feature
representations learned previously can accelerate learning new, more complex,
instances. In effect, we show how curriculum learning, from simple to complex
tasks, works in Sokoban. Furthermore, feature representations learned in
simpler instances are more general, and thus lead to positive transfers
towards more complex tasks, but not vice versa. We have also studied which
part of the knowledge is most important for transfer to succeed, and identify
which layers should be used for pre-training.111We open-sourced all the code
we used. It can be found, after the review, at
https://anonymous.org/blind-review

###### Keywords:

Reinforcement learning Transfer learning Sokoban.

##  1 Introduction

Humans are good at reusing prior knowledge when facing new problems. As a
consequence, we learn new tasks quickly, a skill of great interest in machine
learning. In the human brain, information received by our sensors is first
transformed into different forms, and different types of transformed
information are stored in different areas of our brain. When another problem
arrives later on, we retrieve useful information and adjust it to better suit
solving this new problem. The knowledge stored in artifical neural networks is
also re-usable and transferable [31]. In supervised learning, pre-trained
networks are commonly applied in computer vision [17, 25] and natural language
processing [3, 9]. Feature representations learned from images or words
overlap to some extent, which makes such feature representations reusable and
transferable. In reinforcement learning (RL), transfer learning is relatively
new, although with the spread of deep neural networks, reusing pre-trained
models becomes possible in RL as well [1, 7]. Transfer learning works well in
RL for recognition tasks, but tasks that rely heavily on planning are harder.

Figure 1: An example instance of Sokoban.

In this paper, we study transfer learning of behavior in Sokoban, a popular RL
game in which planning is important [10, 12]. It has already been proved that
Sokoban is PSPACE-complete [8] and NP-hard problem [10]. An example instance
from [22] is shown in Fig. 1. The goal of Sokoban is to control a warehouse
worker that pushes all boxes onto targets. Sokoban is a challenging game where
one wrong move can lead to a dead end (after a box has been pushed, it can not
be pulled, and we cannot undo an inadvertent push). This non-reversibility is
known to make games harder for AI agents [5]. Learning to solve Sokoban tasks
is a challenge, especially in the multi-box scenario. For humans, if we have
learned the basics of Sokoban (what is a box, what can an agent do), and if we
are faced with a new, more complex instance, then we immediately focus on the
new challenges in the instance, rather than re-learning the basics again. This
building on prior knowledge saves time in the problem-solving process.

We investigate if we can achieve this kind of pretraining/fine-tuning learning
in an RL agent. Our main hypothesis is that feature representations learned in
Sokoban instances can be reused to improve solving other instances, and that
features learned in simpler instances are more general and better
transferable. We test this hypothesis by means of different experiments, in
which parts of the neural network that has previously been trained on one type
of instances (e.g. one box one target) are taken over (unchanged) to a new
type of instances (e.g. two boxes two targets), whereas the remaining part of
the network is trained on these new instances from scratch. The overall idea
is that we see successful transfer if the preserved knowledge (in terms of
network layers) leads to a faster learning process on the new problem type.

The main contributions of this paper are as follows: First, we show that
feature representations learned in simple Sokoban instances can accelerate
learning in more complex instances, indicating that curriculum learning can be
used in Sokoban. Second, feature representations of simpler instances are more
general and reusable than features learned in more complex instances. Third,
our results confirm that in RL lower layers learn more general features.
Interestingly, in some cases the best performance is achieved when more
specific features are transfered, when source task and target task are similar
enough to support these more specific features. Fourth, we found negative
transfer from a simple supervised learning task, which tells us that choice
and design of the source tasks are crucial. Fifth, we show that transferring
top-fully-connected layers will not only be unhelpful but also harmful to the
learning. We also used popular visualization techniques to explore potential
reasons for successful transfers, which we explain in detail. Our code and
test environments will be made available after blind review.

The paper is structured as follows: we first briefly review related work on
transfer learning and Sokoban in the next section; then the environment and
methods we are using are described in Section 3; Section 4 shows the
experimental settings and results; in the last section, we conclude our work
and discuss some potential future directions.

##  2 Related Work

De la Cruz et al. [6] studied the reuse of feature representations between two
similar games: Breakout and Pong, using DQN. They used a 3-layer convolutional
network. Weights learned in one game were transferred to improve learning the
other game; results showed positive transfer of features between the different
games. Pong and Breakout do not require planning; in our experiments, in
Sokoban, we study how a curriculum of simpler instances can benefit the
learning of complex instances. Spector et al. [26] used self-transfer in a DQN
grid-world task to identify which parts should be transferred and which parts
should be fixed, showing significant benefit of knowledge transfer.

Sokoban is a planning task that has been used as a benchmark for model-based
reinforcement learning [22, 16]. It It has also been used in model-free RL
[14, 15], achieving performance competitive with model-based methods. The
efficiency of AlphaZero-style curriculum learning has been shown by solving
hard single Sokoban instances [11, 12]. Previous works were aimed at solving
single Sokoban instances; our paper focuses on the transferability of learned
knowledge among _different_ instances.

This transferability of learned feature representations was first studied in
image classification problems [31]. It was shown that bottom layers in CNNs
extract more general features while ones extracted from back layers are more
specific. In this paper, we verify this idea under RL settings.

Reinforcement learning [27, 21] aims to reinforce behaviors of the learning
agent by rewarding signals obtained from interactions with the environment. It
has reached super-human performance in games such as Go [24], StarCraft [29,
20], as well as Atari games [2] and robotic tasks. In this paper we follow the
conventional MDP notation for RL [27].

Transfer learning reuses prior knowledge to improve the learning efficiency or
performance in new tasks [30, 28]. In reinforcement learning, higher-level
knowledge such as macro actions, skills and lower-level knowledge such as
reward functions, policies could be transferred. Transferring learned
knowledge could take different approaches, such as reward shaping [4],
learning from demonstration [19] and policy reuse [13].

##  3 Experimental Setup

Figure 2: Examples of Sokoban instances, increasing in difficulty from 1 box
and 1 target to 3 boxes and 3 targets

The environment used in the paper is the Gym environment for Sokoban [23]; for
the agent algorithms we follow Weber et al. [22]. Examples are shown in Fig.
2. The game is solved by controlling the agent (green sprite) to push all
boxes (yellow squares) onto corresponding targets(red squares). There’s no
hint about which boxes should on which targets, and boxes can only be pushed;
some actions are irreversible, and can leave the game in an unsolvable state.
The difficulty of the game can be increased easily by putting more boxes as
well as targets into generated rooms. The agent can go up, down, left, and
right. The agent gets a final reward of 10 by pushing all boxes on targets.
Pushing a box on a target will result a reward of 1 and a penalty of -1 for
pushing a box off a target. We also give a small penalty of 0.1 for each step
the agent takes.

We perform three types of experiments: (1) related tasks (source and target
tasks are both RL tasks, while source tasks are to solve $n$-boxes Sokoban
instances and target tasks are to solve $m$-boxes Sokoban instances, where
$n\neq m$), (2) different tasks (source tasks are SL tasks and target tasks
are RL tasks), and (3) different texture appearance(source and target tasks
are both RL tasks, while source tasks are to solve original Sokoban instances
and target tasks are to solve Sokoban instances with different texture
appearance). The agent was first pre-trained on source tasks and then fine-
tuned on target tasks. RL tasks are to solve 100 randomly generated $n$-boxes
Sokoban instances. SL tasks are to recognize the location of the agent in
Sokoban instances.

The overall statistics of the maps are shown in Fig. 3. As the number of
objectives increases, the number of steps for the optimal solution also
increases, and so does the difficulty of solving the game.

Figure 3: Distribution of optimal solutions in different Sokoban instances.

###  3.1 Neural Network Architecture

The neural network we employ is taken from the DeepMind baseline [22] directly
without hyper-parameter tuning. The model consists of 3 convolutional (Conv)
layers with kernel size 8x8, 4x4, 3x3, strides of 4, 2, 1, and number of
output channels 32, 64, 64. This is followed by a fully connected (FC) hidden
layer with 512 units. The outputs of this FC layer will be fed into two heads:
one for outputting the policy logits and one for outputting the state value.
This is one of the most commonly-used architectures in RL, we selected it also
in order to show what can be achieved with popular architecture. Details of
architecture and hyper parameters we employ are found in Table 1.

learning rate | $7\cdot 10^{-4}$
---|---
gamma | 0.99
entropy coef | 0.1
value loss coef | 0.5
eps | $10^{-5}$
alpha | 0.99
rollout storage size | 5
No. of environments for collecting trajectories | 30
Table 1: Hyper-parameters of the neural network and training.

###  3.2 Transfer Approach

The main idea of our transfer approach is to reuse feature representations
from source tasks learned by the Conv layers in new unseen target tasks. As
detailed in the last sub-section, our model consists of 3 Conv layers and 2 FC
layers. The feature representations were transferred to new tasks by copying
the weights of the first $k$ Conv layers trained in source tasks (where there
are $n_{s}$ boxes/targets) to initialize the new learning model in target
tasks (where there are $n_{t}$ boxes/targets). Then we froze these weights
(they were no longer trainable) and retrained the remaining part of the model.
In our experiments, $k\in\\{1,2,3\\}$, $n_{s}\in\\{1,2,3\\}$,
$n_{t}\in\\{1,2,3\\}$. Please refer to Fig. 4 for an explanation of this
approach. Different squares represent different layers of our neural network.
The first 3 layers are Conv layers and the last two are FC layers. Reds are
weights taken from pre-trained model and fixed, greens are weights
reinitialized and trainable.

Solved ratios were used for measuring agents’ performances, and they were
calculated every 1,000 environment steps. 20 randomly selected test instances
were performed by the current learning agent. We say the transfer is
_positive_ when the performance with the transfer is better than without
(training from scratch), and _negative_ when the performance with the transfer
is worse than without.

Figure 4: Three different transfer approaches, red layers are fixed while
green layers are trainable. They correspond $k=1,2,3$ from left to right
respectively.

