# Fine-tuning StyleGAN2 for Cartoon Face Generation

Jihye Back
Seoul National University
<EMAIL_ADDRESS>

###### Abstract

Recent studies have shown remarkable success in the unsupervised image to
image (I2I) translation. However, due to the imbalance in the data, learning
joint distribution for various domains is still very challenging. Although
existing models can generate realistic target images, it’s difficult to
maintain the structure of the source image. In addition, training a generative
model on large data in multiple domains requires a lot of time and computer
resources. To address these limitations, we propose a novel image-to-image
translation method that generates images of the target domain by finetuning a
stylegan2 pretrained model. The stylegan2 model is suitable for unsupervised
I2I translation on unbalanced datasets; it is highly stable, produces
realistic images, and even learns properly from limited data when applied with
simple fine-tuning techniques. Thus, in this paper, we propose new methods to
preserve the structure of the source images and generate realistic images in
the target domain. The code and results are available at
https://github.com/happy-jihye/Cartoon-StyleGan2

##  1 Introduction

The style-based generative model [1, 2] is very effective in image generation,
and it generates high-resolution images by learning not only global attributes
but also stochastic details. Several researchers [3, 4, 5, 6, 7, 8, 9, 10]
have used this model to perform several tasks such as image editing and image
translation, noting the powerfulness of style-based architecture. Furthermore,
recent research to apply transfer learning to this model is also active.
Because style-based architecture trains sequentially from low-resolution to
high-resolution, simple fine-tuning techniques make it easier to transfer
models of source domains to target domains. Thus, many unsupervised I2I
translations have adopted a transfer learning to reduce resources, memory and
time.

In this paper, we fine-tune the stylegan2 model for unsupervised I2I
translation. We propose two methods to make source images and target images
similar, such as paired images.

1\. FreezeSG, which freezes the initial blocks of the style vectors and
generator. This is very simple and allows the target image to follow the
structure of the source image.

2\. Structure Loss, which make to reduce the distance between the inital
blocks of the source generator and the inital blocks of the target generator.
The effectiveness of applying Layer Swapping to models trained by this loss is
also remarkable.

##  2 Related Work

#### Generative Adversarial Networks (GANs)

The GAN framework [11] has shown impressive results in image generation [2,
12, 1], image translation [3, 4, 5, 6], image editing [13, 14, 15], and face
image synthesis [2, 1]. GAN is a generative model that trains through the
adversarial process of generator G and discriminator D: The generator creates
a fake image to confuse the discriminator so it cannot recognize whether the
image is real or fake. In this paper, we aim to align the source distribution
to the target domain using the GAN network.

#### Image-to-Image Translation

Recent studies have shown remarkable success in the image to image (I2I)
translation [3, 4, 5, 6] that aims to learn the mapping between an input image
and an output image. Pix2Pix [5] is the first framework for I2I translation,
which uses cGANs [16] to enable successful I2I transformations for a variety
of tasks when there is paired dataset. Because of the limited paired training
data, several methods have been proposed to address unsupervised image-to-
image translation tasks. UNIT [6] assumes a shared latent space, so that image
pairs from different domains can be mapped into one shared latent space. Other
unsupervised I2I methods [7, 8], including CycleGAN [9], introduce the cycle
consistency loss to make a constraint between the source image and the target
image. However, these frameworks must be trained on data from both the source
domain and the target domain simultaneously. Training consists of the mapping
process in multiple domains, which requires a lot of computational resources
and time. If there is a severe imbalance between data of each domain, mode
collapse may also happen. To overcome these limitations, we proposed a new
unsupervised Image-to-Image translation method that generates images of target
domains using the stylegan2 [2] pretrained model.

Figure 1: We compared the results of FreezeSG with existing models (FreezeD,
FreezeG). At this time, we swapped the output of the 4-th style block.

#### Transfer Learning

Transfer learning is a very powerful and effective method in computer vision.
In transfer learning, a new model is trained using the weights of the pre-
trained model. This method can reduce time and computer resources because it
trains by fine-tuning a pre-trained model without starting from random
initialization. Efforts to apply transfer learning to GANs are also active. Mo
et al [17] proposed a FreezeD model for the generator to learn the target
distribution by freezing the highest-resolution layer of the discriminator
during the fine-tuning process. Karras et al [18] proposed an adaptive
discriminator augmentation (ADA) mechanism that stabilizes training in limited
data. They have also proven that ADA performs better with FreezeD. These
studies have shown that applying transfer learning to GANs allows image
translation from the source domain to the target domain more efficiently.

With the effectiveness and powerfulness of stylegan2 [2] and fine-tuning
techniques [17, 18], there has been a recent move to apply them to
unsupervised I2I translation. FreezeG [19] argued that freezing the
generator’s low-level resolution layer helps maintain the structure of the
source domain. In addition, Pinkney et al [20] proposed a layer swapping
method that combines the high-resolution layer of the FFHQ model with the low-
resolution layer of the animation model to generate a photo-realistic face.
Huang et al [21] suggested a FreezeFC mechanism that freezes 8-layer
$w=f(z),\mathbf{w}\in\mathcal{W}$ of the StyleGAN2 network during the fine-
tuning process to increase the similarity between the source and target
domains.

In this paper, we proposed a method for unsupervised image to image(I2I)
translation by applying transfer learning to the stylegan2 pretrained model.
Inspired by these works [19, 20, 21], we also proposed new methods to sustain
the structure of the source images and generate realistic images in the target
domain.

