# ID-Unet: Iterative Soft and Hard Deformation for View Synthesis

Mingyu Yin1 Li Sun1,2 Qingli Li1
1 Shanghai Key Laboratory of Multidimensional Information Processing,
2Key Laboratory of Advanced Theory and Application in Statistics & Data
Science,
East China Normal University, 200241 Shanghai, China
Corresponding author, email<EMAIL_ADDRESS>Supported by the the
Science and Technology Commission of Shanghai Municipality (No.19511120800).

###### Abstract

View synthesis is usually done by an autoencoder, in which the encoder maps a
source view image into a latent content code, and the decoder transforms it
into a target view image according to the condition. However, the source
contents are often not well kept in this setting, which leads to unnecessary
changes during the view translation. Although adding skipped connections, like
Unet, alleviates the problem, but it often causes the failure on the view
conformity. This paper proposes a new architecture by performing the source-
to-target deformation in an iterative way. Instead of simply incorporating the
features from multiple layers of the encoder, we design soft and hard
deformation modules, which warp the encoder features to the target view at
different resolutions, and give results to the decoder to complement the
details. Particularly, the current warping flow is not only used to align the
feature of the same resolution, but also as an approximation to coarsely
deform the high resolution feature. Then the residual flow is estimated and
applied in the high resolution, so that the deformation is built up in the
coarse-to-fine fashion. To better constrain the model, we synthesize a rough
target view image based on the intermediate flows and their warped features.
The extensive ablation studies and the final results on two different data
sets show the effectiveness of the proposed model.
https://github.com/MingyuY/Iterative-view-synthesis

##  1 Introduction

Figure 1: (a) The ID-Unet realizes the translation from the source view to the
target, either existing in the MultiPIE dataset
($-30^{\circ}$,$-15^{\circ}$,$0^{\circ}$), or under a new view (inside the
yellow box) by the linear interpolation between two adjacent view conditions.
(b) Extra results on CelebA from the existing model training on MultiPIE.

Novel view synthesis, also known as view translation, facilitates the computer
to render the same object under arbitrary poses, given an input object image
in a source pose. This is a challenging task, since it requires the model to
understand not only the image content, but also the relation between the
object poses and its appearances showing in the image. The model needs to
figure out the intrinsic shape of the object and keep it stable during the
translation. Meanwhile, it should be able to synthesize the appearance of the
object, conforming to the target view condition.

Recently, learning-based method has been employed broadly for this task.
Particularly, view synthesis is commonly regarded as a multi-domain image-to-
image translation task, which is often modeled by the autoencoder (AE) [6, 45]
or variational autoencoder (VAE) [4, 46]. Both consist of a pair of encoder
and decoder, in which only the last layer of the encoder connects to the
decoder, as shown in Figure 2 (a). However, their limitation has already been
realized [44, 21]. Basically, using the latent code from the last layer is not
enough to represent the content. Since the decoder can only get one latent
code, the source content cannot be kept well in the translated image. A simple
but effective solution is the Unet [32] structure. It utilizes several skipped
connections by making the shortcuts from the encoder to the decoder, therefore
the output can take more features from the source, as shown in Figure 2 (b).
Such as V-Unet[9] is a VAE model with skipped connections and used for person
synthesis. Unet indeed improves the image quality. But directly using the low-
level encoder features makes it difficult to satisfy the domain requirement,
hence the image sometimes fails to be translated into the target domain.

Intuitively, in view translation, the encoder feature needs to be deformed
before giving it to the decoder. A straightforward way is to apply the the
same optical flow on the different resolutions of the feature map. The flow
can be either determined by the priory knowledge [33] or learned by the model
[46], and the structure is shown Figure 2 (c). However, we find that using the
same flow on different resolutions limits the model’s ability for synthesis.
On one hand, the flow is often not accurate enough. It is estimated based on
the feature of a certain resolution, therefore may be inappropriate for other
sizes. On the other hand, the model can already change the view even without
any intentional deformations, which implies that we should give it the
flexibility to determine the deformation on different resolutions.

To properly exploit the encoder features in the view synthesis, this paper
proposes an iterative way to deform them in the coarse-to-fine fashion, so
that they can be aligned with the corresponding part in the decoder. The
deformed features skip several intermediate layers, and are directly given to
the layers in the decoder to complement the content details. Inspired by the
idea of progressively estimating the optical flow for the raw pixels [23, 3],
our model specifies the offset vectors for the encoder features from the low
to the high resolution, and these displacements are accumulated across the
multiple resolutions. Specifically, we first use offsets from the low
resolution as an approximation to coarsely deform the feature, then the
residual offsets are estimated by comparing the roughly deformed result to the
decoder feature of the same size. The residuals refine the coarse flow and
they are applied to give the additional deformation. The refined flow is
further employed by the next block in a larger size. In brief, the encoder
feature is first warped according to the coarse flow, and then the remaining
offsets is estimated and applied, so that the result is better consistent with
the target view.

To compute the initial flow and its following-up residuals, we design the Soft
and Hard Conditional Deformation Modules (SCDM and HCDM) based on the features
from the encoder and decoder. The view label is the extra conditional input to
control the amount of displacement. The idea of the soft flow is to compute
the similarity scores (also known as the attention matrix) between the encoder
and decoder features like [40, 42]. Given the two of them, the spatial and
channel similarities are measured, and then applied onto the encoder features
to align them into the target view. However, the soft flow is not efficient
enough to compute on multiple resolutions. Furthermore, if the target view is
far from the source, the similarity may no longer reflect the spatial
deformation. Our solution is to estimate the optical flow to ”hard” warp the
feature before the spatial and channel attention in SCDM. Moreover, we also
design the HCDM which gives the high resolution residuals onto the previous
small optical flow, and it ”hard” warps the current feature and further aligns
it to the target view.

The contributions lie in following aspects: (1) We propose an iterative view
translation framework which deforms the encoder feature from different layers
and gives them to the decoder to improve the synthesis quality. (2) We design
the SCDM and HCDM and use them to align the encoder feature into the target
view. (3) Extensive experiments on two different datasets show the
effectiveness of the proposed framework and our designed modules.

Figure 2: An illustration of several comparing frameworks. (a) and (b) are
cVAE and Unet, respectively. (c) is the combination of them, and $\mathrm{T}$
realizes the translation from source view $a$ to target view $b$ based on
optical flow. (d) improved from cVAE+Unet, the optical flow is estimated
iteratively. The initial flow $\mathrm{T}_{1}$ is calculated according to the
low-resolution features. As the resolution increases layer by layer, the
residual $\Delta\mathrm{T}_{n}$ is calculated to progressively refine the
previous result.

##  2 Related Works

GAN and its structure design. GAN [10, 27, 5, 17, 26] has shown its ability in
synthesizing high dimensional structured data. The rationale behind GANs is to
learn the mapping from a latent distribution $z\sim N(0,I)$ to mimic the real
data through adversarial training. Because of the instability of the
adversarial training, it often needs to give extra constraints on
discriminator $\mathrm{D}$ [12, 13]. Moreover, by incorporating an encoder
$\mathrm{E}$, GAN can be applied in a variety of I2I translation, either
supervised by the groundtruth [16, 41] or not [50, 6]. In AE, the source image
is first converted into a latent code by $\mathrm{E}$, and then $\mathrm{G}$
takes the code and transforms it back into the image. Since there are multiple
visual domains, the source and target domain labels are given to the AE as the
guide. Variational autoencoder (VAE) [20] has the similar structure with AE,
in which the latent code is assumed to follow the posterior distribution, and
the posterior is to be close to a prior during training. Hence, VAE is not a
deterministic model like AE. It can support sampling from the posterior or
prior, with their corresponding synthesis looking like real images. VAE is
extended to its conditional version cVAE [35, 4] as shown in Figure 2 (a), and
cVAE is suitable for either synthesizing the diverse styles of images [51], or
disentangling the latent code [14, 48].

