#  BrepGen: A B-rep Generative Diffusion Model with Structured Latent Geometry

Xiang Xu 1,2 Joseph G. Lambourne2 Pradeep Kumar Jayaraman2
Zhengqing Wang1 Karl D.D. Willis2 Yasutaka Furukawa1
1Simon Fraser University 2Autodesk Research

###### Abstract

This paper presents BrepGen, a diffusion-based generative approach that
directly outputs a Boundary representation (B-rep) Computer-Aided Design (CAD)
model. BrepGen represents a B-rep model as a novel structured latent geometry
in a hierarchical tree. With the root node representing a whole CAD solid,
each element of a B-rep model (i.e., a face, an edge, or a vertex)
progressively turns into a child-node from top to bottom. B-rep geometry
information goes into the nodes as the global bounding box of each primitive
along with a latent code describing the local geometric shape. The B-rep
topology information is implicitly represented by node duplication. When two
faces share an edge, the edge curve will appear twice in the tree, and a
T-junction vertex with three incident edges appears six times in the tree with
identical node features. Starting from the root and progressing to the leaf,
BrepGen employs Transformer-based diffusion models to sequentially denoise
node features while duplicated nodes are detected and merged, recovering the
B-Rep topology information. Extensive experiments show that BrepGen sets a new
milestone in CAD B-rep generation, surpassing existing methods on various
benchmarks. Results on our newly collected furniture dataset further showcase
its exceptional capability in generating complicated geometry. While previous
methods were limited to generating simple prismatic shapes, BrepGen
incorporates free-form and doubly-curved surfaces for the first time.
Additional applications of BrepGen include CAD autocomplete and design
interpolation. The code, pretrained models, and dataset will be released.

Figure 1: BrepGen is a diffusion-based approach to generate 3D CAD models in
B-rep format. Geometry and topology relations are progressively denoised for
faces, edges, and vertices to form a water-tight solid at the end. Extensive
evaluations demonstrate that BrepGen generates a wide range of complex B-reps.

##  1 Introduction

Nearly every man-made object begins its life as a Computer-Aided Design (CAD)
model. Within the realm of CAD, the Boundary representation (B-rep) is the
predominant format for describing shapes, and is widely used in freeform
surface modeling to express complicated geometry. The B-rep representation
[35] generalizes half-edge meshes [23], allowing parametric curves and
surfaces to replace the planar facets and linear edges used in mesh modeling.
A B-rep consists of sets of interconnected faces, edges, and vertices. A face
is the visible region of a parametric surface, bounded by closed loops formed
by its adjacent edges, while an edge is the visible region of a parametric
curve, trimmed by vertices that define its start and end points. The adjacency
of neighbouring edges and vertices is recorded, allowing the structure to
provide a complete description of the final solid shape. A system capable of
directly generating B-reps would revolutionize the CAD design workflow,
reducing the extensive manual labor required from skilled designers and the
reliance on professional CAD software.

However, it is challenging to directly generate the B-rep. In contrast to
triangle meshes, B-reps contain a variety of parametric surface and curve
types such as toruses and Non-Uniform Rational B-Splines (NURBS). Each
geometry also has a different definition and its own set of parameters, making
it difficult to generate. Furthermore, the topological relations between all
geometry must be correct to form a water-tight solid. Recent CAD generative
models avoid direct B-rep generation and instead focus on sketch and extrude
modeling operations [39, 42, 40, 26, 18, 43] that represent only a limited
range of 3D shapes. Direct B-rep generation methods, like SolidGen [11], are
limited to highly simplified prismatic shapes and non-freeform surfaces.

