# Focus on Content not Noise: Improving Image Generation for Nuclei
Segmentation by Suppressing Steganography in CycleGAN

Jonas Utz1 Tobias Weise1,2 Maja Schlereth1 Fabian Wagner2 Mareike Thies2
Mingxuan Gu2 Stefan Uderhardt3,4,5 Katharina Breininger1 1Department
Artificial Intelligence in Biomedical Engineering, Friedrich-Alexander-
Universität Erlangen-Nürnberg, Germany 2 Pattern Recognition Lab, FAU
Erlangen-Nürnberg, Germany 3 Department of Medicine 3 - Rheumatology and
Immunology, FAU Erlangen-Nürnberg, Germany 4 Deutsches Zentrum für
Immuntherapie, FAU Erlangen-Nürnberg, Germany 5 Exploratory Research Unit,
Optical Imaging Centre Erlangen, FAU Erlangen-Nürnberg, Germany
<EMAIL_ADDRESS>
###### Abstract

Annotating nuclei in microscopy images for the training of neural networks is
a laborious task that requires expert knowledge and suffers from inter- and
intra-rater variability, especially in fluorescence microscopy. Generative
networks such as CycleGAN can inverse the process and generate synthetic
microscopy images for a given mask, thereby building a synthetic dataset.
However, past works report content inconsistencies between the mask and
generated image, partially due to CycleGAN minimizing its loss by hiding
shortcut information for the image reconstruction in high frequencies rather
than encoding the desired image content and learning the target task. In this
work, we propose to remove the hidden shortcut information, called
steganography, from generated images by employing a low pass filtering based
on the discrete cosine transform (DCT). We show that this increases coherence
between generated images and cycled masks and evaluate synthetic datasets on a
downstream nuclei segmentation task. Here we achieve an improvement of $5.4$
percentage points in the F1-score compared to a vanilla CycleGAN. Integrating
advanced regularization techniques into the CycleGAN architecture may help
mitigate steganography-related issues and produce more accurate synthetic
datasets for nuclei segmentation.

##  1 Introduction

(a) Generated nuclei

(b) Real nuclei

Figure 1: 1(a) image with steganography amplified by adaptive histogram
equalization. The generated image contains information in seemingly empty
regions. 1(b) Processed real nuclei image for reference

Detecting individual nuclei and cellular structures in microscopy images is a
common task in biomedical image analysis and provides insights into cellular
processes. Deep neural networks have been successfully used to automate nuclei
detection but are typically trained in a supervised manner with a paired
dataset of nuclei images and manual labels [8, 10]. However, acquiring manual
labels is time-consuming and prone to errors and inconsistencies, especially
in fluorescence microscopy (FM) images, e.g., due to clustered nuclei, bleed-
through, and other artifacts. Unsupervised methods present an alternative,
with two approaches being commonly used: a two-stage approach where a
generative model is used to generate a synthetic dataset for training a
supervised segmentation network and a one-stage approach where a generative
model is directly used to perform nuclei segmentation. In both cases, the
generative model is trained with an unpaired dataset of real microscopy images
and synthetic masks which can be obtained from a suitable synthesis model such
as ellipses [1]. The model is then trained to perform unpaired image-to-image
translation between masks and nuclei images. A commonly chosen approach is the
CycleGAN architecture, introduced by Zhu _et al_. [13]. A CycleGAN employs two
generators, each dedicated to translate between two image domains and two
discriminators for the respective domains. A cycle-consistency loss for the
generators enforces consistency between an original image and its cycled
version to ensure the learned mapping is a bijection. Essentially, the cycle
loss ensures that the generators retain the image content and only transfer
the domain (_e.g_., from mask to nuclei). However, this constraint does not
necessarily lead to a content similarity between the input and the generated
image in the other domain. As a shortcut to minimize the cycle-consistency
loss, generators can embed a compressed version of the input in the generated
image as a high-frequency low-amplitude signal to create a perfect
reconstruction of the input image [4, 7]. This emerging property has been
denoted as CycleGAN _steganography_ since the generator actively hides
information from the discriminator. This can cause a mismatch between the
source image and the generated image leading to label noise in the context of
synthetic dataset generation. The prominent effect observed in nuclei images
could be attributed to the minimal influence of missing instances on the
discriminator’s output. Since the discriminator’s role is to classify whether
an image appears real, the absence of certain nuclei instances does not
considerably impact the realism of the microscopy image. An impression of the
CycleGAN steganography effect can be obtained from Figure 2. By performing
adaptive histogram equalization, which is an image processing technique that
enhances contrast by redistributing pixel intensities based on local regions
the steganography signal we visualize the steganography signal in Figure 1.
For reference, we apply the adaptive histogram equalization not only to the
generated image but also a real nuclei image. The unprocessed version of
Figure 1(a) is also used in Figure 2.

Figure 2: Severe example of CycleGAN steganography: Despite only one nucleus
being visible in the generated image, generator $F$ can reconstruct the input
mask almost perfectly. This is possible due to the embedding of a compressed
version of the input mask in the generated image as a high-frequency low-
amplitude signal by generator $G$.

The CycleGAN steganography property was first reported by Chu _et al_. in [4].
Since the hidden signal is of low amplitude with a high frequency, it has
characteristics similar to noise. Motivated by this observation, Porav _et
al_. [7] propose using DnCNN, an end-to-end differentiable image denoising
network, between the two generators to remove hidden information. The
denoising network is only applied for the cycle-consistency loss and is frozen
during CycleGAN training. They evaluate their method on a day-to-night style
transfer task with semantic segmentation. Löhdefink and Fingscheidt [5] try to
corrupt the hidden signal and break undesired information exchange between the
generators by adding Gaussian noise or discretization noise to the generated
images. They evaluate their approach on a supervised semantic segmentation of
driving scenes.
In this paper, we generate a fully synthetic dataset of FM images of nuclei
with a CycleGAN using transformed ellipses as input masks. To mitigate the
before-mentioned CycleGAN steganography problem, we propose to use the
discrete cosine transform (DCT) to perform low-pass filtering of the generated
images to remove the hidden signal adaptively. While we still observe a
discrepancy in content between the input mask and generated nuclei images
using our approach, the cycled mask matches the content of the generated
image. This allows us to use the generated images with their cycled masks as
training data for the supervised training of segmentation networks. We
evaluate our approach by testing only with synthetic data trained networks on
real data. We observe an increased average precision when DCT filtering is
used due to the improved alignment between the mask and generated image. To
summarize, the contributions of our work are:

  * •

We compare different CycleGAN steganography removal techniques on FM images.

  * •

We propose an efficient way to remove steganography from CycleGAN images by
introducing an additional filter step, based on the DCT.

  * •

We analyze the impact of CycleGAN steganography on downstream segmentation
performance on two public FM nuclei datasets and show improved performance
when using our proposed method.

##  2 Methods

Figure 3: Extended CycleGAN architecture featuring a DCT-based low-pass
filter, effectively removing undesired steganography from generated images.
The downstream segmentation network evaluates the model’s performance and is
trained on generated images and cycled masks.

###  2.1 CycleGAN

Our method is based on CycleGAN[13] which can learn a translation between two
domains, such as the domain of masks $X$ and nuclei images $Y$, with
individual samples from the domains denoted as $\boldsymbol{x}$ and
$\boldsymbol{y}$ respectively. This translation is performed by two generator
networks $G:X\rightarrow Y$ and $F:Y\rightarrow X$. Both generators are
trained with an adversarial loss using discriminator networks $D_{X}$ and
$D_{Y}$ for each domain. For generator $G$, which receives a mask image
$\boldsymbol{x}$ and generates a fake nuclei image $\boldsymbol{\hat{y}}$, the
adversarial loss $\mathcal{L}_{\mathrm{adv}}$ is defined as:

$\displaystyle\mathcal{L}_{\mathrm{adv}}$
$\displaystyle=\mathbb{E}_{\boldsymbol{y}\sim p_{\text{data
}}(\boldsymbol{y})}\left[\log D_{Y}(\boldsymbol{y})\right]$ (1)
$\displaystyle\phantom{={}}+\mathbb{E}_{\boldsymbol{x}\sim p_{\text{data
}}(\boldsymbol{x})}\left[\log\left(1-D_{Y}(G(\boldsymbol{x}))\right]\right.$

The adversarial loss for generator $F$ is computed analogously. An additional
cycle consistency loss is used to ensure that the mapping is bijective by
enforcing $F(G(\boldsymbol{x}))\approx\boldsymbol{x}$ and
$G(F(\boldsymbol{y}))\approx\boldsymbol{y}$. In other words, a sample should
be close to its original after being consecutively processed by both
generators. The cycle-consistency loss for generator $G$ ($F$ again
analogously) is formulated as:

$\displaystyle\mathcal{L}_{\text{cyc}}=\mathbb{E}_{\boldsymbol{y}\sim
p_{\text{data
}}(\boldsymbol{y})}\left[\|G(F(\boldsymbol{y}))-\boldsymbol{y}\|_{1}\right].$
(2)

###  2.2 DCT filtering

The DCT algorithm is best known in the context of the JPEG image compression
algorithm [11], which removes high frequencies from images for compression. We
propose eliminating the high-frequency pixel-level patterns [4] used by
CycleGANs to hide information by setting coefficients associated with higher
frequencies to zero. The DCT is a bijective mapping between an input $I$ and
the frequency domain. In this work, $I$ refers to the generated image by
CycleGAN, _i.e_., $\boldsymbol{\hat{y}}$ or $\boldsymbol{\hat{x}}$. In the
following equations, $I_{xy}$ represents the pixel intensity at position
$(x,y)$ in the image. The coefficients $C$ of the DCT are computed on the
image $I$ of $N\times M$ pixels as follows:

$\displaystyle\operatorname{DCT}(i,j)=\alpha(i)\alpha(j)\sum_{x=0}^{N-1}\sum_{y=0}^{M-1}I_{xy}hv,$
(3) $\displaystyle\text{where
}h=\cos\frac{\pi(2x+1)i}{2N},v=\cos\frac{\pi(2y+1)j}{2M}.$

The functions $h$ and $v$ weight the contributions of different frequency
components in the image $I_{xy}$ to calculate the DCT coefficient at position
$(i,j)$.The scaling factor $\alpha$ can be used to make the transform
orthonormal:

$\alpha(k)=\begin{cases}\frac{1}{\sqrt{2}},&\text{if }k=0\\\
1,&\text{otherwise}\end{cases}.$ (4)

When computed on $2D$ images of size $N\times M$, $N\times M$ coefficients
$C^{M\times N}$ are obtained. In order to eliminate specific frequency bands
from the image, we applied a technique of setting corresponding DCT
coefficients to zero in the frequency domain. This was achieved by element-
wise multiplication of the DCT coefficients matrix with a mask matrix, where
the mask contained zeros at the positions corresponding to higher frequencies.
The purpose of this step is to remove steganography from the generated images,
as the steganography signal, introduced into the images by the generators to
ease up reconstruction, is of low amplitude and high frequency. Specifically,
we performed low-pass filtering by nullifying the coefficients corresponding
to the $50\%$ highest frequencies. To obtain an image
$\boldsymbol{\hat{y}^{*}}$ or $\boldsymbol{\hat{x}^{*}}$ from the filtered
coefficients $C^{*}$, the inverse DCT is applied by computing:

$\displaystyle\operatorname{iDCT}(i,j)=\sum_{i=0}^{N-1}\sum_{j=0}^{M-1}\alpha(i)\alpha(j)C^{*}(i,j)hv.$
(5)

Naturally, the filtering does not only affect the hidden signal but any high-
frequency image information, such as foci inside of the nuclei. However, we
argue that for reconstructing masks from nuclei images, these high-frequency
features are not mandatory, and the differences are subtle. A side-by-side
comparison between an original image and a DCT filtered image and their
absolute difference is depicted in Figure 4(c). As the cycle loss mainly
promotes the CycleGAN steganography, we apply the filtering only to images
used for computation of the cycle loss, _i.e_., the adversarial losses are
computed on the unfiltered generator output. Figure 3 gives an overview of the
CycleGAN data flow with our proposed DCT filtering for synthetic dataset
generation.

(a) Generated image

(b) Filtered image

(c) Difference image

Figure 4: Generated images (a) filtered with our proposed DCT filtering are
almost indistinguishable since the filter is designed only to remove
steganography. In (c) the absolute difference between the filtered and the
generated image is shown.

###  2.3 Mask synthesis

Böhland _et al_. show that ellipses are a simple but suitable model for
representing the shape of nuclei [1] and can be used to create synthetic label
images. However, careful parameterization of the ellipses is required to match
the properties of the nuclei of real microscopy images. A substantial
deviation between synthetic and real distribution might result in CycleGAN
being unable to learn the mapping correctly. The standard equation for an
ellipse is

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,$ (6)

with the semi-major axis $a$ and the semi-minor axis $b$. To express how
elongated an ellipse is, we use the eccentricity $e$, which is for $a>b$
defined as:

$e=\sqrt{1-\left(\frac{b}{a}\right)^{2}}.$ (7)

For mask synthesis, we randomly sample the eccentricity in the range
$[0.4,0.9]$ to compute the minor axis $b$ with:

$b=\sqrt{1-e^{2}}a.$ (8)

The size of the semi-major axis $a$ is randomly sampled from a uniform range
which is determined on a per-dataset basis. Similarly, the number of nuclei
for each image is determined. A random rotation with a random angle from a
uniform distribution in the range [0°, 179°] is applied to each nucleus
individually. Nuclei are placed in the synthetic mask images so that they may
touch but never overlap.

##  3 Experiments

###  3.1 Evaluation setup

To evaluate the influence of DCT filtering on CycleGAN training, we generate
synthetic datasets using a CycleGAN with DCT filtering and an unmodified
CycleGAN as a baseline. Additionally, we implement two methods from
literature: the denoising approach (DnCNN) from [7] and the noise-injection
approach (Noise) using Gaussian noise as proposed in [5]. Instead of using the
synthetic input mask from our mask synthesis model and the generated
microscopy image by CycleGAN for our synthetic dataset, we use the cycled
input mask along with the generated microscopy image. This is motivated by the
fact that our method aims to increase correspondence between generated image
and cycled mask by filtering the generated image. Additionally, ellipses are a
suitable but imperfect approximation of the nuclei shape. If the CycleGAN
deviates slightly from this approximation, the cycled mask allows adaptation
to these deviations. This is under the assumption that CycleGAN steganography
was removed successfully. However, the cycled masks generated by CycleGAN are
close to binary masks that might contain minor imperfections. To convert the
binary masks into instance masks, we close small holes in the masks and
compute the Euclidean distance transform of the binary mask and then apply
Watershed segmentation with morphological operators. An example of this post-
processing step is visualized in Figure 5. The synthetic dataset is then
utilized for training the nuclei segmentation network StarDist [8].
Subsequently, the trained segmentation network is applied to real microscopy
data for testing. We use StarDist’s matching function to compute precision and
recall with IoU thresholds of $\tau=0.5$ and $\tau=0.75$. Additionally, we
report the F1 score. Each dataset is generated three times with random
initialization and results are averaged.

Figure 5: To convert the binary cycled masks into instance masks that can be
used to train segmentation networks, we apply the watershed algorithm on the
distance-transformed cycled masks.

###  3.2 Datasets

Figure 6: Sample images from DSB dataset

Figure 7: Samples from BBBC039v1 dataset

To evaluate our method, we train CycleGAN on two public FM nuclei datasets
from the Broad Bioimage Benchmark Image Collection [2]. For each nuclei image,
we normalize pixel values into the range $[-1,1]$ and extract random crops of
$256\times 256$ pixels. To create synthetic labels, we use the mask synthesis
model described in Section 2.3. The parameters for mask synthesis can be
obtained from Table 1 for the DSB dataset and Table 2 for dataset BBBC039v1.
They were determined based on a small set of three selected nuclei images for
each dataset individually. During the training of CycleGAN, new random masks
are created for each batch. After the training of CycleGAN, a fixed set of
$1500$ synthetic masks is used to generate nuclei images as a training set for
the downstream nuclei segmentation network.

####  3.2.1 Data Science Bowl (DSB) 2018 Nuclei Dataset

The Data Science Bowl (DSB) 2018 Nuclei Dataset was released for the 2018 Data
Science Bowl Kaggle challenge and is listed in the Broad Bioimage Benchmark
Collection as BBBC038v1. The dataset contains thousands of nuclei obtained
from various organisms, including humans, mice, and flies. Nuclei in the
dataset appear in different contexts and states, _e.g_., cell division,
genotoxic stress, and differentiation. While the original dataset contains a
variety of treatments (_e.g_., hematoxylin and eosin stained nuclei), only FM
images were used in this work. The dataset contains two labeled subsets,
stage1_train with 538 FM images and stage1_test with 53 FM images.
Additionally, the dataset contains an unlabelled subset, stage2_test of which
we select 667 FM images. We use stage1_train and stage2_test for CycleGAN
training. The downstream nuclei segmentation networks were tested on
stage1_test. Figure 7 visualizes the variety of the dataset using three
example images.

Table 1: Parameters of the mask synthesis model of DSB dataset. Each synthetic mask’s parameters are chosen from random uniform intervals denoted below. By using multiple subsets of parameter combinations, we allow the mask synthesis model to reflect the diverse characteristics of the DSB dataset. Length major axis | Nuclei count | Eccentricity $e$
---|---|---
$(5,10)$ | $(1,150)$ | $(0.4,0.9)$
$(10,15)$ | $(1,40)$ | $(0.4,0.9)$
$(15,20)$ | $(1,40)$ | $(0.4,0.9)$
$(20,25)$ | $(1,40)$ | $(0.4,0.9)$
$(25,30)$ | $(1,20)$ | $(0.4,0.9)$
$(30,35)$ | $(1,20)$ | $(0.4,0.9)$

####  3.2.2 BBBC039v1 dataset

The dataset BBBC039v1 is a FM dataset, and the images were obtained as part of
a high-throughput chemical screen on U-2 OS cells by Caicedo _et al_. [3]. The
dataset presents a variety of nuclei shapes, different degrees of clustering,
and consists of a total of 230000 single nuclei, which were manually
annotated. However, compared to the DSB dataset, it exhibits much less
variability in shape and nuclei style, as all images were obtained from the
same experiment. The dataset holds 200 unique images, each of size $520\times
696$ pixels. Of these, 150 were used for CycleGAN training. The remaining 50
images were used as the test set for the downstream nuclei segmentation
network. Exemplary images are shown in Figure 7.