In AE or VAE, $\mathrm{E}$ and $\mathrm{G}$ are only connected through the
last latent code, which is not enough to guarantee the synthesis quality.
AdaIN [15], SPADE [29], CIN[8] and CBIN[25] are other ways to inject the
feature into the multiple decoder layers through a side branch, which adjusts
the statistics of features in the main branch. The Unet [32] and its variants
link $\mathrm{E}$ and $\mathrm{G}$ by setting up shortcuts between them. But
it often leads to failures in I2I translation. Xiao _et al._ [44] use
$\mathrm{G}$’s output as the residual added onto the source image to improve
the quality. Li _et al._ [21] designs PONO layer in Unet, normalizing and
adapting source domain features from $\mathrm{E}$ to $\mathrm{G}$. However,
these structures are not designed for view synthesis.

Figure 3: (a) The detailed architecture of the proposed ID-Unet. $X_{a}$ is
mapped to the variable $Z$ through encoder, and it is supplied to the encoder
together with the target view label $C_{b}$. SCDM and HCDM warp the encoder
features to the target view, and give their output to the decoder in a way of
de-normalization (DFNM) [46] to complement the details. And the low resolution
optical flow is as an approximation to change the high resolution feature by
$\Phi_{s}$ and $\Phi$, so that the flow is formed in the coarse-to-fine
fashion. (b) Iteratively update of the view conditional $C_{diff}$ to
$W_{diff3}$, $W_{diff2}$ and $W_{diff1}$.

View synthesis. Traditional approaches [2, 18, 31] for this task are mainly
based on projection geometry, which tries to recover the 3D relation between
the object and its projected image plane. They directly estimate either the
depth and camera pose [2], or 3D model parameters [18, 31], so that the object
can be projected into the target view. Learning-based methods [7, 49] become
increasingly popular nowadays. In [7], a CNN model learns to process the
latent code for object shape and camera pose, and map it into an image. In
[49], the CNN predicts the optical flow to warp the source view into the
target. Recently, due to the great success of GAN [28, 38, 36, 33, 45], the AE
structure plus the adversarial training begins to play the key role in view
synthesis. Meanwhile, VAE and its probabilistic latent vector [37, 46] can be
applied in this task as well, which even better keeps the contents from the
source. However, none of these works consider the coarse-to-fine iterative
deformation on features to perform view synthesis.

##  3 Method

We intend to synthesize object in arbitrary views. Given an image $X_{a}$
containing an object in the source view $C_{a}$, and an expected target view
$C_{b}$ as the inputs, the model outputs $\hat{X}_{b}$, a synthesis of the
same object in the target view. The difficulty of this task lies in accurately
changing the object from the original to the target view, while keeping other
attributes (_e.g._ identity) unchanged during the translation.

Figure 4: Illustration of Soft and Hard Conditional Deformation Module. We
show the SCDM and HCDM on the left and right respectively. Both have 3 inputs,
$F_{gi}$, $F_{ei}$ and $W_{diff}$ from $\mathrm{G}$, $\mathrm{E}$ and view
condition, and 1 output $F_{out}$ given to main branch of $\mathrm{G}$. SCDM
consists of 3 stages, which are KG warp, spatial soft warp and channel soft
warp. HCDM directly estimates the optical flow and warps the encoder feature
$F_{ei}$.

###  3.1 The Framework of Iterative View Translation

A brief framework is given in Figure 2 (d). The idea is to apply multiple
deformations on the shallow layer features in the encoder and give them to the
decoder, which is conducive to maintain the source content irrelevant to the
view. Note that in Figure 2 (c), module T also estimates the optical flow and
is applied on different resolutions, but it is in the independent way. Here
the key improvement is the coarse-to-fine manner to estimate the initial
deformation $\mathrm{T}_{1}$ and refine it through $\Delta\mathrm{T}_{i}$
iteratively, where $i=2,3$ in our setting. Moreover, we find that using the
deformed low-level features in the decoder causes the missing of content
details in the translated image. While cVAE has a better ability to keep
complete objects by introducing the prior distribution as a regularization.
The proposed Figure 2 (d) inherits the cVAE+Unet structure in Figure 2 (c). In
particular, the source view $X_{a}$ is input to the encoder to provide the
content, and is mapped to a posterior distribution, from which the latent $Z$
can be sampled. The decoder takes $Z$ and the target view condition $C_{b}$ to
synthesize the translation.

The proposed ID-Unet, with its specific architecture shown in Figure 3,
accomplishes the iterative view translation on the features $F_{e}$ in
different layers of the encoder, so that they are aligned with their
corresponding part $F_{g}$ in the decoder. Notice that $F_{e}$ have spatial
dimensions and are under the original view $C_{a}$. If the same features under
the target view $C_{b}$ could also be obtained, it would be easy to estimate
the optical flow to deform $F_{e}$. Intuitively, the decoder feature $F_{g}$
can be roughly assumed in the target view $C_{b}$, since the first decoder
feature $F_{g3}$ is computed according to the latent $Z$ and condition
$C_{b}$, which has already been aligned into $C_{b}$. This can be extended to
other decoder features such as $F_{g2}$ and $F_{g1}$. They are closer to
$C_{b}$ than their counterparts $F_{e2}$ and $F_{e1}$, so we employ the pair
$F_{e3}$ and $F_{g3}$ to estimate the initial $\mathrm{T}_{1}$, and the
following pairs to predict $\Delta\mathrm{T}_{i}$.

###  3.2 Soft and Hard Deformation

We design two types of modules, applying the soft and hard deformations on low
and high resolution feature, respectively. Both of them depend on $W_{diff}$,
a $1\times 1$ vector given by MLP, which reflects the view difference. We will
elaborate it in the next section.

Soft Conditional Deformation Module (SCDM)

SCDM estimates the initial deformation $\mathrm{T}_{1}$ based on a pair of
features $F_{e3}$ and $F_{g3}$ at the lowest resolution, as shown in the left
of Figure 4. Instead of directly comparing $F_{e3}$ and $F_{g3}$, a two-
channel flow is first predicted through kernel given conv (KGconv) and applied
onto $F_{e3}$ by the warping operation $\Phi$. Here, the purpose is to align
$F_{e3}$ in the target view direction to form $\tilde{F}_{e3}$, so that the
soft flow can be calculated from two similar features $\tilde{F}_{e3}$ and
$F_{g3}$, preventing from inappropriate matching two views far from each
other. Note that KGconv uses $\text{W}_{diff}$ as conv kernels to generate $x$
and $y$ offsets in the optical flow to assist view translations [46].