##  3 Method

###  3.1 FreezeSG

Noting the effectiveness of FreezeG [17], we analyze the factors that
determine the structure of the generated image and find that not only the
early layers of the generator but also the style vectors injected into the
generator are involved. Inspired by this, we freeze the initial blocks of the
generator and the initial style vectors injected in the fine-tuning process of
stylegan2. We call this simple yet effective method _a Freeze Style vector and
Generator (FreezeSG)_.

Figure 1 shows that FreezeSG reflects the source image better than freezing
the generator alone. In addition, it shows that the structure of the source
image is better preserved if the low-resolution layer of the generator in the
source domain and the high-resolution layer of the generator in the target
domain are integrated by applying the Layer Swapping (LS) technique. When LS
is applied, the generated images by FreezeSG have a higher similarity to the
source image than when FreezeG or the baseline (FreezeD + ADA) were used.

###  3.2 Structure Loss

Figure 2: The architecture of our generator (using structure loss). If we want
to apply a loss for the RGB output of three style blocks(4x4 - 16x16), you
calculate the mse-loss between the output of (b) a fixed source generator and
the output of (c) the target generator being trained. After calculating, and
add them all.(d) shows the integration of the low-resolutoin of the source
generator and the high-resolution of the target generator by applying the
Layer Swapping method. Figure 3: Comparison of training results using
structure loss and baseline(freezeD). We used the Layer Swapping technique for
$l=2$.

In Section 3.1, we showed the effectiveness of FreezeSG. However, since this
fixes the weights of the low-resolution layer of the generator, it is
difficult to obtain meaningful results when layer swapping on the low-
resolution layer. We introduce a simple and effective loss function that we
called _structure loss_ to obtain meaningful results even when layer swapping
is performed for low-resolution layers.

#### Adversarial Loss.

Original GAN [11] is a model that trains through the adversarial process of
generator $G$ and discriminator $D$. The objective of a GAN can be expressed
as

$\mathcal{L}_{adv}=\mathbb{E}_{x\sim p_{data}}\left[\log
D(x)\right]+\mathbb{E}_{z\sim p(z)}\left[\log\left(1-D(G(z))\right)\right],$
(1)

where $G$ tries to minimize this objective to learn the distribution of the
target domain, and $D$ tries to maximize it to discriminate between the real
image $x$ and the fake image $G(z)$.

#### Structure Loss.

We adopted the structure of _input/output skips_ among the three architectures
(MSG-GAN, In-put/output skips, and Residual nets) of the stylegan2 model [2].
Figure 2(a) shows this architecture, which is simplified by upsampling and
summing the contributions of RGB outputs corresponding to different
resolutions. Based on the fact that the structure of the image is determined
at low resolution, we apply _structure loss_ to the values of the low-
resolution layer so that the generated image is similar to the image in the
source domain. The structure loss makes the RGB output of the source generator
to be fine-tuned to have a similar value with the RGB output of the target
generator during training.

Structure loss is available as follows.

1\. If we want to apply _structure loss_ for $n$ style blocks, we must first
extract the RGB outputs of both the source generator and the target generator
for each resolution.

2\. Calculate the mse loss between $G^{s}_{l=k}(w_{s})$ and
$G^{t}_{l=k}(w_{s})$ of each resolution, and add it up to the n-th layer.

$L_{structure}=\sum_{k=1}^{n}{\mathbb{E}\left[G^{s}_{l=k}(w_{s})-G^{t}_{l=k}(w_{t})\right]}$
(2)

where $G^{s}_{l=k}(w_{s})$ is the RGB outputs of the source generator, and
$G^{t}_{l=k}(w_{t})$ is the RGB outputs of the target generator. $k$ is the
index of the style block.

#### Full Objective.

Finally, the objective functions are expressed as

$\mathcal{L}_{D}=-\mathcal{L}_{adv}$ (3)
$\mathcal{L}_{G}=\mathcal{L}_{adv}+\lambda_{structure}\mathcal{L}_{structure}$
(4)

where $\lambda_{structure}$ is a hyper-parameter that control the relative
importance of structure of source domain. We use $\lambda_{structure}=1$ in
all of our experiments.

Figure 3 shows that our model generates images much more naturally than
baseline [17, 2]. Since we processed the initial layer of $G$ to learn the
structure, images of areas such as jaws and heads are well generated.

Figure 4: Comparison between freezeG and our models.

##  4 Experiments

###  4.1 Datasets

#### Source domain dataset.

We used Flickr-Faces-HQ (FFHQ) [1] as a dataset for the source domain. This
has high-quality images of human faces and there are 70,000 images. We used
FFHQ to train the stylegan2 model [2] of 256 resolution.

#### Target domain dataset.

We experimented with a variety of datasets, including Naver Webtoon [22],
Metfaces [18], and Disney [20]. Naver Webtoon Dataset [22] contains facial
images of webtoon characters serialized on Naver. We made this dataset by
crawling webtoons from Naver’s webtoons site and cropping the faces to 256 x
256 sizes. There are about 15 kinds of webtoons and 8,000 images. We trained
the entire Naver Webtoon dataset, and we also trained each webtoon in this
experiment. We also experimented with other dataset [18, 20], and learned
using images from 256 resolution.

###  4.2 Training details

In Section 3.1, we freeze the initial blocks of the generator and style vector
and then train the model by fine-tuning the stylegan2 pre-trained model. In
this process, the same objective function as stylegan2 was used. We find it
effective to freeze two style blocks (4x4 - 8x8) when generating 256 x 256
resolution images. When applying Layer Swapping [20], it is most effective to
integrate the low-resolution layer of the source generator (4x4 - 64x64) and
the high-resolution layer (64x64 - 256x256) of the target generator.

