# Unsupervised Feature Clustering Improves Contrastive Representation Learning
for
Medical Image Segmentation ††thanks: * indicates equal contribution.
978-1-6654-6819-0/22/$31.00 ©2022 IEEE

Yejia Zhang  University of Notre Dame
Notre Dame, IN, 46556, USA
<EMAIL_ADDRESS>Xinrong Hu  University of Notre Dame
Notre Dame, IN, 46556, USA
<EMAIL_ADDRESS>Nishchal Sapkota  Yiyu Shi  University of Notre Dame
Notre Dame, IN, 46556, USA
<EMAIL_ADDRESS>University of Notre Dame
Notre Dame, IN, 46556, USA
<EMAIL_ADDRESS>Danny Z. Chen  University of Notre Dame
Notre Dame, IN, 46556, USA
<EMAIL_ADDRESS>
###### Abstract

Self-supervised instance discrimination is an effective contrastive pretext
task to learn feature representations and address limited medical image
annotations. The idea is to make features of transformed versions of the same
images similar while forcing all other augmented images’ representations to
contrast. However, this instance-based contrastive learning leaves performance
on the table by failing to maximize feature affinity between images with
similar content while counter-productively pushing their representations
apart. Recent improvements on this paradigm (e.g., leveraging multi-modal
data, different images in longitudinal studies, spatial correspondences)
either relied on additional views or made stringent assumptions about data
properties, which can sacrifice generalizability and applicability. To address
this challenge, we propose a new self-supervised contrastive learning method
that uses unsupervised feature clustering to better select positive and
negative image samples. More specifically, we produce pseudo-classes by
hierarchically clustering features obtained by an auto-encoder in an
unsupervised manner, and prevent destructive interference during contrastive
learning by avoiding the selection of negatives from the same pseudo-class.
Experiments on 2D skin dermoscopic image segmentation and 3D multi-class whole
heart CT segmentation demonstrate that our method outperforms state-of-the-art
self-supervised contrastive techniques on these tasks.

###### Index Terms:

Self-supervised learning, Contrastive learning, Medical image semantic
segmentation, Label efficient learning

##  I Introduction

Self-supervised contrastive methods have recently made great strides in
representation learning both in natural scene imaging [3, 4] and biomedical
imaging [16, 1, 13, 18, 6] applications. The core idea is to pull
representations of “similar” images (referred to as _positives_) close while
“dissimilar” images (_negatives_) are contrasted in feature space. Such
methods implemented this idea using an instance discrimination pretext task
where only transformed versions of the same images are taken as positives
while augmented versions of other images are negatives. This approach is sub-
optimal when two different images that contain similar content are treated as
negatives (e.g., see “Harmful Negatives” in Fig. 1). This disproportionately
hinders medical imaging pretraining compared to natural scene datasets since
medical image datasets generally have far fewer semantic categories, less
inter-class as well as intra-class appearance diversity, and considerably
fewer samples to learn meaningful features.

Figure 1: Examples of harmful selections of positives and negatives from our
downstream datasets. On the left, skin lesions with similar appearances and
the same class label are treated as negatives (harmful negatives). On the
right, choosing positive and negative slices based on axial position proximity
leads to both harmful negatives (similar lung tissue in distant CT slices) and
harmful positives (proximal slices across two volumes still vary greatly).

Recent contrastive learning methods for biomedical imaging tasks addressed
these drawbacks by improving the selection of positives and negatives. One
common approach is to designate different images of the same patient as
positives. This can be done either using images of a patient across time in
longitudinal studies [11, 14] or leveraging scans taken at the same time but
from different views [1, 15]. A related idea is to treat the same anatomical
structure captured from different modalities as positives [12]. However, these
methods require the existence of relevant meta-data and additional
corresponding images which are often unavailable. Instead of relying on
instance correspondences, Global & Local Contrastive Learning (GLCL [2]) and
Positional Contrastive Learning (PCL [17]) proposed to use slices in close
proximity as positives by assuming semantic correlations in corresponding
axial positions of 3D volumes. These approaches reasonably avoid some harmful
negatives (although many remain unaddressed; see MMWHS “Harmful Negatives” in
Fig. 1) and benefit from sample diversity in positives. However, they assumed
that images are sensibly aligned (see “Harmful Positives” in Fig. 1 for an
example of poor alignment) and, although they operate in 2D slices, cannot be
applied to general 2D tasks. These advances show promising potential of
contrastive learning beyond instance discrimination, but they either have
stringent data correspondence requirements or rely on binding assumptions of
image structures which limit their generalizability and applicability to other
biomedical tasks. Furthermore, many rely on heuristics that still do not fully
handle the issue of harmful positives & negatives.

To address these challenges, we explore distances between learned image
features as a proxy for appearance similarity. Utilizing feature distances
enables us to cluster images based on appearance in a way that is effective
across tasks, imaging modalities, data dimensionality, and spatial settings.
In related medical imaging works, [19] used variational auto-encoders to
attain features of images to select representative samples for improving
annotation efficiency, and [9] used features from an auto-encoder to select
similar instances to assign corresponding anatomical regions for pretraining.
Our objective is to leverage the extracted pseudo-categories in order to
improve samplings of positives & negatives for contrastive learning. We choose
to use Beta Variational Auto-Encoders ($\beta$-VAEs) [10] for their general
applicability to image data, ease of training, and propensity to learn
disentangled features.

In this paper, we propose a contrastive learning framework based on
unsupervised feature clustering (called UFC), which outperforms instance
discrimination pretraining and can be employed generally across medical
imaging modalities & tasks. First, we extract categorical information from
unlabeled data in an unsupervised manner by hierarchically clustering features
learned by a $\beta$-VAE. Second, we pretrain a convolutional encoder with
contrastive learning guided by pseudo-categories from the predicted clusters
to avoid contrasting similar images while maximizing similarity with more
diverse positive samples. Finally, we transfer the encoder to fine-tune on a
downstream segmentation task to assess the quality of features learned. We
evaluate our framework on a skin lesion dataset (ISIC) and a heart CT dataset
(MMWHS) for segmentation. The results show that our framework outperforms
state-of-the-art contrastive learning methods irrespective of the percentages
of annotations used for fine-tuning.

Figure 2: The overall workflow of our proposed framework.  TABLE I:
Comparisons against state-of-the-art contrastive methods in Dice score %’s
(STD) using different amounts of training annotations, averaged over 5 runs.
LGCL & PCL are designed for 3D volumes and are not applicable to ISIC.

Method | ISIC Skin Dermoscopy | MMWHS Heart CT
---|---|---
| 5% | 10% | 20% | 100% | 10% | 20% | 40% | 100%
Rand. Init. | 79.3 ($\pm$ 0.3) | 81.7 ($\pm$ 0.2) | 83.3 ($\pm$ 0.2) | 84.7 ($\pm$ 0.5) | 38.4 ($\pm$ 6.2) | 62.1 ($\pm$ 7.0) | 73.8 ($\pm$ 4.9) | 85.9 ($\pm$ 2.7)
SimCLR [3] | 80.1 ($\pm$ 0.8) | 82.7 ($\pm$ 0.3) | 83.7 ($\pm$ 0.2) | 84.9 ($\pm$ 0.3) | 44.5 ($\pm$ 8.4) | 67.2 ($\pm$ 8.3) | 80.6 ($\pm$ 2.9) | 86.6 ($\pm$ 2.5)
MoCov2 [4] | 79.9 ($\pm$ 0.7) | 82.2 ($\pm$ 0.7) | 83.3 ($\pm$ 0.2) | 84.8 ($\pm$ 0.2) | 42.4 ($\pm$ 7.2) | 64.7 ($\pm$ 9.7) | 75.9 ($\pm$ 5.3) | 86.4 ($\pm$ 2.3)
BYOL [7] | 79.4 ($\pm$ 0.5) | 80.6 ($\pm$ 0.4) | 82.5 ($\pm$ 0.4) | 84.8 ($\pm$ 0.3) | 42.0 ($\pm$ 5.0) | 58.5 ($\pm$ 10.7) | 69.5 ($\pm$ 9.1) | 86.1 ($\pm$ 2.0)
LGCL [2] | - | - | - | - | 35.5 ($\pm$ 2.5) | 62.4 ($\pm$ 12.1) | 74.0 ($\pm$ 3.8) | 85.2 ($\pm$ 2.4)
PCL [17] | - | - | - | - | 38.2 ($\pm$ 6.5) | 60.8 ($\pm$ 8.3) | 71.6 ($\pm$ 9.2) | 85.5 ($\pm$ 2.8)
Ours | 81.9 ($\pm$ 0.4) | 82.8 ($\pm$ 0.3) | 84.6 ($\pm$ 0.4) | 85.3 ($\pm$ 0.2) | 46.9 ($\pm$ 5.8) | 69.9 ($\pm$ 8.2) | 81.4 ($\pm$ 2.6) | 87.3 ($\pm$ 1.5)

Our main contributions can be summarized below.

  1. 1.

We propose a novel three-stage self-supervised contrastive learning framework
that is broadly applicable, effective for diverse medical image data, and
easily incorporated into existing deep learning pipelines.

  2. 2.

Given the limited works that address the difficulties of harmful negatives in
medical imaging tasks, we show the potential of using categorical information
extracted for “free” with only unlabeled data to improve representation
learning for medical image segmentation, and highlight the disadvantages of
instance discrimination.

  3. 3.

We achieve state-of-the-art performances on the skin lesion segmentation and
multi-class CT heart segmentation tasks over known contrastive methods.