Table 2: Parameters of the mask synthesis model of BBBC039v1 dataset. Each synthetic mask’s parameters are chosen from random uniform intervals denoted below. Compared to DSB, the BBBC039v1 dataset is less diverse concerning nuclei shape and distribution, reflected by our selected parameters. Length major axis | Nuclei count | Eccentricity $e$
---|---|---
$(10,20)$ | $(20,60)$ | $(0.6,0.9)$
$(20,40)$ | $(20,30)$ | $(0.6,0.9)$

###  3.3 Network training

####  3.3.1 CycleGAN

All compared CycleGAN variants are based on the network architecture described
in the original CycleGAN paper [13] and we used the official PyTorch
implementation.111Available at https://github.com/junyanz/pytorch-CycleGAN-
and-pix2pix/. We trained all networks for 100 epochs with a learning rate of
2e-4, which was linearly decayed to 0 for another 100 epochs, using an Adam
optimizer ($\beta_{1}=0.5$ and $\beta_{2}=0.999$) with a batch size of $1$. To
stabilize the training, the discriminator was updated from an image pool of 50
images [9]. The weight $\lambda$ for the cycle-consistency loss was set to
$\lambda_{F}=15.0$ for generator $F$ and to $\lambda_{G}=10.0$ for generator
$G$. We set $\lambda_{G}$ lower since the mapping from masks to nuclei images
is not unique, as it is not directly possible to infer the texture from a
nucleus based on a binary mask. For our comparison with [7], we use the
reference implementation of DnCNN [12], which is trained for blind Gaussian
denoising with a large range of noise levels $(\sigma\in[0,55])$. The
denoising network weights were frozen during training and inference runs.
Denoising is only applied for computation of the cycle loss.

####  3.3.2 StarDist

We use StarDist [8] as downstream nuclei segmentation network. It is a popular
choice in the field due to its effectiveness, versatility, and state-of-the-
art nuclei segmentation performance. However, in this work, the focus is not
on achieving the best possible result but on investigating the applicability
of differently generated synthetic datasets for a downstream task. We train
StarDist for 50 epochs with a batch size of 16 and a learning rate of $3\times
10^{-4}$. We increased the number of rays used for the star-convex polygon
representation from the default 32 to 64 in order to obtain more detailed
polygon shapes. All remaining parameters of StarDist were left unchanged.

##  4 Results

###  4.1 Assessment of image fidelity

While the primary goal of our proposed methods is to improve the downstream
task of nuclei segmentation, we report the results of the fidelity comparison
for generated and original images in Table 3 for completeness. To this end, we
compute the Fréchet inception distance (FID) as proposed in [6]. The CycleGAN
with DCT filtering yields the best FID score ($96.4$ for both datasets. The
low standard deviation ($\pm 0.7$ for the three repetitions indicates that DCT
filtering can also help to produce more stable results. We generally observe
unusually high FID scores among all datasets. However, the FID score
computation is based on a feature extractor trained on natural images. This
representation might not be optimal for quality assessment of FM images,
leading to higher scores and limited validity.

Table 3: FID score (lower is better) is chosen as metric to compare the image fidelity of generated data for datasets DSB and BBBC039v1 with original images from the respective datasets Model | DSB | BBBC039v1
---|---|---
Orig. CycleGAN | $110.1\,\pm\,3.0$ | $380.2\,\pm\,12.9$
w. DCT (ours) | $\bf{96.4}\,\pm\,\bf{0.7}$ | $328.8\,\pm\,13.8$
w. noise [5] | $104.9\,\pm\,4.6$ | $376.6\,\pm\,39.7$
w. DnCNN [7] | $101.3\,\pm\,3.5$ | $\bf{327.1}\,\pm\,\bf{4}$

###  4.2 Qualitative results of synthetic images

A representative sample of generated images using the discussed methods is
depicted along with (from left to right) the generating input mask, the
respective cycled mask, and the post-processed instance mask as an overlay in
Figure 8. For all four analyzed methods, we observe a deviation from the
original input mask. This deviation is not necessarily negative as long as it
is represented in the cycled mask. This is not the case for the unmodified
CycleGAN. Multiple nuclei instances are not rendered into the image but occur
in the cycled mask. n this case, it is almost a pixel-perfect copy of the
original input mask. The reason for this is the before-mentioned CycleGAN
steganography. If our proposed DCT filtering is used during training, we
observe no steganography during test time. The cycled mask resembles the
generated image closely. We also observe that the nuclei do not exhibit an
exact ellipse shape, which we consider a positive effect. This allows the
model to adapt to the shape of nuclei found in the training set and overcome
shortcomings of the ellipse synthesis representations. For the approaches from
literature, specifically the filtering with the denoising network DnCNN [7],
we observe that steganography was reduced but not fully removed as there are
nuclei in the cycled mask that were not present in the generated image. For
the noise injection approach from [5], we observe the largest deviation from
the input mask in the generated image. However, since this is reflected in the
cycled mask, we conclude that removing the steganography was successful. We
also observe a slightly less realistic reproduction of the typical nuclei
texture found in the remaining images when noise injection is used.

(a) Original input mask

Generated Image

Cycled Mask

Overlay

(b) Original CycleGAN

(c) with DCT filtering (ours)

(d) with DnCNN [7]

(e) with Gaussian noise injection [5]

Figure 8: Generated images based on the DSB dataset along with the cycled
mask. The final segmentation mask used for StarDist training is shown in the
third column as an overlay over the generated image.

###  4.3 Segmentation performance on DSB

The results for the segmentation network tested on the real-data test split
textitstage1_test of the DSB dataset are reported in Table 4 for an IoU
threshold of $\tau=0.5$ and in Table 5 for an IoU threshold $\tau=0.75$. The
best results in terms of precision are achieved with the training set created
by our proposed CycleGAN with DCT filtering, outperforming the baseline
CycleGAN by $19.7$ percentage points for $\tau=0.5$ and $38.7$ percentage
points for $\tau=0.75$. For the approach using DnCNN we observe comparable
performance at $\tau=0.5$, with a larger gap ($8.3$ percentage points less
precision compared to DCT) for the more challenging threshold of $\tau=0.75$.
The noise-injection approach suffers from a high variance leading to a
slightly worse average performance.

Table 4: StarDist segmentation performance in % on DSB stage1_test split, trained on synthetic datasets generated from different CycleGAN methods. Scores are computed for $\tau=0.5$ Model | Precision | Recall | F1-Score
---|---|---|---
Orig. CycleGAN | $74.3\,\pm\,5$ | $63.7\,\pm\,3$ | $68.6\,\pm\,4$
w. DCT (ours) | $94.0\,\pm\,1$ | $61.1\,\pm\,1$ | $74.0\,\pm\,1$
w. noise [5] | $87.0\,\pm\,14$ | $39.8\,\pm\,31$ | $50.3\,\pm\,37$
w. DnCNN [7] | $92.3\,\pm\,1$ | $62.8\,\pm\,3$ | $\bf{74.7}\,\pm\,\bf{2}$
Table 5: StarDist segmentation performance in % on DSB stage1_test, trained on synthetic datasets generated from different CycleGAN methods. Scores are computed for $\tau=0.75$, which requires more precise predictions compared to $\tau=0.5$ Model | Precision | Recall | F1-Score
---|---|---|---
Orig. CycleGAN | $49.9\,\pm\,5$ | $40.2\,\pm\,3$ | $43.3\,\pm\,4$
w. DCT (ours) | $85.6\,\pm\,1$ | $55.6\,\pm\,1$ | $\bf{67.4}\,\pm\,\bf{1}$
w. noise [5] | $67.7\,\pm\,32$ | $35.5\,\pm\,29$ | $44.5\,\pm\,36$
w. DnCNN [7] | $77.3\,\pm\,11$ | $52.4\,\pm\,5$ | $62.4\,\pm\,7$

###  4.4 Segmentation performance on BBC039v1

We report results for the downstream segmentation task on the BBBC039v1 test
set for $\tau=0.5$ and $\tau=0.75$ in Table 6 and Table 7 respectively. We
observe a high precision for all methods except the noise injection,
indicating a low number of false positives and a high number of true
positives. Compared to the DSB dataset, we observe higher scores among all
methods and metrics due to the lower diversity and therefore complexity of the
data set. The lower complexity results in a lower amount of steganography.
Consequently, the methods differ only slightly in their F1-score, with DCT
filtering being the best method by a margin of 2 percentage points
($\tau=0.5$) compared to the baseline. The margin between the proposed DCT
filtering and the baseline CycleGAN methods increases for a higher threshold
($\tau=0.75$) to $8.4$ percentage points in terms of F1-score. This indicates
that DCT filtering leads to more realistic nuclei shapes. Similar to the
results of the DSB dataset, we observe a high variability in performance for
the noise-injection results. The reason of this is liekly due to noise
injection causing the GAN training to be less stable, generating more
deficient samples.