Then, to measure the similarity between source $\tilde{F}_{ei}$ and target
$F_{gi}$, we compute the $\text{Soft flow}\in\mathbb{R}^{HW\times HW}$ by the
inner product between $\hat{e}_{v}$ and $\hat{g}_{u}$: $\text{Soft
flow}(u,v)=\hat{g}_{u}^{T}\hat{e}_{v}$, where $\hat{e}_{v}$ and
$\hat{g}_{u}\in\mathbb{R}^{C}$ represent the channel-wise centralized feature
of $\tilde{F}_{ei}$ and $F_{gi}$ at position $v$ and $u$,
$\hat{e}_{v}=e_{v}-\mu(e_{v})$ and $\hat{g}_{u}=g_{u}-\mu(g_{u})$. $\text{Soft
flow}(u)\in\mathbb{R}^{HW}$ represents the similarity between $F_{g3}$ at
position $u$ and $\tilde{F}_{e3}$ at all position, so the weighted
$\tilde{F}_{ei}$ is the output feature element $F_{sp}(u)$. The weight,
$\text{Soft flow}(u)$, is normalized by the Softmax function and multiplied on
each position of $\tilde{F}_{ei}$.

$F_{sp}(u)=\text{softmax}(\frac{1}{\tau}\cdot\text{Soft
flow}(u))\cdot\tilde{F}_{ei}.$ (1)

Different from the classical flow warp (hard warp), $F_{sp}$ in (1) is the
weighted sum of the feature at multiple positions in $\tilde{F}_{ei}$.
However, smooth weights may change image contents like colors or styles. In
order to maintain them, we balance the soft and hard warp by incorporating a
temperature $\tau<1$ in (1), which increases the impact of the high-weight
position (which is more relevant) on the output.

Finally, based on $F_{sp}$ and $F_{g3}$, we obtain the similarity matrix
$Cov_{c}$ along the channel in the same way of spatial dimension, and ”Soft
warp” is also performed on $F_{sp}$ to maintain more valid information in the
channel dimension.

Hard Conditional Deformation Module (HCDM)

Basically, HCDM utilizes the results of SCDM, and refines the deformation for
larger size $F_{e2}$ and $F_{e3}$. Once the soft flow is obtained, the globe
deformation $\Phi_{s}$ can be approximated. For the high-resolution features,
as shown in Figure 3 (a), $\Phi_{s}$ also takes effect in HCDM. It first makes
the coarse deformation on $F_{e2}$ and $F_{e1}$. Due to the size mismatch
between Soft Flow and feature $F_{e2}$ or $F_{e1}$, one element in Soft Flow
matrix is scaled and applied to the corresponding square area in the feature
of larger size, simplifying as $F_{e2^{\prime}}=\Phi_{s}(F_{e2})$. Then the
residual optical flow at high resolution is further estimated by the deformed
results $F_{e2^{\prime}}$, the target view features $F_{g2}$ and $W_{diff2}$
together. They are concatenated to learn the residual flow. The residual (Res
hard flow) can be superimposed, giving
$F_{e1^{\prime}}=\Phi(\Phi_{s}(F_{e1}))$, in which $\Phi$ denotes the hard
warping operation by the optical flow. Therefore, with the increase on
resolution, the optical flow for translation is gradually refined by HCDM.

###  3.3 Iteratively Update View Difference Condition

With the gradual refinement of optical flow, the features $F_{e2^{\prime}}$
and $F_{e1^{\prime}}$ have been converted to the target view to a certain
extent. Then the actual view of the current features ($F_{e2^{\prime}}$ or
$F_{e1^{\prime}}$) is no longer the same as the source, and the condition
$W_{diff}$ should also be adapted, since it no longer translates from the
source to the target, but from the current view to the target. In our model,
$W_{diff}$ is updated iteratively together with the feature. Specifically, we
use the current flow to measure the amount of the translation, and learn how
to update $W_{diff}$ by the model itself. In Figure 3 (b), the view label
difference $C_{diff}$ is passed through an MLP, to get $W_{diff}$. $W_{diff3}$
used for the first warp is directly obtained from $W_{diff}$ through one fc
layer. During the further operation, the mean of optical flow
$(\mu(\mathrm{d}x),\mu(\mathrm{d}y))$ is concatenated with $W_{diff}$ to
determine the next conditional vector ($W_{diff2}$ or $W_{diff1}$) for the
further deformation.

###  3.4 Training Details and Loss Functions

Adversarial and Reconstruction Loss

We use adversarial loss $L_{\mathrm{E},\mathrm{G}}^{adv}$ and
$L_{\mathrm{D}}^{adv}$ [22] to ensure the translated image approximates the
true distribution like in (2). As shown in Figure 3 (a), the final
$\hat{X}_{b}$ is mixed by two parts. One is the $\hat{X}^{warp}_{b}$, obtained
by the soft and hard deformation on the source $X_{a}$, and the other
$\hat{X}^{g}_{b}$ is the output of the generator. The model learns a single
channel mask to weight and combine the two results. The mask is computed based
on the output and the optical flow in the last HCDM.

$\displaystyle L_{\mathrm{D}}^{adv}=$
$\displaystyle\mathbb{E}_{{X}}[\max(0,1-\mathrm{D}(X,C_{b}))]$ (2)
$\displaystyle+$
$\displaystyle\mathbb{E}_{\hat{X}_{b}}[\max(0,1+\mathrm{D}(\hat{X}_{b},C_{b}))],$
$\displaystyle L_{\mathrm{E},\mathrm{G}}^{adv}=$
$\displaystyle\mathbb{E}_{\hat{X}_{b}}[\max(0,1-\mathrm{D}(\hat{X}_{b},C_{b}))]$

Like ACGAN [27], we use classification losses $L_{\mathrm{C}}^{cls}$ and
$L_{\mathrm{E},\mathrm{G}}^{cls}$ in (3). The classifier $\mathrm{C}$ shares a
part of its weights with discriminator $\mathrm{D}$.

$\displaystyle
L_{\mathrm{C}}^{cls}=-\mathbb{E}_{X_{b}}\sum_{c}\mathbb{I}(c=C_{b})\log\mathrm{C}(c|X_{b}),$
(3) $\displaystyle
L_{\mathrm{E},\mathrm{G}}^{cls}=-\mathbb{E}_{\hat{X}_{b}}\sum_{c}\mathbb{I}(c=C_{b})\log\mathrm{C}(c|\hat{X}_{b})$

In addition, by combining the reconstruction loss in image domain
$L_{\mathrm{E},\mathrm{G}}^{pixel}=||X-\hat{X}_{j}||_{1}$ and feature domain
$L_{\mathrm{E},\mathrm{G}}^{content}=\sum_{i}||\phi^{i}({X})-\phi^{i}({\hat{X}_{j}})||_{1}$,
the image quality is guaranteed more faithfully. Here $\phi$ indicates $i$-th
layer of a pre-trained VGG [34] network, and $j=b,a,aa$. $\hat{X}_{a}$ and
$\hat{X}_{b}$ are the fake images at target view A and B. $\hat{X}_{aa}$ the
cyclic translation result, which is translated back from the synthesised image
in view B.

Figure 5: Besides the normal translated image $\hat{X}_{b}$, image
$\hat{X}_{R}$ is generated from $F_{g3}$, $F_{g2}$ and $F_{g1}$ for the rough
loss.

Disentangling Loss

The source image $X_{a}$ is mapped to a code $Z\sim\mathrm{E}(Z|X_{a})$ where
$\mathrm{E}(Z|X_{a})$ is a posterior depending on the source $X_{a}$. $Z$ is
fed directly into $\mathrm{G}$, so it should keep the content of the object,
and be irrelevant to views [45, 46]. To prevent $Z$ from taking view relevant
factors, we add two auxiliary classifier losses for $\mathrm{E}$. One computes
the classification loss $L_{E}^{clsC}$ which tries to predict
$\hat{C}_{a}=\mathrm{E}(c|X)$ to approximate view label $C_{a}$, as is defined
in the first term in (4). Another adversarial constraint
$L_{\mathrm{E}}^{cls}$ in (4) makes the view classification based on $Z$ by
the hidden layer classifier $\mathrm{DAC}$, which is the last two terms in
(4).