##  4 Experiments

We designed experiments with different source, target tasks and $k$, in order
to verify the hypotheses we proposed. We experimented with Sokoban instances
with 1, 2, and 3 boxes. All experiments were run for 1 million environment
steps. Experimental details are shown in Table 2. We use abbreviations for
each experiment. For instance, s1t1k1 means source tasks are 1-box instances,
target tasks are 1-box instances and we transfer and fix the 1(first) layer.
The neural networks were trained using A2C, a single threaded variant of A3C
[18]. All experiments were performed 5 times with different random seeds, and
figures were drawn using averaged results with 0.95 confidence interval. Heavy
fluctuations were caused by irreversible actions, one irreversible action
during the game could make the whole game unsolvable.

abbreviation | source task | target task | $k$ | abbreviation | source task | target task | $k$
---|---|---|---|---|---|---|---
s1t1k3 | 1-box | 1-box | 3 | s1t2k2 | 1-box | 2-boxes | 2
s2t1k1 | 2-boxes | 1-box | 1 | s1t1k3 | 1-box | 2-boxes | 3
s2t1k2 | 2-boxes | 1-box | 2 | s2t2k3 | 2-boxes | 2-boxes | 3
s2t1k3 | 2-boxes | 1-box | 3 | s1t3k1 | 1-box | 3-boxes | 1
s3t1k1 | 3-boxes | 1-box | 1 | s1t3k2 | 1-box | 3-boxes | 2
s3t1k2 | 3-boxes | 1-box | 2 | s1t3k3 | 1-box | 3-boxes | 3
s3t1k3 | 3-boxes | 1-box | 3 | s2t3k3 | 2-boxes | 3-boxes | 3
sPt1k1 | prediction | 3-boxes | 1 | s1t2k1 | 1-box | 2-boxes | 1
s1t1fc_game2 | 1-box | 1-box_game2 | fc |  |  |  |
Table 2: Experimental design details. $k$ is the number of fixed layers.

###  4.1 Transfer Among Related Tasks

_Related tasks_ are tasks where the only difference between source and task is
the difficulties of instances, i.e. the number of boxes and targets. (Recall
that both source and task are trained on 100 different map-layouts, in all
experiments.)

Figure 5: Performance of transferring feature representations learned in
1-box, 2-boxes, 3-boxes instances to learning in 1-box with $k=3$.
$n_{s}=1,2,3$, $n_{t}=1$, $k=3$. Pre-training on 1-box instances is much
better than pre-training on 2 or 3 box instances when training new 1-box
instances.

Figure 6: Performance of transferring feature representations learned in
1-box, 2-boxes, 3-boxes instances to learning in 2-boxes (left) and 3-boxes
(right) with $k=3$. $n_{s}=1,2,3$, $n_{t}=2,3$, $k=3$.

Fig. 5 and Fig. 6 show results for training on 1-box, 2-boxes, 3-boxes
instances with reusing features learned in different tasks, and we fix $k=3$.
All results showed that transferring feature representations learned in
single-box instances is positive. Performance of agents (s1t1k3, s1t2k3,
s1t3k3) who are using features learned from single-box instances always
outperform other agents, including agents training from scratch and using
features learned from other instances. The transfer, however, is not ’bi-
directional’, feature representations learned in multiple-box instances could
not be successfully transferred to the learning in single-box instances. Their
performance (s2t1k3, s3t1k3) converged to a relatively low solved ratio, which
indicates that transferred features are not suitable for single-box instances.
Just as humans learn more general knowledge in simpler cases, our agents also
showed that the knowledge learned from single-box instances is more general
and transferable than ones learned in multiple-box instances.

To further enhance performances of transferring features learned in single-box
instances, we tried different $k$. We expected that the performance will be
the best when $k=1$ since the first layer learn the most general features.
However, the results in Fig. 7 instead show that not $k=1$ but $k=2$ (s1t2k2,
s1t3k2) perform the best. Similar to [6], features learned in the first 2
layers are still general enough for transfer; in addition, the difference
between source tasks and target tasks is not as large as expected, and
features learned between different instances are more overlapping than
expected.

It is also interesting to see the influence of how many layers are fixed on
the success of the transfer. In particular, we want to know whether a smaller
$k$ could change the negative transfer from multiple-box instances to single-
box instances into positive. (We believe features from multiple and single-box
instances are overlapping to some extent.) Results are shown in Fig. 8. We see
that indeed the first layer (s2t1k1, s3t1k1) did learn enough general features
from multiple-boxes instances to solve the single-box instances. Although
agents with features only learned by the first layer could converge to decent
performance in the end, the transfer is still negative. An interesting point
is that $k=3$ (s2t1k3) performs better than $k=2$ (s2t1k2) when source tasks
are 2-boxes instances. Note that $k=2$ (s3t1k2) performs better than $k=3$
(s3t1k3) when source tasks are 2-boxes instances. There are more overlapping
features between the 2-boxes instances and single instances.

Figure 7: Performance of transferring feature representations learned in 1-box
instances to learning in 2-boxes (left) and 3-boxes (right) with different
$k$. $n_{s}=1$, $n_{t}=2,3$, $k=1,2,3$.

Figure 8: Performance of transferring feature representations learned in
2-boxes (left) and 3-boxes (right) instances to learning in 1-box instances
with different $k$. $n_{s}=2,3$, $n_{t}=1$, $k=1,2,3$.

###  4.2 Transfer Among Different Tasks (SL/RL)

Feature representations learned from previous tasks can either be helpful or
harmful. In the previous subsection we saw some positive transfer to related
Sokoban tasks, in this subsection we study if transfer between supervised and
reinforcement learning tasks works. We follow prior work, Anderson et al. [1]
showed that features can be transfered from hand-crafted supervised learning
tasks to reinforcement learning. Their model was first trained to predict
state dynamics of the environment, and then pre-trained hidden layers were
helpful to accelerate solving RL tasks.

For transfer to different (randomly chosen) instances in Sokoban, we also
formed a supervised task, which was to train a prediction model to recognize
the location of the agent, shown in Fig. 9(a). When humans are solving
Sokoban, we first need to know where the agent is before we draw up a plan. If
we already know the location of objectives, the solving process could be
faster. After the prediction model could correctly recognize where the agent
is, we took feature representations of the trained model and plug them into a
new agent. The first layer of learned features is fixed, and we only train the
remaining part. Fig. 9(b) shows the performance of transferring and training
from scratch. We find negative transfer for (sPt1k1): the performance is much
worse compare with training from scratch. The bad performance is due to the
choice of the source task, which is too different for the target tasks, and
the expressive power that is learned is inadequate to overcome this
difference.

(a) (b)

Figure 9: (a): How SL tasks work. Input states and neural network will learn
to predict locations of the agent. (b): Performance of training from scratch
and training with transferred feature representations from SL tasks.

###  4.3 Transfer To Different Appearance

Experiments we described in previous subsections were all trying to transfer
Conv layers which learned feature representations. In the next experiment, we
try to make the agent utilize another part of the learned model, which are
back FC layers of the whole model. The source and target tasks were both
single-box instances, but the target tasks were instances with different
appearances. Fig. 10(b) is an example. The maps used for two groups of tasks
were the same, the only difference was how they look like, the appearance was
changed, with different textures, and we call it Game2. Fig. 10(a) shows the
transfer approach. We took FC layers trained in source tasks and fixed them,
and retrained the remaining Conv layers. Since maps were the same, solutions
of the instances were the same. When Conv layers learn new feature
representations successfully, instances are solved then.

Fig. 11(a) shows the performance. One would expect that transferred FC
layers(s1t1fc_game2) are faster because the agent only needs to learn new
feature representations. However, the experiments did not show this result.
Apparently, when the whole model is trained jointly, it has more flexibility
to be trained into the final shape; when the last part of the model is fixed,
the learning of the first part will be trying to cater for the last part in
order to solve the problem, which made the learning slower.

(a) (b)

Figure 10: (a): Transfer approach for transfer to Game2. FC layers are taken
from previously training and fixed, only conv layers will be retrained. (b):
An example instance in Game2. We changed appearances in Game2 with different
textures of objectives.

###  4.4 Visualizing Agent Detection

In order to better understand what the network learned, we provide a
visualization. We follow Yosinski et al. who showed that convolutional neural
networks can detect latent objectives without explicit labels [31]. Fig. 11(b)
shows a latent ’agent detector’ for Sokoban. The neural network automatically
learned to detect the agent without giving any labels or information. Left
rows are pixel inputs, right rows are outputs of one specific feature map.
Yellow-green units are detected agents. We note that although the network was
trained in single-box instances, it still performed quite well in multiple-box
instances, which is a potential reason for the successful transfer. The
agent’s abilities that were learned in source tasks are useful in target
tasks.

(a) (b)

Figure 11: (a): Training on Game2 using transferred FC layers. Its performance
is worse than training from scratch. (b): The agent detector. Outputs of the
twenty third feature map of the first convolutional layer, which is an agent
detector learned from 1-box instances, and it’s still usable in multiple-boxes
scenarios.

##  5 Conclusion and Future Work

Our experiments showed that in a reinforcement learning setting the agent in
Sokoban can learn four characteristics that are similar to humans. (1) Feature
representations learned previously can accelerate the new learning in other
Sokoban instances. Knowledge learned in previous related tasks could be reused
to accelerate new learning, transfer learning is occurring, creating an
implicit learning curriculum. (2) Feature representations learned in single-
box instances are more general, and are more effective for learning in
multiple-boxes instances, but not vice versa. Knowledge learned in simpler
tasks is more general and more effective, even in more complex tasks. Further
experiments showed negative learning, that confirms these results. (3) Feature
representations learned in unrelated supervised learning tasks can hurt fine-
tuning performance. If the learned knowledge is required to be helpful in new
coming tasks, it’s better to learn from similar tasks, otherwise the choice of
tasks needs to be careful. (4) Fixing the top-fully-connected layers and
retraining the bottom convolutional layers slows down learning and hurts
performance. We conclude that learning should have explicit order, less
flexibility will not only be unhelpful but also hurt the learning process and
the performance.

Our experiments showed that with a simple 5-layer convolutions/fully connected
network (based on DeepMind’s baseline [22]), transfer learning and curriculum
learning of behavior to occur in Sokoban. This is surprising, since Sokoban is
a planning-heavy problem, for which one would expect more elaborate network
architectures to be necessary. Reusing pre-trained feature representations in
RL fields is not well studied, and to the best of our knowledge, these are the
first results show transfer learning and curriculum learning with such a
simple network in such a planning-heavy behavioral task. In the future, we
would like to see more utilization of pre-trained feature representations and
of the enire pre-trained model in RL. We believe that reusing pre-trained
model can significantly improve data-efficient reinforcement learning.