To this end, we introduce BrepGen, a generative approach that directly outputs
diverse B-rep CAD models using Denoising Diffusion Probabilistic Models (DDPM)
[8]. Key to our method is the use of a structured latent geometry
representation that transforms any B-rep into a tree data structure.
Concretely, BrepGen encodes the geometry as node features in a tree, where the
root node identifies the CAD solid, and child-nodes, at the following three
levels, define the global position and latent local geometry for every face,
edge, and vertex. For the local geometry, we follow UV-Net [12] and use a grid
of points uniformly-sampled along the UV domain as a substitute for its
parametric function. Mating and association topologies are encoded through the
duplication of tree nodes in two ways. First, every shared edge (resp. vertex)
in a B-rep model turns into multiple nodes in a tree with the same node
features, establishing connections between adjacent geometries and their
corresponding mating duplicates. Second, the number of faces constituting a
solid and the number of edges bounding a face also varies, an additional
duplication is performed to pad child nodes at each parent to a predefined
length, leading to uniform width trees.

This node duplication is the key in our structured latent geometry
representation, encoding both the B-rep geometry and topology information in a
unified tree format, where continuous geometry regression implicitly recovers
discrete topology. Concretely, a Transformer-based diffusion model generates
node features top to bottom (i.e., face to edge to vertex), while nodes with
similar features are detected and merged 1) across different parents to
recover mating and adjacency relations, and 2) within each parent to restore
unique geometric elements associated with it. By merging the duplicated nodes,
the topology of each face is reconstructed, first connecting edges into closed
loops and then using them to trim the parent face. The resulting trimmed faces
are joined to output the CAD model as a watertight B-rep. In summary, we make
the following contributions:

  * •

A structured latent geometry representation, whose hierarchical tree, with
node duplication, encodes the B-rep geometry and topology in a unified format.

  * •

A latent diffusion module, capable of generating free-form surfaces and
trimming curves.

  * •

A newly collected Furniture B-rep Dataset with high-quality B-reps of indoor
objects across 10 categories.

  * •

Direct B-rep generation with SOTA performance.

Figure 2: Our structured latent geometry tree representation of a B-rep CAD
model. Shape feature is sampled from the parametric surface or curve using
grid in the parameter domain (left). Position and local geometry are extracted
for the face, edge, and vertex, and then encoded as node features in the tree
(mid). Topology is encoded by mating and association duplication. Node feature
is represented by color values, where duplicated nodes are of the same color
(right).

##  2 Related Work

Learning from B-reps. Due to their well defined topology, B-reps can be
representing as a graph [3] and subsequently fed to graph neural networks for
classification and segmentation [5, 12, 36] or used with custom convolutions
[16] or hierarchical graph structures [13, 4, 14]. We build upon prior work
encoding B-rep data, in particular the UV-based point sampling introduced by
UV-Net [12], and develop a novel generative diffusion model for direct B-rep
synthesis.

Sequential CAD Generation. Sequences of modeling operations stored in
parametric CAD files can be used as supervision to train generative models
[39, 42, 26, 43, 40]. Self-supervised learning can also infer the operations
that reconstruct a model [18, 41]. While significant progress has been made on
improving the fidelity and controllability aspects of the generated results,
sequential CAD generative models are limited to producing sketches comprised
of line, arc, circle, and the extrude modeling operation. Our method focuses
on direct synthesis of the B-rep data structure with emphasis on supporting
complex freeform curves and surfaces that cannot be achieved by previous
methods.

Direct B-rep Generation. Direct B-rep generation involves the creation of
geometric and topological primitives that are part of the B-rep data
structure. This includes parametric curves and surfaces, and vertices, edges,
and faces that can be used to trim and stitch the surfaces patches into a
solid model. While previous approaches can generate parametric curves [34] and
surfaces [29, 17], the problem of generating the topology that trims the
surfaces and connects them together to form a watertight solid, rather than
relying on predefined templates [30], is less explored. PolyGen [24]
demonstrates that it is possible to jointly synthesize geometry and topology
of n-gon meshes using Transformers [31] and pointer networks [32]. [33]
applied a pointer network to identify planar and cylindrical faces given the
2D projection of edges. [7] reconstructed B-reps from point clouds using a
neural network to predict the rough B-rep geometry and topology, that were
refined using a combinatorial optimization. Closest to our work is SolidGen
[11], that can synthesize entire B-reps, first with the synthesis of vertices,
followed by conditional construction of edge topology using a pointer network,
finally followed by another pointer network that conditionally connected the
edges to form faces. While this approach can produce plausible B-reps, it is
restricted to prismatic primitives which limit the complexity of the results,
and the amount of data that it can be trained on. Our method is more general
and can produce prismatic-looking, as well as freeform doubly-curved geometry.