$\displaystyle L_{\mathrm{E}}^{clsC}=$ $\displaystyle-\mathbb{E}_{X\sim
X_{a}}\sum_{c}\mathbb{I}(c=C_{a})\log\mathrm{E}(c|X),$ (4) $\displaystyle
L_{\mathrm{E}}^{clsZ}=$
$\displaystyle-\mathbb{E}_{Z\sim\mathrm{E}(Z|X_{a})}\sum_{c}\frac{1}{C}\log\mathrm{DAC}(c|Z),$
$\displaystyle L_{\mathrm{DAC}}^{clsZ}=$
$\displaystyle-\mathbb{E}_{Z\sim\mathrm{E}(Z|X_{a})}\sum_{c}\mathbb{I}(c=C_{a})\log\mathrm{DAC}(c|Z)$

Here $L_{\mathrm{DAC}}^{clsZ}$ is the penalty to train $\mathrm{DAC}$,
ensuring the accuracy of the view classification. $L_{\mathrm{E}}^{clsZ}$ is
the adversarial loss applied on $\mathrm{E}$ to make $\mathrm{DAC}$ confused
to predict the uniform value on each view. Furthermore, via the constraint of
KL loss $L_{KL}=D_{\text{KL}}[\mathrm{E}(Z|X_{a})||{N}({0},{I})]$, the latent
code $Z$ from the encoder is close to the standard normal distribution and has
no category-related information.

Rough Loss

We design the rough loss on the deformed features in SCDM and HCDM, to make
the features conform to the target view. As is described in section 3.1 and
Figure 3, the decoder features $F_{g3}$, $F_{g2}$ and $F_{g1}$ are assumed
under target view $C_{b}$. To better ensure that they are in target view,
$F_{g3}$, $F_{g2}$ and $F_{g1}$ are combined and fed to a layer $\psi$ to
generate an image $\hat{X}_{R}=\psi(F_{g3},F_{g2},F_{g1})$ as shown in Figure
5. The image $\hat{X}_{R}$ is constrained by pixel-wise L1 loss and
classification loss of the classifier $\mathrm{C}$, like in (5).

$L^{rough}_{\mathrm{E},\mathrm{G}}=||X_{b}-\hat{X}_{R}||_{1}+\sum_{c}\mathbb{I}(c=C_{b})\log\mathrm{C}(\hat{X}_{R})$
(5)

Overall Objective. The total optimization loss is a weighted sum of the above.
Generators $\mathrm{E},\mathrm{G}$, discriminator $\mathrm{D}$, classifier
$\mathrm{C}$, and the latent classifier $\mathrm{DAC}$ are trained by
minimizing (6).

$\displaystyle L_{\mathrm{E},\mathrm{G}}=$ $\displaystyle
L_{\mathrm{E},\mathrm{G}}^{adv}+L_{\mathrm{E},\mathrm{G}}^{cls}+\alpha_{1}L_{\mathrm{E},\mathrm{G}}^{content}+\alpha_{2}L_{\mathrm{E},\mathrm{G}}^{pixel}$
(6) $\displaystyle+$
$\displaystyle\alpha_{3}L_{KL}+L_{\mathrm{E}}^{clsC}+L_{\mathrm{E}}^{clsZ}+\alpha_{4}L^{rough}_{\mathrm{E},\mathrm{G}},$
$\displaystyle L_{\mathrm{D}}=$ $\displaystyle L_{\mathrm{D}}^{adv},\quad
L_{\mathrm{C}}=L_{\mathrm{C}}^{cls},\quad
L_{\mathrm{DAC}}=L_{\mathrm{DAC}}^{clsZ}$

The loss weights $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$, $\alpha_{4}$ = 5,
5, 0.1, 10.

| Method |  | MultiPIE |  | 3D chair
---|---|---|---|---|---
|  |  | $L_{1}\downarrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | FID$\downarrow$ | id-acc$\uparrow$ |  | $L_{1}\downarrow$ | SSIM$\uparrow$ | LPIPS$\downarrow$ | FID$\downarrow$
| MV[36] |  | $15.21$ | $0.489$ | $0.217$ | $29.85$ | 0.742 |  | $13.86$ | $0.779$ | $0.224$ | $104.49$
| Unet[32] |  | $14.03$ | $0.619$ | $0.164$ | $49.86$ | 0.396 |  | $21.75$ | $0.697$ | $0.255$ | $86.74$
| cVAE[4] |  | $12.82$ | $0.635$ | $0.119$ | $28.99$ | 0.651 |  | $8.93$ | $0.828$ | $0.102$ | $27.79$
| CRGAN[37] |  | $14.12$ | $0.627$ | $0.141$ | $26.77$ | $0.868$ |  | $13.33$ | $0.788$ | $0.196$ | $28.23$
| VIGAN[45] |  | $12.96$ | $0.638$ | $0.117$ | $29.05$ | $0.686$ |  | $12.13$ | $0.781$ | $0.133$ | $33.18$
| PONO[21] |  | $13.63$ | $0.621$ | $0.126$ | $23.77$ | $0.862$ |  | $12.74$ | $0.780$ | $0.148$ | $37.85$
| CDVAE[46] |  | $13.49$ | $0.623$ | $0.125$ | $23.95$ | $0.917$ |  | $13.38$ | $0.773$ | $0.148$ | $40.81$
| cVAE+Unet |  | $12.37$ | $0.658$ | $0.113$ | $28.98$ | $0.689$ |  | $11.32$ | $0.790$ | $0.123$ | $32.04$
| A:cVAE+Unet+Iterative |  | $12.14$ | $0.676$ | $0.100$ | $27.50$ | $0.893$ |  | $10.64$ | $0.801$ | $0.120$ | $27.76$
| B:A+ImageMix |  | $12.01$ | $0.679$ | $0.101$ | $26.55$ | $0.928$ |  | $9.30$ | $0.819$ | $0.104$ | $36.18$
| C:B+IterativeC |  | $11.11$ | $0.684$ | $0.095$ | $24.55$ | 0.913 |  | $9.055$ | $0.826$ | $0.102$ | $29.06$
| D:C+rough loss |  | $10.72$ | $0.694$ | $0.093$ | $25.12$ | $0.911$ |  | $7.57$ | $0.847$ | $0.089$ | $28.87$
Table 1: Comparison on the MultiPIE and the 3D chair datasets.

##  4 Experiments

###  4.1 Datasets and Quantitative Metrics.

Datasets. We validate the proposed ID-Unet on face dataset MultiPIE[11] and 3D
chair[1] object dataset. MultiPIE contains about 130,000 images, with 13
viewing angles, spanning $180^{\circ}$. Nine of central viewing angles are
used for training and testing. The 3D chair contains 86,304 images, covering a
total of 62 angles. For all the datasets, 80% are used for training and the
rest 20% for testing.

Quantitative Metrics.  To give the evaluation on different methods, we use
following metrics during the test. We calculate L1 error and LPIPS [47] to
measure the difference at pixel level and feature level between the generated
and ground truth image. SSIM [43] is calculated to compare the similarity of
image structure. FID represents the distance between the generated image
distribution and the real image distribution, so as to measure the
authenticity of the generated image. At the same time, on the MultiPIE dataset
[11], we use the face identity recognition network pretrained on VGGface [30]
dataset to calculate the identity accuracy of generated image. Table 1 lists
all the metrics for the ablation and comparison models. More specific training
details are given in the supplementary materials.