In Section 3.2, We train by applying structure loss to the _input/output skip_
architecture of the stylegan2 model [2]. We used this loss for the three low-
resolution layers of the source generator and target generator, and we adopted
mse-loss as a loss function. Furthermore, the structural loss enables
effective layer swapping even for low-resolution layers, unlike freezeSG. So
we swapped different low-resolution layers through various experiments.
(Figure 5)

###  4.3 Experiment Results

We showed that our model was more effective than the baseline (FreezeD + ADA)
[17, 18] and FreezeG [19]. In addition, when the layer swapping technique was
applied, the structure of the source domain was more successfully maintained.
We also conducted additional experiments on the metface [18] and disney [20]
dataset, and the highly quality images were generated when structural loss and
layer swapping were used together. FreezeSG also maintained the structure of
the source image well but it produced an unnatural image than the image of
applying structure loss.

##  5 Conclusions and Future works

In this paper, we proposed Cartoon-StyleGAN: Fine-tuning StyleGAN2 for Cartoon
Face Generation, a novel unsupervised Image-to-Image translation methods.
Experimental results show that these methods are effective in making source
image and target image similar. Compared to previous studies, it generates a
more realistic image.

However, our studies need to adjust layers to optimize for each dataset. In
future works, we expect to be more stable and performance-enhancing.

Figure 5: The results of applying structure loss with Layer Swapping. Figure
6: The results of FreezeSG with Layer Swapping.

## References

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Figure 7:  Comparison between baseline(FreezeD + ADA) and our model applying
the structure loss.

##  Appendix A Appendix

###  A.1 Change Facial Expression and Pose.

Figure 8: The architecture of StyleCLIP. Figure 9: The result images of text
guided manipulation. We additionally introduce _styleclip strength_ to
increase image variation. The third image is the one generated by StyleCLIP,
and the 6-th image is the one applied with styleclip strength, $\alpha=2$.

#### StyleCLIP

Inspired by StyleCLIP [23] that manipulates generated images with text, we
change the faces of generated cartoon characters by text. We used the latent
optimization method among the three methods of StyleCLIP and additionally
introduced _styleclip strength_. It allows the latent vector to linearly move
in the direction of the optimized latent vector, making the image change
better with text.

$w^{\prime}=\alpha(w-w_{s})+w)$ (5)

where $\alpha$ is a styleclip strength, $w$ is a trainable latent vector, and
$w_{s}$ is a fixed latent vector.

We first found a latent vector optimized according to text using a source
generator and styleclip. Then we entered this latent vector into a fine-tuned
target generator to change the facial expression of the cartoon face. Figure 9
shows that StyleCLIP is effective in changing the expression of a webtoon
character.


# Analytic continuation and supersymmetry

Nikita Nekrasov

###### Abstract

Real-valued parameters of quantum field theory, such as Planck constant
$\hbar$, the coupling constants $g^{-2},{\theta},\ldots$, temperature
${\beta}^{-1}$ and spacetime metric ${\bf g}_{\mu\nu}$, chemical potentials or
background gauge fields ${\bf A}_{\mu}^{a}$, can be made complex. In
perturbative string theory, the worldsheet metric is sometime Lorentzian,
somewhere Euclidean, and likely complex in between. Less often, integral, or
quantized, parameters, such as the number $n$ of flavors or species (as in
replica method), the number of colors $N$ (as in matrix models, master field),
spacetime dimension $D$ (dimensional regularization), or angular momentum $l$
(Regge poles) are continued to complex values. We observe that many, if not
all of these analytic continuations can be realized with conventional
supersymmetric quantum field (or M-) theory. The supersymmetry is softly
broken near a defect supporting the theory being analytically continued in
some of its parameters.

Somewhat abridged talk at Strings-Math’2022 in Warsaw.

Simons Center for Geometry and Physics,

Stony Brook University, Stony Brook, NY 11794-3636, USA

##  1 Introduction

It is interesting to study analytic continuations. It is especially
interesting to study analytic continuations of quantum field theory. Usually
the real-valued parameters, such as Planck constant $\hbar$, the coupling
constants $g^{-2},{\theta},\ldots$, temperature ${\beta}^{-1}$ and spacetime
metric ${\bf g}_{\mu\nu}$, chemical potentials or background gauge fields
${\bf A}_{\mu}^{a}$, can be made complex. For example, to a quantum mechanical
system with discrete spectrum Hamiltonian $\hat{H}$ we associate the thermal
partition function

${\mathcal{Z}}({\beta})={\rm
Tr}\,e^{-{\beta}{\hat{H}}}=\oint_{C}\,dze^{-{\beta}z}\,{\rm Tr}G_{\hat{H}}(z)$
(1)

where $G_{\hat{H}}(z)$ is the resolvent [39], and $C$ is the contour circling
around the spectrum ${\sigma}({\hat{H}})$. Using path integral

${\mathcal{Z}}({\beta})=\int_{L{\mathcal{P}}}\,Dp(t)Dq(t)\,e^{{\mathrm{i}}\oint
pdq-\oint H\left(p(t),q(t)\right)d{\beta}}\ ,$ (2)

we represent the partition function as an integral over the space
$L{\mathcal{P}}$ of loops on a phase space, with the measure defined using a
complex-valued one form $d{\beta}={\beta}^{\prime}(t)dt$ on $S^{1}$. If we
eliminate momentum by its equations of motion, e.g. for $H$ which is quadratic
in $p$, the saddle point dominating (2), does not, typically, fit in
${\mathcal{P}}$, since