DM for Geometry Generation. Diffusion models are good at generating geometry
when topology is given. HouseDiffusion [28] generates 2D floor-plan that
follows a given room connectivity. CAGE [19] generates 3D articulated object
conditioned on the ground-truth part connections. More recent methods like
PolyDiff [1] do not require topology as input conditioning but are also
limited to generating triangle soups of polygonal meshes. Using diffusion
model to unconditionally generate well-structured geometry data together with
the correct topology relations remains a challenge, which is what our method
aims to solve.

B-rep Datasets. Datasets containing 3D models in the B-rep format have grown
in number over the last several years. These include synthetically created
datasets with class labels, such as FabWave [2], SolidLetters [12], and MFCAD
[6], as well as datasets of human designed 3D models without canonical class
labels, such as ABC [15], Fusion 360 Gallery [36, 16, 38], Automate [13], and
DeepCAD [39]. Our Furniture B-rep Dataset, to the best of our knowledge, is
the first dataset to contain human-designed 3D models in the B-rep format
across a standard set of classes (e.g. tables, chairs etc).

##  3 Structured Latent Geometry

A B-rep model consists of geometric elements (faces, edges, vertices) with
pairwise topological relationships (face-edge, edge-vertex adjacency matrix).
The challenge lies in generating these two distinct data representations in a
general graph topology. To solve this, our approach unifies the geometry and
topology of a B-rep model as a hierarchical tree with a fixed graph topology,
where node features encode the geometry information and duplicated nodes
(i.e., nodes with near identical features during generation) implicitly encode
the topology information. With the root node representing a whole CAD solid,
the tree has three levels: faces, edges, and vertices from top to bottom. The
unified tree effectively bridges the two formats and becomes the foundation
for training our diffusion models. This section first explains the node
features encoding geometry and then node duplication schemes encoding
topology.

###  3.1 Geometry Encoding by Node Features

A face, an edge, or a vertex node encodes its geometry information into a node
feature with 1) a global position as bounding box parameters or a point and 2)
local shape details as a latent code.

Face ($F$). Underlying each face is a parametric surface with a function
$S(u,v):\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$, mapping a UV coordinate to a
3D point on the unbounded surface on which the face lies (Fig. 2 left). The
shape feature $F_{s}$ is a 2D array of 3D points sampled on the parametric
surface, and the position feature $F_{p}$ is the axis-aligned bounding box
enclosing the points: $F_{p}=[x_{1},y_{1},z_{1},x_{2},y_{2},z_{2}]$ encoding
the bottom-left and top-right corners. Following UV-Net [12], let
$[u_{\text{min}},u_{\text{max}}]\times[v_{\text{min}},v_{\text{max}}]\in\mathbb{R}^{2}$
be a 2D axis-aligned bounding box along the UV axes. Points are sampled from a
$N\times N$ equally spaced grid along the UV axes with step size $\delta
u=\frac{u_{\text{max}}-u_{\text{min}}}{N}$, $\delta
v=\frac{v_{\text{max}}-v_{\text{min}}}{N}$. 3D coordinates at all grid
locations are concatenated to form $F_{s}\in\mathbb{R}^{N\times N\times 3}$ as
the shape details. Capitalizing on the success of latent diffusion models for
image generation [27], a variational autoencoder (VAE) with a UNet backbone
further compresses $F_{s}$ into a latent code $F_{z}$, after normalizing the
3D coordinates by an affine transformation that maps the bounding box to the
canonical cube (i.e., $[-1,1]^{3}$). The feature of a face node is then
defined as $F=[F_{p},F_{z}]$. Note that this feature does not include the
outer trimming boundary or inner holes, which are given by the associated
edges. The ‘Face’ row in Fig. 2 (middle) illustrates two faces with different
bounding box positions and normalized sampled points, revealing the local
shape details.