###  4.2 Ablation Study

In this section, we compare the results in several different ablation settings
to verify the effectiveness of every component in the proposed method.

A: cVAE+Unet+Iterative. Setting A is based on the two common models Unet and
cVAE, combining them and then sending the encoder features to the
corresponding decoder layer after iterative view translation. In Figure 6 and
8, the 2nd, 3rd and 4th rows are generated images from Unet, cVAE and model A,
respectively. We observe that the object from Unet appears incomplete
(disappeared chair part or eyes). For cVAE, the face identity and the chair
color have changed to a large extent. While the setting A can ensure the
integrity of the image and the invariance of the information irrelevant to the
view. Meanwhile, as shown in Table 1, compared with Unet and cVAE, all results
under setting A are significantly improved, especially the id-acc increases
from 0.396 (Unet), 0.651 (cVAE) to 0.893.

B: A+ImageMix. Based on A, setting B combines the output of the generator
$\hat{X}^{g}_{b}$ with the deformation of the original image
$\hat{X}^{warp}_{b}$, which is conducive to maintain more valid content of the
original image and generating more realistic images, as shown in the 5th row
in Figure 6, with the id-acc reaching 0.928.

C: B+IterativeC. The experimental setting C further extends on B. In Figure 6
and 8, the view translation is more accurate and better handled in detail.
Because the view difference condition $W_{diffi}$, where $i=1,2,3$, is updated
iteratively according to the degree of deformation of current features, the
view condition is better adjusted and controlled. The result in Table 1 also
verifies the conclusion.

D: C+rough loss. In setting D, the effectiveness of rough loss is validated.
From the last row in Figure 8, it can be seen that the chairs are not only
close to the targets on pixel, but also have stable shape at different views.
It is obvious that this model can better understand the intrinsic shape of the
chairs. This is also supported by Table 1.

Figure 6: Ablation study on MultiPIE dataset. The source and the ground truth
targets are provided in the first row. Please zoom in for details. Figure 7:
Visualization of optical flow on different layers. We list 4 deformation flows
from the bottom to the top. The direction of the flow points from the target
to source. Figure 8: Ablation study on 3D chair dataset. The source and ground
truth targets are given in the 1st and 2nd rows.

Figure 9: Comparison on MultiPIE. For each image, the top row is the ground
truth while the 2nd to 6th rows are generated by MV [36], CRGAN[37],
VIGAN[45], PONO[21] and CDVAE[46] respectively. The last row is generated by
our ID-Unet. Figure 10: Comparison on 3D chair. The 1st and 2nd rows are the
source and target images while the 3rd to 7th rows are generated by MV [36],
CRGAN [37], VIGAN [45], PONO [21] and CDVAE [46] respectively. The last row is
generated by ID-Unet.

###  4.3 Visualizations

Optical flow. In Figure 7, the source image is translated into 3 target views.
The 3rd row is the result from soft flow, which is converted into a 2-channel
hard flow for visualization by taking out the most relevant coordinate. We
find that the absolute value of the soft flow is larger than the rest of the
hard flow in the 1st, 2nd and 4th rows, which can be used to achieve overall
deformation. The 4th row of KG flow in SCDM has the clear direction. The
magnitudes of residual hard flow in the 1st and 2nd row are smaller, showing
that the feature progressively approaches the target view.

Continuous view synthesis by interpolation. To translate an image in an unseen
view, we linearly interpolate the two conditions to get an arbitrary angle
image that does not exist in the dataset. Figure 1 shows that our model is
smooth enough to achieve view morphing.

Visual comparisons with previous works. As shown in Figure 9 and 10, ID-Unet
can accurately achieve the view synthesis while effectively maintain the
source contents, _e.g._ , the face ID and the chair style. The quantitative
results in Table 1 can also confirm the effectiveness. The results from MV
[36] are excessively bright, and it has problems such as ghosting for
difficult samples. VIGAN [45] and CRGAN [37] have good results on simple
samples, but they can not maintain the original structure for complex chairs,
and synthesize the facial details like eyes in the 2nd facial image. PONO [21]
and CDVAE [46] have good ability to keep the source content, but their models
do not understand the structure of complex objects. For example, the 3rd face
in Figure 9 cannot achieve reasonable translation.

##  5 Conclusion

This paper presents the ID-Unet to perform the view synthesis. It iteratively
makes the deformation on the encoder features from different layers, and
connects them into the decoder to complement the content details. To achieve
the view translation, we design the SCDM and HCDM to align the feature from
the source view to the target. Both the modules take the encoder and decoder
features as well as the view condition vector as the inputs, compare the
features to give either the soft or hard flow, and warp the encoder feature
according to it. Since the flows are computed from features of different
sizes, we accumulate them across resolutions and use the current flow to
coarsely align the encoder feature first, and then estimate the residuals flow
to refine it. Experiments show the effectiveness of the proposed model on two
different datasets.

##  Appendix A More Details on Network Architecture

In this section, we give the specific details of network structure. Figure 11,
12, 13 and 14 are the network structures of the encoder $\mathrm{E}$ , the
decoder $\mathrm{G}$, the iterative view difference condition branch and the
discriminator $\mathrm{D}$, respectively. In Conv and Residual block, F, K and
S respectively represent the number of kernels, the size of the convolution
kernel and the stride. We use the ADAM [19] with learning rates 0.0002 and set
$\beta_{1}$=0, $\beta_{2}$=0.9. We will release our code if this paper is
accepted.

Figure 11: The structure of Encoder. In $\mathrm{E}$, except the PN
(positional normalization) [21] used in the shallow layers ($F_{ei},i=1,2,3$),
the rest adopt the IN (instance normalization) [39]. Figure 12: The structure
of Decoder. The final $\hat{X}$ is mixed by two parts. One is the
$\hat{X}^{warp}$, obtained by the soft ($\Phi_{s}$) and hard ($\Phi$)
deformation on the source $X$, and the other $\hat{X}^{g}$ is the output of
the generator. Where the results of SCDM deformation ($F_{3}$) and HCDM
deformation ($F_{2},F_{1}$) affects $F_{gi},i=1,2,3$ in the form of DFNM [46].
Figure 13: The structure of Iterative conditional branch. Here RM is the
reduce mean operation along the two spatial dimensions. Figure 14: The
structure of Discriminator. In $\mathrm{D}$, the SN (spectral normalization)
[24] is applied to all layers.

##  Appendix B More Visualized Results

###  B.1 The results of view translation

Plenty of results of our method on the MultiPIE [11] dataset are shown in
Figures 15 and 16. Extra results on the 3D chair [11] dataset are shown in
Figures 17, 18, 19, 20, 21 and 22. Note that for all Figures, the 1st column
is the source image, and the remaining columns are the generated images under
different target views.

Figure 15: More results on MultiPIE dataset[11]. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 16: More results on MultiPIE dataset[11]. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 17: More results on 3D chair[1] dataset. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 18: More results on 3D chair[1] dataset. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 19: More results on 3D chair[1] dataset. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 20: More results on 3D chair[1] dataset. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 21: More results on 3D chair[1] dataset. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

Figure 22: More results on 3D chair[1] dataset. The 1st column is the source
image, and the remaining columns are the generated images under different
target views.

###  B.2 The visualization for the flow

As shown in Figure LABEL:fig:flow1, LABEL:fig:flow2 and LABEL:fig:flow3, the
source image (1st column) is translated into 9 target views (2nd to 10th
columns). We visualize the optical flows from various target views, namely Res
hard flow, Soft flow and KG flow.