$p^{\rm
saddle}_{i}(t)=\frac{1}{{\mathrm{i}}{\beta}^{\prime}(t)}\,g_{ij}\frac{dq^{j}}{dt}\,,\qquad
g^{ij}=\frac{{\partial}^{2}H}{\partial p_{i}\partial p_{j}}$

is complex unless ${\beta}^{\prime}(t)\in{\mathrm{i}}{\mathbb{R}}$. Once
$p(t)$ can become complex, so can $q(t)$. One is led to the idea that path
integral contour $L{\mathcal{P}}$ can be deformed into the complex domain
$L{\mathcal{P}}_{\mathbb{C}}$, as a middle dimensional contour ${\Sigma}$,
which splits, in the appropriate homology group, as a sum of Lefschetz thimble
cycles associated with the critical points of the exponential in (2).

The classification of such cycles in one- and two-dimensional quantum field
theories has recently been revived in e.g. [25, 18, 19].

String worldsheet metric is sometime Lorentzian, somewhere Euclidean, and
likely complex in between [42]. Less often, integral, or quantized,
parameters, such as the number $n$ of flavors or species (replica method), the
number of colors $N$ (matrix models, master field), spacetime dimension $D$
(dimensional regularization), or angular momentum $l$ (Regge poles) are
continued into the complex domain. For example, the character (twisted
partition function) of a particle on $S^{2}$ (phase space)

${\chi}_{j}({\theta})=\frac{{\rm
sin}\left((j+\frac{1}{2}){\theta}\right)}{{\rm sin}(\frac{\theta}{2})}$ (3)

admits analytic continuation to complex $\theta$, $j$, etc. It was I.
Gelfand’s dream to understand the dimensional regularization used in
computations of Feynman diagrams in quantum field theory and the subsequent
renormalization, through the representation theory of the non-compact Lie
group $SL(2,{\mathbb{C}})$ (whose real form is the Lorentz group of the four
dimensional Minkowski space which looks like our space-time).

Theoretical physics often uses a trick of analytic continuation. In studying
the disordered systems, where a free energy of a statistical mechanical model

$F(J)=-\frac{1}{\beta}{\rm log}Z({\beta},J)$ (4)

is to be averaged, e.g.

$\langle F(J)\rangle_{J}=\int\,DJ\,e^{-\|J\|^{2}}\,F(J)$ (5)

over some manifold of parameters $J$, e.g. in the spin glass models, where

$Z({\beta},J)=\sum_{\\{{\sigma}_{i}=\pm 1\\}}e^{-{\beta}\sum\limits_{\langle
i,j\rangle}J_{ij}{\sigma}_{i}{\sigma}_{j}}$ (6)

In replica trick, one computes the averages of $n$’th powers of $Z$, then
takes the limit $n\to 0$ to extract the free energy. Another popular
application of the replica trick is in the computation of von Neumann entropy
${\rm Tr}{\rho}{\rm log}{\rho}$ which is extracted in the $n\to 1$ limit of
${\rm Tr}{\rho}^{n}$. Thus, one studies the statistical mechanics of $n$
copies of the original system. For integer positive $n$ this makes sense, but
what is the meaning of the $n$ complex?

Curiously, we find several examples of systems for which these analytic
continuations can be realized with conventional supersymmetric quantum field
(or M-) theory. The supersymmetry is softly broken near a defect supporting
the theory being analytically continued in some of its parameters. We are
interested in analytic continuations of spins, dimensions, temperatures,
fugacities etc.

##  2 Models with $O(N)$ symmetry

In our first model we study the conformally coupled scalar living on
$S^{N-1}$. Conformally coupled scalar field living on an $N$-dimensional
Riemanian manifold $M^{N}$ is described by the action

$S=\frac{1}{2}\int_{M^{N}}d^{N}{\xi}\sqrt{g}\left(g^{mn}{\partial}_{m}{\varphi}{\partial}_{n}{\varphi}+\frac{N-2}{2N-2}R(g){\varphi}^{2}\right)$
(7)

In this paper we are interested in the conformally flat manifolds of the form
$S^{N-1}\times S^{1}$ (more general case $S^{N_{1}-1}\times S^{N_{2}-1}$ with
$N_{1},N_{2}\geq 3$ is also connected to supersymmetric systems, but the story
is more complicated). We endow it with the metric

$ds^{2}_{M^{N}}={\beta}^{2}dt^{2}+R^{2}d{\Omega}_{S^{N-1}}^{2}$ (8)

with $t\sim t+1$,

$\frac{{\beta}R^{N-1}}{2}\int_{S^{N-1}\times
S^{1}}dtd^{N-1}{\Omega}\,\left[\frac{1}{{\beta}^{2}}\left({\partial}_{t}{\varphi}\right)^{2}+\frac{1}{R^{2}}\left({\nabla}{\varphi}\cdot{\nabla}{\varphi}+\left(\frac{N}{2}-1\right)^{2}{\varphi}^{2}\right)\right]$
(9)

We can decompose
${\varphi}(t,x)=\sum_{l=0}^{\infty}\sum_{A=1}^{d_{l}(N)}{\varphi}_{A,l}(t)Y_{A,l}(x)$
into the spherical functions, where

$-{\Delta}_{S^{N-1}_{R}}Y_{A,l}=\frac{l(l+N-2)}{R^{2}}Y_{A,l}\ .$ (10)