Edge ($E$). Underlying each edge is a parametric curve with a function
$C(u):\mathbb{R}\rightarrow\mathbb{R}^{3}$, mapping a $u$-coordinate to a 3D
point on the unbounded curve on which the edge lies (Fig. 2 left). The shape
feature $E_{s}$ is a 1D array of 3D points sampled along the parametric curve.
The position feature $E_{p}$ is again defined as the bounding box parameters
enclosing the points. Let $[u_{\text{min}},u_{\text{max}}]\in\mathbb{R}$ be
the minimum and the maximum U-coordinate value of an edge. Points are sampled
from an $N$ equally spaced grid along the u-axis with step size $\delta
u=\frac{u_{\text{max}}-u_{\text{min}}}{N}$. 3D coordinates at all grid
locations are concatenated to form $E_{s}\in\mathbb{R}^{N\times 3}$ as the
shape details. Akin to the face feature construction, a VAE compresses $E_{s}$
into $E_{z}$, where $E=[E_{p},E_{z}]$ becomes the feature of an edge node. The
‘Edge’ row in Fig. 2 (middle) visualizes the position and the shape details.

Vertex ($V$). A vertex is a 3D point without additional shape details. The
feature of a vertex node is its point coordinate $V=(x,y,z)$. Vertex row in
Fig. 2 (middle) shows the location of two vertices.

###  3.2 Topology Encoding by Node Duplication

Two node duplication schemes encode B-rep topology into the unified tree
structure with a fixed graph topology.

Mating Duplication. A shared geometric element (i.e., an edge shared by faces
or a vertex shared by edges) is duplicated, which creates one copy under each
parent and turns a B-rep graph structure into a tree. For example, when a
shared edge is duplicated, its child vertices are replicated to maintain
information about the loop connectivity. The top-right tree in Fig. 2 shows
the resulting tree with node feature represented by color values, and the
shared blue vertex and the purple edge duplicated. Mating relations can be
later recovered by merging edge or vertex nodes with similar geometry across
different parents.

Association Duplication. The number of edges bounding a face and the faces
forming a solid vary, which is the last ambiguity in the graph topology of our
tree representation. Our idea is to pick a predetermined maximum branching
factor for each tree level and pad randomly selected nodes at each parent
node. The bottom-right tree in Fig. 2 illustrates this process with the tree
randomly padded to two faces and three edges per face. The yellow edge is
duplicated once and the purple edge is duplicated twice. Association relations
can be recovered by removing child nodes with the same geometry under each
parent. Note that every edge is always connected to two vertices of its start
and end, making the vertex association duplication unnecessary.

Figure 3: Top row: Generation of the face bounding box position and face
latent geometry. Nodes are split in half to represent each. Bottom row:
Generation of the edge bounding box position and edge-vertex joint latent
geometry, both conditioned on the parent face. Gray color represents noisy
feature values. Points are decoded from the latent geometry for visualization.

##  4 Method

BrepGen consists of two VAEs that compress the face and edge shape feature
into geometry latent vectors, and a latent diffusion module that sequentially
denoises the node latent features in a tree from root to leaf, enabling
effective training and inference.