## Acknowledgement

The financial support to Zhao Yang is from the China Scholarship Council(CSC).
Computation support is from ALICE and DSLab. The authors thank Hui Wang,
Matthias Müller-Brockhausen, Michiel van der Meer, Thomas Moerland and all
members from the Leiden Reinforcement Learning Group for helpful discussions.

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Lightlike reduction of the M5 brane
Andreas Gustavsson

Physics Department, University of Seoul, Seoul 02504 KOREA

<EMAIL_ADDRESS>

###### Abstract

##  1 Introduction

The lightlike dimensional reduction of the M5 brane results in instanton
moduli space dynamics [10]. In this paper we will revisit a technical issue
regarding the way that selfduality is implemented in these theories. By
lightlike reduction, selfduality of the six-dimensional tensor field is
inherited as a four-dimensional selfduality condition on the Yang-Mills field
strength. There are two independent lightlike directions and the lightlike
dimensional reduction results in a five-dimensional theory where one direction
is singled out as it originates from the other lightlike direction. We are now
facing the problem of how to implement the selfduality equation of motion from
a Lagrangian. It turns out that there is a Lagrangian that does precisely this
through an auxiliary Lagrange multiplier field $G_{ij}$ that is antiselfdual
offshell.

The Lagrangian can not be straightforwardly obtained by dimensional reduction
of some Lagrangian for the abelian M5 brane. The reason can be traced to the
selfduality of the tensor gauge field in six dimensions, and to the
selfduality of the Yang-Mills field in the reduced theory. Let us assume that
we have an abelian M5 brane on $\mathbb{R}^{1,5}$. If we denote the
nonselfdual six-dimensional tensor field as $H_{MNP}$, the two lightlike
directions as $x^{\pm}$, and define

$\displaystyle F_{ij}$ $\displaystyle=$ $\displaystyle H_{ij-}$ (1.1)
$\displaystyle G_{ij}$ $\displaystyle=$ $\displaystyle H_{ij+}$ (1.2)

then selfduality of $H_{MNP}$ implies selfduality of $F_{ij}$ and
antiselfduality of $G_{ij}$. But whereas selfduality of $H_{MNP}$ is an
equation of motion in six dimenions, in the dimensionally reduced theory, we
will take antiselfduality of $G_{ij}$ as a constraint that holds offshell and
selfduality of $F_{ij}$ arises as an equation of motion by varying $G_{ij}$ in
the term $\frac{1}{2}G_{ij}F_{ij}$ in the Lagrangian. If dimensional reduction
is carried out straightforwardly, then we will get antiselfduality of $G_{ij}$
as an onshell equation of motion. This explains why we need to proceed by some
amount of guesswork, if we want to have offshell antiselfduality of $G_{ij}$.

By making a general ansatz with some arbitary coefficients, we find that for
the supersymmetry variation of the fermion we shall have

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\Gamma_{ij}\Gamma^{+}{\cal{\varepsilon}}G_{ij}+\frac{1}{4}\Gamma_{ij}\Gamma^{-}{\cal{\varepsilon}}F_{ij}+....$
(1.3)

The minus sign for the first term is counterintuitive. If we start from the
supersymmetry variation in six dimensions

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{12}\Gamma^{MNP}{\cal{\varepsilon}}H_{MNP}+...$ (1.4)

and decompose this variation into components, then we get

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{4}\Gamma_{ij}\Gamma^{+}{\cal{\varepsilon}}G_{ij}+\frac{1}{4}\Gamma_{ij}\Gamma^{-}{\cal{\varepsilon}}F_{ij}+...$
(1.5)

Instead of reducing the supersymmetry variations, we reduce the selfduality
equation of motion

$\displaystyle H_{MNP}$ $\displaystyle=$
$\displaystyle\frac{1}{6}{\cal{\varepsilon}}_{MNP}{}^{RST}H_{RST}$ (1.6)

along $x^{-}$. We then find a Lagrangian that gives those corresponding
equations of motion that we want to get. That is, the selfduality equation for
$F_{ij}$ (but not for $G_{ij}$) and another equation of motion to be presented
below. It turns out that there exists a Lagrangian that gives all the desired
equations of motion. It is given by [1], [2]

$\displaystyle{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}G_{ij}F_{ij}+\frac{1}{2}F_{i+}F_{i+}$ (1.7)

It is important to note the very different roles that these various fields
play. While $F_{ij}$ and $F_{i+}$ are usual field strength components in five
dimensions, $G_{ij}$ shall be viewed as an auxiliary field that we do not
express in terms of some gauge potential. We then supersymmetrize this
Lagrangian. This strategy is similar in spirit to what was done in [3] for a
spatial dimensional reduction of the M5 brane on a circle bundle. It is
somewhat similar in spirit to Sen’s construction of the selfdual tensor field
Lagrangian that also uses an auxiliary offshell selfdual tensor field to
implement selfduality on $H_{MNP}$ as an equation of motion [4], [5], [6],
[7]. We review Sen’s construction in the Appendix A.

In section 2 we present the Lagrangian that we get by lightlike reduction of
the M5 brane on $\mathbb{R}^{1,5}$. We subsequently show that this can be
generalized to a nonabelian gauge group. In section 3 we covariantize our flat
space result for a rather generic six-manifold with two lightlike Killing
vector fields. Here computations are a bit complicated if one chooses an
explicit lightcone coordinate system on the six-manifold and of the
corresponding five-dimensional base-manifold. Instead we will introduce two
lightlike Killing vector fields in the six-manifold and formulate the theory
covariantly in six-dimensions. We then impose as a constraint on top of the
Lagrangian that all the fields have a vanishing Lie derivative along one of
the two lightlike directions. In sections 4 we study reduction of
supersymmetry by a lightlike Weyl projection. A detailed computation of the
supersymmetry variation of the Lagrangian is presented in Appendix B.

##  2 Minkowski space

We begin with putting the M5 brane on the Minkowski space $\mathbb{R}^{1,5}$.
We assume that the metric is

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dt^{2}+(dx^{5})^{2}+dx_{i}dx_{i}$ (2.1) $\displaystyle=$ $\displaystyle
2dx^{+}dx^{-}+dx_{i}dx_{i}$ (2.2)

where the two lightlike coordinates are chosen as

$\displaystyle x^{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(x^{5}\pm t\right)$ (2.3)

We define

$\displaystyle F_{ij}$ $\displaystyle=$ $\displaystyle H_{ij-}$ (2.4)
$\displaystyle F_{i+}$ $\displaystyle=$ $\displaystyle H_{i+-}$ (2.5)
$\displaystyle G_{ij}$ $\displaystyle=$ $\displaystyle H_{ij+}$ (2.6)

and

$\displaystyle{\cal{\varepsilon}}^{ijkl}$ $\displaystyle=$
$\displaystyle{\cal{\varepsilon}}^{ijkl+-}$ (2.7)

Dimensional reduction is performed by putting the derivative along $x^{-}$ to
zero on all fields. The selfduality equation of motion
$H_{MNP}=\frac{1}{6}{\cal{\varepsilon}}_{MNP}{}^{RST}H_{RST}$ implies the
following equations of motion

$\displaystyle F_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal{\varepsilon}}_{ijkl}F_{kl}$ (2.8)
$\displaystyle G_{ij}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{\cal{\varepsilon}}_{ijkl}G_{kl}$ (2.9)
$\displaystyle H_{ijk}$ $\displaystyle=$
$\displaystyle-{\cal{\varepsilon}}_{ijkl}F_{l+}$ (2.10)

The Bianchi identity $\partial_{[Q}H_{MNP]}=0$ implies

$\displaystyle\partial_{+}F_{ij}+2\partial_{[i}F_{j]+}$ $\displaystyle=$
$\displaystyle 0$ (2.11) $\displaystyle
3\partial_{[i}G_{jk]}-\partial_{+}H_{ijk}$ $\displaystyle=$ $\displaystyle 0$
(2.12)

These Bianchi identities are of course satisfied offshell, but in addition we
will now also assume that the following selfduality equations

$\displaystyle G_{ij}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{\cal{\varepsilon}}_{ijkl}G_{kl}$ (2.13)
$\displaystyle H_{ijk}$ $\displaystyle=$
$\displaystyle{\cal{\varepsilon}}_{ijkl}F_{l+}$ (2.14)

are satisfied offshell as constraints. Then the remaining equations

$\displaystyle F_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\cal{\varepsilon}}_{ijkl}F_{kl}$ (2.15)
$\displaystyle\partial_{i}G_{ij}$ $\displaystyle=$
$\displaystyle\partial_{+}F_{j+}$ (2.16) $\displaystyle\partial_{i}F_{i+}$
$\displaystyle=$ $\displaystyle 0$ (2.17)

will be viewed as ordinary equations of motion that we would like to derive
from a Lagrangian and Euler-Lagrange equations. Indeed there is such a
Lagrangian. It is given by

$\displaystyle{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle\frac{k}{2}\left(G_{ij}F_{ij}+F_{i+}F_{i+}\right)$ (2.18)

To this we add the matter part Lagrangian

$\displaystyle{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(D_{i}\phi^{A})^{2}+\frac{i}{2}\bar{\psi}\Gamma_{i}D_{i}\psi+\frac{i}{2}\bar{\psi}\Gamma_{-}D_{+}\psi$
(2.20)
$\displaystyle+\frac{e}{2}\bar{\psi}\Gamma_{-}\Gamma^{A}[\psi,\phi^{A}]$

where we assume the gauge group is nonabelian and the covariant derivative is
$D_{M}\Phi=\partial_{M}-ie[A_{M},\Phi]$. The inner product on the gauge Lie
algebra, that is the trace, is implicit in our notation. For the supersymmetry
variations we make the ansatz