Res hard flow
The 2nd and 6th rows are the results in full resolution. The 3rd and 7th rows
are in the half resolution. Both of them have more details, and the flow
amplitude is small. This indicates that they are used for the refinement and
supplement local details. Through observations, it can be found that the
directions for pixels are not exactly the same. However, most of them are
still consistent with the overall rotation. _E.g._ the face region becomes
light blue when it turns to the left, and light red when turning to the right.

Soft flow
The Soft flow (in the quarter resolution) is shown in the 4th and 8th rows. It
(the 8th row) has a large amplitude, which can better realize the whole rough
view deformation. Due to the lack of image details, the background pixels may
need the large displacement to find their corresponding position. Since the
Soft flow values are normalized for displaying, it makes the color of the face
area lighter (in the 4th row). But in fact, their magnitudes are larger than
other hard flows, which can be seen from the 8th row.

GK flow
The KG flow (also in the quarter resolution) in SCDM are displayed on the 5th
and 9th rows. Their magnitudes are small, but they have more obvious direction
information. _E.g._ , when the face turns left and right, it is shown in blue
and red, respectively. It demonstrates that the view difference information of
$C_{diff}$ has been effectively applied.

###  B.3 Visualization for two components of the final generated images

Here We show the final generated image, and its two components of $X^{g}$ and
$X^{warp}$. They are combined by $1-mask$ and $mask$, respectively. Note that
$X^{warp}$ clearly indicates the effectiveness of the flow, since it directly
deforms the raw pixels. As shown in Figure 26, the 1st, 2nd, 3rd, and 4th rows
are $X$, $X^{g}$, $X^{warp}$ and $mask$, respectively.

The darker the color of the $mask$, the lower the weight of $X^{warp}$ (Gray
means the value is close to 0). It can be found that $X^{g}$ is good enough
compared to $X^{warp}$, and the weight (1-$mask$) on $X^{g}$ is also larger.

It is observed that $X^{warp}$ maintains the brightness, color and identity of
the original image to a large extent. At the same time, for the invisible
areas in the source image, some areas will be missing in the deformed image
$X^{warp}$. Although not in high quality, its view is still correct,
therefore, it can better assist the generation of $X^{g}$ in the way of DFNM.

Figure 26: The 1st,2dn and 3rd rows are the final generated image, and its two
components of $X^{g}$ and $X^{warp}$. The 4th row is the $mask$ for weighting.
The darker the color of the $mask$, the lower the weight of $X^{warp}$.

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# Baryonic states in supersymmetric Yang-Mills theory

Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9,
D-48149 Münster
Department of Physics, Government College University Lahore, Lahore 54000,
Pakistan
E-mail  Georg Bergner, Camilo Lopez
Theoretisch-Physikalisches Institut, Universität Jena, Max-Wien-Platz 1,
D-07743 Jena
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Henning Gerber,
Gernot Münster
Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9,
D-48149 Münster
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Istvan Montvay
Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22603 Hamburg, Germany
E-mail<EMAIL_ADDRESS>Stefano Piemonte
Institute for Theoretical Physics, Universität Regensburg, Universitätsstr.
31, D-93053 Regensburg
E-mail<EMAIL_ADDRESS>Philipp Scior
Fakultät für Physik Universität Bielefeld Universitätsstr. 25, D-33615
Bielefeld
E-mail<EMAIL_ADDRESS>
###### Abstract:

In $\mathcal{N}$=1 supersymmetric Yang-Mills theory the superpartner of the
gluon is the gluino, which is a spin 1/2 Majorana particle in the adjoint
representation of the gauge group. Combining three gluinos, it is possible to
form colour neutral bound states, analogous to baryons in QCD. The correlation
functions of the corresponding baryonic operators contain a contribution
represented by a “sunset diagram”, and in addition, unlike in QCD, another
contribution represented by a “spectacle diagram”. We present first results
from an implementation and calculation of these objects, obtained from
numerical simulations of supersymmetric Yang-Mills theory.

##  1 Introduction

Despite being an extremely successful effective theory, the Standard Model of
particle physics needs some extensions. $\mathcal{N}=1$ supersymmetric Yang-
Mills theory (SYM) is the supersymmetric extension of the gluonic sector of
the Standard Model. The superpartner of the gluon is the gluino. Gauge
invariance together with supersymmetry requires gluinos to transform according
to the adjoint representation of the gauge group $\text{SU}(N)$. SYM is
similar to QCD in some aspects. It is asymptotically free at high energies,
and due to confinement, the low-energy degrees of freedom are expected to be
colour-neutral bound states of gluons and gluinos [1]. Their masses can be
computed non-perturbatively using powerful computing machines.

Supersymmetry (SUSY) is the essential feature distinguishing SYM from QCD. If
SUSY is unbroken in the continuum, as non-perturbative studies of the
supersymmetric Ward identities indicate [2], the bound states are arranged in
supermultiplets, see [3, 4] for numerical investigations for gauge groups
$\text{SU}(2)$ and $\text{SU}(3)$. Effective actions for the low-energy
degrees of freedom in SYM have been constructed in [5] and extended in [6, 7].
They predict supermultiplets consisting of mesons, glueballs, and gluino-glue
bound states. It is, however, also possible to construct colour singlet bound
states of three gluinos, analogous to baryons in QCD. These baryonic states
are not described by the low-energy effective actions [5, 6] and it is unknown
how they fit into the supermultiplets. In this paper we investigate these
interesting objects.

##  2 $\mathcal{N}=1$ supersymmetric Yang-Mills theory

The continuum Lagrangian of SYM in Euclidean space-time is given by

$\mathcal{L}=\frac{1}{4}F^{a}_{\mu\nu}F^{a}_{\mu\nu}+\frac{1}{2}\bar{\lambda}^{a}\gamma_{\mu}(\mathcal{D}_{\mu}\lambda)^{a}+\frac{m_{g}}{2}\bar{\lambda}^{a}\lambda^{a},$
(1)

where $F^{a}_{\mu\nu}$ is the non-Abelian field strength tensor built from the
gauge field $A_{\mu}(x)$, which represents the gluon. The gluino field
$\lambda(x)$, the fermion superpartner of the gluon, is minimally coupled to
the gauge field. The gluino satisfies the Majorana condition

$\overline{\lambda}(x)=\lambda^{T}(x)\,C\,,$ (2)

where $C$ is the charge conjugation matrix. The gluino transforms under the
adjoint representation of the gauge group. The covariant derivative thus acts
as

$(\mathcal{D}_{\mu}\lambda)^{a}=\partial_{\mu}\lambda^{a}+g_{0}\,f^{a}_{bc}A^{b}_{\mu}\lambda^{c}\,.$
(3)

A non-zero mass $m_{g}$ of the gluino in supersymmetric Yang-Mills theory
causes a soft breaking of supersymmetry. For vanishing gluino mass it is
expected that supersymmetry is unbroken in the continuum [8].