###  4.1 Shape Geometry VAE

Following Stable Diffusion [27], we use two VAEs to compress the shape feature
$F_{s},E_{s}$ into lower-dimensional latent $F_{z},E_{z}$. Shape feature is
first normalized to $[-1,1]$. A VAE with 2D convolution UNet backbone
compresses the normalized face feature. A similar VAE with 1D convolution
compresses the edge feature. Both VAEs are trained with MSE reconstruction
loss and a KL regularization term. Concretely, the face VAE encoder compresses
$F_{s}$ into latent vector $F_{z}=E(F_{s})$. A decoder then reconstructs
$\hat{F}_{s}=D(F_{z})=D(E(F_{s}))$, Reconstruction loss is between
$\hat{F}_{s}$, $F_{s}$. In practice, we set $N=32$ to densely sample the UV
parameter domain, and downsample the input by a factor of $8$ while retaining
the feature depth, resulting in $F_{z}\in\mathbb{R}^{4\times 4\times 3}$ and
$E_{z}\in\mathbb{R}^{4\times 3}$. To further reduce computation, the two
endpoint vertices are concatenated with an edge to form joint latent
$E_{zv}\in\mathbb{R}^{4\times 3+6}$.

###  4.2 Latent Diffusion Module

Our latent diffusion module is trained with the latent geometry and global
position as node features in the tree. The overall architecture of our
generative latent diffusion module is illustrated in Fig. 3. We follow the
DDPM [8] training scheme and use four Transformer-based denoisers to
sequentially remove the noise added to the nodes.

Diffusion Process. Given a tree with node features $\mathbf{x}_{0}$, the
forward diffusion process adds Gaussian noise to all the nodes in $\mathbf{T}$
steps. The noisy tree node features $\mathbf{x}_{t}$ at time step $t$ are
sampled as:

$\displaystyle q(\mathbf{x}_{t}|\mathbf{x}_{0})$
$\displaystyle=\mathcal{N}(\mathbf{x}_{t};\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0},(1-\bar{\alpha}_{t})\mathbf{I}),$
(1) $\displaystyle\mathbf{x}_{t}$
$\displaystyle=\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\boldsymbol{\epsilon}_{t},$

where noise $\boldsymbol{\epsilon}_{t}\sim\mathcal{N}(0,\mathbf{I})$ and
$\bar{\alpha}_{t}=\prod_{i=1}^{t}\alpha_{i}$, $\alpha_{t}=1-\beta_{t}$. The
$\beta_{t}$ is determined by the noise variance scheduler, a linear scheduler
by default. After the noise injection, the encoded B-rep geometry and topology
are corrupted. For example, duplicated nodes have different features and no
longer exhibit the correct topological relations.

Sequential Denoising. Generating all geometry at once is difficult. We use
sequential generation to denoise the face, edge, and vertex progressively.
Concretely, we factored the distribution of node features $\mathbf{x}$ into a
product of conditional distributions:

$p(\mathbf{x})=p(E_{zv}|E_{p},F)p(E_{p}|F)p(F_{z}|F_{p})p(F_{p}|\emptyset).$
(2)

The ordering reflects a top-down generation process with edge conditioned on
face, and latent geometry conditioned on global position. Fig. 3 shows the
conditional denoising process with a denoising network for each distribution.

The four denoising networks in our latent diffusion module share a common
Transformer backbone. Without loss of generality, we will use edge position
denoiser (Fig. 3 bottom-left) as an example. Input to the denoiser are edge
tokens with noisy bounding box parameters. Network is conditioned on the
previously denoised face tokens from Fig. 3 (top-row), which are embedded as:

$\boldsymbol{F}\leftarrow\text{MLP}(W_{p}F_{p})+\text{MLP}(W_{z}F_{z}).$ (3)

$F_{p}\in\mathbb{R}^{6}$ is the face bounding box coordinates and
$F_{z}\in\mathbb{R}^{48}$ is the flattened latent geometry vector.
$W_{p}\in\mathbb{R}^{d\times 6}$, $W_{z}\in\mathbb{R}^{d\times 48}$ are the
two $d$-dimension embedding matrices. MLP is fully connected layers with SiLU
activation. The noisy edge bounding boxes are encoded in a similar manner as
$\boldsymbol{E_{p}}=\text{MLP}(W_{p}E_{p})$.