##  II Method

Our method consists of three main stages: unsupervised feature clustering
(UFC), cluster-guided contrastive learning, and segmentation fine-tuning (see
Fig. 2). First, we train a $\beta$-VAE to learn feature representations in an
unsupervised manner and apply a hierarchical clustering method to divide those
image-level features into clusters. Next, we assign features in each cluster
the same pseudo-label, which guides the contrastive learning of our encoder
$Enc$. The above two stages serve as the pretraining process. Finally, we keep
the weights in $Enc$ and concatenate it with a decoder $Dec$ to fine-tune the
downstream segmentation task. We detail UFC and cluster-guided contrast
learning in the following subsections.

###  II-A Unsupervised Feature Clustering

To group images that share similar anatomical structures more easily, the
first step is to map the images to a low-dimensional feature space. Many
unsupervised methods can be utilized, including a pretext task, contrastive
learning (CL), an auto-encoder (AE), and a variational auto-encoder (VAE).
Regarding the pretext task, training the model to predict the rotation or
positions of input images results in extracting task-relevant features with
essential information missed. For CL, previous works like [2] demonstrated
reasonable initializations for segmentation tasks, but the biases incurred
from harmful samples could magnify if we use their features directly for
clustering. Hence, we mainly focus on the AE & VAE, which both comprise of an
encoder $E(\cdot)$ and a decoder $D(\cdot)$. For an input image $m_{i}$, the
encoder gives a feature representation of $m_{i}$, $z_{i}=E(m_{i})$, and the
decoder further outputs a reconstructed image, $\hat{m}_{i}=D(E(m_{i}))$. AE
learns the embeddings only by minimizing the mean square error between $m_{i}$
and $\hat{m}_{i}$, while VAE assumes the distribution of the embeddings to be
a Gaussian distribution $p(z)=G(\mu,\theta)$ and seeks to force the real
distribution of $z$ to be close to this posterior distribution. More
specifically, the encoder of VAE generates both a mean vector $\hat{\mu}$ and
a variance vector $\hat{\theta}$. After that, the feature vector $z$ is
sampled from the distribution $q(z|x)=G(\hat{\mu},\hat{\theta})$ for image
reconstruction. The overall loss function for VAE is:

$\begin{split}L_{vae}&=L_{rec}+L_{prior}\\\
&=\mathbb{E}_{q(z|x)}[\log(p(x|z)]-\beta*D_{KL}(q(z|x)\|p(z)),\end{split}$ (1)

where $D_{KL}$ is the Kullback-Leibler divergence and $\beta$ is a weight to
adjust how strong the constraint should be put on the latent distribution
[10]. If $\beta=1$, then the loss in Eq. (1) corresponds to the original VAE;
otherwise, it is $\beta$-VAE.

After features are obtained, clustering them into appearance-based pseudo-
categories is a key step in UFC. We use the agglomerative clustering method
[19] that hierarchically combines features in the closest proximity so that
images that look the most similar would be grouped first. We avoid centroid-
based clustering since they have an inherent disadvantage of having to select
the number of centers which is very task-dependent. This issue is mitigated in
density-based clustering; however, we must assume that data form coherent
density patterns in embedding space which may not be the case given imbalanced
categories with limited examples for classes of interest or data distributions
with large intra-class appearance variations. Furthermore, for distribution-
based techniques, prior knowledge of the distribution is needed, which is
difficult in high-dimensional objects like images. Thus, we find agglomerative
clustering to be well-suited for our task given its flexibility in selecting
cluster sizes as well as its tendency to prioritize the most visually-similar
images for grouping.

In order to decide the number of clusters, $k$, we draw a curve of mean
distance within each cluster vs. $k$, and the elbow point roughly gives a
range for a $k$ candidate. Actually, we find that the value of $k$ largely
depends on the dataset, and there is no optimal value, rather there is just an
optimal range for most cases. We assign every image embedding to a cluster in
the feature space. Based on that, we have pseudo-labels $\hat{y}_{i}$
representing the cluster indices for all the images. These pseudo-labels
provide additional prior knowledge about the dataset, which help with
formulating more reasonable positive pairs and negative pairs.

###  II-B Cluster-guided Contrastive Learning

Here, we aim to train a model to learn image representations that take
advantage of the clustering information above. Given a batch of input images
$\left\\{x_{i}|i=1,2,\ldots,b\right\\}$ and their pseudo-labels
$\left\\{\hat{y}_{i}|i=1,2,\ldots,b\right\\}$, we apply two different
augmentations $\hat{x}_{i}=aug_{1}(x_{i})$ and $\tilde{x}_{i}=aug_{2}(x_{i})$,
combining spatial and appearance transformations, to generate two different
views. We then use an encoder to map the images to one-dimensional features
and deploy an additional multi-layer perceptron (MLP) to increase the
representation ability, $f_{i}=MLP(Enc(\hat{x_{i}}))$. We also employ $L_{2}$
normalization on the features $f_{i}$ for training stability. So far, the
above processes are the same as SimCLR [3], but the biggest difference in our
method is our selection strategy for positive pairs. Since we obtain a pseudo-
label for each image, we extend the concept of positive pairs to features that
correspond to images categorized as in the same cluster. The contrastive loss
for a batch is defined as follows:

$\small{L_{con}=-\frac{1}{2b}\sum_{f_{i}\in
B}\frac{1}{\left|P(f_{i})\right|}\log\frac{\sum_{f_{j}\in
P(f_{i})}\exp(f_{i}\cdot f_{j}/\tau)}{\sum_{f_{k}\in
B-P(f_{i})}\exp(f_{i}\cdot f_{k}/\tau)}}$ (2)

where $B$ is the feature set of size $2*b$ combining the two augmentation
views’ features, $\tau$ is a temperature parameter, and
$P(f_{i})=\left\\{{f_{j}|\hat{y}_{j}}=\hat{y}_{i}\right\\}$ is the collection
of features that are viewed as positive pairs of $f_{i}$. In comparison,
$P(f_{i})$ in SimCLR [3] and MoCo [4] contains only a feature from the same
image, which might result in many harmful negative pairs (e.g., images
belonging to the same cluster). We also notice that BYOL [7] and SimSiam [5]
inherently avoid the issue of harmful negatives by only utilizing positive
pairs. However, we elect to use a contrastive loss over these approaches
because 1) they were designed for large-scale natural scene datasets while the
number of images or slices in a medical image segmentation dataset is often
much smaller, and 2) negative samples benefit medical images more due to more
limited appearance variations & semantic categories.

##  III Experiments and Results

###  III-A Datasets and Implementation Details

Skin Lesion Segmentation. For ISIC’2017 [8], we use the provided 2000 train,
150 validation, and 600 test image splits with a single foreground class.
During pretraining, we additionally use training images from ISIC’s 2018
(10015 images) and 2020 (33126 images) challenges for a total of 45141
unlabeled samples after removing duplicate images. For preprocessing and
augmentations, we normalize pixel intensities to have zero mean and unit
variance, apply a random resized crop with a network input size of
192$\times$192, and follow additional augmentations used by BYOL [7].

Whole Heart Segmentation. MMWHS’2017 [20] contains 20 CT volumes with 7
foreground classes. We perform 4-fold cross-validation on the 20 volumes with
a 2:1:1 train:validation:test ratio. For preprocessing, we first crop the 3D
images to keep only the areas of interest, and then we resize the shape of
each slice along the axial plane to 256$\times$256 along with intensity
normalization to values between 0 and 1.

Implementation. We run all experiments (coded in PyTorch) on NVIDIA P100s. For
our backbone network, we use a vanilla UNet . Only the down-sampling path of
the architecture is pretrained, while we fine-tune using the transferred
encoder weights with a randomly initialized decoder. For the two-layer MLP
head, we use 2048 intermediate and 256 output dimensions. For network training
on ISIC (resp., MMWHS) data, we employ AdamW with $\beta_{1}=0.9$,
$\beta_{2}=0.999$ for feature extraction, contrastive learning, and fine-
tuning tasks with learning rates of $1\times 10^{-3}$ (resp., $3\times
10^{-4}$), $2.5\times 10^{-4}$ ($3\times 10^{-4}$), and $1\times 10^{-4}$
($5\times 10^{-4}$), while training for 200 (200), 400 (200), and 200 (120)
epochs, with batch sizes of 60 (48), 55 (48), and 24 (8), respectively. We use
cosine learning rate decay for both datasets and $\tau$=0.1.

###  III-B Comparison with State-of-the-arts

For each experiment in Tab. I, we pretrain exclusively on images in the
training split and evaluate on the test split. When evaluating the
effectiveness of our proposed method versus state-of-the-art contrastive
methods, we note the following observations. 1) Our framework _beats recent
natural scene contrastive methods_ and improves the quality of representations
for both datasets. This supports our hypothesis that reducing harmful
positives & negatives through appearance priors would be beneficial over
vanilla instance discrimination. 2) Our method _outperforms contrastive
methods for 3D medical imaging_ even when spatial assumptions reasonably hold
(e.g., on the MMWHS dataset). Here, LGCL and PCL are adversely impacted by
both harmful positives when contents from corresponding slices are poorly
aligned and harmful negatives where slices present similar tissues (e.g.,
disparate regions of large organs; see right side of Fig. 1). 3) _Negative
samples play an important role in metric learning_. When comparing with BYOL,
a method that uses only positive samples, we can see that the performance
struggles to even beat training from scratch. We reiterate our point that
negative samples are key since medical images tend to have less appearance
diversity, categories, and amount of overall samples.