$\displaystyle\delta\phi^{A}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma^{A}\psi$ (2.21) $\displaystyle\delta A_{i}$
$\displaystyle=$ $\displaystyle i\bar{\cal{\varepsilon}}\Gamma_{i-}\psi$
(2.22) $\displaystyle\delta A_{+}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma_{+-}\psi$ (2.23) $\displaystyle\delta\psi$
$\displaystyle=$
$\displaystyle\frac{a}{4}\Gamma_{ij}\Gamma_{-}{\cal{\varepsilon}}G_{ij}+\frac{b}{4}\Gamma_{ij}\Gamma_{+}{\cal{\varepsilon}}F_{ij}-c\Gamma_{i}\Gamma_{+-}{\cal{\varepsilon}}F_{i+}+\Gamma_{i}\Gamma^{A}{\cal{\varepsilon}}D_{i}\phi^{A}+\Gamma_{-}\Gamma^{A}{\cal{\varepsilon}}D_{+}\phi^{A}$
(2.24) $\displaystyle\delta G_{ij}$ $\displaystyle=$
$\displaystyle\frac{if}{2}\bar{\cal{\varepsilon}}\Gamma_{k}\Gamma_{ij+}D_{k}\psi+\frac{if}{2}\bar{\cal{\varepsilon}}\Gamma_{-}\Gamma_{ij+}D_{+}\psi$
(2.26)
$\displaystyle+\frac{e}{2}\bar{\cal{\varepsilon}}\Gamma_{-}\Gamma_{ij+}\Gamma^{A}[\psi,\phi^{A}]$

There are five scalar fields $\phi^{A}$ and fermions $\psi$. We use eleven-
dimensional gamma matrices satisfying in particular
$\\{\Gamma^{M},\Gamma^{A}\\}=0$, and the spinors satisfy the eleven-
dimensional Majorana condition $\bar{\psi}=\psi^{T}C$ and the six-dimensional
Weyl projection $\Gamma\psi=\psi$ and
$\Gamma{\cal{\varepsilon}}=-{\cal{\varepsilon}}$ where
$\Gamma=\Gamma^{012345}$. When we then make a supersymmetry variation of the
Lagrangian, we get the following conditions for the parameters,

$\displaystyle a+k$ $\displaystyle=$ $\displaystyle 0$ (2.27) $\displaystyle
b+kf$ $\displaystyle=$ $\displaystyle 0$ (2.28) $\displaystyle c-b$
$\displaystyle=$ $\displaystyle 0$ (2.29)

from the noncommutator terms, and from the commutator terms we get additional
conditions

$\displaystyle e$ $\displaystyle=$ $\displaystyle ec$ (2.30) $\displaystyle k$
$\displaystyle=$ $\displaystyle b$ (2.31) $\displaystyle 1$ $\displaystyle=$
$\displaystyle k$ (2.32)

From the latter three conditions, we again get $b=c$, which is in agreement
with the condition we get in the abelian case, but in addition we get $c=k=1$
that we do not find in the abelian case. So demanding supersymmetry in the
nonabelian generalization gives a more restrictive solution for the
coefficients. With $c=k=1$ the remaining coefficients are then also uniquely
fixed as

$\displaystyle a$ $\displaystyle=$ $\displaystyle-1$ (2.33) $\displaystyle b$
$\displaystyle=$ $\displaystyle 1$ (2.34) $\displaystyle c$ $\displaystyle=$
$\displaystyle 1$ (2.35) $\displaystyle f$ $\displaystyle=$ $\displaystyle-1$
(2.36) $\displaystyle k$ $\displaystyle=$ $\displaystyle 1$ (2.37)

Within our ansatz, we have found a unique set of parameters for which we have
the following nonabelian Lagrangian

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(G_{ij}F_{ij}+F_{i+}F_{i+}\right)-\frac{1}{2}(D_{i}\phi^{A})^{2}$
(2.39)
$\displaystyle+\frac{i}{2}\bar{\psi}\Gamma_{i}D_{i}\psi+\frac{i}{2}\bar{\psi}\Gamma_{-}D_{+}\psi+\frac{e}{2}\bar{\psi}\Gamma_{-}\Gamma^{A}[\psi,\phi^{A}]$

This is invariant under the supersymmetry variations

$\displaystyle\delta\phi^{A}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma^{A}\psi$ (2.40) $\displaystyle\delta A_{i}$
$\displaystyle=$ $\displaystyle i\bar{\cal{\varepsilon}}\Gamma_{i-}\psi$
(2.41) $\displaystyle\delta A_{+}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma_{+-}\psi$ (2.42) $\displaystyle\delta\psi$
$\displaystyle=$
$\displaystyle-\frac{1}{4}\Gamma_{ij}\Gamma_{-}{\cal{\varepsilon}}G_{ij}+\frac{1}{4}\Gamma_{ij}\Gamma_{+}{\cal{\varepsilon}}F_{ij}-\Gamma_{i}\Gamma_{+-}{\cal{\varepsilon}}F_{i+}+\Gamma_{i}\Gamma^{A}{\cal{\varepsilon}}D_{i}\phi^{A}+\Gamma_{-}\Gamma^{A}{\cal{\varepsilon}}D_{+}\phi^{A}$
(2.43) $\displaystyle\delta G_{ij}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\bar{\cal{\varepsilon}}\Gamma_{k}\Gamma_{ij+}D_{k}\psi-\frac{i}{2}\bar{\cal{\varepsilon}}\Gamma_{-}\Gamma_{ij+}D_{+}\psi+\frac{e}{2}\bar{\cal{\varepsilon}}\Gamma_{-}\Gamma_{ij+}\Gamma^{A}[\psi,\phi^{A}]$
(2.44)

##  3 Covariant formulation

We now turn to a generic Lorentzian six-manifold that we assume has two
independent lightlike Killing vectors $u^{M}$ and $v^{M}$ respectively. We
assume that their inner product

$\displaystyle v_{M}u^{M}$ $\displaystyle=$ $\displaystyle\lambda$ (3.1)

is everywhere nonvanishing. The two lightlike Killing vectors satisfy the
equations,

$\displaystyle v^{M}v_{M}$ $\displaystyle=$ $\displaystyle 0$ (3.2)
$\displaystyle u^{M}u_{M}$ $\displaystyle=$ $\displaystyle 0$ (3.3)
$\displaystyle\nabla_{M}v_{N}+\nabla_{N}v_{M}$ $\displaystyle=$ $\displaystyle
0$ (3.4) $\displaystyle\nabla_{M}u_{N}+\nabla_{N}u_{M}$ $\displaystyle=$
$\displaystyle 0$ (3.5)

Let us now start by the Killing vector equation

$\displaystyle\nabla_{M}v_{N}+\nabla_{N}v_{M}$ $\displaystyle=$ $\displaystyle
0$ (3.6)

If we contract the left-hand side by $u^{M}$, then we have

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
u^{M}\nabla_{M}v_{N}+u^{M}\nabla_{N}v_{M}$ (3.7) $\displaystyle=$
$\displaystyle u^{M}\nabla_{M}v_{N}+\partial_{N}\lambda-v^{M}\nabla_{N}u_{M}$
(3.8) $\displaystyle=$ $\displaystyle
u^{M}\nabla_{M}v_{N}+v^{M}\nabla_{M}u_{N}+\partial_{N}\lambda$ (3.9)

and if we further contract by $u^{N}$, then the first term vanishes by the
Killing equation and the second term vanishes by using $u^{N}u_{N}=0$. We thus
find that

$\displaystyle{\cal{L}}_{u}\lambda$ $\displaystyle=$ $\displaystyle 0$ (3.10)
$\displaystyle{\cal{L}}_{v}\lambda$ $\displaystyle=$ $\displaystyle 0$ (3.11)

In the Abelian case, we define

$\displaystyle A_{M}$ $\displaystyle=$ $\displaystyle B_{MN}v^{P}$ (3.12)

which satisfies $A_{M}v^{M}=0$. Let us now examine the implications of
selfduality equation of motion

$\displaystyle H_{MNP}$ $\displaystyle=$
$\displaystyle\frac{1}{6}{\cal{\varepsilon}}_{MNPRST}H^{RST}$ (3.13)

on the component fields

$\displaystyle F_{MN}$ $\displaystyle=$ $\displaystyle H_{MNP}v^{P}$ (3.14)
$\displaystyle G_{MN}$ $\displaystyle=$ $\displaystyle H_{MNP}u^{P}$ (3.15)
$\displaystyle K_{M}$ $\displaystyle=$ $\displaystyle H_{MNP}u^{N}v^{P}$
(3.16)

that we may further expand in traceless and traceparts as

$\displaystyle F_{MN}$ $\displaystyle=$
$\displaystyle\widetilde{F}_{MN}+K_{M}v_{N}-K_{N}v_{M}$ (3.17) $\displaystyle
G_{MN}$ $\displaystyle=$
$\displaystyle\widetilde{G}_{MN}-K_{M}u_{N}+K_{N}u_{M}$ (3.18)

while $K_{M}$ is already traceless. Then we may expand

$\displaystyle H_{MNP}$ $\displaystyle=$
$\displaystyle\widetilde{H}_{MNP}+\frac{3}{\lambda}\widetilde{F}_{MN}u_{P}+\frac{3}{\lambda}\widetilde{G}_{MN}v_{P}-\frac{6}{\lambda^{2}}K_{M}u_{N}v_{P}$
(3.19)

The induced selfduality relations are then given by

$\displaystyle\frac{1}{36}{\cal{\varepsilon}}_{MNP}{}^{RST}\widetilde{H}_{RST}$
$\displaystyle=$ $\displaystyle-\frac{1}{\lambda^{2}}K_{M}u_{N}v_{P}$ (3.20)
$\displaystyle\frac{1}{6}{\cal{\varepsilon}}_{MNP}{}^{RST}\widetilde{F}_{RS}u_{T}$
$\displaystyle=$ $\displaystyle\widetilde{F}_{MN}u_{P}$ (3.21)
$\displaystyle\frac{1}{6}{\cal{\varepsilon}}_{MNP}{}^{RST}\widetilde{G}_{RS}v_{T}$
$\displaystyle=$ $\displaystyle\widetilde{G}_{MN}v_{P}$ (3.22)

If one defines

$\displaystyle{\cal{E}}^{MNRS}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}{\cal{\varepsilon}}^{MNRSTP}v_{T}u_{P}$ (3.23)

then one may express these relations as

$\displaystyle\widetilde{H}^{RST}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2}}{\cal{E}}^{RSTM}K_{M}$ (3.24)
$\displaystyle\frac{1}{2}{\cal{E}}_{MN}{}^{RS}\widetilde{F}_{RS}$
$\displaystyle=$ $\displaystyle\widetilde{F}_{MN}$ (3.25)
$\displaystyle\frac{1}{2}{\cal{E}}_{MN}{}^{RS}\widetilde{G}_{RS}$
$\displaystyle=$ $\displaystyle-\widetilde{G}_{MN}$ (3.26)