Table 6: StarDist segmentation performance in % on BBBC039v1 test set, trained on synthetic datasets generated from different CycleGAN methods. Scores are computed for $\tau=0.5$ Model | Precision | Recall | F1-Score
---|---|---|---
Orig. CycleGAN | $99.9\,\pm\,1$ | $86.3\,\pm\,6$ | $92.1\,\pm\,4$
w. DCT (ours) | $97.9\,\pm\,1$ | $90.7\,\pm\,1$ | $\bf{94.1}\,\pm\,\bf{1}$
w. noise [5] | $82.1\,\pm\,29$ | $76.3\,\pm\,28$ | $79.1\,\pm\,28$
w. DnCNN [7] | $99.9\,\pm\,1$ | $82.4\,\pm\,2$ | $90.3\,\pm\,2$
Table 7: StarDist segmentation performance on BBBC039v1 test set, trained on synthetic datasets generated from different CycleGAN methods. Scores are computed for $\tau=0.75$ Model | Precision | Recall | F1-Score
---|---|---|---
Orig. CycleGAN | $82.8\,\pm\,11$ | $72.6\,\pm\,15$ | $77.3\,\pm\,13$
w. DCT (ours) | $89.2\,\pm\,10$ | $82.5\,\pm\,8$ | $\bf{85.7}\,\pm\,\bf{9}$
w. noise [5] | $65.2\,\pm\,54$ | $61.1\,\pm\,50$ | $61.1\,\pm\,52$
w. DnCNN [7] | $91.4\,\pm\,11$ | $75.5\,\pm\,10$ | $82.6\,\pm\,10$

##  5 Discussion

In this work, we proposed a new anti-steganography approach for CycleGANs and
evaluated existing methods to improve CycleGAN consistency for FM nuclei
generation. We further showed the performance of nuclei segmentation trained
solely on synthetic data. Using our proposed method, we observe a qualitative
improvement in the generated images and masks and outperform existing anti-
steganography approaches quantitatively on a downstream segmentation task. A
considerable advantage of DCT filtering compared to using a DnCNN is the
reduced computational overhead. Instead of feeding the generated images
through a neural network (_i.e_., the DnCNN), we make use of the efficient and
computationally cheap implementation of the DCT algorithm. By pairing the
cycled mask with the generated image for synthetic dataset creation, we allow
the network to deviate from the input mask, allowing CycleGAN to overcome
potential shortcomings of the mask synthesis model. This can be a crucial
factor in enabling CycleGAN to learn the desired mapping. In [1] Böhland _et
al_. show that CycleGAN can only produce satisfactory results if the shape and
occurrence of synthetic masks are similar to the ground-truth distribution. In
unsupervised settings, the ground-truth distribution is not available, and
estimating the parameters of the mask synthesis model relies on a limited
number of samples, which can introduce challenges and uncertainties. A
limitation of using the cycled masks is the necessity of introducing post-
processing steps to convert them into instance masks. This in turn might
induce an unknown bias into the processed masks and is thus prone to errors.
This could be avoided by directly generating instance masks in CycleGAN,
_e.g_., by rendering additional boundaries of masks or using a StarDist-like
mask formulation based on star-convex polygons.
In this work, we did not consider the segmentation capabilities of CycleGAN
itself. Due to its architectural design, CycleGAN is not only trained to
generate microscopy images from masks but also to segment microscopy images
into masks. In future work, we want to explore to what extent this allows this
allows CycleGAN to be used as an unsupervised nuclei segmentation network,
_e.g_., by using segmentation losses such as the Dice loss for cycle
consistency. Exploring the generalizability of the proposed steganography
removal approach to other cycle-loss-based image-to-image translation tasks
could open up new possibilities for improving various GAN-based applications.

##  6 Acknowledgments

We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - Projektnummer 405969122 (J. U., S. U., K. B.)
and Dhip campus - Bavarian aim (K. B., M. S., T. W.) M. T., F. W., M. G., and
S. U. received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (ERC Grant No.
810316).

## References

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  * [4] Casey Chu, Andrey Zhmoginov, and Mark Sandler.  CycleGAN, a master of steganography.  arXiv preprint arXiv:1712.02950, 2017.
  * [5] Jonas Löhdefink and Tim Fingscheidt.  Improving performance of semantic segmentation cyclegans by noise injection into the latent segmentation space.  arXiv preprint arXiv:2201.06415, 2022.
  * [6] Gaurav Parmar, Richard Zhang, and Jun-Yan Zhu.  On aliased resizing and surprising subtleties in gan evaluation.  In CVPR, 2022.
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# A new approach to string theory.

Albert Schwarz  A. Schwarz
Department of Mathematics
University of California
Davis, CA 95616, USA,
schwarz @math.ucdavis.edu

###### Abstract

In the present paper we consider quantum theories obtained by quantization of
classical theories with first-class constraints assuming that these
constraints form a Lie algebra. We show that in this case, one can construct
physical quantities of a new type. We apply this construction to string
theory. We find that scattering amplitudes in critical bosonic closed string
theory can be expressed in terms of physical quantities of the new type. Our
techniques can be applied also to superstring and heterotic string.

Keywords Operator frormalism; conformal field theory; BRST formalism

##  1 Introduction

In BRST formalism we can construct physical quantities by taking correlation
functions of BRST-closed operators in a physical (BRST-closed) state. (These
correlation functions can be considered as polylinear functions on BRST-
cohomology.) In the present paper, we consider quantum theories obtained by
quantization of classical theories with first-class constraints assuming that
these constraints form a Lie algebra. We show that in this case, one can
construct physical quantities of a new type (Section 2). We apply this
construction to string theory ( Sections 5 and 6). We find that scattering
amplitudes in critical bosonic closed string theory can be expressed in terms
of physical quantities of the type described in Section 2. Our techniques can
be applied also to superstring and heterotic string; this will be shown in a
separate paper [1].

Our results about scattering amplitudes in string theory are based on a
comparison with the expression of these amplitudes in operator formalism [2].
The operator formalism is closely related to Segal’s definition of conformal
field theory [3]. We remind this definition (or, more precisely, the
modification of this definition that is used in operator formalism) and the
main ideas of operator formalism (Sections 3 and 4).

One of the main takeaways from our results: Knowing the one-string space in
BRST formalism one can calculate physical quantities describing interacting
strings.

The present paper is a byproduct of my attempts to formulate string theory in
algebraic and geometric approaches to quantum theory [4], [5], [6], [7]. The
results of this paper show the way to solve this problem: it is sufficient to
work in the one-string space.

##  2 General considerations

For every supermanifold $M$ we can construct a supermanifold $\Pi TM$
reversing the parity in fibers of tangent bundle $TM$. If $(x^{1},...,x^{m})$
are coordinates in $M$ then the coordinates in $\Pi TM$ are
$(x^{1},...,x^{m},\xi^{1},...,\xi^{m})$ where the parity of $\xi^{k}$ is
opposite to the parity of $x^{k}.$ Polynomial functions on $\Pi TM$ are
identified with differential forms on $M$, more general functions with
pesudodifferential forms. The formula $Q=\xi^{k}\frac{\partial}{\partial
x^{k}}$ specifies an odd vector field on $\Pi TM$; the anticommutator of this
vector field with itself vanishes. We can say that $Q$ specifies a structure
of $Q$-manifold on $\Pi TM$; in other words, $Q$ is a homological vector
field. It defines an odd derivation $d$ of the algebra of functions on $\Pi
TM$. The oprator $d$ obeys $d^{2}=0$ and can be identified with de Rham
differential.

There exists an invariant definition of $\Pi TM$ that shows that the
construction of $\Pi TM$ is functorial. In other words, a map $M\to N$ induces
a map $\Pi TM\to\Pi TN$; the induced map agrees with de Rham differential.
Namely, $\Pi TM$ can be identified with the space of maps of
$(0,1)$-dimensional superspace $\mathbb{R}^{0,1}$ into $M$. Every vector field
on the space $\mathbb{R}^{0,1}$ induces a vector field on the space of maps.
The Lie superalgebra of vector fields on $\mathbb{R}^{0,1}$ is
$(1,1)$-dimensional; an odd vector field on $\mathbb{R}^{0,1}$ induces a
homological vector field $Q$ on the space of maps, an even vector field
induces a grading on the algebra of functions on this space.

This remark allows us to say that $\Pi T\textgoth{g}$ where $\textgoth{g}$ is
a Lie superalgebra is equipped with the structure of differential Lie
superalgebra. We denote this Lie superalgebra by $\textgoth{g}^{\prime}$ and
the differential in it by $Q$. Sometimes it is convenient to consider a semi-
direct product $\textgoth{g}^{\prime\prime}$ of the Lie superalgebra and the
Lie superalgebra of vector fields on $\mathbb{R}^{0,1}$.

Similarly, if $G$ is a supergroup then $G^{\prime}=\Pi TG$ is also a
supergroup (the multiplication in $G$ induces a multiplication in the space of
maps $\mathbb{R}^{0,1}\to G$). The Lie superalgebra of $G^{\prime}$ can be
identified with $\textgoth{g}^{\prime}$ where $\textgoth{g}$ stands for the
Lie superalgebra of $G.$ The homological vector field on $G^{\prime}$ induces
the differential $Q$ on $\textgoth{g}^{\prime}.$

If $\textgoth{g}$ is a Lie algebra with generators $T_{k}$ and commutation
relations $[T_{k},T_{l}]=f_{kl}^{r}T_{r}$ the Lie superalgebra
$\textgoth{g}^{\prime}$ has even generators $T_{k}$, odd generators $b_{k}$
and commutation relations $[T_{k},T_{l}]=f_{kl}^{r}T_{r}$,
$[T_{k},b_{l}]=f_{kl}^{r}b_{r}$, $[b_{k},b_{l}]_{+}=0.$ The generators $T_{k}$
are $Q$-exact (acting by $Q$ on $b_{k}$ we obtain $T_{k}$).

Every element of $G^{\prime}$ can be represented in the form
$g\exp(\mu^{k}b_{k})$ where $\mu_{k}$ are odd parameters, $g\in G$, and $\exp$
stands for the exponential map of Lie superalgebra into the corresponding
supergroup.