###  III-C Study on Clustering Effects

The quality of extracted features and their clustering is central to our
approach. For feature learning, we empirically observe that the type of auto-
encoder architecture (e.g., AE, VAE, $\beta$-VAE) had minimal effects on
downstream performance. For clustering, we see more pronounced effects from
both clustering type and granularity (see Tab. II). Granularity modulates the
difficulty of negative samples (raising k eventually reduces to instance
discrimination) and the number of harmful negatives present. To initialize
$k$, selecting the elbow point on the curve of mean distance within each
cluster seems to be a reasonable heuristic that outperforms SimCLR.

##  IV Conclusions

In this paper, we proposed a pretraining framework that uses appearance priors
through unsupervised feature clustering to facilitate contrastive learning of
representations for medical images. Our approach requires no additional
metadata or stringent spatial priors, and still outperforms recent state-of-
the-art contrastive methods on both 2D skin lesion segmentation & 3D CT
cardiac structure segmentation.

TABLE II: Study on the effects of clustering type and categorical granularity
on 5$\%$ annotated ISIC skin segmentation. DBSCAN uses the conventional
minpoints=$4$.

| k-Means | DBSCAN | Agglomerative
---|---|---|---
$k$=3 |  $k$=20 |  $k$=100 |  $\epsilon$=0.9 |  $\epsilon$=0.5 |  $\epsilon$=0.1 |  $k$=3 |  $k$=20 |  $k$=100
Dice % | 79.1 |  80.2 | 80.4 | 78.5 | 79.6 | 80.0 | 79.0 | 81.9 | 80.5

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# Towards higher super-$\sigma$-model categories

Rafał R. Suszek  R.R.S.: Katedra Metod Matematycznych Fizyki, Wydział Fizyki
Uniwersytetu Warszawskiego, ul. Pasteura 5, PL-02-093 Warszawa, Poland
<EMAIL_ADDRESS>
###### Abstract.

A simplicial framework for the gerbe-theoretic modelling of supercharged-loop
dynamics in the presence of worldsheet defects is discussed whose
equivariantisation with respect to global supersymmetries of the bulk theory
and subsequent orbit decomposition lead to a natural stratification of and
cohomological superselection rules for target-space supergeometry, expected to
encode essential information on the quantised 2$d$ field theory. A physically
relevant example is analysed in considerable detail.

##  1\. Introduction

Models of geometrodynamics of extended distributions of (super)charge, known
as nonlinear $\sigma$-models with topological Wess–Zumino terms, remain an
active field of physical and mathematical research, not least because of their
applicability in the description of a wide range of dynamical systems, from
the critical field theory of collective excitations of quantum spin chains all
the way to classical superstring theory, and because of the great wealth of
the underlying mathematics. The hierarchical higher-geometric structures –
a.k.a. $p$-gerbes and their morphisms – associated with background
$(p+2)$-form fields in these models, which couple to the worldvolume
(super)charge current through distinguished Cheeger–Simons differential
characters generalising the standard line holonomy (as derived for $\,p=1\,$
in [Gaw88, GR02, FSW08, RS09]), have long been known not only to give rise to
a canonical prequantisation of the models, as in [Gaw88, Sus11], and to
provide a natural cohomological classification of the models themselves, their
boundary conditions and generic defects, but also – to lead to a
categorification of their prequantisable group-theoretic symmetries, including
the gauged ones, and more general dualities (e.g., $T$-duality). These
correspondences can oftentimes be implemented by certain defect networks
embedded in the $\sigma$-model spacetime, to which sheaf-theoretic data of
$k>0$-cells of the weak $(p+1)$-categories with $p$-gerbes as 0-cells are
pulled back along the $\sigma$-model field in order to render the
Dirac–Feynman amplitude of the $\sigma$-model with defects well-defined, cp.
[RS09, RS11, Sus11, Sus12]. In particular, the distinguished _topological_
defects, studied at some length in [FFRS07, FSW08, RS09, RS11, Sus11], enter
the generalised world-sheet orbifold construction of [FFRS07, RS09] of the
$\sigma$-model on the space $\,M/{\rm G}_{\sigma}\,$ of orbits of an action of
a rigid-symmetry group $\,{\rm G}_{\sigma}\,$ on the target space $\,M\,$ of
the original $\sigma$-model, in which they serve to model the so-called ${\rm
G}_{\sigma}$-twisted sector of the orbifold theory, instrumental in the
gauging of $\,{\rm G}_{\sigma}$. Finally, in some highly symmetric settings,
such as, e.g., that of the Wess–Zumino–Witten (WZW) $\sigma$-model, there are
defects realising the _generating_ symmetries of the $\sigma$-model [FSW08,
RS09, RS11, Sus22] which encode highly nontrivial information on the ensuing
quantum field theory, such as, e.g., its fusion rules and Moore–Seiberg data.
Therefore, it is clear that understanding the structure of the fully fledged
higher categories behind the topological couplings of poly-phase
$\sigma$-models, and constructing concrete examples of their $k$-cells, is a
goal of fundamental relevance to the study of these important field theories.

This note gives a concise account of a proposal, advanced in [Sus22] on the
basis of the earlier studies [RS09, RS11, Sus11, Sus12], for an effective
structurisation of the stratified target spaces and the higher-geometric
objects over them defining the 2$d$ $\sigma$-model in the presence of defects
compatible with configurational symmetries of the bulk field theory. The
proposal is formulated in the most general (target-space)
${\mathbb{Z}}/2{\mathbb{Z}}$-graded setting of the Green–Schwarz-type
super-$\sigma$-model and uses mixed group-theoretic, simplicial and
cohomological tools. It paves the way to a systematic construction of
maximally supersymmetric defects in the flat
${\mathbb{Z}}/2{\mathbb{Z}}$-graded target geometry, and leads to interesting
novel predictions for a higher-geometric target-space realisation of the non-
perturbative data of the bulk theory in the highly (super)symmetric setting of
the WZW(-type) models with Lie-(super)group targets.

##  2\. A bicategory for the super-$\sigma$-model with defects

The 2$d$ super-$\sigma$-model is a lagrangean theory of (super)fields from the
mapping supermanifold $\,[\Sigma,M]=\underline{\rm Hom}_{\rm{\bf
sMan}}(\Sigma,M)\,$ defined for a closed oriented (Graßmann-)even manifold
$\,\Sigma\,$ (the ‘spacetime’ of the model) and a supermanifold $\,M$. For
$\,M\,$ even, $\,[\Sigma,M]\,$ is represented by $\,C^{\infty}(\Sigma,M)\,$
and we obtain a theory of embeddings shaped by an interplay between forces
sourced by two tensor fields on the target space $\,M$: a metric tensor
$\,{\rm g}\,$ and a closed 3-form Kalb–Ramond field $\,\chi\,$ with periods
from $\,2\pi{\mathbb{Z}}$. For $\,M\,$ properly graded, the inner-${\rm Hom}$
functor $\,[\Sigma,M]\,$ is to be evaluated on the nested family of
superpoints $\,{\mathbb{R}}^{0|N},\ N\in{\mathbb{N}}^{\times}$, both tensors
are even, and the ‘metric’ $\,{\rm g}\,$ typically degenerates in the
Graßmann-odd directions. In either setting, the 3-form geometrises, in the
sense of [Mur96], as a gerbe $\,\mathcal{G}\,$ with connective structure of
curvature $\,\chi$. The gerbe trivialises upon pullback to the 2$d$ worldsheet
$\,\Sigma$, thereby defining – for $\,\partial\Sigma=\emptyset\,$ – the
topological Wess–Zumino (WZ) coupling of $\,\chi\,$ to the charged-loop
current in $\,M$, given by the surface holonomy [Gaw88] of $\,\mathcal{G}\,$
along the image of $\,\Sigma\,$ in $\,M$.

In the boundary $\sigma$-model, with $\,\partial\Sigma\neq\emptyset$,
specifying the gerbe $\,\mathcal{G}\,$ alone is _not_ sufficient to obtain a
consistent field theory as the topological term becomes ill-defined. Instead,
a distinguished 1-cell from the bicategory $\,\mathfrak{BGrb}_{\nabla}(M)\,$
of (1-)gerbes over $\,M\,$ is required to exist over a submanifold
$\,\iota_{D}:D\hookrightarrow M\,$ into which $\,\partial\Sigma\,$ is
constrained to map, to wit, a trivialisation
$\,\mathcal{T}_{D}:\iota_{D}^{*}\mathcal{G}\cong\mathcal{I}_{\omega}\,$ in
terms of a trivial gerbe $\,\mathcal{I}_{\omega}\,$ associated with a _global_
de Rham primitive $\,\omega\in\Omega^{2}(D)\,$ of $\,\chi$, cp. [GR02]. The
trivialisation is to be pulled back to $\,\partial\Sigma\,$ by the
$\sigma$-model field.