We see (9) is a Lagrangian describing an infinite set
$\left({\varphi}_{A,l}(t)\right)_{l=0}^{\infty}{}_{A=1}^{d_{l}(N)}$ of
harmonic oscillators with frequencies $\frac{l+\frac{N}{2}-1}{R}$. The
multiplicity $d_{l}(N)$ is computed from the generating function

$\sum_{l=0}^{\infty}x^{l}d_{l}(N)=\frac{1+x}{(1-x)^{N-1}}$ (11)

(this is $1-x^{2}$ times the character of the space of polynomials in $N$
variables, with $x$ being the fugacity for dilatation symmetry). Thus

$d_{l}(N)=\left(\begin{matrix}N+l-2\\\
l\end{matrix}\right)+\left(\begin{matrix}N+l-3\\\ l-1\end{matrix}\right)\ ,\\\
d_{l}(2)=2-{\delta}_{l,0}\,,\ d_{l}(3)=2l+1\,,\ d_{l}(4)=(l+1)^{2}\,,\ldots$
(12)

Note that $d_{l}(N)$ admits analytic continuation to complex values of $N$, as

$D_{r}(N)\,=\,\frac{2r}{{\Gamma}(N-1)}\frac{{\Gamma}(r+N/2-1)}{{\Gamma}(r-N/2+2)}\,,\qquad
r=N/2-1+l$ (13)

from which the symmetry

$D_{r}(N)=(-1)^{N}D_{-r}(N)$ (14)

is readily seen. This is related to the symmetry $x\mapsto x^{-1}$ of (11)
accompanied by the $(-1)^{N}x^{N-2}$ prefactor.

We can thus write the partition function of conformal scalar on
$S^{N-1}_{R}\times S^{1}_{\beta}$:

$Z_{N}(q)={\rm
Tr}_{{\mathcal{H}}_{S^{N-1}_{R}}}\,e^{-{\beta}{\hat{H}}}=q^{c_{0}}\prod_{r\geq\frac{N}{2}-1}^{\infty}\frac{1}{\left(1-q^{r}\right)^{D_{r}(N)}}=q^{c_{0}}{\exp}\,\sum_{n=1}^{\infty}\frac{1}{n}\frac{q^{-n/2}+q^{n/2}}{(q^{-n/2}-q^{n/2})^{N-1}}$
(15)

where $q$ is the conformal invariant:

$q=e^{-\frac{\beta}{R}}$ (16)

and $c_{0}/R$ is the Casimir energy.111One can compute $c_{0}$ as a
regularized sum
$c_{0}=\frac{1}{2}\sum_{l=0}^{\infty}d_{l}(N)\left(l+N/2-1\right):=-\frac{1}{2}{\rm
Coeff}_{\tau^{1}}\frac{e^{\tau/2}+e^{-\tau/2}}{(e^{\tau/2}-e^{-\tau/2})^{N-1}}\Biggr{|}_{\tau=0}$
(17) See [1] for more systematic treatment We can refine (15) by introducing
the general $O(N)$ twisted boundary conditions:

${\varphi}(t+1,x)={\varphi}(t,{\bf g}\cdot x)$ (18)

where ${\bf g}\in O(N)$ is characterized by $[N/2]$ angles
${\theta}_{1},\ldots,{\theta}_{[N/2]}$ and, for odd $N$, in addition, by
${\xi}=\pm 1$:

$\begin{cases}N=2m\,,\qquad\qquad
Z_{N}(q,{\theta}_{1},\ldots,{\theta}_{m})={\exp}\,\sum\limits_{n=1}^{\infty}\,\,\frac{q^{-n}-q^{n}}{n}\prod\limits_{a=1}^{m}\frac{1}{q^{n}-2{\rm
cos}(n{\theta}_{a})+q^{-n}}\\\ N=2m+1\,,\qquad
Z_{N}(q,{\theta}_{1},\ldots,{\theta}_{m},{\xi})={\exp}\,\sum\limits_{n=1}^{\infty}\frac{q^{-n/2}+q^{n/2}{\xi}^{n}}{n}\prod\limits_{a=1}^{m}\frac{1}{q^{n}-2{\rm
cos}(n{\theta}_{a})+q^{-n}}\\\ \end{cases}$ (19)

For three dimensional conformally coupled scalar this formalism gives:

$Z(q,{\theta},{\xi})={\exp}\,\sum\limits_{n=1}^{\infty}\frac{1}{n}\frac{q^{-n/2}+q^{n/2}{\xi}^{n}}{q^{n}+q^{-n}-2{\rm
cos}(n{\theta})}$ (20)

One can get similar formulas for the conformally coupled fermions. These
three-dimensional partition functions are closely related to MacMahon function

$M(q)=\prod_{n=1}^{\infty}\frac{1}{(1-q^{n})^{n}}\,=\,{\exp}\,\sum_{n=1}^{\infty}\frac{1}{n}\frac{q^{n}}{(1-q^{n})^{2}}$
(21)

which counts plane partitions [34], is featured in representation theory of
$W_{1+\infty}$-algebra [3, 2], and in the theory of topological strings [6,
35]. Recall [32] that the supersymmetric partition function of the
$\Omega$-deformed maximally supersymmetric $U(1)$ Yang-Mills theory on
noncommutative ${\mathbb{R}}^{6}$ (or $D5-D(-1)$ system) is equal to