Let us consider now a classical system that after quantization can be
described by Hilbert space $\cal E$. If a new classical system is obtained
from this system by means of constraints obeying a Lie algebra $\textgoth{g}$
of group $G$ then the quantized system can be described in BRST-formalism by
the space ${\cal E}^{\prime}$ obtained by adding ghosts to $\cal E$.( To
obtain ${\cal E}^{\prime}$ we take tensor product of $\cal E$ by
representation space of canonical anticommutation relations
$[\hat{c}^{k},\hat{b}_{l}]_{+}=\delta^{k}_{l},[\hat{c}^{k},\hat{c}_{r}]_{+}=0,[\hat{b}_{l},\hat{b}_{r}]_{+}=0.$)
The constraints induce operators $T_{k}$ in $\cal E$; the BRST-operator
$\hat{Q}$ has the form
$\hat{Q}=T_{k}c^{k}+\frac{1}{2}f^{r}_{kl}\hat{c}^{k}\hat{c}^{l}\hat{b}_{r}$
where $f^{r}_{kl}$ are structure constants of the algebra $\textgoth{g}$ and
$\hat{c}^{k},\hat{b}_{l}$ are ghosts obeying canonical anticommutation
relations (in the case of infinite number of degrees of freedom we should use
normal ordering; this can lead to anomalies). The operators
$\hat{T}_{k}=T_{k}+f^{r}_{kl}\hat{c}^{l}\hat{b}_{r}$ are BRST-trivial in
${\cal E}^{\prime}$; this follows from relation
$\hat{T}_{k}=[\hat{Q},\hat{b}_{k}]_{+}.$ Together with operators $\hat{b}_{k}$
they specify a representation $\psi$ of the Lie superalgebra
$\textgoth{g}^{\prime},$ i.e. a homomorphism of $\textgoth{g}^{\prime}$ into
the space $\cal L$ of linear operators acting in ${\cal E}^{\prime}$; this
homomorphism agrees with differentials (in this statement the space $\cal L$
is considered as Lie superalgebra).

We assume that the representation $\psi$ is integrable(=can be exponentiated),
i.e. it can be obtained from a representation $\Psi$ of the group
$G^{\prime}.$ ( Recall that $\textgoth{g}$ is the Lie algebra of the group
$G.$)

The representation $\Psi$ induces a map $\Psi^{*}$ of ${\cal L}^{*}$ ( of the
superspace of linear functionals on $\cal L$) into the space of functions on
$G^{\prime}$ ( the space of (pseudo)differential forms on $G$). This map
agrees with differentials; this means, in particular, that it transforms
$Q$-closed element $\sigma\in{\cal L}^{*}$ into a closed (in general
inhomogeneous) form $\Psi^{*}(\sigma)$ on $G$. (The BRST-operator acts on
$\cal L$ as (anti)commutator with $\hat{Q}$, this action induces a BRST-
operator on ${\cal L}^{*}$ .)

Integrating $\Psi^{*}(\sigma)$ over a cycle in $G$ we obtain a physical
quantity (the integral does not change if we add to $\sigma$ a $Q$-exact term,
hence it depends only on BRST cohomology class of $\sigma$).

If $K$ is a subgroup of $G$ and the form $\Psi^{*}(\sigma)$ descends to $G/K$
we can integrate the form on $G/K$ over a cycle in $G/K$. ( Here $G/K$ stands
for the space of right cosets= space of orbits of left action of $K$ on $G$.)
This construction leads to a more general class of physical quantities.

Let us consider a special case when a $Q$-closed element $\sigma\in{\cal
L}^{*}$ is specified by the formula

$\sigma(A)=\langle\rho|A|\chi\rangle$ (1)

where $A\in\cal L$, $\langle\rho|\in(\cal E^{\prime})^{*}$ and
$|\chi\rangle\in{\cal E^{\prime}}$ are $Q$-closed. Taking $A$ as
$\Psi(g^{\prime})$ where $g^{\prime}\in G^{\prime}$ we obtain a $Q$-closed
function

$(\Psi^{*}\sigma)(g^{\prime})=\langle\rho|\Psi(g^{\prime})|\chi\rangle$ (2)

on $G^{\prime}$ (a non-homogeneous closed form on $G$). Representing
$g^{\prime}\in G^{\prime}$ as $g\exp(\mu^{k}b_{k}))$ where $g\in G$ we obtain

$(\Psi^{*}\sigma)(g\exp(\mu^{k}b_{k}))=\langle\rho|\Psi(g\exp(\mu^{k}b_{k})))|\chi\rangle=\langle\rho|\Psi(g)\exp(\mu^{k}\hat{b}_{k})|)\chi\rangle$
(3)

( We use the fact that $G$ is embedded into $G^{\prime}$, hence $\Psi$ is
defined on $G.$)

The function (2) descends to $G^{\prime}/K^{\prime}=(G/K)^{\prime}$
(equivalently the corresponding closed (pseudo)differential form descends to
$G/K$) if $\langle\rho|$ is a $K^{\prime}$-invariant element of $({\cal
E}^{\prime})^{*}$ (The relation $\langle\rho|\Psi(k^{\prime})=\langle\rho|$
for $k^{\prime}\in K^{\prime}$ implies that
$(\Psi^{*}\sigma)(k^{\prime}g^{\prime})=(\Psi^{*}\sigma)(g^{\prime}).$)

Homogeneous components of the form (3) are closed forms that can be
represented as

$\langle\rho|\Psi(g)B|\chi\rangle$ (4)

where $B$ is a homogeneous polynomial with respect to $\hat{b}_{k}.$

Notice that our constructions can be applied to the case when $\cal E$ and
${\cal E}^{\prime}$ are replaced by their $n$-th tensor powers; then the
groups $G$ and $G^{\prime}$ should be replaced by direct products of $n$
copies of these groups.

One can consider a more general situation when we have two subgroups of the
group $G$ denoted by $K$ and $H$, the element $|\chi\rangle$ is a
$H^{\prime}$-invariant element of ${\cal E}^{\prime}$ (i.e.
$\Psi(h^{\prime})|\chi\rangle=|\chi\rangle$ for all $h^{\prime}\in
H^{\prime}$) and the element $\langle\rho|$ is a $K^{\prime}$-invariant
element of the dual space. Then the function (2) descends to
$H^{\prime}\backslash G^{\prime}/K^{\prime}$ (to the space of double cosets).

Our consideration can be generalized to the case when $\textgoth{g}$ is a Lie
algebra of semigroup $G.$ In this case one should assume that the
representation $\psi$ is semiintegrable, i.e. it can be obtained from
representation of semigroup $G^{\prime}$ having Lie algebra
$\textgoth{g}^{\prime}.$

Another important generalization: it is sufficient to assume that
$\langle\rho|$ is $\textgoth{k}^{\prime}$-invariant (i.e.
$\langle\rho|\psi(\textgoth{k}^{\prime})=0$.) Here $\textgoth{k}$ is a Lie
subalgebra of the Lie algebra $\textgoth{g}$. If $\textgoth{k}$ is a Lie
algebra of a connected subgroup $K$ of semigroup $G$ this assumption is
equivalent to $K^{\prime}$-invariance of $\langle\rho|$; we come back to the
situation considered above. However, in the situation of the next sections the
Lie algebra $\textgoth{k}$ cannot be considered as a Lie algebra of some
group.

It is essy to check that $\textgoth{k}^{\prime}$-invariance of $\langle\rho|$
implies that the function (2) descends to $G^{\prime}/\textgoth{k}^{\prime}$
(equivalently the corresponding form descends to $G/\textgoth{k}$).

To define the space of cosets $G/\textgoth{k}$ we consider the left action of
the Lie algebra $\textgoth{k}$ on the semigroup $G.$ This action specifies a
foliation of $G$; one can define $G/\textgoth{k}$ as the space of leaves of
the foliation.

Alternatively, $G/\textgoth{k}$ can be defined as a connected manifold $M$
where the semigroup $G$ acts transitively with Lie stabilizer $\textgoth{k}.$
(We say that action of $G$ on $M$ is transitive if it induces a surjective map
$\tau_{m}$ of the Lie algebra $\textgoth{g}$ to the tangent space of $M$ at
any point $m\in M.$ The Lie stabilizer at the point $m$ is defined as the
kernel of $\tau_{m}$; we assume that there exists a point with a Lie
stabilizer $\textgoth{k}.$)

More generally,if $|\chi\rangle$ is $\textgoth{h}^{\prime}$-invariant and
$\langle\rho|$ is $\textgoth{k}^{\prime}$-invariant then the function(2)
descends to a function on the space of double cosets
$\textgoth{h}^{\prime}\backslash G^{\prime}/\textgoth{k}^{\prime}$ ( to a
(pseudo)differential form on the space of double cosets
$\textgoth{h}\backslash G/\textgoth{k}$).

In this statement $\textgoth{k}$ and $\textgoth{h}$ are Lie subalgebras of
$\textgoth{g}.$

##  3 CFT, TCFT, SCFT, TSFT

Let us start with a reminder of some basic constructions that are used in two-
dimensional conformal field theory (CFT) and in operator formalism of string
theory.

Recall that two Riemannian manifolds are conformally equivalent (specify the
same conformal manifold) if there exists a diffeomorphism between these
manifolds preserving the Riemannian metric up to multiplication by a function.

A two-dimensional oriented conformal manifold can be identified with a complex
manifold of complex dimension $1$. Maps preserving conformal structure are
either holomorphic or antiholomorphic maps of complex manifolds.

We consider moduli space of complex curves (= one-dimensional compact
connected complex manifolds) of genus g with boundary consisting of $n$
parametrized circles.( We assume that these circles are ordered.) This moduli
space denoted by ${\cal P}({\rm g},n)$ can be regarded as an infinite-
dimensional complex manifold. Equivalently one can define ${\cal P}({\rm
g},n)$ as the moduli space of complex curves of genus g with $n$ embedded
standard discs.