The boundary $\sigma$-model can be viewed as a two-phase field theory in which
the bulk phase, defined by the triple $\,(M,{\rm g},\mathcal{G})\,$ abuts onto
the empty phase $\,({\mathbb{R}}^{0|0},0,\mathcal{I}_{0})\,$ across the
boundary domain wall (or boundary _defect_) $\,\partial\Sigma$. In this
picture, the target space is the _stratified_ supermanifold
$\,\widetilde{M}\equiv M\sqcup{\mathbb{R}}^{0|0}$, and the two limiting field
configurations at the defect are modelled by the two mappings
$\,\iota_{1}\equiv\iota_{D}:D\longrightarrow M\subset\widetilde{M}\,$ and
$\,\iota_{2}:D\dashrightarrow{\mathbb{R}}^{0|0}\subset\widetilde{M}$. This is
the point of departure for a far-reaching generalisation, contemplated in
[FSW08, RS09], in which an arbitrary number of phases coexist over 2$d$
patches within $\,\Sigma$, separated by _defect lines_ which, in turn,
intersect at a discretuum of _defect junctions_ , graded by their valence. The
physical prototype of this situation is the decomposition of a demagnetised
ferromagnetic medium into Weiss domains of uniform magnetisation which jumps
across domain walls between them, and the most natural application of defects
in the field-theoretic setting under consideration is the modelling of the
_twisted sector_ in the theory with the target space given by the orbispace of
an action (not necessarily free or proper) of a symmetry group $\,{\rm
G}_{\sigma}\subset{\rm Isom}(M,{\rm g})\,$ of the $\sigma$-model for
$\,[\Sigma,M]\,$ – this orbifold $\sigma$-model can be _defined_ in terms of
(${\rm G}_{\sigma}$-classes of) patchwise continuous field configurations in
$\,M\,$ with ${\rm G}_{\sigma}$-jump discontinuities localised at an
arbitrarily fine mesh of defect lines carrying data of a ${\rm
G}_{\sigma}$-equivariant structure on $\,\mathcal{G}\,$ [JK06, RS09, FFRS09,
Sus12].

The above considerations set the stage for a precise definition of the poly-
phase super-$\sigma$-model: It starts with a splitting of
$\,\Sigma\in\partial^{-1}\emptyset\,$ by an embedded defect graph
$\,\Gamma\subset\Sigma\,$ into a family $\,\\{D_{i}\\}_{i\in I}\,$ of
(topologically) closed 2$d$ domains $\,D_{i}\,$ which compose the extended
worldsheet $\,\widehat{\Sigma}=\sqcup_{i\in I}\,D_{i}\,$ and intersect at a
family $\,\\{L_{(i,j)}\\}_{(i,j)\in I_{\Gamma}\subset I^{\times 2}}\,$ of
closed oriented defect lines $\,L_{(i,j)}$. The latter form the extended
defect graph $\,\widehat{\Gamma}=\sqcup_{(i,j)\in
I_{\Gamma}}\,L_{i,j}\subset\widehat{\Sigma}\times_{\Sigma}\widehat{\Sigma}\equiv\widehat{\Sigma}{}^{[2]}\,$
and join transversally at defect junctions whose ensemble $\,V=\sqcup_{\nu\geq
3}\,V_{\nu}\,$ decomposes into subsets
$\,V_{\nu}\subset\widehat{\Sigma}{}^{[\nu]}\,$ of junctions of a fixed valence
$\,\nu$, further split as
$\,V_{\nu}=\sqcup_{\varepsilon_{\nu}\in({\mathbb{Z}}/2{\mathbb{Z}})^{\times\nu}}\,V_{\varepsilon_{\nu}},\
\varepsilon_{\nu}=(\varepsilon^{(\nu)}_{k,k+1}),$ $k\in\overline{1,n}\,$ into
subcomponents $\,V_{\varepsilon_{\nu}}\,$ with distinct cyclic
($(\nu,\nu+1)\equiv(\nu,1)$) sequences of _in_ -coming
($\varepsilon^{(\nu)}_{k,k+1}=+1$) and _out_ -going
($\varepsilon^{(\nu)}_{k,k+1}=-1$) defect lines, cp. [RS09]. There are
canonical projections
$\,p_{1},p_{2}:\widehat{\Gamma}\longrightarrow\widehat{\Sigma}\,$ (assigning
to a given defect line the corresponding components of the boundaries of the
two domains separated by it) and
$\,p^{\varepsilon_{\nu}}_{k,k+1}:V_{\varepsilon_{\nu}}\longrightarrow\widehat{\Gamma}\,$
(assigning to a given defect junction the endpoints of the defect lines
converging at it, in anti-clockwise cyclic order), satisfying the obvious
identities of order 2, e.g., $\,p_{2}\circ p^{(+++)}_{1,2}=p_{1}\circ
p^{(+++)}_{2,3}\,$ for two defect lines $\,L_{1,2}\,$ and $\,L_{2,3}\,$ _in_
-coming at $\,\upsilon\in V_{(+++)}$. To the manifolds (with boundary)
$\,\widehat{\Sigma},\widehat{\Gamma}\,$ and $\,V_{\nu}$, we associate the
respective strata $\,M_{0},M_{1}\,$ and
$\,M_{\nu-1}=\sqcup_{\varepsilon_{\nu}\in({\mathbb{Z}}/2{\mathbb{Z}})^{\times\nu}}\,T_{\varepsilon_{\nu}}\,$
of the target supermanifold (possibly further stratified, as in the boundary
example above), which determine a natural decomposition
$\,\mathcal{F}_{\sigma}\equiv[\widehat{\Sigma},M_{0}]\sqcup[\widehat{\Gamma},M_{1}]\sqcup\sqcup_{\nu\geq
3}\,\sqcup_{\varepsilon_{\nu}\in({\mathbb{Z}}/2{\mathbb{Z}})^{\times\nu}}\,[V_{\varepsilon_{\nu}},T_{\varepsilon_{\nu}}]\,$
of the space of $\sigma$-model fields. These target strata are endowed with
smooth structure maps $\,\iota_{1},\iota_{2}:M_{1}\longrightarrow M_{0}\,$ and
$\,\pi^{\varepsilon_{\nu}}_{k,k+1}:T_{\varepsilon_{\nu}}\longrightarrow
M_{1}\,$ subject to relations of order 2 mirroring those satisfied by their
worldsheet counterparts, requisite for the consistency of the field-theoretic
framework. The structure maps give rise to a family of pullback operators:
$\,\Delta=\iota_{2}^{*}-\iota_{1}^{*}\,$ and
$\,\Delta_{\varepsilon_{\nu}}=\sum_{k=1}^{\nu}\,\varepsilon^{(\nu)}_{k,k+1}\,\pi^{\varepsilon_{v}\,*}_{k,k+1}\,$
which obey the identities $\,\Delta_{\varepsilon_{\nu}}\circ\Delta=0\,$ and
thus establish a relative-cohomological structure on $\,M\equiv M_{0}\sqcup
M_{1}\sqcup\sqcup_{\nu\geq
3}\,\sqcup_{\varepsilon_{\nu}\in({\mathbb{Z}}/2{\mathbb{Z}})^{\times\nu}}\,T_{\varepsilon_{\nu}}\,$
in which the form
$\,\Omega\equiv(\chi,\omega,0)\in\Omega^{3}(M_{0})\oplus\Omega^{2}(M_{1})\oplus\Omega^{1}(M_{\nu\geq
3})\,$ acquires the status of a relative de Rham 3-cocycle, cp. [Sus12, Sec.
7.2], and the defect strata $\,M_{1}\,$ and $\,T_{\varepsilon_{\nu}}\,$ play
the rôle of correspondence spaces: The former supports a trivialisation of the
$\Delta$-image of the ‘bulk’ gerbe $\,\mathcal{G}\,$ on $\,M_{0}$, whereas the
latter carries a secondary trivialisation of the
$\Delta_{\varepsilon_{\nu}}$-image of that primary trivialisation. In order to
motivate this result of the in-depth analysis reported in [RS09], let us
consider the special case in which
$\,\iota_{D^{\times}}:D^{\times}\equiv(\iota_{1},\iota_{2})(M_{1})\hookrightarrow
M_{0}^{\times 2}\,$ is an embedding, and the connected component
$\,{\mathbb{S}}^{1}\subset\Gamma\,$ mapped to $\,M_{1}\,$ separates
diffeomorphic domains $\,D_{1}\,$ and $\,D_{2}$. Invoking the Wong–Affleck
‘folding trick’ [WA94], we may then regard the defect line as a boundary
defect in the $\sigma$-model on $\,D_{1}\,$ (onto which $\,D_{2}\,$ has been
‘folded’) with the target supermanifold $\,M_{0}^{\times 2}\,$ and the gerbe
$\,{\rm pr}_{1}^{*}\mathcal{G}\otimes{\rm pr}_{2}^{*}\mathcal{G}^{*}\,$ (the
dualisation of $\,{\rm pr}_{2}^{*}\mathcal{G}\,$ reflects the flip of the
orientation accompanying the ‘folding’, cp. [GSW11]), and with the boundary
$\,{\mathbb{S}}^{1}\,$ sent to $\,D^{\times}$. The reasoning of [GR02],
referred to previously, now calls for a trivialisation $\,{\rm
pr}_{1}^{*}\mathcal{G}\otimes{\rm
pr}_{2}^{*}\mathcal{G}^{*}\cong\mathcal{I}_{\omega}$, or, equivalently, a so-
called gerbe bi-module
$\,\Phi:\iota_{1}^{*}\mathcal{G}\cong\iota_{2}^{*}\mathcal{G}\otimes\mathcal{I}_{\omega}\,$
over $\,M_{1}$. This turns out to be the right structure for an _arbitrary_
choice of the correspondence space $\,(M_{1},\iota_{\cdot})$, cp. [RS09]. Note
in the passing that there is always the distinguished _identity defect_ ,
mapping to the component $\,M_{0}\subset M_{1}\,$ with
$\,\iota_{A}\mathord{\restriction}_{M_{0}}\equiv{\rm id}_{M_{0}}\,$ and
$\,\Phi\mathord{\restriction}_{M_{0}}\equiv{\rm id}_{\mathcal{G}}$. There is
no equally straightforward and general argument elucidating the gerbe-
theoretic structure to be pulled back to the $\,V_{\varepsilon_{\nu}}$, but we
may give a heuristic reasoning which emphasises the main idea behind the
rigorous construction while circumnavigating its technicalities, given ibidem.
In it, we assume all lines to be _in_ -coming at $\,v\,$ for the sake of
simplicity and drop the sign labels. Thus, whenever
$\,I_{(\nu)}\equiv(\pi_{1,2},\pi_{2,3},\ldots,\pi_{\nu-1,\nu},\pi_{\nu,1})(T_{(++\cdots+)})\subset
M_{1}^{\times\nu}\,$ is a submanifold,  _and_ the value of the action
functional of the $\sigma$-model is invariant under homotopy moves of the
defect within $\,\Sigma$, i.e., when we are dealing with a _topological_
defect in a conformally invariant $\sigma$-model, we may – upon cutting out a
disc centred on a given $\,\upsilon\in V_{(++\cdots+)}\,$ whose boundary
intersects each line emanating from $\,\upsilon\,$ once, and subsequently
cutting out from it another disc of radius $\,r\approx 0\,$ with the same
properties – deform the defect lines $\,L_{k,k+1}\,$ on the ensuing annulus in
such a way that the angular distance between the nearest neighbours
$\,(L_{k,k+1},L_{k+1,k+2})\,$ is $\,\varepsilon\approx
0(\approx\nu\varepsilon)$. Now we should be able to read off the sought-after
vertex structure from the limit $\,\varepsilon\searrow 0\,$ succeeded by
$\,r\searrow 0$. The former leaves us with a single defect line mapped into
$\,M_{1}^{(\nu)}=\\{(q_{1},q_{2},\ldots,q_{\nu})\in M_{1}^{\times\nu}\ |\
\iota_{2}(q_{k})=\iota_{1}(q_{k+1}),\ k\in\overline{1,\nu-1}\\}\,$ and
carrying the data of the composite 1-isomorphism $\,(\Phi_{\nu}\otimes{\rm
id}_{\mathcal{I}_{\omega_{1}+\omega_{2}+\cdots+\omega_{\nu-1}}})\circ\cdots\circ(\Phi_{3}\otimes{\rm
id}_{\mathcal{I}_{\omega_{1}+\omega_{2}}})\circ(\Phi_{2}\otimes{\rm
id}_{\mathcal{I}_{\omega_{1}}})\circ\Phi_{1}\,$ with
$\,(\Phi_{k},\omega_{k})={\rm pr}^{*}_{k}(\Phi,\omega)$, representing the
_fusion_ of the $\,\nu\,$ gerbe bi-modules. The latter reproduces the endpoint
$\,\widetilde{\upsilon}\,$ of the fused defect, or, equivalently, a junction
between it and the identity defect stretching from $\,\widetilde{\upsilon}\,$
to the boundary of the big disc (and mapping to $\,M_{0}\subset
M_{1}^{(\nu)}\,$ in a natural manner). In order to understand what ought to be
put at the junction, we fold the disc along the fused defect _and_ its
identity extension, whereupon we obtain a half-disc worldsheet with a
_piecewise_ boundary condition at the fold (the other part of its boundary has
a different status, and so we do not consider it here). At this stage, we may
apply a dimensionally reduced variant of the folding trick to the boundary
(Chan–Paton) degrees of freedom at the defect junction
$\,\widetilde{\upsilon}\,$ mapped to $\,T_{(++\cdots+)}$, whereby it
transpires that $\,\widetilde{\upsilon}\,$ should carry a trivialisation of
$\,(\Phi_{\nu,1}\otimes{\rm
id}_{\omega_{1,2}+\omega_{2,3}+\cdots+\omega_{\nu-1,\nu}})\circ\cdots\circ(\Phi_{3,4}\otimes{\rm
id}_{\mathcal{I}_{\omega_{1,2}+\omega_{2,3}}})\circ(\Phi_{2,3}\otimes{\rm
id}_{\mathcal{I}_{\omega_{1,2}}})\circ\Phi_{1,2}\,$ for
$\,(\Phi_{k,k+1},\omega_{k,k+1})=\pi^{*}_{k,k+1}(\Phi,\omega)\,$ (this makes
sense as $\,\Phi\,$ is represented by a principal
${\mathbb{C}}^{\times}$-bundle). The heuristic argument does not fix the
(global) connection of the trivialisation, it is only the detailed computation
of [RS09] which shows that it should be null. Altogether, we end up with the
superstring background
$\,\mathfrak{B}=(\mathcal{M},\mathcal{B},\mathcal{J})\,$ composed of: the bulk
target $\,\mathcal{M}=(M_{0},{\rm g},\mathcal{G})\,$ in which $\,(M_{0},{\rm
g})\,$ is a quasi-metric supermanifold with a gerbe $\,\mathcal{G}\,$ of
curvature $\,\chi\,$ over it; the $\mathcal{G}$-_bi-brane_
$\,\mathcal{B}=(M_{1},\iota_{\cdot},\omega,\Phi)\,$ with the bimodule
$\,\Phi:\iota_{1}^{*}\mathcal{G}\cong\iota_{2}^{*}\mathcal{G}\otimes\mathcal{I}_{\omega}$;
and the $(\mathcal{G},\mathcal{B})$-_inter-bi-brane_
$\,\mathcal{J}=\sqcup_{\nu\geq
3}\,\sqcup_{\varepsilon_{\nu}\in({\mathbb{Z}}/2{\mathbb{Z}})^{\times\nu}}\,(T_{\varepsilon_{\nu}},\pi^{\varepsilon_{\nu}}_{\cdot,\cdot},\varphi_{\varepsilon_{\nu}})\,$
with the component fusion 2-isomorphisms