$Z_{6}(q,{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3})=M(q)^{-\frac{({\varepsilon}_{1}+{\varepsilon}_{2})({\varepsilon}_{1}+{\varepsilon}_{3})({\varepsilon}_{2}+{\varepsilon}_{3})}{{\varepsilon}_{1}{\varepsilon}_{2}{\varepsilon}_{3}}}$
(22)

where $q$ is the $D(-1)$-instanton fugacity, and
${\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3}$ are the three
parameters of the $U(3)$ $\Omega$-deformation. The group $U(3)\subset O(6)$
rotates the space-time $\mathbb{R}^{6}$ and the transverse $\mathbb{R}^{2}$
via determinant $U(1)\approx O(2)$. Thus, the replica of three dimensional
bosons and fermions can be realized physically as six dimensional gauge theory
by playing with the $\Omega$-deformation parameters. Once the
$\Omega$-deformation preserves more supersymmetry, the exponent in (22)
becomes $1$.

Similarly, two dimensional boson partition function ($N=2$)

$Z(q)=q^{-\frac{1}{12}}\prod_{n=1}^{\infty}\frac{1}{(1-q^{n})^{2}}$ (23)

is related to Euler function

${\phi}(q)=\prod_{n=1}^{\infty}(1-q^{n})$ (24)

The replica of the two dimensional chiral bosons is given by the
$\Omega$-deformed ${\mathcal{N}}=2^{*}$ four dimensional $U(1)$ super-Yang-
Mills theory:

$Z_{4}(q,{\varepsilon}_{1},{\varepsilon}_{2},{\varepsilon}_{3})=\left(\frac{1}{{\phi}(q)}\right)^{\frac{({\varepsilon}_{1}+{\varepsilon}_{3})({\varepsilon}_{1}+{\varepsilon}_{2})}{{\varepsilon}_{1}{\varepsilon}_{2}}}$
(25)

where $\varepsilon_{3}$ is the mass of the adjoint hypermultiplet [28, 31].
One can refine the “replica theories” by lifting them to the supersymmetric
theories in one dimension higher, compactified on a circle. The
$\varepsilon$-parameters become $q$-parameters. For example, the two
dimensional example refines to

$Z_{5}(q,q_{1},q_{2},q_{3})\,=\,{\exp}\,\sum_{n=1}^{\infty}\frac{q^{n}}{n}\frac{(1-q_{1}^{n}q_{3}^{n})(1-q_{2}^{n}q_{3}^{n})}{(1-q_{1}^{n})(1-q_{2}^{n})(1-q^{n})}$
(26)

which has two interpretations: as Witten index of five dimensional maximally
super-symmetric Yang-Mills theory compactified on a circle with the Lorentz
and R-symmetry twists compatible with a fraction of supersymmetry, or as
elliptic genus of a six dimensional $(2,0)$-superconformal tensor theory
compactified on a torus [32]. The refinement of the replica of three
dimensional theory brings about the (linearized) eleven dimensional
supegravity on ${\mathbb{R}}^{10}\times S^{1}$, subject to an $SU(5)$ twist
[32, 29]. Recently this partition function was reproduced using a twisted
version of supergravity [36].

Our second problem related to $S^{N-1}$ is the classical two dimensional
$O(N)$-model, i.e. the sigma model with $S^{N-1}$ as a target. We found, in
[18, 19], that the complex critical points of the action

$\int_{T^{2}}\,\sqrt{h}h^{ab}{\nabla}_{a}{\vec{n}}\cdot{\nabla}_{b}{\vec{n}}$
(27)

with ${\vec{n}}\in{\mathbb{C}}^{N}$,
${\vec{n}}\cdot{\vec{n}}=\sum_{i=1}^{N}n_{i}^{2}=1$, and complex flat metric
$h_{ab}$ on $T^{2}$ are described in terms of an algebraic integrable system.
The vector ${\vec{n}}$ with $N$ components is recovered from the so-called
Baker-Akhiezer $\psi(k)$ function, which is an analytic function on an
algebraic curve (called the Fermi-curve). The components $n_{i}$ are related
to the evaluation of $\psi$ at $N$ special points, singled out by an
additional structure. In this way all $O(N)$ models are unified, and in a
sense $N$ is complexified (replaced by an algebraic curve).

##  3 Interpolating between $SU$, $SO$, $Sp$

In Dyson’s original work on ensembles of random matrices with various time-
reversal properties [13] one finds a discrete series of measures on the space
of $N$ indistinguishable particles on a real line (or a circle), which is
naturally generalized into a continuous family of $\beta$-ensembles. The
points $\beta=1,2,4$ correspond to the unitary, orthogonal, and symplectic
groups. The interpolating model is no longer a matrix model, yet it can be
viewed, rather artificially, as a double-trace deformation of a more
conventional matrix model.

In a simplest setting, the partition function of the $\beta$-ensemble is given
by the integral

$Z_{\beta}=\int_{{\mathbb{R}}^{N}/S(N)}dx_{1}\ldots
dx_{N}\,\prod_{i<j}|x_{i}-x_{j}|^{2\beta}\,e^{-\sum_{i=1}^{N}V(x_{i})}$ (28)

with some single-particle potential $V$. The beauty of (28) is that these
models share many features with the fractional quantum Hall systems, they are
amenable to the analysis via powerful Ward identities, more recently they were
connected [38, 8, 30] to instanton partition functions [24] and topological
recursion [14]. The connection between the partition sums of supersymmetric
gauge theory and random matrix models was pointed out in [28] for
${\varepsilon}_{1}=-{\varepsilon}_{2}$ background, where it was suggested that
the ${\mathcal{N}}=1$ Dijkgraaf-Vafa theories [11] solved by matrix models
could potentially be analyzed systematically starting with the
${\mathcal{N}}=2$ instanton count. It is however puzzling that the matrix
models, at least in the asymptotic large $\hat{N}$ expansion, admit the
$\beta$-deformation, while the $\mathcal{N}=1$ theory seems to be confined to
the anti-self-dual $\Omega$-backgrounds. In a parallel series of developments,
[12] proposed to describe the _refined topological strings_ using
$\beta$-deformed matrix models.