It is easy to construct a natural map $\phi_{m,n}:{\cal P}({\rm
g},n)\times{\cal P}({\rm g^{\prime}},n^{\prime})\to{\cal P}({\rm g}+{\rm
g}^{\prime},n+n^{\prime}-2)$ identifying the last circle in the first factor
with the first circle in the second factor. Similarly one can construct a map
$\phi_{n}:{\cal P}({\rm g},n)\to{\cal P}({\rm g}+1,n-2)$ identifying two last
circles.

In particular, the map ${\cal P}({\rm 0},2)\times{\cal P}({\rm 0},2)\to{\cal
P}({\rm 0},2)$ specifies a structure of semigroup on ${\cal P}({\rm 0},2)$ .
This semigroup was introduced independently by Neretin, Konntsevich, and
Segal; we call it the semigroup of annuli and denote it by $\mathcal{A}.$

The map ${\cal P}({\rm 0},2)\times{\cal P}({\rm g},n)\to{\cal P}({\rm g},n)$
specifies an action of $\mathcal{A}$ on ${\cal P}({\rm g},n).$

Notice that the Lie algebra of $\mathcal{A}$ can be identified with diff (
with the complexification of the Lie algebra of vector fields on a circle); in
other words, this is a complex Lie algebra with generators $l_{n}$ obeying
$[l_{m},l_{n}]=(m-n)l_{m+n}.$

In Segal’s approach, a CFT having central charge $c=0$ specifies a map
$\sigma_{{\rm g},n}:{\cal P}({\rm g},n)\to{\cal H}^{n}$ where $\cal H$ is a
vector space equipped with bilinear inner product ${\cal H}\otimes{\cal
H}\to\mathbb{C}.$ Using this inner product one can construct maps
$\tilde{\phi}_{m,n}:{\cal H}^{m}\otimes{\cal H}^{n}\to{\cal H}^{m+n-2}$ and
$\tilde{\phi}_{n}:{\cal H}^{n}\to{\cal H}^{n-2}.$ Segals’s axioms are
compatibility conditions for maps $\sigma_{{\rm g},n}$,
$\phi_{m,n},\phi_{n},\tilde{\phi}_{m,n},\tilde{\phi}_{n}.$

The action of semigroup $\mathcal{A}$ on ${\cal P}({\rm g},1)$ and complex
conjugate action generate an action of $\mathcal{A}\times\mathcal{A}$ and
corresponding Lie algebra diff$\times$diff on $\cal H.$

A CFT having central charge $c\neq 0$ specifies a map sending a point of
${\cal P}({\rm g},n)$ into a point of ${\cal H}^{n}$ defined up to a
multiplication by a number. In this case, we have a projective representation
of diff$\times$diff in $\cal H$, i.e. a representation of the central
extension of this algebra in $\cal H.$

The central extension of diff is called Virasoro algebra; we denote it by Vir.

Let us consider CFT with a central charge $c.$ The Lie algebra Vir $\times$
Vir acts on its space of states $\cal H$. In other words, we have operators
$L_{m},\tilde{L}_{n}$ obeying

$[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n},$

$[\tilde{L}_{m},\tilde{L}_{n}]=(m-n)\tilde{L}_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n},$
$[L_{m},\tilde{L}_{n}]=0.$

There exist many important analogs of these constructions. In particular, one
can consider spaces ${\cal P}^{\prime}({\rm g},n)=\Pi T{\cal P}({\rm g},n)$
instead of ${\cal P}({\rm g},n)$. It is obvious that analogs of maps
$\phi_{m,n}$ and $\phi_{n}$ exist for these spaces. It follows that
$\mathcal{A}^{\prime}={\cal P}^{\prime}(0,2)$ is a semigroup acting on ${\cal
P}^{\prime}({\rm g},n).$

Let us fix a $\mathbb{Z}_{2}$-graded vector space $\cal H$ equipped with inner
product and parity reversing differential $q$ respecting this product. Then
topological conformal field theory (TQFT) is specified by maps ${\cal
P}^{\prime}({\rm g},n)\to{\cal H}^{n}.$ 111 Notice that such a map specifies a
differential form on ${\cal P}({\rm g},n)$ with values in the space ${\cal
H}^{n}$. The more standard definition of TQFT is formulated in terms of these
forms.We impose compatibility conditions of these maps with analogs of maps
$\phi_{m,n},\phi_{n},\tilde{\phi}_{m,n},\tilde{\phi}_{n}.$ as well as
compatibility conditions with the differential $q$ and homological vector
field on ${\cal P}^{\prime}({\rm g},n).$ It follows that the semigroup
$\mathcal{A}^{\prime}\times\mathcal{A}^{\prime}$ and its Lie algebra
diff${}^{\prime}\times{\rm diff}^{\prime}$ act in $\cal H.$

Replacing in the definition of CFT conformal manifolds with superconformal
manifolds we obtain a definition of superconformal field theory (SCFT). One
can define also topological superconformal field theory (TSFT); the
modification that leads from SCFT to TSFT is very similar to the modification
leading from CFT to TCFT.

##  4  Subalgebras, stabilizers, invariants

The Lie algebra diff consists of complex vector fields on a circle. A very
general way to construct Lie subalgebras of diff is based on the consideration
of embedding of the circle into a complex manifold $M$. Then complex vector
fields on the circle that can be holomorphically extended to $M$ constitute a
Lie subalgebra of diff. We can get a smaller Lie subalgebra assuming that the
extended vector field vanishes on some subset of $M.$

A more concrete realization of this construction can be obtained if we take as
$M$ a one-dimensional connected complex manifold ( a complex curve) with $n$
parametrized boundary components ($n$ circles $B_{1},..,B_{n}$), $p$ punctures
($p$ deleted points $x_{1},...,x_{p}$) and $m$ marked points
$u_{1},...,u_{m}.$ ( Equivalently one can consider a complex curve
$\underline{M}$ with $n$ embedded disks, $p$ punctures and $m$ marked points,
then we take as $M$ the curve $\underline{M}$ with deleted disks.) The moduli
space of objects of this kind will be denoted by ${\cal P}(n,p,m)$, its
connected components (labeled by genus g of $\underline{M}$) will be denoted
by ${\cal P}({\rm g},n,p,m)$. (If $p=0,m=0$ we obtain the space ${\cal P}({\rm
g},n)$ considered in preceding section.) The direct product of $n$ copies of
the semigroup $\mathcal{A}$ (hence also the direct product of $n$ copies of
Lie algebra diff) acts on these moduli spaces.

Let us fix one of boundary components (say the first one ) and consider the
action of corresponding semigroup $\mathcal{A}$ on ${\cal P}({\rm g},n,p,m)$.
The Lie stabilizer $\textgoth{k}_{M}\subset$diff at the point $M\in{\cal
P}({\rm g},n,p,m)$ can be described as the Lie algebra of complex vector
fields on the boundary component $S=B_{1}$ that have a meromorphic extension
to $M$ with zeros at the marked points and singularities only in the
punctures.

Taking the product of $n$ copies of the semigroup $\mathcal{A}$ corresponding
to all boundary components and considering the Lie stabilizer
$\textgoth{k}_{M}\subset$ diff $\times...\times$diff we obtain

${\cal P}({\rm
g},n,p,m)=(\mathcal{A}\times...\times\mathcal{A})/\textgoth{k}_{M}.$

(We used the fact that $\mathcal{A}\times...\times\mathcal{A}$ acts
transitively on ${\cal P}({\rm g},n,p,m).$) The Lie stabilizer
$\textgoth{k}_{M}$ at the point $M\in{\cal P}({\rm g},n,p,m)$ consists of
vector fields on the boundary of $M$ that have a meromorphic extension to $M$
with zeros at the marked points and singularities only in the punctures.

Let us consider now CFT with central charge $c=0$ in Segal’s approach. In this
approach, we assign a vector $\phi_{M}\in{\cal H}^{n}$ to every point
$M\in{\cal P}({\rm g},n.)$ Here $\cal H$ stands for linear space equipped with
non-degenerate inner product.The semigroup $\mathcal{A}^{n}$, hence its Lie
algebra diffn acts on ${\cal H}^{n}$. If $f\in$diff is complex vector field on
a circle then the corresponding operator acting on $i$-th factor of ${\cal
H}^{n}$ is donoted $L^{(i)}(f)$.The Virasoro generators acting on $i$-th
factor (operators corresponding to vector fields $z^{k+1}\frac{d}{dz}$) are
denoted by $L^{(i)}_{k}.$ The Lie stabilizer $\textgoth{k}_{M}\subset{\rm
diff}^{n}$ consists of complex vector fields on the boundary that can be
holomorphically extended to $M.$

It is easy to check that $\phi_{M}$ is $\textgoth{k}_{M}$-invariant.