$\displaystyle\varphi_{\varepsilon_{\nu}}\ :\
\bigl{(}\Phi^{(\nu)}_{\nu,1}\otimes{\rm
id}_{\mathcal{I}_{\Delta_{\varepsilon_{\nu}}\omega-\omega^{(\nu)}_{\nu,1}}}\bigr{)}\circ\cdots\circ\bigl{(}\Phi^{(\nu)}_{3,4}\otimes{\rm
id}_{\mathcal{I}_{\omega^{(\nu)}_{1,2}+\omega^{(\nu)}_{2,3}}}\bigr{)}\circ\bigl{(}\Phi^{(\nu)}_{2,3}\otimes{\rm
id}_{\mathcal{I}_{\omega^{(\nu)}_{1,2}}}\bigr{)}\circ\Phi^{(\nu)}_{1,2}\cong{\rm
id}_{\pi_{1}^{(\nu)\,*}\mathcal{G}}\,,$

where
$\,(\Phi^{(\nu)}_{k,k+1},\omega^{(\nu)}_{k,k+1})\equiv\pi^{(\varepsilon_{\nu})\,*}_{k,k+1}(\Phi^{\varepsilon^{(\nu)}_{k,k+1}},\varepsilon^{(\nu)}_{k,k+1}\,\omega)$,
and where
$\,\pi_{1}^{(\varepsilon_{\nu})}=\iota_{1}\circ\pi^{(\varepsilon_{\nu})}_{1,2}\,$
if $\,\varepsilon^{(\nu)}_{1,2}=+1$, and
$\,=\iota_{2}\circ\pi^{(\varepsilon_{\nu})}_{1,2}\,$ if
$\,\varepsilon^{(\nu)}_{1,2}=-1$. The superfield theory is determined by the
Dirac–Feynman amplitude $\,\mathcal{A}_{\rm DF}[\xi]={\rm
exp}({\mathsf{i}}\,S_{\sigma}[\xi])\,$ on $\,\mathcal{F}_{\sigma}\,$ in which
the action ‘functional’ splits $\,S_{\sigma}[\xi]=S_{\rm metr}[\xi]+S_{\rm
WZ}[\xi]\,$ into the ‘metric’ term $\,S_{\rm metr}[\xi]=\int_{\Sigma}\,{\rm
Vol}(\Sigma,\xi^{*}{\rm g})\,$ and the ‘topological’ WZ term given by the
decorated-surface holonomy $\,{\rm exp}({\mathsf{i}}\,S_{\rm WZ}[\xi])={\rm
Hol}_{(\mathcal{G},\Phi,(\varphi_{\cdot}))}(\xi|\Gamma)\,$ of [RS09].

The higher-supergeometric elements $\,\mathcal{G},\Phi\,$ and
$\,\varphi_{\varepsilon_{\nu}}\,$ of $\,\mathfrak{B}\,$ are distinguished 0-,
1- and 2-cells, respectively, of $\,\mathfrak{BGrb}_{\nabla}(M)$, cp. [Ste00,
Wal07]. The fundamental property of and the main rationale for physical
interest in these objects, discussed at length in, i.a., [Gaw88, GR02, Sus11],
is that they canonically determine – via cohomological transgression,
originally proposed in [Gaw88] – a prequantisation of the above superfield
theory and encode a lot of non-trivial information on its (nonperturbative)
structure, cp. [Sus21] for a recent review.