These definitions (3.12) and (3.14) are mutually consistent only if we require
that ${\cal{L}}_{v}F_{MN}=0$. We impose (3.26) as well as (3.24) as an
offshell constraints. In addition, we have the usual Bianchi identity
$D_{[M}F_{NP]}=0$ that holds offshell. In Minkowski space we also had the
Bianchi identity (2.11) that we should now covariantize as well. To this end,
we start by computing the following Lie derivative,

$\displaystyle{\cal{L}}_{u}F_{MN}$ $\displaystyle=$
$\displaystyle{\cal{L}}_{v}G_{MN}-\nabla_{M}K_{N}+\nabla_{N}K_{M}+H_{MNP}{\cal{L}}_{v}u^{P}$
(3.27)

To get this result, we have used the Bianchi identity $\nabla_{[Q}H_{MNP]}=0$.
We now see that if ${\cal{L}}_{v}G_{MN}=0$ and ${\cal{L}}_{v}u^{M}=0$, then we
have the Bianchi identity

$\displaystyle{\cal{L}}_{u}F_{MN}+2\nabla_{[M}K_{N]}$ $\displaystyle=$
$\displaystyle 0$ (3.28)

It is easy to show that ${\cal{L}}_{v}u^{M}=0$ is equivalent to the condition
that the Lie derivatives for the two Killing vectors commute,
$[{\cal{L}}_{u},{\cal{L}}_{v}]=0$, which means that we have a torus fibration,
with the only peculiar feature being that the vectors that span this torus are
lightlike vectors. With these assumptions, we find that the following
Lagrangian ${\cal{L}}={\cal{L}}_{A}+{\cal{L}}_{CS}+{\cal{L}}_{m}$, where

$\displaystyle{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda}\widetilde{G}^{MN}\widetilde{F}_{MN}+\frac{1}{2\lambda}K^{M}K_{M}$
(3.29) $\displaystyle{\cal{L}}_{CS}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\lambda}{\cal{\varepsilon}}^{MNPQRS}\omega(A)_{MNP}\Omega_{QR}u_{S}$
(3.30) $\displaystyle{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(D_{M}\phi^{A})^{2}-\frac{R}{10}(\phi^{A})^{2}$
(3.32)
$\displaystyle+\frac{i}{2}\bar{\psi}\Gamma^{M}D_{M}\psi+\frac{e}{2}\bar{\psi}\Gamma_{M}\Gamma^{A}[\psi,\phi^{A}]v^{M}$

is invariant under the following supersymmetry variations

$\displaystyle\delta\phi^{A}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma^{A}\psi$ (3.33) $\displaystyle\delta A_{M}$
$\displaystyle=$ $\displaystyle i\bar{\cal{\varepsilon}}\Gamma_{MN}\psi v^{N}$
(3.34) $\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\Gamma^{MNP}{\cal{\varepsilon}}\left(-\frac{1}{4\lambda}\widetilde{G}_{MN}v_{P}+\frac{1}{4\lambda}\widetilde{F}_{MN}u_{P}-\frac{1}{\lambda}K_{M}u_{N}v_{P}\right)$
(3.36)
$\displaystyle+\Gamma^{M}\Gamma^{A}{\cal{\varepsilon}}D_{M}\phi^{A}-4\Gamma^{A}\eta\phi^{A}-\frac{ie}{2}\Gamma_{M}\Gamma^{AB}{\cal{\varepsilon}}[\phi^{A},\phi^{B}]v^{M}$
$\displaystyle\delta G_{MN}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}D_{Q}\left(\bar{\cal{\varepsilon}}\Gamma^{Q}\Gamma_{MNP}\psi\right)u^{P}+\frac{e}{2}\bar{\cal{\varepsilon}}\Gamma_{Q}\Gamma_{MNP}\Gamma^{A}[\psi,\phi^{A}]u^{P}v^{Q}$
(3.37)

Here we normalize the Chern-Simons three-form such that its variation is given
by $\delta\omega(A)_{MNP}=\delta A_{M}F_{NP}$ and we define
$\Omega_{MN}=\nabla_{M}\left(\frac{1}{\lambda}u_{N}\right)-\nabla_{N}\left(\frac{1}{\lambda}v_{N}\right)$.
When we make a supersymmetry variation, we get the result

$\displaystyle\delta{\cal{L}}$ $\displaystyle=$
$\displaystyle\frac{i}{2\lambda}\bar{\cal{\varepsilon}}\Gamma^{MN}\psi{\cal{L}}_{v}G_{MN}$
(3.40)
$\displaystyle+2e\bar{\psi}\Gamma^{AB}({\cal{L}}_{v}{\cal{\varepsilon}})[\phi^{A},\phi^{B}]+2e\bar{\psi}\Gamma^{AB}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A},\phi^{B}]-e\bar{\psi}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A},\phi^{A}]$
$\displaystyle+\frac{i}{\lambda}\bar{\cal{\varepsilon}}\Gamma^{MN}\psi{\cal{L}}_{v}\left(K_{M}u_{N}\right)$

Here the Lie derivative of a fermion is defined as
${\cal{L}}_{v}{\cal{\varepsilon}}=v^{M}\nabla_{M}{\cal{\varepsilon}}+\nabla_{M}v_{N}\Gamma^{MN}{\cal{\varepsilon}}$.
We see that the Lagrangian is invariant if we impose ${\cal{L}}_{v}=0$ on all
fields, but in addition we need to assume that ${\cal{L}}_{v}u^{M}=0$. We also
need to impose that ${\cal{L}}_{v}{\cal{\varepsilon}}=0$ which is a
restriction on the geometry. The details of the computation of the
supersymmetry variation of the Lagrangian are presented in Appendix B.

##  4 A lightlike Weyl projection

We may reduce the amount of supersymmetry by half by imposing the following
lightlike Weyl projection

$\displaystyle\Gamma_{M}{\cal{\varepsilon}}v^{M}$ $\displaystyle=$
$\displaystyle 0$ (4.1)

Then all the explicit commutator terms will drop out from the supersymmetry
variations,

$\displaystyle\delta\phi^{A}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma^{A}\psi$ (4.2) $\displaystyle\delta A_{M}$
$\displaystyle=$ $\displaystyle iv_{M}\bar{\cal{\varepsilon}}\psi$ (4.3)
$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{4\lambda}\Gamma^{MNP}{\cal{\varepsilon}}\widetilde{F}_{MN}u_{P}+\Gamma^{M}{\cal{\varepsilon}}K_{M}+\Gamma^{M}\Gamma^{A}{\cal{\varepsilon}}D_{M}\phi^{A}$
(4.4) $\displaystyle\delta G_{MN}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}D_{Q}\left(\bar{\cal{\varepsilon}}\Gamma^{Q}\Gamma_{MNP}\psi\right)u^{P}$
(4.5)

One may notice that (4.1) implies that

$\displaystyle v^{M}\nabla_{M}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\nabla_{M}v_{N}\Gamma^{MN}{\cal{\varepsilon}}$ (4.6)

which can be used to show that when we make a supersymmetry variation of the
Lagrangian, one gets

$\displaystyle\delta{\cal{L}}$ $\displaystyle=$
$\displaystyle\frac{i}{2\lambda}\bar{\cal{\varepsilon}}\Gamma^{MN}\psi{\cal{L}}_{v}G_{MN}$
(4.9)
$\displaystyle+2e\bar{\psi}\Gamma^{AB}({\cal{L}}_{v}{\cal{\varepsilon}})[\phi^{A},\phi^{B}]+2e\bar{\psi}\Gamma^{AB}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A},\phi^{B}]-e\bar{\psi}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A},\phi^{A}]$
$\displaystyle+\frac{i}{\lambda}\bar{\cal{\varepsilon}}\Gamma^{MN}\psi{\cal{L}}_{v}\left(K_{M}u_{N}\right)$

Although the explicit commutators dropped out from the supersymmetry
variations by the lightlike Weyl projection of the supersymmetry parameter,
the Lagrangian is of course the same as before, and we can now also see that
its supersymmetry variation under the lightlike Weyl projected supersymmetries
is also of the same form as before. In particular the Lagrangian can be
supersymmetric only if ${\cal{L}}_{v}{\cal{\varepsilon}}=0$, which is a
condition that severly restricts the possible six-manifolds.

###  4.1 Further reduction of supersymmetry

We can avoid having to impose the condition
${\cal{L}}_{v}{\cal{\varepsilon}}=0$ by further reducing down to the $(1,0)$
tensor multiplet. To this end, we impose a second Weyl projection that breaks
the $SO(5)$ R-symmetry down to $SU(2)$ R-symmetry. We define the chirality
matrix $\widehat{\Gamma}=\widehat{\Gamma}^{1234}$ where we put hats on 11d
gamma matrices associated with $SO(5)$. We then project the supersymmetry
parameter as

$\displaystyle\widehat{\Gamma}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle-{\cal{\varepsilon}}$ (4.10)

We also have the 6d Weyl projection
$\Gamma{\cal{\varepsilon}}=-{\cal{\varepsilon}}$. If we define
$\widehat{\Gamma}^{5}$ such that
$\Gamma\widehat{\Gamma}\widehat{\Gamma}^{5}=1$, then we find that
$\widehat{\Gamma}^{5}{\cal{\varepsilon}}={\cal{\varepsilon}}$. We also notice
that $\widehat{\Gamma}^{T}=C\widehat{\Gamma}C^{-1}$ and consequently
$\bar{\cal{\varepsilon}}\widehat{\Gamma}=-\bar{\cal{\varepsilon}}$ and
$\bar{\cal{\varepsilon}}\Gamma^{5}=-\bar{\cal{\varepsilon}}$. The $(2,0)$
tensor multiplet spinor gets separated into two Weyl components. The component
with the same chirality as ${\cal{\varepsilon}}$ goes into the vector
multiplet spinor that we will continue to denote as $\psi$,

$\displaystyle\widehat{\Gamma}\psi$ $\displaystyle=$ $\displaystyle-\psi$
(4.11)