More generally, let us take $M\in{\cal P}({\rm g},n,p=0,m).$ Fixing
holomorphic coordinates at marked points (=holomorphic disks with centers at
these points) we obtain a point $\tilde{M}\in{\cal P}({\rm g},n+m)$ and a
vector $\phi_{\tilde{M}}\in{\cal H}^{n+m}.$

If $\chi=\chi_{1}\otimes...\otimes\chi_{m}\in{\cal H}^{m}$ we can define
$\psi(\chi)\in{\cal H}^{n}$ as inner product of $\phi_{\tilde{M}}$ and $\chi$.
(We use inner product in $\cal H$ to calculate the pairing of last $m$ factors
in ${\cal H}^{n+m}$ with $\chi.$) If $L^{(i)}_{k}\chi_{i}=0$ for $k\geq 0$
then $\psi(\chi)$ does not depend on the choice of coordinate systems at
marked points; it is $\textgoth{k}_{M}$ -invariant. Here $\textgoth{k}_{M}$
stands for the Lie algebra of complex vector fields on the boundary of $M$
that can be extended to holomorphic vector fields on $M$ vanishing at marked
points. (It can be characterized also as Lie stabilizer of $\mathcal{A}^{n}$
at the point $M\in{\cal P}({\rm g},n,p=0,m.$)

Let us formulate a similar statement in the case when we work with TCFT
instead of CFT. In this case we have maps ${\cal P}^{\prime}({\rm
g},n)\to{\cal H}^{n}$ where ${\cal P}^{\prime}({\rm g},n)=\Pi T{\cal P}{\rm
g},n)$ and $\cal H$ is equipped by a differential $q$. The algebra diff′ is
represented in $\cal H$ by operators $L(f),b(f)$ where $f\in$diff. They obey
$[L(f),L(g)]=L([f,g],[L(f),b(g)]=b([f,g]),[b(f),b(g)]_{+}=0,L(f)=[q,b(f)]_{+}$.
This action induces an action of diff′n on ${\cal H}^{n}$; the operators
acting on $i$-th factor are denoted by $L^{(i)}(f),b^{(i)}(f)$ or by
$L^{(i)}_{k},b^{(i)}_{k}$ if $f=z^{k+1}\frac{d}{dz}.$

Let us consider $M\in{\cal P}({\rm g},n,p=0,m)$ and a vector
$\kappa=\kappa_{1}\otimes...\otimes\kappa_{m}\in{\cal H}^{m}$ obeying
$q\kappa_{i}=0$ and

$L^{(i)}_{k}\kappa_{i}=0,b^{(i)}_{k}\kappa_{i}=0$ (5)

for $k\geq 0$. Then slightly modifying the above construction we can define a
vector $\tau(\kappa)\in{\cal H}^{n}$. This vector is
$\textgoth{k}^{\prime}_{M}$ -invariant where $\textgoth{k}_{M}$ is the Lie
stabilizer of $\mathcal{A}^{n}$ at the point $M\in{\cal P}({\rm g},n,p=0,m).$
( Considering $M$ as a point of ${\cal P}^{\prime}({\rm g},n,p=0,m)$ we can
say that $\textgoth{k}^{\prime}_{M}$ is the Lie stabilizer of
$\mathcal{A}^{\prime n}$ at this point.)

Using the inner product in $\cal H$ we can define the bra-state
$\langle\tau(\kappa)|$.This state is also $\textgoth{k}^{\prime}_{M}$
-invariant.

##  5 String theory

Let us consider classical CFT that gives CFT having central charge $c$ after
quantization. To obtain the corresponding string theory we impose constraints
$L_{n}=0,\tilde{L}_{n}=0$ where $L_{n},\tilde{L}_{n}$ are classical analogs of
Virasoro generators. Using the general construction of Section 2 we see that
one can get the space of states of string theory (more precisely, one-string
space in BRST-formalism) by adding ghosts. In other words, we should take the
tensor product of Hilbert space $\cal E$ of CFT by the space of ghosts ${\cal
E}_{gh}$ that can be considered as a space of states of CFT with central
charge $c_{gh}=-26.$ ( The space of ghosts is a tensor product of spaces of
states of $bc$-system and $\tilde{b}\tilde{c}$-system.) We obtain the space
${\cal E}^{\prime}={\cal E}\otimes{\cal E}_{gh}$. Let us consider critical
closed bosonic string. This means that we assume that $c=26$. Then the space
${\cal E}^{\prime}$ is a space of states of CFT with zero central charge.
Generators of Virasoro algebra of this CFT will be denoted by
$\hat{L}_{n},\tilde{\hat{L}}_{m}.$ We need the following relations between
operators $\hat{L}_{n},\tilde{\hat{L}}_{m},b_{n},\tilde{b}_{n},Q$ acting in
this space:

$\begin{split}[\hat{L}_{m},\hat{L}_{n}]=(m-n)\hat{L}_{m+n}\\\
[\tilde{\hat{L}}_{m},\tilde{\hat{L}}_{n}]=(m-n)\tilde{\hat{L}}_{m+n}\\\
[\hat{L}_{m},b_{n}]=(m-n)b_{m+n},[b_{m},b_{n}]_{+}=0\\\
[\tilde{\hat{L}}_{m},\tilde{b}_{n}]=(m-n)\tilde{b}_{m+n},[\tilde{b}_{m},\tilde{b}_{n}]_{+}=0\\\
\hat{L}_{n}=[Q,b_{n}],\tilde{\hat{L}}_{n}=[Q,\tilde{b}_{n}],[Q,Q]_{+}=0\end{split}$
(6)

These relations indicate that by adding ghosts to CFT with critical central
charge $c=26$ we obtain TCFT (topological CFT) on the space ${\cal
E}^{\prime}$. Our results can be extended to any TCFT, the assumption that
TCFT is obtained from CFT adding ghosts is irrelevant.

The Lie superalgebra $\rm diff^{\prime}$ is represented in ${\cal E}^{\prime}$
by linear operators $L({\bf v}),b({\bf v})$ obeying

$[L({\bf v}),L({\bf v}^{\prime})]=L([{\bf v},{\bf v}^{\prime}]),[L({\bf
v}),b({\bf v}^{\prime})]=b([{\bf v},{\bf v}^{\prime}]),[b({\bf v}),b({\bf
v}^{\prime})]_{+}=0.$

Operators $\tilde{L}({\bf v}),\tilde{b}({\bf v})$ obey similar relations, they
give a second representation of $\rm diff^{\prime}$, commuting with the first
one. (Here ${\bf v},{\bf v}^{\prime}$ are complex-valued vector fields on
circle:${\bf v},{\bf v}^{\prime}\in\rm diff$.)

The first four lines of (6) describe the representation of generators of ${\rm
diff}^{\prime}\times{\rm diff}^{\prime}$ in ${\cal E}^{\prime}.$ This
representation can be extended to a representation of ${\rm
diff}^{\prime\prime}\times{\rm diff}^{\prime\prime}.$ (Recall that one can
obtain ${\rm diff}^{\prime\prime}$ adding nilpotent generator and ghost number
to the generators $L_{n},b_{n}$ of ${\rm diff}^{\prime}$.)

Let us consider the diagonal part of the Lie algebra ${\rm
diff}^{\prime}\times{\rm diff}^{\prime}$ ( the Lie subalgebra generated by
operators $L_{n}+\tilde{L}_{n},b_{n}+\tilde{b}_{n}$).

We assume that the action of the diagonal part of the Lie algebra ${\rm
diff}^{\prime}\times{\rm diff}^{\prime}$ in ${\cal E}^{\prime}$ can be
integrated and gives an action of $\mathcal{A}^{\prime}$ on $\cal E^{\prime}$
( a homomorphism $\Psi$ of the semigroup $\mathcal{A}^{\prime}$ into the space
$\cal L$ of linear operators in $\cal E^{\prime}$).

This is a standard assumption that lies in the basis of Segal’s definition of
CFT ( see Section 3).

We can apply general considerations of Section 2 taking $G=\mathcal{A}.$

Let us consider the case when a form on $G=\mathcal{A}$ ( a function on
$G^{\prime}=\mathcal{A}^{\prime}$) is specified by (3). The semigroup
$\mathcal{A}$ is homotopy equivalent to $S^{1}$ therefore an integral of the
closed form (3) over any cycle of of dimension $>1$ vanishes. To get non-
trivial physical quantities we construct the form (3) in such a way that it
descends to $G/\textgoth{k}$ where $\textgoth{k}$ is an appropriate Lie
subalgebra of the Lie algebra diff of the semigroup $G=\mathcal{A}$ (or, more
generally, a Lie subalgebra of the Lie algebra ${\rm diff}^{n}$ of the
semigroup $G=\mathcal{A}^{n}$).

Examples of subalgebras $\textgoth{k}$ and corresponding quotient spaces were
constructed in Section 3.

##  6 String amplitudes

We start the construction of string amplitudes fixing a one-dimensional
compact complex manifold $P_{0}\in{\cal P}({\rm g},1)$ ( a complex curve of
genus g with parametrized boundary diffeomorphic to a circle $S^{1}$). Let us
denote by $\textgoth{k}$ a Lie algebra consisting of vector fields on the
boundary that can be extended to holomorphic vector fields on $P_{0}.$ The
semigroup $\mathcal{A}$ acts on the moduli space ${\cal P}({\rm g},1)$, hence
we can consider the corresponding action of its Lie algebra diff on this
space. The Lie algebra $\textgoth{k}$ can be characterized as a Lie stabilizer
of this action at $P_{0}.$ The action of $\mathcal{A}$ on ${\cal P}({\rm
g},1)$ is transitive, hence ${\cal P}({\rm g},1)$ can be identified with
$\mathcal{A}/\textgoth{k}$. The Lie stabilizer $\textgoth{k}_{P}$ of diff at
the point $P\in{\cal P}({\rm g},1)$ is a Lie subalgebra of diff consisting of
vector fields that can be holomorphically extended from boundary to $P.$

This construction can be generalized to the case when $P_{0}\in{\cal P}({\rm
g},n)$ (i.e. it has a boundary consisting of $n$ parametrized circles; we
assume that the orientation of boundary circles agrees with the orientation of
$P_{0}$). The group $\mathcal{A}^{n}$ and its Lie algebra ${\rm diff}^{n}$
(the direct sum of $n$ copies of the Lie algebra diff) act on ${\cal P}({\rm
g},n)$ The Lie algebra $\textgoth{k}_{P}$ can be defined as the Lie stabilizer
of this action at $P$; if $P=P_{0}$ we use the notation
$\textgoth{k}_{P}=\textgoth{k}.$ The Lie algebra $\textgoth{k}_{P}$ consists
of complex vector fields on the boundary that can be holomorphically extended
from boundary to $P.$ The action of $\mathcal{A}^{n}$ on ${\cal P}({\rm g},n)$
is transitive, hence ${\cal P}({\rm g},n)$ can be identified with
$\mathcal{A}^{n}/\textgoth{k}.$

All these statements are particular cases of statements formulated in Section
3.