##  3\. The Trinity: Simpliciality, Symmetry and Semisimplicity

The construction of the poly-phase super-$\sigma$-model from the previous
section features submanifolds of Segal’s nerve of the $\Sigma$-fibred pair
groupoid $\,{\rm Pair}_{\Sigma}(\widehat{\Sigma})\,$ of the extended
worldsheet. The nerve is a canonical example of a _simplicial manifold_ , and
the identities of order 2 satisfied by the structure maps
$\,p^{\varepsilon_{\nu}}_{k,k+1}\,$ and $\,p_{A}\,$ are readily seen to
_follow_ from the elementary simplicial identities obeyed by the face maps of
$\,{\rm Pair}_{\Sigma}(\widehat{\Sigma})$. The order-2 relations between the
structure maps $\,\pi^{\varepsilon_{\nu}}_{k,k+1}\,$ and $\,\iota_{A}\,$ of
the background $\,\mathfrak{B}\,$ are a target-space realisation of their
worldsheet counterparts. They give rise to subsets in another simplicial
supermanifold, to wit, the nerve of the pair groupoid $\,{\rm Pair}(M_{0})\,$
of the bulk target space (cp., $\,D^{\times}\,$ and $\,I_{(\nu)}$). While
there is no a priori reason to expect that components of the stratified target
$\,M\,$ form a simplicial supermanifold, there are important circumstances in
which they do: This happens, e.g., in $\sigma$-models with topological defects
with _induction_ , studied in [RS09] in the context of orbifolding, in which
defect junctions of valence $\,\nu>3\,$ can be obtained from binary trees of
defect junctions of valence 3 in a limiting procedure in which the lengths of
all _internal_ edges are sent to 0, at no cost in the value of the action
functional (owing to the topological nature of the defects). As a result,
fusion 2-isomorphisms for the junctions of higher valence are induced, through
(vertical) composition, from the elementary ones for trivalent junctions which
define the binary-tree resolutions, cp. [RS09] and our heuresis in the
previous section. Such an intrinsically _semi_ -simplicial structure is
promoted to a fully fledged simplicial one through adjunction of the identity
defect, encountered previously, whose gerbe-theoretic data $\,{\rm
id}_{\mathcal{G}}\,$ are provided by the bulk gerbe itself [Wal07]. This
defect can be drawn anywhere in the worldsheet (in particular, it can be
attached to any defect junction, whereby the valence of the junction is
increased by 1). Its existence suggests the incorporation of the degeneracy
maps of a simplicial target.

Our hitherto considerations lead us to the definition of a _simplicial
superstring background_ with the target given by (a submanifold in) a
stratified simplicial supermanifold
$\,(M_{\bullet},d^{(\bullet)}_{\cdot},s^{(\bullet)}_{\cdot})\,$ with face maps
$\,d^{(n+1)}_{i}:M_{n+1}\longrightarrow M_{n}\,$ and degeneracy maps
$\,s^{(n)}_{i}:M_{n}\longrightarrow M_{n+1}\,$ defined for
$\,i\in\overline{0,n+1}\,$ and for all $\,n\in{\mathbb{N}}$, and subject to
the standard simplicial identities. The $\,d^{(n+1)}_{i}\,$ reproduce the
previously considered structure maps _uniquely_ as
$\,(\iota_{1},\iota_{2})=(d^{(1)}_{1},d^{(1)}_{0})\,$ and
$\,(\pi^{(3)}_{1,2},\pi^{(3)}_{2,3},\pi^{(3)}_{1,3})=(d^{(2)}_{2},d^{(2)}_{0},d^{(2)}_{1})$,
and – for $\,v>3\,$ – also $\,\pi^{(v)}_{1,v}=d^{(2)}_{1}\circ
d^{(3)}_{1}\circ\cdots\circ d^{(v-1)}_{1}\,$ and
$\,\pi^{(v)}_{k,k+1}=d^{(2)}_{2}\circ d^{(3)}_{2}\circ\cdots\circ
d^{(v-k)}_{2}\circ d^{(v-k+1)}_{0}\circ d^{(v-k+2)}_{0}\circ d^{(v-1)}_{0}\,$
for $\,k\in\overline{1,v-1}$, consistently with the identities of order 2
mentioned earlier. The $\,s^{(n)}_{i}\,$ account for the existence of the flat
identity (sub-)bi-brane $\,s^{(0)\,*}_{0}\Phi\equiv{\rm
id}_{\mathcal{G}}:s^{(0)\,*}_{0}d^{(1)\,*}_{1}\mathcal{G}=\mathcal{G}\cong\mathcal{G}=s^{(0)\,*}_{0}d^{(1)\,*}_{0}\mathcal{G}\otimes\mathcal{I}_{s^{(0)\,*}_{0}\omega}\,$
and allow to write down fusion 2-isomorphisms for defect junctions with
identity defect lines attached.

As simpliciality seems to be favoured by topological defects, which are
transmissive to the Virasoro currents of a _conformal_ $\sigma$-model, the
Segal–Sugawara realisation of the Virasoro algebra within the universal
enveloping algebra of a Kač–Moody algebra of a simple Lie algebra, known,
e.g., from the study of the WZW $\sigma$-model, suggests a natural direction
of enhancement of our construction: Focusing on $\sigma$-models with a rich
configurational symmetry (e.g., those with targets given by homogeneous spaces
of Lie supergroups), we may combine simpliciality with symmetry to further
constrain the geometry of the supertarget and the simplicial gerbe over it.
The point of departure is the identification of the bicategorial realisation
of rigid symmetries of the $\sigma$-model, which is readily achieved for
symmetries induced by isometries of the metric bulk target $\,(M_{0},{\rm
g})$. Thus, we consider a Lie supergroup $\,{\rm G}_{\sigma}\,$ together with
actions $\,M_{n}\lambda:{\rm G}_{\sigma}\times M_{n}\longrightarrow M_{n},\
n\in{\mathbb{N}}$, and so also with the fundamental vector fields
$\,\mathcal{K}^{n}_{X}=-(X\otimes{\rm id}_{\mathcal{O}_{M_{n}}})\circ
M_{n}\lambda^{*}\,$ over the $\,M_{n}\equiv(|M_{n}|,\mathcal{O}_{M_{n}})$,
labelled by elements of the tangent Lie superalgebra
$\,\mathfrak{g}_{\sigma}\ni X\,$ of $\,{\rm G}_{\sigma}$. The structure maps
$\,\pi^{\varepsilon_{\nu}}_{k,k+1}\,$ and $\,\iota_{A}\,$ are assumed ${\rm
G}_{\sigma}$-equivariant to ensure a natural alignment of the bulk and defect-
quiver variations of the action functional engendered by the
$\,\mathcal{K}^{n}_{X}$, cp. [RS09, GSW13]. We then say that $\,{\rm
G}_{\sigma}\,$ is a Lie supergroup of prequantisable rigid (configurational)
symmetries of the super-$\sigma$-model if (i) $\,\Omega\,$ is ${\rm
G}_{\sigma}$-invariant; (ii) the action admits a generalised relative
(co)momentum
$\,\kappa_{\cdot}:\mathfrak{g}_{\sigma}\longrightarrow\Omega^{1}(M_{0})\oplus\Omega^{0}(M_{1})\,$
such that $\,{\rm D}_{M_{\cdot}}\kappa_{X}=-\imath_{\mathcal{K}_{X}}\Omega$,
where $\,{\rm D}_{M_{\cdot}}\,$ is the relative de Rham differential [GSW13,
Prop. 2.8]; and (iii) it lifts to the higher-geometric components of
$\,\mathfrak{B}\,$ as a ${\rm G}_{\sigma}$-indexed111We put our discussion in
the so-called $\mathcal{S}$-point picture for the sake of simplicity. family
of 1-isomorphisms
$\,\Lambda_{g}:M_{0}\lambda_{g}^{*}\mathcal{G}\cong\mathcal{G}\,$ and
2-isomorphisms $\,\lambda_{g}:(\iota_{2}^{*}\Lambda_{g}\otimes{\rm
id}_{\mathcal{I}_{\omega}})\circ
M_{1}\lambda_{g}^{*}\Phi\cong\Phi\circ\iota_{1}^{*}\Lambda_{g}\,$ satisfying
the coherence identities
$\,M_{\nu-1}\lambda_{g}^{*}\varphi_{\varepsilon_{\nu}}=(\varphi_{\varepsilon_{\nu}}\circ{\rm
id})\bullet({\rm id}\circ\lambda^{(\nu)}_{g\,1,2})\bullet({\rm
id}\circ\lambda^{(\nu)}_{g\,2,3}\circ{\rm id})\bullet\cdots\bullet({\rm
id}\circ\lambda^{(\nu)}_{g\,\nu-1,\nu}\circ{\rm
id})\bullet(\lambda^{(\nu)}_{g\,\nu,1}\circ{\rm id})\,$ (in which some
_canonical_ 2-isomorphisms have been dropped for brevity). The existence of
such a coherent lift is a necessary and sufficient condition for the
invariance of the decorated-surface holonomy under the symmetry
transformations $\,\xi\mapsto M_{\cdot}\lambda_{g}\circ\xi$, and can be read
off from [GSW13, Thm. 4.4]. The marriage between simpliciality and symmetry
thus defined can be neatly established by declaring the structure maps of
$\,(M_{\bullet},d^{(\bullet)}_{\cdot},s^{(\bullet)}_{\cdot})\,$ ${\rm
G}_{\sigma}$-equivariant and, accordingly, by propagating a _given_ bulk
symmetry $\,M_{0}\lambda\,$ over $\,M_{\bullet}\,$ to engender a simplicial
${\rm G}_{\sigma}$-space with an action $\,M_{\bullet}\lambda\ :\ {\rm
G}_{\sigma}\times M_{\bullet}\longrightarrow M_{\bullet}$. This yields
_maximally (super)symmetric defects_.