We will not consider hypermultiplets. For the vector mulitiplet we have the
supersymmetry variations (using the notation $\varphi=\phi^{5}$ for the fifth
scalar field)

$\displaystyle\delta\varphi$ $\displaystyle=$
$\displaystyle-i\bar{\cal{\varepsilon}}\psi$ (4.12) $\displaystyle\delta
A_{M}$ $\displaystyle=$ $\displaystyle iv_{M}\bar{\cal{\varepsilon}}\psi$
(4.13) $\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{4\lambda}\Gamma^{MNP}{\cal{\varepsilon}}\widetilde{F}_{MN}u_{P}+\Gamma^{M}{\cal{\varepsilon}}K_{M}+\Gamma^{M}{\cal{\varepsilon}}D_{M}\varphi$
(4.14) $\displaystyle\delta G_{MN}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}D_{Q}\left(\bar{\cal{\varepsilon}}\Gamma^{Q}\Gamma_{MNP}\psi\right)u^{P}$
(4.15)

The vector multiplet Lagrangian is

$\displaystyle{\cal{L}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda}\widetilde{G}^{MN}F_{MN}+\frac{1}{2\lambda}K^{M}K_{M}+{\cal{L}}_{CS}-\frac{1}{2}(D_{M}\varphi)^{2}-\frac{R}{10}\varphi^{2}$
(4.17)
$\displaystyle+\frac{i}{2}\bar{\psi}\Gamma^{M}D_{M}\psi+\frac{e}{2}\bar{\psi}\Gamma_{M}[\psi,\varphi]v^{M}$

We may now introduce a new gauge field

$\displaystyle{\cal{A}}_{M}$ $\displaystyle=$ $\displaystyle
A_{M}+v_{M}\varphi$ (4.18)

that is a supersymmetry invariant. If we define

$\displaystyle{\cal{D}}_{M}\Phi$ $\displaystyle=$
$\displaystyle\partial_{M}\Phi-ie[{\cal{A}}_{M},\Phi]$ (4.19)

then the standard gauge variations may be expressed in terms of
${\cal{A}}_{M}$ and ${\cal{D}}_{M}$ as

$\displaystyle\delta A_{M}$ $\displaystyle=$
$\displaystyle{\cal{D}}_{M}\lambda$ (4.20) $\displaystyle\delta\Phi$
$\displaystyle=$ $\displaystyle-ie[\Phi,\lambda]$ (4.21)

So we see that ${\cal{A}}_{M}$ behaves just like a gauge field, with its
corresponding field strength

$\displaystyle{\cal{F}}_{MN}$ $\displaystyle=$ $\displaystyle
F_{MN}+{\cal{D}}_{M}(v_{N}\varphi)-{\cal{D}}_{N}(v_{M}\varphi)$ (4.22)

The vector multiplet Lagrangian can be recast in the following form

$\displaystyle{\cal{L}}_{vector}$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda}\widetilde{G}^{MN}\left({\cal{F}}_{MN}-2{\cal{D}}_{M}(v_{N}\varphi)\right)+{\cal{L}}_{CS}(A)-\frac{1}{2}({\cal{D}}_{M}\varphi)^{2}-\frac{R}{10}\varphi^{2}$
(4.24) $\displaystyle+\frac{i}{2}\bar{\psi}\Gamma^{M}{\cal{D}}_{M}\psi$

with the supersymmetry variations

$\displaystyle\delta\varphi$ $\displaystyle=$
$\displaystyle-i\bar{\cal{\varepsilon}}\psi$ (4.25)
$\displaystyle\delta{\cal{A}}_{M}$ $\displaystyle=$ $\displaystyle 0$ (4.26)
$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{4\lambda}\Gamma^{MNP}{\cal{\varepsilon}}\left({\cal{F}}_{MN}-2{\cal{D}}_{M}(v_{N}\varphi)\right)u_{P}-\frac{1}{\lambda}\Gamma^{M}{\cal{\varepsilon}}G_{MN}v^{N}+\Gamma^{M}{\cal{\varepsilon}}{\cal{D}}_{M}\varphi$
(4.27) $\displaystyle\delta G_{MN}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}{\cal{D}}_{Q}\left(\bar{\cal{\varepsilon}}\Gamma^{Q}\Gamma_{MNP}\psi\right)u^{P}$
(4.28)

The Chern-Simons term can be expressed in terms of
${\cal{A}}_{M}-v_{M}\varphi$ in place of $A_{M}$ and then covariant
derivatives with respect to ${\cal{A}}_{M}$ appear that act on $v_{M}\varphi$.
All terms are quadratic in the fields, except for when ${\cal{D}}_{M}$ acts on
some fields that gives in addition a factor of ${\cal{A}}_{M}$ that however is
a supersymmetry single. What we have now achieved is that we have written the
Lagrangian and the supersymmetry variations entirely in terms of the
supersymmetry invariant gauge field ${\cal{A}}_{M}$ and there are no explicit
commutator terms. The commutators that do appear are all implicit in the gauge
covariant derivatives where the gauge field is a supersymmetry singlet. This
means that we can now truncate supersymmetry in a different way from what we
usually do under dimensional reduction, such that we can have a supersymmetric
Lagrangian, even when ${\cal{L}}_{v}{\cal{\varepsilon}}$ is not vanishing [9].
To this end we keep the zero modes of the bosonic fields. For the fermionic
field, we keep the nonzero modes that correspond to the nonzero modes of the
supersymmetry parameter ${\cal{\varepsilon}}$. To understand why this works,
let us imagine that we have a six-dimensional action where we integrate the
Lagrangian over all six directions, including the direction generated by
$v^{M}$. Let us assume this direction is parametrized by a coordinate
$\sigma\sim\sigma+2\pi$. If we now keep the bosonic zero mode along $\sigma$,
and if we make a supersymmetry variation of the Lagrangian, then the integral
over $\sigma$ will select the modes of $\psi$ that match the modes of
${\cal{\varepsilon}}$ such that any fermionic bilinear combination
$\bar{\cal{\varepsilon}}(...)\psi$ that appears in the variation of the action
is a zero mode. For the supersymmetry variations of bosonic zero mode fields,
we again integrate bilinear expressions $\bar{\cal{\varepsilon}}(...)\psi$
that selects the bosonic zero mode variation on the left-hand side, and those
nonzero modes of $\psi$ that correspond to the modes of ${\cal{\varepsilon}}$
that makes the right-hand side a zero mode. For the supersymmetry variation of
the fermionic field, the modes of the fermionic field will have to match with
the mode of the supersymmetry parameter ${\cal{\varepsilon}}$ since these
variations are on the form $\delta\psi={\cal{\varepsilon}}\times$(bosonic zero
mode field). We thus have a consistent truncation of the modes to the zero
modes for the bosons and to the nonzero modes of the fermionic field that
match the modes of the ${\cal{\varepsilon}}$. This is a consistent truncation
of modes when the Lagrangian is abelian, or when the Lagrangian is nonabelian
and all interactions come from covariant derivatives where the gauge field is
a supersymmetry singlet. If there are other interaction terms in the
Lagrangian, then when we make a supersymmetry variation of the Lagrangian we
may encounter the problematic term that is cubic in the fermionic field
$\sim\bar{\cal{\varepsilon}}\psi\psi\psi$. For such cubic terms, integration
over $\sigma$ will not just select one nonzero mode of the fermionic field,
but it will select all modes in the entire Kaluza-Klein tower. In such cases
there are no consistent truncations to finitely many modes, except for the
trivial truncation down to both bosonic and fermionic zero modes, which is the
usual dimensional reduction scenario. But if
${\cal{L}}_{v}{\cal{\varepsilon}}$ is non-zero, then the usual dimensional
reduction of course breaks all supersymmetry.

We now have got a supersymmetric Lagrangian, although just for the $(1,0)$
tensor multiplet, for any geometry restricted only by the existence of a
solution to the six-dimensional conformal Killing spinor equation. But even
then, we still have to restrict to the Lorentzian signature. It would
therefore be interesting to see if lightlike dimensional reduction can be
applied also in Euclidean signature. In Euclidean signature we can not impose
the six-dimensional chiral Weyl projection since this is not compatible with
the $SO(6,5)$ eleven-dimensional Majorana condition. So we are forced to
consider the nonchiral $(2,2)$ tensor multiplet. Then we may obtain results in
the $(2,0)$ theory by holomorphic factorization of the $(2,2)$ theory.
Lightlike dimensional reduction would in Euclidean signature correspond to
dimensional reduction along a complex coordinate $\bar{z}=x-it$, which means
that the fields would depend holomorphically on $z$.

## Acknowledgements

This work was supported in part by NRF Grant RS-2023-00208011 and NRF Grant
2020R1I1A1A01052462.

##  Appendix A Review of Sen’s construction

Let us assume the gauge group is abelian. Then Lagrangian that was first
proposed by Sen is given by111We do not consider tangent space selfduality for
$Q_{MNP}$, so our approach coincides with Sen’s approach only in flat
Minkowski space. [4], [5], [6], [7], [8]

$\displaystyle{\cal{L}}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{24}H_{MNP}^{2}+\frac{1}{6}Q^{MNP}H_{MNP}$ (A.1)

where $Q^{MNP}=\frac{1}{6}{\cal{\varepsilon}}^{MNPRST}Q_{RST}$ is a selfdual
Lagrange multiplier field that is imposing selfduality on $H_{MNP}$ as an
equation of motion obtained by varying $Q^{MNP}$. The kinetic term for
$H_{MNP}$ in this Lagrangian has the wrong sign. Nevertheless, we will take
$H_{MNP}$ as the field strength of a two-form,

$\displaystyle H_{MNP}$ $\displaystyle=$ $\displaystyle 3\nabla_{[M}B_{NP]}$
(A.2)

If we vary $B_{MN}$, then we get the equation of motion

$\displaystyle\nabla_{M}\left(h^{MNP}+2Q^{MNP}\right)$ $\displaystyle=$
$\displaystyle 0$ (A.3)

To understand the meaning of this equation of motion, we decompose
$H_{MNP}=H_{MNP}^{+}+H_{MNP}^{-}$ into selfdual and antiselfdual parts, and
then we dualize

$\displaystyle{\cal{\varepsilon}}^{MNPRST}\nabla_{M}\left(H_{RST}^{+}-H_{RST}^{-}+2Q_{MNP}\right)$
$\displaystyle=$ $\displaystyle 0$ (A.4)

Now we can compare this with the Bianchi identity

$\displaystyle{\cal{\varepsilon}}^{MNPRST}\nabla_{M}\left(H_{RST}^{+}+H_{RST}^{-}\right)$
$\displaystyle=$ $\displaystyle 0$ (A.5)

By adding these two equations, we get

$\displaystyle\nabla_{[M}\left(Q_{RST]}+H_{RST]}^{+}\right)$ $\displaystyle=$
$\displaystyle 0$ (A.6)

that we solve by taking

$\displaystyle Q_{RST}$ $\displaystyle=$ $\displaystyle-
H_{RST}^{+}+C_{RST}^{+}$ (A.7)

for some closed selfdual three-form $C_{MNP}^{+}$ satisfying

$\displaystyle\nabla_{[M}C_{RST]}^{+}$ $\displaystyle=$ $\displaystyle 0$
(A.8)

Inserting this solution back into the Lagrangian, it becomes

$\displaystyle{\cal{L}}_{B}$ $\displaystyle=$
$\displaystyle-\frac{1}{24}H_{MNP}^{2}-\frac{1}{12}H^{MNP}C_{MNP}^{+}$ (A.9)

and now we get the right sign for the kinetic term.