All these examples above get embedded in a conventional supersymmetric gauge
theory setup (and string/M-theory via [29]) via identification

${\beta}=-\frac{\varepsilon_{1}}{\varepsilon_{2}}\,,{\rm or}\
-\frac{\varepsilon_{2}}{\varepsilon_{1}}$ (29)

The simplest way to understand the emergence of matrix-model like structure is
by looking at the instanton measure written in the plethystic form [31], and
observe that the contribution

$I_{\rm vm}={\sf
E}\left[-\frac{S_{12}S_{12}^{*}}{(1-q_{1}^{-1})(1-q_{2}^{-1})}\right]$ (30)

of the vector multiplets can be written as:

$I_{\rm vm}={\sf
E}\left[-\frac{(1-q_{1})M_{12}M_{12}^{*}}{(1-q_{2}^{-1})}\right]$ (31)

where we write

$S_{12}=N_{12}-K_{12}(1-q_{1})(1-q_{2})=M_{12}(1-q_{1})$ (32)

(see [31] for explanations of the notations) where, importantly, $M$ is the
pure character:

$M_{12}=\sum_{\alpha=1}^{n}\sum_{j=1}^{\infty}e^{a_{\alpha}}q_{2}^{j-1}q_{1}^{{\lambda}^{({\alpha})t}_{j}}\sim\sum_{I}e^{x_{I}}$
(33)

where $x_{I}$ will play the role of the eigenvalues of some _effective matrix_
${\mathcal{X}}$. In the limit where all $\varepsilon$’s, $a$’s are therefore
$x_{I}$’s are uniformly scaled to zero, the measure (31) approaches the
$\beta=-{\varepsilon}_{1}/{\varepsilon}_{2}$-deformed matrix ensemble for
$\mathcal{X}$:

${\Delta}_{\mathcal{X}}^{2\beta}:=\prod_{I\neq
J}(x_{I}-x_{J})^{-{\varepsilon}_{1}/{\varepsilon}_{2}}$ (34)

where we are being cavalier with the infinite-dimensional products. Similar
considerations apply to the contributions of the bi-fundamental and
fundamental hypermultiplets. Thus quiver gauge theories become multi-matrix
models.

One application of this imprecise dictionary is the matrix interpretation of
the folded and crossed instantons of [31] and followup papers. For example,
the folded instanton setup fuses ${\mathcal{N}}=2^{*}$ gauge theory on
${\mathbb{R}}^{4}={\mathbb{C}}^{2}$ with $\Omega$-background with parameters
${\varepsilon}_{1},{\varepsilon}_{2}$, and mass of the adjoint hypermultiplet
${\varepsilon}_{3}$ with the ${\mathcal{N}}=2^{*}$ theory on
${\mathbb{C}}^{2}$ with $\Omega$-background with parameters
${\varepsilon}_{2},{\varepsilon}_{3}$ and adjoint mass ${\varepsilon}_{1}$. A
typical instanton measure in this case looks like

${\sf
E}\left[-(1-q_{3})\frac{S_{12}S_{12}^{*}}{(1-q_{1}^{-1})(1-q_{2}^{-1})}-(1-q_{1})\frac{S_{23}S_{23}^{*}}{(1-q_{2}^{-1})(1-q_{3}^{-1})}+q_{3}\frac{1-q_{1}^{-1}q_{2}^{-1}q_{3}^{-1}}{1-q_{2}^{-1}}S_{12}S_{23}^{*}\right]$
(35)

Now, introducing the effective matrices ${\mathcal{X}},{\mathcal{Y}}$ whose
eigenvalues $(x_{I})$, $(y_{J})$ are defined through

$\sum_{I}e^{x_{I}}=\frac{S_{12}}{1-q_{1}}\,,\ \sum_{J}=\frac{S_{23}}{1-q_{3}}$
(36)

we rewrite (35) as

$\frac{{\Delta}_{\mathcal{X}}^{2\beta}{\Delta}_{\mathcal{Y}}^{2/{\beta}}}{{\rm
Det}(ad({\mathcal{X}})+{\varepsilon}_{3})^{\beta}\,{\rm
Det}(ad({\mathcal{Y}})+{\varepsilon}_{3})^{1/{\beta}}}\times\frac{{\rm
Det}({\mathcal{X}}\otimes 1-1\otimes{\mathcal{Y}}+{\varepsilon}_{3}){\rm
Det}({\mathcal{X}}\otimes 1-1\otimes{\mathcal{Y}}-{\varepsilon}_{3})}{{\rm
Det}({\mathcal{X}}\otimes 1-1\otimes{\mathcal{Y}})^{2}}$ (37)

in other words, a $\beta$-deformed supermatrix $3$-matrix model (cf. [16])
with the superpotential

${\rm
Tr}\left({\mathcal{Z}}[{\mathcal{X}},{\mathcal{Y}}]+{\varepsilon}_{3}{\mathcal{Z}}^{2}+V({\mathcal{X}})+W({\mathcal{Y}})\right)$
(38)