Drawing further motivation from the WZW $\sigma$-model, with its generating
bi-chiral loop-group symmetries and the corresponding maximally symmetric bi-
branes of [FSW08, RS09, RS11] which are supported over _orbits_ of the bulk-
group action, we come to the final stage of the rather natural structurisation
of the super-$\sigma$-model. It boils down to imposing the requirement of
_semisimplicity_ upon the simplicial target ${\rm G}_{\sigma}$-supermanifold,
by which we mean a (disjoint-sum) decomposition of
$\,(M_{\bullet},d^{(\bullet)}_{\cdot},s^{(\bullet)}_{\cdot},M_{\bullet}\lambda)\,$
into orbits of the simplicial ${\rm G}_{\sigma}$-action. The demand that these
support – as a token of quantum-mechanical consistency of the construction – a
simplicial gerbe described previously is then anticipated to give rise to
cohomological superselection rules for the _admissible_ orbits in the
decomposition, which are the only ones that we choose to keep. For
$\,M_{\bullet}\,$ topologically nontrivial, this is bound to yield powerful
constraints on the ensuing target geometry, coming from the standard Dirac-
quantisation argument. Compactness of the target geometry should then result
in a rationalisation of the (super)background.

##  4\. The maximally supersymmetric simplicial Lie superbackgrounds

A prime example of a super-$\sigma$-model to which the above principles may be
applied _constructively_ , and in which their consequences may be explored, is
the WZW(-type) $\sigma$-model, with the bulk target given by a (Kostant–)Lie
supergroup $\,{\rm G}$. The latter is endowed with a canonical bi-invariant
Cartan 3-cocycle $\,\chi_{\rm C}^{q}=q\,{\rm tr}_{\mathfrak{g}}(\theta_{\rm
L}\wedge[\theta_{\rm L}\overset{\wedge}{,}\theta_{\rm L}]),\
\mathfrak{g}\equiv{\rm sLie}{\rm G}\,$ which geometrises, in physically
interesting cases (e.g., for $\,{\rm G}\,$ even compact and connected, and on
the super-Minkowski group) and for a suitable choice of the loop charge
$\,q\in{\mathbb{R}}$, as a supersymmetric gerbe $\,\mathcal{G}_{\rm C}$. The
target is to be seen as an orbit of an action $\,{\rm G}_{0}\lambda\,$ of a
subgroup of the group $\,{\rm G}\times{\rm G}\,$ of left and right
translations on $\,{\rm G}$. The corresponding maximally supersymmetric
defects implement the _right_ regular action $\,\wp\,$ of $\,{\rm G}\,$ on
itself, cp. [FSW08, RS09, RS11], and so here the stratified target
$\,M_{\bullet}\,$ embeds in the nerve $\,{\rm N}({\rm G}\rtimes_{\wp}{\rm
G})\,$ of the right action groupoid $\,{\rm G}\rtimes_{\wp}{\rm G}$. The
morphism composition of the latter groupoid represents supergroup
multiplication $\,{\rm m}_{\rm G}\ :\ {\rm G}\times{\rm G}\longrightarrow{\rm
G}$, which admits a gerbe-theoretic realisation in the form of a (generalised)
supersymmetric _multiplicative structure_ , instrumental in the construction
of the bi-brane for the said multiplicative defect. The structure is a
simplicial gerbe over Segal’s model of the classifying space $\,B{\rm G}\,$ of
$\,{\rm G}\,$ (containing $\,{\rm N}({\rm G}\rtimes_{\wp}{\rm G})\,$ as a
simplicial sub-manifold), and as such comprises a distinguished 1-isomorphism
$\,\mathcal{M}:{\rm pr}_{1}^{*}\mathcal{G}_{\rm C}\otimes{\rm
pr}_{2}^{*}\mathcal{G}_{\rm C}\cong{\rm m}_{\rm G}^{*}\mathcal{G}_{\rm
C}\otimes\mathcal{I}_{\varrho_{\rm PW}}\,$ over $\,{\rm G}^{\times 2}$,
written in terms of a Polyakov–Wiegmann 2-form $\,\varrho_{\rm PW}$, and a
coherent (quasi-)associator 2-isomorphism $\,\alpha\,$ over $\,{\rm G}^{\times
3}$, cp. [CJM+05, Sus22] for details. As the defect of interest maps to
$\,{\rm G}^{\times 2}\,$ and has structure maps $\,(\iota_{1},\iota_{2})=({\rm
pr}_{1},\wp\equiv{\rm m}_{\rm G})$, we may invoke the existence of
$\,\mathcal{M}\,$ in the Wong–Affleck argument of Sec. 2 to conclude that
$\,\mathcal{G}_{\rm C}\,$ must trivialise over the second cartesian factor
$\,\iota_{D}:D\hookrightarrow{\rm G}\,$ in the relevant bi-brane geometry
$\,\widetilde{\iota}{}_{D}:{\rm G}\times D\hookrightarrow{\rm G}^{\times 2}$.
This, however, implies that the bi-brane arises as a product
$\,\mathcal{B}_{\rm maxym}\equiv({\rm G}\times D,{\rm
pr}_{1},\wp,\widetilde{\iota}{}_{D}^{*}\varrho_{\rm PW}-{\rm
pr}_{2}^{*}\omega_{D},\Phi)\,$ of another kind of _fusion_ :
$\,\Phi=(\widetilde{\iota}{}_{D}^{*}\mathcal{M}\otimes{\rm
id}_{\mathcal{I}_{-{\rm pr}_{2}^{*}\omega_{D}}})\circ({\rm id}_{{\rm
pr}_{1}^{*}\mathcal{G}_{\rm C}}\otimes{\rm
pr}_{2}^{*}\mathcal{T}_{D}^{-1}\otimes{\rm id}_{\mathcal{I}_{-{\rm
pr}_{2}^{*}\omega_{D}}})\,$ between
$\,\widetilde{\iota}{}_{D}^{*}\mathcal{M}\,$ and the _boundary_ bi-brane
$\,\mathcal{B}_{\partial}\equiv(D,\iota_{D},\ast,\omega_{D},\mathcal{T}_{D})\,$
introduced before. By a similar argument [Sus22], the existence of
$\,\alpha\,$ turns the problem of constructing
$\,\varphi_{\varepsilon_{\nu+1}}\,$ into a search – over (the second factor
in) a disjoint union of ${\rm G}_{\nu}\lambda$-orbits within $\,{\rm
G}\times\widetilde{D}{}_{\nu}\equiv{\rm G}\times(D^{\times\nu}\cap{\rm
m}_{1\\_\nu}^{-1}(D))\,$ with $\,{\rm m}_{1\\_\nu}\equiv{\rm m}_{\rm
G}\circ({\rm m}_{\rm G}\times{\rm id}_{\rm G})\circ\cdots\circ({\rm m}_{\rm
G}\times{\rm id}_{{\rm G}^{\times\nu-2}})\,$ – for _boundary_ fusion
2-isomorphisms $\,\varphi^{(\partial)}_{\nu+1}:\otimes_{i=1}^{\nu}\,{\rm
pr}_{i}^{*}\mathcal{T}_{D}\cong({\rm
m}_{1\\_\nu}^{*}\mathcal{T}_{D}\otimes{\rm
id})\circ\mathcal{M}_{1\\_\nu-1,\nu}\mathord{\restriction}_{\widetilde{D}{}_{\nu}}$,
defined in terms of the 1-isomorphism
$\,\mathcal{M}_{1\\_\nu-1,\nu}=\mathcal{M}_{12\cdots\nu-1,\nu}\circ\cdots\circ(\mathcal{M}_{12,3}\otimes{\rm
id})\circ(\mathcal{M}_{1,2}\otimes{\rm id})\,$ in which
$\,\mathcal{M}_{12\cdots k-1,k}=(({\rm m}_{1\\_k-1}\times{\rm id}_{\rm
G})\circ({\rm pr}_{1},{\rm pr}_{2},\ldots,{\rm pr}_{k}))^{*}\mathcal{M}$.
(Incidentally, this gives a physical meaning to the seemingly meaningless
concept of ‘fusion of branes’ discussed in the literature [CW08].)

In the quantisation of the $\sigma$-model determined by the gerbe, states are
represented by disc Dirac–Feynman amplitudes [Gaw00, Sec. 4.1], and so it
stands to reason that the boundary bi-brane $\,\mathcal{B}_{\partial}\,$
provides a geometric realisation of the spectrum of the quantum theory.
Accordingly, we may anticipate that fusion 2-isomorphisms carry information on
its Verlinde fusion ring. We conclude this note with a review of evidence
which corroborates these expectations and a recapitulation of conjectures
based firmly thereon in the setting of the bosonic WZW $\sigma$-model. For
further details, as well as novel bicategorial constructions for the
Green–Schwarz super-$\sigma$-model with the super-Minkowskian target, we refer
the Reader to the extensive study [Sus22].