Let us now study supersymmetry. We make the following ansatz for the
supersymmetry variations,

$\displaystyle\delta B_{MN}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma_{MN}\psi$ (A.10) $\displaystyle\delta Q_{MNP}$
$\displaystyle=$
$\displaystyle-\frac{i}{2}\nabla_{Q}\left(\bar{\cal{\varepsilon}}\Gamma^{Q}\Gamma_{MNP}\psi\right)$
(A.11)

where $\delta Q_{MNP}$ is selfdual as is necessary if we assume that $Q_{MNP}$
is selfdual offshell. The minus sign in the supersymmetry variation of
$Q_{MNP}$ is chosen so that the equation (A.6) is invariant. The supersymmetry
variation of the Lagrangian becomes

$\displaystyle\delta{\cal{L}}_{B}$ $\displaystyle=$
$\displaystyle-\frac{i}{2}\bar{\cal{\varepsilon}}\Gamma_{NP}\psi\nabla_{M}Q^{MNP}$
(A.12)

by using the Bianchi identity for $H_{MNP}$. For the matter part, we have the
Lagrangian

$\displaystyle{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}(\nabla_{M}\phi^{A})-\frac{R}{10}(\phi^{A})^{2}+\frac{i}{2}\bar{\psi}\Gamma^{M}\nabla_{M}\psi$
(A.13)

Its supersymmetry variation becomes

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-\frac{i}{12}\bar{\psi}\Gamma^{Q}\Gamma^{MNP}{\cal{\varepsilon}}\nabla_{Q}Q_{MNP}$
(A.14)

by assuming the supersymmetry variation of the fermion is

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\Gamma^{MNP}{\cal{\varepsilon}}Q_{MNP}+{\mbox{matter}}$
(A.15)

where we suppress the contributions from matter fields. Their contribution to
$\delta{\cal{L}}_{m}$ cancel out among themselves in a standard manner. By
using selfduality of $Q_{MNP}$ the variation can also be written as

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\cal{\varepsilon}}\Gamma_{NP}\psi\nabla_{M}Q^{MNP}$
(A.16)

and now we see that the full Lagrangian is supersymmetric,
$\delta{\cal{L}}_{H}+\delta{\cal{L}}_{m}=0$.

We may also work with the Lagrangian (A.9). We shall then take $w^{+}_{MNP}$
to be a supersymmetry singlet. The supersymmetry variation then becomes

$\displaystyle\delta{\cal{L}}_{B}$ $\displaystyle=$
$\displaystyle\frac{i}{4}\bar{\cal{\varepsilon}}\Gamma_{NP}\psi\nabla_{M}\left(H^{MNP}+w^{+MNP}\right)$
(A.17)

For this to cancel against $\delta{\cal{L}}_{m}$, we shall change the
supersymmetry variation to be

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\frac{1}{12}\Gamma^{MNP}{\cal{\varepsilon}}\left(H_{MNP}+w_{MNP}^{+}\right)+{\mbox{matter}}$
(A.18)

We would now like to point out that there are minus signs here that are
associated to $Q_{MNP}$, in equations (A.11) and (A.15) that are analogues to
the minus signs that we found for $G_{ij}$ in this paper at those
corresponding places.

##  Appendix B The supersymmetry variation of the Lagrangian

In this appendix we present the detailed computation for the supersymmetry
variation of the Lagrangian in the covariant formulation. For the matter part,

$\displaystyle{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(D_{M}\phi^{A}\right)^{2}-\frac{R}{10}(\phi^{A})^{2}$
(B.2)
$\displaystyle+\frac{i}{2}\bar{\psi}\Gamma^{M}D_{M}\psi+\frac{e}{2}\bar{\psi}\Gamma_{M}\Gamma^{A}[\psi,\phi^{A}]v^{M}$

by omiting contributions from $G_{MN}$ and $F_{MN}$ in the variation of the
fermion, we get

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma^{A}\psi D_{M}^{2}\phi^{A}$ (B.16)
$\displaystyle-\frac{iR}{5}\bar{\cal{\varepsilon}}\Gamma^{A}\psi\phi^{A}$
$\displaystyle+i\bar{\psi}\Gamma^{M}\Gamma^{N}\Gamma^{A}\nabla_{M}{\cal{\varepsilon}}D_{N}\phi^{A}$
$\displaystyle+i\bar{\psi}\Gamma^{M}\Gamma^{N}\Gamma^{A}{\cal{\varepsilon}}D_{M}D_{N}\phi^{B}$
$\displaystyle-4i\bar{\psi}\Gamma^{M}\Gamma^{A}\nabla_{M}\eta\phi^{A}$
$\displaystyle-4i\bar{\psi}\Gamma^{M}\Gamma^{A}\eta D_{M}\phi^{A}$
$\displaystyle+\frac{e}{2}\bar{\psi}\Gamma^{M}\Gamma_{N}\Gamma^{AB}\nabla_{M}{\cal{\varepsilon}}[\phi^{A},\phi^{B}]v^{N}$
$\displaystyle+e\bar{\psi}\Gamma^{M}\Gamma_{N}\Gamma^{AB}{\cal{\varepsilon}}[D_{M}\phi^{A},\phi^{B}]v^{N}$
$\displaystyle+\frac{e}{2}\bar{\psi}\Gamma^{MN}\Gamma^{AB}[\phi^{A},\phi^{B}]\nabla_{M}v_{N}$
$\displaystyle-e\bar{\psi}\Gamma_{M}\Gamma^{N}\Gamma^{A}\Gamma^{B}{\cal{\varepsilon}}[D_{N}\phi^{B},\phi^{A}]v^{M}$
$\displaystyle-4e\bar{\psi}\Gamma_{M}\Gamma^{AB}\eta[\phi^{B},\phi^{A}]v^{M}$
$\displaystyle+\frac{ie^{2}}{2}\bar{\psi}\Gamma_{M}\Gamma_{N}\Gamma^{A}\Gamma^{BC}{\cal{\varepsilon}}[[\phi^{B},\phi^{C}],\phi^{A}]v^{P}v^{M}$
$\displaystyle-
eD^{M}\phi^{A}[\bar{\cal{\varepsilon}}\Gamma_{MN}\psi,\phi^{A}]v^{N}$
$\displaystyle+\frac{ie}{2}\left(\bar{\psi}\Gamma_{N}\Gamma^{A}[\psi,\bar{\cal{\varepsilon}}\Gamma^{A}\psi]-\bar{\psi}\Gamma^{M}[\psi,\bar{\cal{\varepsilon}}\Gamma_{MN}\psi]\right)v^{N}$

We assume that ${\cal{\varepsilon}}$ is anticommuting. We expand (B.16),

$\displaystyle-i\bar{\cal{\varepsilon}}\Gamma^{A}\psi\phi^{A}+\frac{e}{2}\bar{\psi}\Gamma^{MN}\Gamma^{A}{\cal{\varepsilon}}[F_{MN},\phi^{A}]$
(B.17)

and then the first term cancels (B.16). We expand (B.16),

$\displaystyle-4i\bar{\psi}\Gamma^{A}\Gamma^{N}\eta D_{N}\phi^{A}$ (B.18)

which cancels against (B.16). On (B.16) we apply
$\Gamma^{M}\nabla_{M}\eta=-\frac{R}{20}{\cal{\varepsilon}}$, and get

$\displaystyle-\frac{iR}{5}\bar{\psi}\Gamma^{A}\eta\phi^{A}$ (B.19)

which cancels against (B.16). We notice that (B.16) and (B.16) combine into

$\displaystyle
2e\bar{\psi}\Gamma^{AB}(v^{M}\nabla_{M}{\cal{\varepsilon}})[\phi^{A},\phi^{B}]$
(B.20)

and this combines with (B.16) into

$\displaystyle
2e\bar{\psi}\Gamma^{AB}({\cal{L}}_{v}{\cal{\varepsilon}})[\phi^{A},\phi^{B}]$
(B.21)

where the Lie derivative is given by

$\displaystyle{\cal{L}}_{v}{\cal{\varepsilon}}$ $\displaystyle=$
$\displaystyle
v^{M}\nabla_{M}{\cal{\varepsilon}}+\frac{1}{4}\nabla_{M}v_{N}\Gamma^{MN}{\cal{\varepsilon}}$
(B.22)

We rewrite (B.16) as

$\displaystyle
e\bar{\psi}\Gamma^{AB}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A}-ie[v^{M}A_{M},\phi^{A}],\phi^{B}]+e\bar{\psi}\Gamma^{MN}\Gamma^{AB}{\cal{\varepsilon}}[D_{M}\phi^{A},\phi^{B}]v_{N}$
(B.23)

where the Lie derivative is

$\displaystyle{\cal{L}}_{v}\phi^{A}$ $\displaystyle=$ $\displaystyle
v^{M}\nabla_{M}\phi^{A}$ (B.24)