For non-perturbative studies of SYM the theory is regularised on a Euclidean
space-time lattice. The introduction of a space-time lattice regulator
necessarily breaks supersymmetry [9]. Models that preserve part of an extended
superalgebra are discussed in Ref. [10]. In our simulations, we use the
“Curci-Veneziano” lattice action [11]

$S=S_{g}+S_{f}\,,$ (4)

where the gauge part ($S_{g}$) of the full action is given by the standard
Wilson action

$S_{g}=\frac{\beta}{2}\sum_{x}\sum_{\mu\neq\nu}\left(1-\frac{1}{N_{c}}{\rm
Re}\,\mathrm{tr}\,U_{\mu\nu}(x)\right).$ (5)

Here $\beta\equiv 2N_{c}/g_{0}^{2}$ is the bare lattice gauge coupling for the
SU($N_{c}$) gauge field and $U_{\mu\nu}(x)$ is the standard plaquette
variable. The fermion part of the action in equation (4) is

$\begin{array}[]{rcl}S_{f}&\equiv&\frac{1}{2}\sum_{x}\left\\{\overline{\lambda}^{a}(x)\lambda^{a}(x)-\kappa\sum_{\mu=1}^{4}\left[\overline{\lambda}^{a}(x+\hat{\mu})V_{\mu}^{ab}(x)(1+\gamma_{\mu})\lambda^{b}(x)+\overline{\lambda}^{a}(x)V_{\mu}^{abT}(x)(1-\gamma_{\mu})\lambda^{b}(x+\hat{\mu})\right]\right.\\\\[5.69054pt]
&&+\left.\frac{\mathrm{i}}{4}g_{0}\kappa
c_{SW}\overline{\lambda}^{a}(x)\sigma_{\mu\nu}P^{(cl)}_{\mu\nu;ab}(x)\lambda^{b}(x)\right\\}.\end{array}$
(6)

Here $\kappa$ is the hopping parameter, which is related to the bare gluino
mass by $\kappa=1/(2m_{g}+8)$, $\gamma_{\mu}$ denotes a Dirac matrix, and
$\sigma_{\mu\nu}=-\frac{1}{2}[\gamma_{\mu},\gamma_{\nu}]$. The links
$V_{\mu}^{ab}(x)$ are the gauge field variables in the adjoint representation,
obtained from links $U_{\mu}(x)$ in the fundamental representation by

$V_{\mu}^{ab}(x)\equiv
2\,\mathrm{tr}\left(U^{\dagger}_{\mu}(x)T^{a}U_{\mu}(x)T^{b}\right),$ (7)

where $T^{a}$ are the group generators of SU($N_{c}$). Finally, the last term
in the fermion action contains the clover-symmetrised lattice field strength
tensor:

$P^{(cl)}_{\mu\nu}(x)=\frac{1}{4a}\sum_{i=1}^{4}\frac{1}{2\mathrm{i}g_{0}a}\bigg{(}U^{(i)}_{\mu\nu}(x)-U^{(i)\dagger}_{\mu\nu}(x)\bigg{)}\,.$
(8)

The coefficient $c_{SW}$ is tuned up to one-loop order in perturbation theory
to improve the convergence of on-shell observables to the continuum limit
[12].

##  3 Baryons in Supersymmetric Yang-Mills theory

By combining three gluino fields $\lambda(x)$, it is possible to construct a
colourless object, provided all the colour indices are contracted with indices
of invariant colour tensors $t_{abc}$ for $a,b,c=1,\ldots,N_{c}^{2}-1$. The
corresponding particles are analogous to baryons in Quantum Chromodynamics
(QCD), and will therefore also be called “baryons”. Due to the fact that
gluinos are in the adjoint representation of the gauge group, these objects
can be constructed for both SU(2) and SU(3).

Interpolating fields for baryons will generally be of the form

$W(x)=t_{abc}\lambda_{a}(x)\big{(}\lambda^{T}_{b}(x)\Gamma\lambda_{c}(x)\big{)}$
(9)

with an invariant colour tensor $t_{abc}$ and a matrix $\Gamma$ in Dirac
space. In the case of $\text{SU}(2)$ one has $t_{abc}=\epsilon_{abc}$ and
$\Gamma$ has to be symmetric. For $\text{SU}(3)$ the $t_{abc}$ can be chosen
as the completely antisymmetric structure constants $f_{abc}$ or as the
completely symmetric tensor $d_{abc}$. The spin matrix $\Gamma$ then has to
have the opposite symmetry type in order that $W\neq 0$. Depending on the
choice of $\Gamma$, the baryon field $W(x)$ carries either spin 1/2 only or
both spin 1/2 and spin 3/2. These parts can be singled out with suitable
projection operators [13]. A possibility, which we considered, is the choice
$t_{abc}=\epsilon_{abc}$ and $\Gamma=C\gamma_{4}$ for gauge group
$\text{SU}(2)$.

The conjugate field is

$\overline{W}(x)=-\big{(}C\,W(x)\big{)}^{T}.$ (10)

The masses of baryons can be determined from the corresponding correlation
functions

$\displaystyle D(x,y)$ $\displaystyle=\langle W(x)\overline{W}(y)\rangle$
$\displaystyle D^{\alpha\delta}(x,y)$ $\displaystyle=\langle
W^{\alpha}(x)\overline{W}^{\delta}(y)\rangle$ $\displaystyle=-\langle
W^{\alpha}(x)C^{\delta\alpha^{\prime}}W^{\alpha^{\prime}}(y)\rangle$
$\displaystyle=-t_{abc}t_{a^{\prime}b^{\prime}c^{\prime}}\Gamma^{\beta\gamma}\Gamma^{\beta^{\prime}\gamma^{\prime}}C^{\delta\alpha^{\prime}}\langle\lambda_{a}^{\alpha}(x)\lambda_{b}^{\beta}(x)\lambda_{c}^{\gamma}(x)\lambda_{a^{\prime}}^{\alpha^{\prime}}(y)\lambda_{b^{\prime}}^{\beta^{\prime}}(y)\lambda_{c^{\prime}}^{\gamma^{\prime}}(y)\rangle.$
(11)

The fermionic expectation value can be decomposed into gluino propagators by
means of Wick’s theorem. In SYM, in contrast to QCD, every pair of fermion
fields contributes due to the Majorana nature of the gluino. In order to
obtain the Wick contractions, it is convenient to express the correlation
function in terms of six $\lambda$, by converting factors $\bar{\lambda}$ into
$\lambda$ by means of the Majorana condition. The Wick contractions are

$\langle\lambda^{\alpha}_{a}(x)\lambda^{\beta}_{b}(y)\rangle=K^{\alpha\beta}_{ab}(x,y)=-(\bigtriangleup(x,y)C)^{\alpha\beta}_{ab},$
(12)

where $\bigtriangleup=Q^{-1}$ is the fermion propagator. Taking into account
the fermionic signs, we get 15 terms:

$\displaystyle D^{\alpha\alpha^{\prime}}(x,y)=$
$\displaystyle\,t_{a^{\prime}b^{\prime}c^{\prime}}t_{abc}\Gamma^{\beta\gamma}\Gamma^{\beta^{\prime}\gamma^{\prime}}\times\\{$
$\displaystyle+2\bigtriangleup^{\alpha\alpha^{\prime}}_{aa^{\prime}}(x,y)\bigtriangleup^{\beta\beta^{\prime}}_{bb^{\prime}}(x,y)\bigtriangleup^{\gamma\gamma^{\prime}}_{cc^{\prime}}(x,y)$
$\displaystyle+4\bigtriangleup^{\alpha\beta^{\prime}}_{ab^{\prime}}(x,y)\bigtriangleup^{\beta\gamma^{\prime}}_{bc^{\prime}}(x,y)\bigtriangleup^{\gamma\alpha^{\prime}}_{ca^{\prime}}(x,y)$
$\displaystyle+2\bigtriangleup^{\alpha\beta}_{ab}(x,x)\bigtriangleup^{\delta\alpha^{\prime}}_{ca^{\prime}}(x,y)\bigtriangleup^{\delta^{\prime}\beta^{\prime}}_{c^{\prime}b^{\prime}}(y,y)C^{\gamma\delta}C^{\gamma^{\prime}\delta^{\prime}}$
$\displaystyle+4\bigtriangleup^{\alpha\beta}_{ab}(x,x)\bigtriangleup^{\beta^{\prime}\gamma}_{b^{\prime}c}(y,x)\bigtriangleup^{\gamma^{\prime}\alpha^{\prime}}_{c^{\prime}a^{\prime}}(y,y)$
$\displaystyle+1\bigtriangleup^{\alpha\alpha^{\prime}}_{aa^{\prime}}(x,y)\bigtriangleup^{\beta\delta}_{bc}(x,x)\bigtriangleup^{\delta^{\prime}\beta^{\prime}}_{c^{\prime}b^{\prime}}(y,y)C^{\gamma\delta}C^{\delta^{\prime}\gamma^{\prime}}$
$\displaystyle+2\bigtriangleup^{\alpha\delta^{\prime}}_{ac}(x,y)\bigtriangleup^{\beta\delta}_{bc}(x,x)\bigtriangleup^{\beta^{\prime}\alpha^{\prime}}_{b^{\prime}a^{\prime}}(y,y)C^{\gamma\delta}C^{\gamma^{\prime}\delta^{\prime}}\\}.$
(13)

Figure 1: Contributions to the propagator of a three-gluino state. The second
contribution, which we call “spectacle graph” comes from contractions of the
form $\lambda(x)\lambda(x)$, which are allowed for Majorana fermions.

Depending on the topology of the associated fermion line diagrams, the
contributions can be divided into two types called “Sunset” and “Spectacle”,
which are represented in Fig.1.

The corresponding algebraic expressions are given by

$\displaystyle D_{\text{sunset}}(x,y)=$
$\displaystyle\,t_{a^{\prime}b^{\prime}c^{\prime}}t_{abc}\Gamma^{\beta\gamma}\Gamma^{\beta^{\prime}\gamma^{\prime}}P_{\raise
0.8264pt\hbox{$\scriptstyle\pm$}}^{\alpha\alpha^{\prime}}\times\\{$ (14)
$\displaystyle+2\bigtriangleup^{\alpha\alpha^{\prime}}_{aa^{\prime}}(x,y)\bigtriangleup^{\beta\beta^{\prime}}_{bb^{\prime}}(x,y)\bigtriangleup^{\gamma\gamma^{\prime}}_{cc^{\prime}}(x,y)$
(15)
$\displaystyle+4\bigtriangleup^{\alpha\beta^{\prime}}_{ab^{\prime}}(x,y)\bigtriangleup^{\beta\gamma^{\prime}}_{bc^{\prime}}(x,y)\bigtriangleup^{\gamma\alpha^{\prime}}_{ca^{\prime}}(x,y)\\}$
(16) $\displaystyle D_{\text{spectacle}}(x,y)=$
$\displaystyle\,t_{a^{\prime}b^{\prime}c^{\prime}}t_{abc}\Gamma^{\beta\gamma}\Gamma^{\beta^{\prime}\gamma^{\prime}}P_{\raise
0.8264pt\hbox{$\scriptstyle\pm$}}^{\alpha\alpha^{\prime}}\times\\{$
$\displaystyle+2\bigtriangleup^{\alpha\beta}_{ab}(x,x)\bigtriangleup^{\delta\alpha^{\prime}}_{ca^{\prime}}(x,y)\bigtriangleup^{\delta^{\prime}\beta^{\prime}}_{c^{\prime}b^{\prime}}(y,y)C^{\gamma\delta}C^{\gamma^{\prime}\delta^{\prime}}$
$\displaystyle+4\bigtriangleup^{\alpha\beta}_{ab}(x,x)\bigtriangleup^{\beta^{\prime}\gamma}_{b^{\prime}c}(y,x)\bigtriangleup^{\gamma^{\prime}\alpha^{\prime}}_{c^{\prime}a^{\prime}}(y,y)$
$\displaystyle+1\bigtriangleup^{\alpha\alpha^{\prime}}_{aa^{\prime}}(x,y)\bigtriangleup^{\beta\delta}_{bc}(x,x)\bigtriangleup^{\delta^{\prime}\beta^{\prime}}_{c^{\prime}b^{\prime}}(y,y)C^{\gamma\delta}C^{\delta^{\prime}\gamma^{\prime}}$
$\displaystyle+2\bigtriangleup^{\alpha\delta^{\prime}}_{ac}(x,y)\bigtriangleup^{\beta\delta}_{bc}(x,x)\bigtriangleup^{\beta^{\prime}\alpha^{\prime}}_{b^{\prime}a^{\prime}}(y,y)C^{\gamma\delta}C^{\gamma^{\prime}\delta^{\prime}}\\}$
(17)

In contrast to mesonic or glueball correlation functions, the baryonic
correlation functions are not symmetric or antisymmetric with respect to
Euclidean time, if no parity projection is included, see e. g. [14]. Therefore
we include the projection onto parity eigenstates by means of the parity
projector $P_{\raise 0.8264pt\hbox{$\scriptstyle\pm$}}$, which is defined as
$P_{\raise 0.8264pt\hbox{$\scriptstyle\pm$}}=\frac{1}{2}(1\raise
1.18056pt\hbox{$\scriptstyle\pm$}\gamma_{4})$ for zero momentum.

We have implemented the contributions to the baryonic correlation function
into the measurement code of our SYM collaboration. The numerically quite
expensive part is the evaluation of the spectacle piece. We have calculated it
by means of combining the exact contribution of the lowest eigenmodes of the
Wilson-Dirac matrix with the stochastic estimator technique.

Figure 2: Numerical results for correlation functions at each time slice for
gauge group SU(2), lattice volume, $V=24^{3}\cdot 48$, $\beta=1.75$ and
$\kappa=0.14925$.

Figure 3: Effective mass plots for fixed APE pre-smearing with level 40 and
different levels of Jacobi smearing of the Dirac fermion matrix.

Presently some preliminary results have been obtained. Fig. 2 shows the two
pieces of the baryonic correlation function and its sum for gauge group
$\text{SU}(2)$. In this case the sunset piece appears to dominate the
correlation function for not too large times. Fig. 3 shows the effective mass
for different smearings.

##  4 Conclusion and outlook

In this paper we constructed the correlation function for a three-gluino state
in supersymmetric Yang-Mills theory and presented preliminary estimates of its
mass using Monte Carlo simulation at non-zero gluino mass and at finite
inverse gauge coupling $\beta$. With better numerical results we expect to be
able to perform chiral and continuum extrapolations in the future, and to find
out how the baryonic states fit into the supermultiplet found in Ref. [3].

## Acknowledgements

The authors gratefully acknowledge the Gauss Centre for Supercomputing e. V.
(www.gauss-centre.eu) for funding this project by providing computing time on
the GCS Supercomputer JUQUEEN and JURECA at Jülich Supercomputing Centre (JSC)
and SuperMUC at Leibniz Supercomputing Centre (LRZ). Further computing time
has been provided on the compute cluster PALMA of the University of Münster.
This work is supported by the Deutsche Forschungsgemeinschaft (DFG) through
the Research Training Group “GRK 2149: Strong and Weak Interactions-from
Hadrons to Dark Matter”. G. Bergner and C. Lopez acknowledge support from the
Deutsche Forschungsgemeinschaft (DFG) under Grant No. BE 5942/2-1. S. Ali
acknowledges financial support from the Deutsche Akademische Austauschdienst
(DAAD).

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