The bosonic WZW defect & another link to the CS theory. The bulk WZW
$\sigma$-model for the compact 1-connected Lie group $\,{\rm G}\,$ is defined
by the Cartan–Killing metric $\,{\rm g}=-6q\,{\rm
tr}_{\mathfrak{g}}(\theta_{\rm L}\otimes\theta_{\rm L})\,$ and the Cartan
3-form $\,\chi_{\rm C}^{q}\,$ with $\,24\pi
q\equiv{\mathsf{k}}\in{\mathbb{N}}^{\times}$, co-normalised in a manner which
ensures non-anomalous conformality of the (quantised) field theory.
Integrality of the _level_ $\,{\mathsf{k}}\,$ implies the existence of a
unique (isoclass of) gerbe geometrising $\,\chi_{\rm C}^{q}\,$ – the
${\mathsf{k}}$-th tensor power of the Gawȩdzki–Hitchin–Meinrenken basic gerbe
$\,\mathcal{G}_{\rm C}\,$ over $\,{\rm G}$. The rigid symmetries of the bulk
theory make up $\,{\rm G}_{\sigma}={\rm G}\times{\rm G}\,$ and lift to
$\,{\mathsf{N}}_{\bullet}({\rm G}\rtimes_{\wp}{\rm G})\,$ as $\,{\rm
G}_{n}\lambda:{\rm G}_{\sigma}\times{\rm G}^{\times
n+1}\ni((x,y),g,h_{i})\longmapsto(x\cdot g\cdot y^{-1},$ ${\rm
Ad}_{y}(h_{i}))\in{\rm G}^{\times n+1}$. Cohomological arguments localise
$\,\mathcal{B}_{\partial}\,$ over the disjoint union
$\,D=\sqcup_{\lambda\in{\mathsf{P}}^{\mathsf{k}}_{+}}\mathcal{C}_{\lambda}\,$
of the conjugacy classes $\,\mathcal{C}_{\lambda}={\rm Ad}_{\rm G}({\rm
e}^{2\pi{\mathsf{i}}\,\lambda/{\mathsf{k}}})\,$ labelled by weights
$\,\lambda\,$ from the fundamental affine Weyl alcove
$\,{\mathsf{P}}^{\mathsf{k}}_{+}\,$ at level $\,{\mathsf{k}}$, with
$\,\omega_{\partial}\mathord{\restriction}_{\mathcal{C}_{\lambda}}\,$ fixed
_uniquely_ by the bi-chiral loop-group extension of $\,{\rm G}_{1}\lambda$,
cp. [GR02, Sus22]. The existence of $\,(\mathcal{M},\alpha)\,$ is
unobstructed, and so $\,\mathcal{B}_{\partial}\,$ induces the non-boundary
maximally symmetric bi-brane as discussed above. Thus, the connected
components $\,{\rm G}\times\mathcal{C}_{\lambda}\,$ of the defect-line target
are in a 1-1 correspondence with chiral sectors of the bulk Hilbert space,
furnishing _integrable_ highest-weight representations
$\,\mathcal{V}_{\lambda,{\mathsf{k}}}\,$ of the Kač–Moody algebra
$\,\widehat{\mathfrak{g}}{}_{\mathsf{k}}$.

The localisation of $\,\mathcal{B}_{\rm maxym}\,$ conforms with the ‘triple-S’
argument of Sec. 3, and the last remark strengthens the expectation that the
associated inter-bi-brane carries _geometric_ information on the Verlinde
fusion ring of the WZW $\sigma$-model. Recall that the ring structure is
encoded by the multiplicity spaces in the decomposition, into the
$\,\mathcal{V}_{\lambda_{3},{\mathsf{k}}}$, of the Hilbert space of the
_boundary_ WZW $\sigma$-model on a strip with boundaries carrying the data of
$\,\mathcal{B}_{\partial}$, or, equivalently, by the spaces
$\,\mathscr{C}(\mathcal{V}_{\lambda_{1},{\mathsf{k}}}\otimes\mathcal{V}_{\lambda_{2},{\mathsf{k}}},\mathcal{V}_{\lambda_{3},{\mathsf{k}}})\,$
of rank-3 conformal blocks of the (chiral) bulk WZW theory, cp. [GTTNB04].
Each such space is the Hilbert space of the 3$d$ Chern–Simons (CS) theory on
the time cylinder $\,{\mathbb{R}}\times{\mathbb{C}}{\mathsf{P}}^{1}\,$ over
$\,{\mathbb{C}}{\mathsf{P}}^{1}$, coupled to vertical Wilson lines
$\,{\mathbb{R}}\times\\{\sigma_{i}\\},\ i\in\\{1,2,3\\}\,$ with holonomies
along the respective non-contractible loops, encircling simply the punctures
$\,\sigma_{i}$, valued in the $\,\mathcal{C}_{\lambda_{i}}$, cp. [Gaw00]. We
may now look for imprints of these structures in the _boundary_ component
$\,T_{++-}^{\partial}\,$ of the elementary inter-bi-brane geometry
$\,T_{++-}={\rm G}\times T_{++-}^{\partial}\,$ (e.g.), which the ‘triple-S’
argument predicts to be (a subset in) the disjoint union of ${\rm Ad}_{\rm
G}$-orbits in
$\,T_{\lambda_{1},\lambda_{2}}^{\lambda_{3}}\equiv(\mathcal{C}_{\lambda_{1}}\times\mathcal{C}_{\lambda_{2}})\cap{\rm
m}_{\rm G}^{-1}(\mathcal{C}_{\lambda_{3}})\,$ for any
$\,\lambda_{1},\lambda_{2}\,$ and $\,\lambda_{3}$. And, remarkably, we find
them! Indeed, the necessary condition for the existence of
$\,\varphi^{(\partial)}_{3}\,$ on
$\,T_{\lambda_{1},\lambda_{2}}^{\lambda_{3}}\,$ is the vanishing of
$\,\Delta_{++-}\omega\,$ (which restricts to $\,T^{\partial}_{++-}$), and the
latter 2-form turns out to be…the partially symplectically reduced
presymplectic form on the state space of the CS theory described above. Its
partial reduction, due to Alekseev and Malkin [AM95], with respect to the
‘pointed’ gauge group associated with a homological decomposition of
$\,{\mathbb{C}}{\mathsf{P}}_{(3)}\equiv{\mathbb{C}}{\mathsf{P}}\setminus\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}\,$
relative to a point $\,\sigma_{*}\in{\mathbb{C}}{\mathsf{P}}_{(3)}$, leaves us
with the tangent of the residual gauge group $\,[\sigma_{*},{\rm G}]\equiv{\rm
G}\,$ as the characteristic distribution of $\,\Delta_{++-}\omega$, which is
_just_ the symmetry group $\,{\rm G}\,$ of the ${\rm Ad}_{\rm G}$-orbits in
$\,T_{++-}^{\partial}$, whose disjoint union composes the _classical_ state
space of the CS theory in the parametrisation of [AM95]. In the light of the
interpretation of the conformal blocks as intertwiners $\,{\rm
Hom}_{\widehat{\mathfrak{g}}{}_{\mathsf{k}}}(\mathcal{V}_{\lambda_{1},{\mathsf{k}}}\otimes\mathcal{V}_{\lambda_{2},{\mathsf{k}}},\mathcal{V}_{\lambda_{3},{\mathsf{k}}})\,$
between current-symmetry sectors of the chiral bulk theory222More accurately,
we should speak of quantum intertwiners between
$\,U_{q({\mathsf{k}})}(\mathfrak{g})$-modules, cp. [GTTNB04]., this result is
also in keeping with the identification of the _transgressed_ fusion
2-isomorphisms transmissive to rigid bulk symmetries (which the WZW ones are,
cp. [GSW13]) as intertwiners of the symmetry representations on the (twisted)
state spaces fusing at the defect junction, cp. [Sus11, Sus12].

The structure of the simplicial WZW target is ultimately determined by the
requirement of existence of the 2-isomorphisms $\,\varphi^{(\partial)}_{\nu}$,
expected to distinguish a subfamily within the disjoint union of ${\rm
Ad}_{\rm G}$-orbits in the boundary factors
$\,T_{\varepsilon_{\nu}}^{\partial}\subset T_{\varepsilon_{\nu}}$. Taking into
account the above highly nontrivial result (which generalises to arbitrary
$\,\nu$), we are led to the following conjectures:

  1. (1)

The fusion 2-isomorphisms $\,\varphi^{(\partial)}_{\nu}\,$ exist only over
manifolds $\,\times_{i=1}^{\nu-1}\,\mathcal{C}_{\lambda_{i}}\cap{\rm
m}_{1\\_\nu-1}^{-1}(\mathcal{C}_{\lambda_{\nu}})\,$ with non-vanishing
Verlinde numbers $\,{\rm
dim}\,\mathscr{C}(\otimes_{i=1}^{\nu-1}\,\mathcal{V}_{\lambda_{i},{\mathsf{k}}},\mathcal{V}_{\lambda_{\nu},{\mathsf{k}}})\,$
(for conformal blocks of arbitrary rank).

  2. (2)

Given such a manifold, the number of those ${\rm Ad}_{\rm G}$-orbits in its
decomposition which support $\,\varphi^{(\partial)}_{\nu}\,$ is given by the
corresponding Verlinde number.

One may also anticipate that the fusion 2-isomorphisms of valence $\,\nu>3\,$
are induced from the elementary ones with $\,\nu=3\,$ due to simpliciality of
the WZW background, and that the inter-bi-brane fusing matrices defined, as
described in detail in [Sus22], by the associator move of [RS09] relating
inequivalent such induction schemes for $\,\varphi^{(\partial)}_{4}$, are
intimately related to the standard fusing matrices of the bulk WZW
$\sigma$-model. The first conjecture was corroborated for the special case of
$\,{\rm G}={\rm SU}(2)\,$ in [RS11], and the last expectation hinges on the
highly nontrivial cohomological evidence from the simple-current sector
gathered in [RS09] as well as on simple considerations of the topologicality
of the maximally symmetric defect. Various geometric, algebraic and field-
theoretic arguments in favour of the second conjecture were given in [Sus22].

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