##  4 Complex spins

Spin chains, originally introduced to model the magnetic phenomena in metals,
later generalized to models of ice etc. typically involve finite-dimensional
representations of the spin group, e.g. $SU(2)$. In recent years an interest
to spin chains with infinite dimensional representations has been growing. One
source of interest is the integrable structure of two-dimensional conformal
field theory [4], quantum hydrodynamics [33, 22]. Another source of interest
is the integrable structure discovered in the planar ${\mathcal{N}}=4$ super-
Yang-Mills in four dimensions, where a spin chain based on the non-compact
supergroup $PSU(2,2|4)$ is detected [5]. Finally, a growing body of evidence
is provided by the supersymmetric sector of the $\Omega$-deformed
${\mathcal{N}}=2$ theories in four dimensions (and their five and six
dimensional lifts). Namely, a mass deformed superconformal quiver theory in
four dimensions is expected to be related to an integrable spin chain based on
the Yangian of the Kac-Moody algebra associated with the quiver. Instead of
stating the general conjecture, let us give a specific example: a super-QCD
${\mathcal{N}}=2$ theory with $SU(N)$ gauge group, with $2N$ hypermultiplets
in fundamental representation, with $\Omega$-deformation in two out of four
dimensions, is related to the $\mathfrak{sl}_{2}$ spin chain, with the space
of states being a subspace of the tensor product of $N$ infinite-dimensional
modules :

$H=V_{s_{1},a_{1}}(u_{1})\otimes V_{s_{2},a_{2}}(u_{2})\otimes\ldots\otimes
V_{s_{N},a_{N}}(u_{N})$ (39)

Here $V_{s,a}$ denotes a representation of the algebra $\mathfrak{sl}_{2}$,
which has a eigenbasis $|n\rangle$ labelled by $n\in\mathbb{Z}$, with

$L_{0}|n\rangle=(n+a)|n\rangle\,,\ L_{+}|n\rangle=(n+a-s)|n+1\rangle\,,\
L_{-}|n\rangle=(n+a+s)|n-1\rangle$ (40)

The parameter $s\in\mathbb{C}$ is the spin of the representation, related to
the Casimir by
$L_{0}^{2}-\frac{1}{2}\left(L_{+}L_{-}+L_{-}L_{+}\right)=s(s+1)$. The
parameter $a\in\mathbb{C}$ determines a mid-point in the spectrum of $L_{0}$.
The familiar finite-dimensional representations of $SU(2)$ are found for half-
integer $s$ and $a-s$, $a+s$ both non-negative integers.

The gauge theory corresponds to $A_{1}$ quiver. The $a_{i}$’s are the Coulomb
moduli, i.e. the eigenvalues of the adjoint scalar in the vector multiplet,
while the spins $s_{i}$’s are related to the masses by
$m_{i}^{+}-m_{i}^{-}=2s_{i}$, $i=1,\ldots,N$. The sums
$u_{i}=\frac{1}{2}\left(m_{i}^{+}+m_{i}^{-}\right)$ are the inhomogeneities,
the evaluation parameters of the Yangian modules based on the
$\mathfrak{sl}_{2}$ modules. Although the permutations of the $2N$ masses is a
symmetry of the four dimensional theory, the specific two dimensional
subsector defined by $\Omega$-deforming and adding a surface defect, depends
on the ordering of masses, which, in turn, defines the spins etc. See [20, 15,
21] for more details.

Another view on the same surface defect in the fully $\Omega$-deformed theory
reveals the connection to Knizhnik-Zamolodchikov equation borrowed from the
$\widehat{\mathfrak{sl}_{N}}$-representation theory [26, 15, 21]. Again,
complex spins and infinite-dimensional modules feature prominently.

##  5 Future directions

It would be nice to turn on interactions in the two and three dimensional
theories. The supersymmetric methods of a different kind (Wegner and Efetov)
are used in the study of disordered systems for many decades [43]. It would be
nice to find a relation between our higher dimensional theories with
conventional supersymmetry and the supercoset sigma models used in the studies
of integer Hall plateau transitions [44].

The supersymmetric interfaces [10] in one, two, and three dimensional
supersymmetric gauge theories with eight supercharges realize finite
dimensional spin chains. A $T$-dual construction realizes the same systems in
a four dimensional version of Chern-Simons theory [9]. It would be useful to
find a four dimensional analogue of the interfaces [10] providing a geometric
realization of $R$-matrix suitable for infinite dimensional representations of
quiver Yangians and quantum loop algebras.

The most interesting and mysterious question is the role of complex metrics in
string theory. In addition to the arguments of [42] one would like to use the
complex worldsheet metric to avoid unphysical poles in the bulk of
${\mathcal{M}}_{g}$ which pop up in time-dependent cosmological string
backgrounds [23], as well as the spacetime considerations of [41, 17, 40]. We
hope to return to this question in the future (whatever it means in the
complex world).

##  6 Acknowledgements

Discussions with M. Dedushenko, A. Grekov, S. Jeong, N. Lee, A. Okounkov, N.
Seiberg, O. Tsymbaliuk, are gratefully acknowledged. Many thanks to the
organizers of String-Math’2002 in Warsaw, especially to Piotr Sulkowski, for
providing a stimulating working environment in troubled times. Research was
partly supported by the National Science Foundation by award NSF PHY Award
2310279. Any opinions expressed are solely our own and do not represent the
views of the National Science Foundation.

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