Notice that these objects appear in operator formalism in string theory. The
main object of operator formalism is an element of $\cal E^{\prime}$ depending
on $P\in{\cal P}({\rm g},1)$ (more generally, we have a map ${\cal P}({\rm
g},n)\to{\cal E^{\prime}}^{n}$, where ${\cal E^{\prime}}^{n}$ stands for
tensor product of $n$ copies of $\cal E^{\prime}$). In notations of [2] this
map sends $P$ into $\phi_{P}$.

It is well known that $\phi_{P}$ is $\textgoth{k}^{\prime}_{P}$-invariant (see
formula (5.1) of [2] or formula (7.33) of [8]). Notice that $\phi_{P}$ appears
also in Segal’s approach to CFT, $\textgoth{k}^{\prime}_{P}$-invariance of
$\phi_{P}$ follows immediately from this approach (see Section 4).

In what follows we apply the considerations of Section 2 to the case when
$G=\mathcal{A}^{n},P\in{\cal P}({\rm g},n)$ , and $\Psi_{n}$ denotes the map
of ${\mathcal{A}^{\prime}}^{n}$ into into the space of linear operators in
${{\cal E}^{\prime}}^{n}$. ( We have a representation $\psi_{n}$ of the Lie
algebra $({\rm diff}^{\prime}\oplus{\rm diff}^{\prime})^{n}$ in this space.
This representation is a homomorphism $\psi_{n}$ of the Lie algebra $({\rm
diff}^{\prime}\oplus{\rm diff}^{\prime})^{n}$ into the space of linear
operators in ${{\cal E}^{\prime}}^{n}$ considered as Lie algebra. It is
obtained as tensor product of $n$ copies of the homomorphism $\psi$ of ${\rm
diff}^{\prime}\oplus{\rm diff}^{\prime}$ into the space of linear operators in
$\cal E^{\prime}$. On the diagonal part of $({\rm diff}^{\prime}\oplus{\rm
diff}^{\prime})^{n}$ the homomorphism $\psi_{n}$ is specified by operators
$L_{k}({\bf v})+\tilde{L}_{k}({\bf v}),b_{k}({\bf v})+\tilde{b}_{k}({\bf v})$
where $k=1,...,n.$ The representation $\psi_{n}$ can be integrated to give a
representation $\Psi_{n}$ of the diagonal part of $\mathcal{A}^{\prime
n}\times\mathcal{A}^{\prime n}$; later we are using the notations
$\mathcal{A}^{n}$ and $\mathcal{A}^{\prime n}$ for diagonal parts.)

One can verify that $P=gP_{0}$ where $g\in\mathcal{A}^{n}$ implies

$\phi_{P}=(\Psi_{n}(g))(\phi_{P_{0}})$ (7)

The CFT with space of states $\cal E^{\prime}$ has central charge $c=0.$ The
map ${\cal P}({\rm g},n)\to{\cal E^{\prime}}^{n}$ of operator formalism is the
Segal’s map $\sigma_{{\rm g},n}:{\cal P}({\rm g},n)\to{\cal H}^{n}$ in the
case ${\cal H}={\cal E}^{\prime}.$ The formula (7) immediately follows from
Segal’s axioms.

Let us consider the form (3) obtained from (1) where $\langle\rho|$ is
$\textgoth{k}^{\prime}$-invariant. As a $\textgoth{k}^{\prime}$-invariant
element $\langle\rho|$ we take the bra-state corresponding to $\phi_{P_{0}}$
where $P_{0}\in{\cal P}({\rm g},n)$.

Then the expression (3) looks as follows

$\begin{split}(\Psi_{n}^{*}\sigma)(g\exp(\mu^{r}_{k}(b^{(k)}_{r}+\tilde{b}^{(k)}_{r})))=\\\
\langle\rho|\Psi_{n}(g)\Psi_{n}(\exp(\mu^{r}_{k}(b^{(k)}_{r}+\tilde{b}^{(k)}_{r})|\chi\rangle=\\\
\langle\phi_{P}|\exp(\mu^{r}_{k}(b^{(k)}_{r}+\tilde{b}^{(k)}_{r}))|\chi\rangle\end{split}$
(8)

( We used (7).)

The expression (8) can be considered as inhomogeneous closed differential form
on $G=\mathcal{A}^{n}$; it descends to $G/\textgoth{k}={\cal P}({\rm g},n)$
because $\phi_{P}$ is $\textgoth{k}^{\prime}_{P}$-invariant.

Homogeneous components of the form (8) are closed forms on
$G/\textgoth{k}={\cal P}({\rm g},n)$ that can be represented as

$\langle\phi_{P}B|\chi\rangle$ (9)

where $B$ is a homogeneous polynomial with respect to
$b^{(k)}_{r}+\tilde{b}^{(k)}_{r}$

The expression (9) coincides with formulas of operator formalism.

It follows from the considerations above that this expression descends to a
closed form on ${\cal P}({\rm g},n)$. Moreover, by imposing some conditions on
$\chi$ one can prove that it descends further to a closed form $\omega_{B}$ on
$\hat{\cal P}({\rm g},n)={\cal P}({\rm g},n)/(S^{1})^{n}$. (The action of the
group $(S^{1})^{n}$ on ${\cal P}({\rm g},n)$ is defined in terms of rotations
of boundary circles.) Namely, we should assume that $\chi\in{{\cal
E}^{\prime}}^{n}$ can be represented as a tensor product
$\chi=\chi^{(1)}\otimes...\otimes\chi^{(n)}$ where
$(L^{(k)}_{0}-\tilde{L}^{(k)}_{0})\chi^{(k)}=0,$
$(b^{(k)}_{0}-\tilde{b}^{(k)}_{0})\chi^{(k)}=0.$ This condition means that we
can apply the statement at the very end of Section 2 taking the Lie algebra of
the group $(S^{1})^{n}$ as $\textgoth{h}.$

There exists a natural map $\hat{\cal P}({\rm g},n)\to{\cal M}({\rm g},n)$
where ${\cal M}({\rm g},n)$ is the moduli space of complex curves (one-
dimensional compact complex manifolds) of genus g with $n$ marked points. This
map is a homotopy equivalence, hence it induces an isomorphism of homology
groups. This allows us to integrate forms $\omega_{B}$ over homology classes
of ${\cal M}({\rm g},n).$ (Of course we can get a non-zero answer only if the
dimension of the form is equal to the dimension of the homology class. Notice
that equivalently we can integrate the original non-homogeneous form; the
answer depends only on the homogeneous component of degree equal to the
dimension of the integration cycle.)

We obtain a formal expression for string amplitudes integrating $\omega_{B}$
over the fundamental homology cycle of ${\cal M}({\rm g},n)$ (one should take
$B$ having degree equal to the dimension of ${\cal M}({\rm g},n)$). This is a
formal divergent expression; the physical explanation of divergence is the
presence of tachyon in the spectrum of bosonic string. From mathematical
viewpoint, the problem lies in the non-compactness of ${\cal M}({\rm g},n)$
(fundamental homology class is a locally finite cycle, to guarantee
convergence we should integrate over a finite cycle or to work with Deligne-
Mumford compactification). However, integrals of forms $\omega_{B}$ over
genuine homology classes of ${\cal M}({\rm g},n)$ exist. (Notice that these
forms where used in string field theory [8].)

##  7 Modifications

The above constructions can be modified in various ways.

Our considerations were based on the statement at the end of Section 2; we
assumed that $G={\mathcal{A}}^{n}$, the Lie subalgebra $\textgoth{k}$ is a Lie
stabilizer of $G$ at the point of ${\mathcal{P}}({\rm g},n)$ and the Lie
subalgebra $\textgoth{h}$ is the Lie algebra of $(S^{1})^{n}.$ One can take
other subalgebras $\textgoth{k},\textgoth{h}.$; in particular, one can take
one or both of these subalgebras as Lie stabilizers of $G$ at the points of
${\cal P}({\rm g},n,p,m).$( For example, in the situation described at the end
of Section 4 we can take $\textgoth{k}=\textgoth{k}_{M}$ and
$\langle\rho|=\langle\tau(\kappa)|$.)

One can hope to get closed forms with integrals related to interesting
physical quantities (for example, to inclusive cross-sections or to mass
renormalization [9]).

Other modifications allow us to consider scattering in superstring and
heterotic string. They are based on the consideration of superconformal
manifolds and supersymmetric analog of the semigroup $\mathcal{A}.$ Notice
that in the present paper, we tacitly assumed that we consider left-right
symmetric conformal field theories; of course, considering heterotic string
and other theories with independent left and right sectors we se should drop
this assumption. (In these cases it is useful to apply the ideas of [10],
[11].) More details will be given in the follow-up paper [1].

Acknowledgements I am deeply indebted to M. Movshev and A. Rosly for very
useful discussions.

## References

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  * [5] Schwarz A. Geometric and algebraic approaches to quantum theory. Nucl.Phys. B,973, 2021
  * [6] Schwarz, A., 2021. Scattering in algebraic approach to quantum theory. Associative algebras. arXiv:2107.08553.
  * [7] Schwarz A. Scattering in geometric approach to quantum theory arxiv:2107.08557
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