Rather than using cross-attention, the predefined parent-child relations from
the tree are used to directly inject the face condition. Let edge $j$ be the
child node of face $i$. Embedding from the $i$th face token is added to the
$j$th edge bounding box position token as:

$\boldsymbol{\hat{E}_{p,\hskip 1.0ptj}}\leftarrow\boldsymbol{E_{p,\hskip
1.0ptj}}+\boldsymbol{F_{i}}.$ (4)

During top-down generation, a known number of child nodes are added for each
parent at the next stage, leading to simple token addition for the condition.
Edge embedding $E_{zv}$ is similarly embedded as
$\boldsymbol{E_{zv}}=\text{MLP}(W_{zv}E_{zv})$ with
$W_{zv}\in\mathbb{R}^{d\times 18}$. The previously denoised edge position and
parent face condition are added to every $j$th edge token as
$\boldsymbol{E_{zv,\hskip 1.0ptj}}+\boldsymbol{\hat{E}_{p,\hskip 1.0ptj}}$
following Eq. 4. Final input to the denoiser also adds the time and condition
embedding as $f(t,c)$. Note that we do not use learnable positional encoding.

Loss Function. All four Transformer-based denoisers are trained separately to
predict the L2-norm regression loss of the added sampled noise as in DDPM [8].
The general loss term is:

$L=\mathbb{E}_{t,\mathbf{x}_{0},\boldsymbol{\epsilon}_{t}}\Big{[}\|\boldsymbol{\epsilon}_{t}-\boldsymbol{\epsilon}_{\theta}(\sqrt{\bar{\alpha}_{t}}\mathbf{x}_{0}+\sqrt{1-\bar{\alpha}_{t}}\boldsymbol{\epsilon}_{t},t)\|^{2}\Big{]},$
(5)

where $\boldsymbol{\epsilon}_{t}$ is the Gaussian noise added at time $t$ in
the forward pass, $\mathbf{x}_{t}$ is the corrupted data following Eq. 1. In
our case, $\mathbf{x}_{0}$ is the clean node features
$F_{p},F_{z},E_{p},E_{zv}$, and $\mathbf{x}_{t}$ is the noisy node features
after forward pass. During sampling, the predicted noise at every time step is
used to denoise the data from random Gaussian noise.

###  4.3 B-rep Post-Processing

A set of heuristics is used to find duplicated nodes in the tree and
explicitly decode the generated B-rep topology. For efficiency, we use early
pruning to recover association as the tree is being generated. Given the face
nodes, we detect duplicates as those with bounding box corners within 0.08
Euclidean distance of each other and averaged point-wise difference less than
0.2 for the decoded points from the latent shape feature. Duplicate faces are
removed to recover the unique faces belonging to a solid. We avoid
regenerating edges for the duplicated faces in the next stage.

A similar procedure recovers the unique edges associated under every face, a
0.08 edge bounding box threshold and 0.2 threshold for the decoded shape
feature points are used to merge edges under the same parent. The vertex/edge
association is always two and does not require deduplication. Finally, we
traverse the tree again from leaf to root to find duplicated child nodes at
the same level but associated with different parents, which are the shared
edges and vertices for determining the final mating relations.

###  4.4 Implementation Details

We implement BrepGen in PyTorch and trained it on 4 NVIDIA RTX A5000 GPUs. The
diffusion model backbone is a standard Transformer module with pre-layer
normalization, 12 self-attention layers with 12 heads, a hidden dimension of
1024, a feature dimension of 768 and dropout rate of 0.1. AdamW [21] with
$5e-4$ learning rate, gradient clipping of 5 is used for optimization. Weight
for KL regularization is set to $1e-6$. The two face denoisers are trained
with a batch size of 512, and the two edge denoisers are trained with a batch
size of 64.