We rewrite (B.16) as

$\displaystyle-e\bar{\psi}\Gamma^{MN}\Gamma^{AB}{\cal{\varepsilon}}[D_{M}\phi^{A},\phi^{B}]v_{N}-e\bar{\psi}\Gamma^{AB}{\cal{\varepsilon}}[v^{M}D_{M}\phi^{A},\phi^{B}]$
(B.25)
$\displaystyle-e\bar{\psi}\Gamma^{MN}{\cal{\varepsilon}}[D_{M}\phi^{A},\phi^{A}]v_{N}-e\bar{\psi}{\cal{\varepsilon}}[v^{M}D_{M}\phi^{A},\phi^{A}]$
(B.26)

Then if we add (B.16) and (B.16) together, we get

$\displaystyle
2e\bar{\psi}\Gamma^{AB}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A}-ie[v^{M}A_{M},\phi^{A}],\phi^{B}]-e\bar{\psi}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A}-i[v^{M}A_{M},\phi^{A}],\phi^{A}]$
(B.27)
$\displaystyle+e\bar{\psi}\Gamma^{MN}{\cal{\varepsilon}}[D_{M}\phi^{A},\phi^{A}]v_{N}$
(B.28)

We rewrite the last term as

$\displaystyle
e\bar{\cal{\varepsilon}}\Gamma^{MN}\psi[D_{M}\phi^{A},\phi^{A}]v_{N}$ (B.29)

and then we see that it cancels the last term in (B.28). Line (B.16) is
identically zero by a Fierz identity [8]. Summing up what we have got,

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$ $\displaystyle
2e\bar{\psi}\Gamma^{AB}({\cal{L}}_{v}{\cal{\varepsilon}})[\phi^{A},\phi^{B}]$
(B.33)
$\displaystyle+2e\bar{\psi}\Gamma^{AB}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A}-ie[v^{M}A_{M},\phi^{A}],\phi^{B}]$
$\displaystyle-e\bar{\psi}{\cal{\varepsilon}}[{\cal{L}}_{v}\phi^{A}-i[v^{M}A_{M},\phi^{A}],\phi^{A}]$
$\displaystyle+\frac{e}{2}\bar{\psi}\Gamma^{MN}\Gamma^{A}{\cal{\varepsilon}}[F_{MN},\phi^{A}]$

To this we shall add the contributions from $F_{MN}$ and $G_{MN}$ that we
omitted from $\delta\psi$, and then we shall add the supersymmetry variation
of ${\cal{L}}_{A}$.

We start with obtaining the supersymmetry variation of

$\displaystyle{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2\lambda}\widetilde{G}^{MN}F_{MN}+\frac{1}{2\lambda}K^{M}K_{M}$
(B.34)

The variation can be expressed as

$\displaystyle\delta{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\widetilde{G}^{MN}D_{M}\delta
A_{N}+\frac{1}{2\lambda}\delta
G^{MN}\widetilde{F}_{MN}+\frac{2}{\lambda}K^{M}u^{N}D_{[M}\delta A_{N]}$
(B.35)

where in the second term, we have moved the tilde from $\widetilde{G}^{MN}$ to
$\widetilde{F}_{MN}$, since it does not matter on which factor we remove the
trace part. But at the level of computations, the difference
$(\delta\widetilde{G}^{MN})F_{MN}-\delta G^{MN}\widetilde{F}_{MN}$ turns out
to be proportional to the Bianchi identity (3.28). We get

$\displaystyle\delta{\cal{L}}_{A}$ $\displaystyle=$ $\displaystyle-\delta
A_{N}D_{M}\left(\frac{1}{\lambda}\widetilde{G}^{MN}\right)$ (B.39)
$\displaystyle+\frac{1}{2\lambda}\delta G^{MN}F_{MN}$
$\displaystyle-\frac{1}{\lambda}\delta G^{MN}K_{M}v_{N}$
$\displaystyle+\frac{1}{\lambda}K^{M}u^{N}D_{M}\delta A_{N}$

where we can now apply the supersymmetry variation of $G^{MN}$. The explicit
commutator term that is being produced by $K_{M}$ is now given by

$\displaystyle\delta{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle-\frac{e}{2\lambda}\bar{\cal{\varepsilon}}\Gamma_{Q}\Gamma_{MNP}\Gamma^{A}\psi[K^{M},\phi^{A}]u^{N}v^{P}v^{Q}$
(B.40)

From $\delta{\cal{L}}_{m}$ we get a similar looking contribution by making the
ansatz

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\Gamma^{MNP}{\cal{\varepsilon}}\alpha K_{M}u_{N}v_{P}$ (B.41)

for some parameter $\alpha$. Then that term gives the commutator contribution

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-e\alpha\bar{\cal{\varepsilon}}\Gamma_{MNP}\Gamma_{Q}\Gamma^{A}\psi[K^{M},\phi^{A}]u^{N}v^{P}v^{Q}$
(B.42)

The sum of these two contributions can be zero by choosing
$\alpha=-\frac{1}{2\lambda}$ since then one finds the combination
$\Gamma_{QMNP}v^{P}v^{Q}=0$. We conclude that we should have

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\Gamma^{MNP}\left(\frac{1}{4\lambda}F_{MN}u_{P}-\frac{1}{4\lambda}\widetilde{G}_{MN}v_{P}-\frac{1}{2\lambda}K_{M}u_{N}v_{P}\right)$
(B.43)

If we express this in terms of $\widetilde{F}_{MN}$ it becomes

$\displaystyle\delta\psi$ $\displaystyle=$
$\displaystyle\Gamma^{MNP}\left(\frac{1}{4\lambda}\widetilde{F}_{MN}u_{P}-\frac{1}{4\lambda}\widetilde{G}_{MN}v_{P}-\frac{1}{\lambda}K_{M}u_{N}v_{P}\right)$
(B.44)

Let us next isolate all contributions involving $\widetilde{G}^{MN}$. We have

$\displaystyle\delta{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle-i\bar{\cal{\varepsilon}}\Gamma_{NP}\psi
v^{P}D_{M}\left(\frac{1}{\lambda}\widetilde{G}^{MN}\right)$ (B.45)

and

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$ $\displaystyle
i\bar{\cal{\varepsilon}}\Gamma_{NP}\psi
v^{P}D_{M}\left(\frac{1}{\lambda}\widetilde{G}^{MN}\right)+\frac{i}{2\lambda}\bar{\cal{\varepsilon}}\Gamma_{MN}\psi{\cal{L}}_{v}\widetilde{G}_{MN}$
(B.46)

so they cancel up to a Lie derivative.

Let us now list the remaining terms in $\delta{\cal{L}}_{A}$,

$\displaystyle\delta{\cal{L}}_{A}$ $\displaystyle=$
$\displaystyle\frac{i}{4}\bar{\cal{\varepsilon}}\Gamma^{Q}\Gamma^{MNP}\psi
D_{Q}\left(\frac{1}{\lambda}F_{MN}u_{P}+\frac{2}{\lambda}K_{M}u_{N}v_{P}\right)$
(B.48) $\displaystyle-i\bar{\cal{\varepsilon}}\Gamma^{NP}\psi
v_{P}D^{M}\left(\frac{2}{\lambda}K_{[M}u_{N]}\right)$

From $\delta{\cal{L}}_{m}$ with the above $\delta\psi$, we get

$\displaystyle\delta{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle-\frac{i}{4}\bar{\cal{\varepsilon}}\Gamma^{MNP}\Gamma^{Q}\psi
D_{Q}\left(\frac{1}{\lambda}F_{MN}u_{P}\right)+\frac{i}{2}\bar{\cal{\varepsilon}}\Gamma^{MNP}\Gamma^{Q}D_{Q}\left(\frac{1}{\lambda}K_{M}u_{N}v_{P}\right)$
(B.49)

Adding them, we get

$\displaystyle\delta{\cal{L}}_{A}+\delta{\cal{L}}_{m}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\bar{\cal{\varepsilon}}\Gamma^{QMNP}\psi
F_{MN}D_{Q}\left(\frac{1}{\lambda}u_{P}\right)$ (B.51)
$\displaystyle+\frac{i}{\lambda}\bar{\cal{\varepsilon}}\Gamma^{MN}\psi{\cal{L}}_{v}\left(K_{M}u_{N}\right)$

We define

$\displaystyle\Omega_{MN}$ $\displaystyle=$
$\displaystyle\nabla_{M}\left(\frac{1}{\lambda}u_{N}\right)-\nabla_{N}\left(\frac{1}{\lambda}u_{M}\right)$
(B.52)

We can avoid the restriction to manifolds with $\Omega_{MN}=0$ if we add a
Chern-Simons term

$\displaystyle{\cal{L}}_{CS}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\lambda}{\cal{\varepsilon}}^{MNPQRS}\omega(A)_{MNP}\Omega_{QR}u_{S}$
(B.53)

To see that, we notice the identity

$\displaystyle\Omega_{MN}v^{N}$ $\displaystyle=$
$\displaystyle-\frac{1}{\lambda}{\cal{L}}_{v}u_{M}$ (B.54)

so then we just need to assume that ${\cal{L}}_{v}u^{M}=0$, which is a natural
condition to impose on the manifold for the purpose of dimensional reduction,
in order to have $\Omega_{MN}v^{N}=0$. We choose the normalization such that
$\delta\omega(A)_{MNP}=\delta A_{M}F_{NP}$. Then

$\displaystyle\delta{\cal{L}}_{CS}$ $\displaystyle=$
$\displaystyle-\frac{1}{4\lambda}{\cal{\varepsilon}}^{MNPQRS}i\bar{\cal{\varepsilon}}\Gamma_{MT}\psi
F_{NP}\Omega_{QR}u_{S}v^{T}$ (B.55) $\displaystyle=$
$\displaystyle-\frac{i}{4\lambda}\bar{\cal{\varepsilon}}\Gamma_{MT}\Gamma^{MNPQRS}\psi
F_{NP}\Omega_{QR}u_{S}v^{T}$ (B.56) $\displaystyle=$
$\displaystyle-\frac{i}{4\lambda}\bar{\cal{\varepsilon}}\Gamma^{NPQR}\psi
F_{NP}\Omega_{QR}$ (B.57)

where we have used the identity

$\displaystyle\Gamma_{MT}\Gamma^{MNPQRS}$ $\displaystyle=$
$\displaystyle-5\delta_{T}^{[N}\Gamma^{PQRS]}$ (B.58)

Adding up all the contributions, we get (3.40).

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