For training the latent diffusion module, we normalized the bounding box to
[-3, 3], and multiply the latent geometry by 0.5 to reduce the value range to
roughly match that of a unit Gaussian. To reduce train and test domain
difference, we perform cross-model augmentation where the input to the
denoisers are randomly augmented following the scheme proposed in [9]. We also
randomly shuffle the training input sequence to avoid over-fitting to a
particular order. Node padding is done by randomly selecting a child node
(with repetition) and use it for padding until reaching the predefined length.
It is separately performed for every geometry in the CAD model at every
iteration.

At inference, we use PNDM [20] of 200 steps, with the exception of the face
and edge position where we switch to a slower DDPM of 1000 steps for the last
250 time steps. Empirically we find this coarse to fine denoising generate
more precise bounding box position. We also find it helpful to sample more
tokens than the training maximum. This helps to avoid the case where noisy
nodes are incorrectly merged early in the denoising process, leading to a lack
of available tokens to fill in missing region at later stages.

##  5 Furniture B-Rep Dataset

To promote future B-rep generation research, we introduce Furniture B-rep
Dataset containing 6,171 B-rep CAD models across 10 common furniture
categories. To our knowledge, this is the first dataset in the B-rep format
containing 3D models with freeform surfaces in addition to canonical category
labels. We reached out to OnShape, a web-based CAD tool and repository [25],
and obtained an agreement to export their publicly available design assets
into our B-rep dataset. We used keywords to identify 3D models in each
category and searched inside both partstudio and assembly documents. We
perform a number of manual filtering steps on the raw data to remove
duplicates, verify correct categories, remove low quality data, and rotate the
3D models in each category into a canonical orientation. A visual overview of
the dataset is provided in Fig. 4.

Figure 4:  Furniture B-rep Dataset overview colored by category. Figure 5:
Unconditional generation results on mechanical parts by (a) DeepCAD, (b)
SolidGen and (c) BrepGen. Our method generates more realistic-looking CAD
models with fewer broken geometry. Topological connections are also correct
even on complicated objects.

##  6 Experiments

This section presents unconditional and conditional generation results.
Extensive analysis demonstrates that BrepGen consistently produces high
quality B-reps with complex topology and geometry, while providing enhanced
control over the generation process.

###  6.1 Experiment Setup

Datasets. Generation is evaluated on the 1) DeepCAD dataset [39] of mechanical
parts made from sketch and extrude operations, and 2) the Furniture B-rep
Dataset with more complicated furniture models. We use the original
train/val/test split from DeepCAD and remove duplicated models in the training
set following [37]. Closed faces (cylinders, etc.) are split on the seams
following SolidGen [11]. B-reps with more than 30 faces or 20 edges per face,
and made from multiple bodies are removed. After filtering, a total of 87,815
B-reps are used for training the VAEs and the latent diffusion module. We also
split Furniture B-rep Dataset into 90/5/5 and train on 1,198 B-reps with a
maximum of 50 faces and 30 edges per face.

Evaluation Metrics. We use two sets of metrics to quantitatively measure
generation quality: Distribution Metrics and CAD Metrics. For Distribution
Metrics we use 3,000 B-reps randomly-sampled from the generated data and 1,000
B-reps from the reference test set. For each B-rep, we sample 2,000 points
from the solid surface and compute the following metrics:

$\bullet$ Coverage (COV) is the percentage of reference data with at least one
match after assigning every generated data to its closest neighbor in the
reference set based on Chamfer Distance (CD).

$\bullet$ Minimum Matching Distance (MMD) is the averaged CD between a
reference set data and its nearest neighbor in the generated data.

$\bullet$ Jensen-Shannon Divergence (JSD) measures the distribution distance
between reference and generated data after converting point clouds into
$28^{3}$ discrete voxels.

For CAD Metrics the same 3,000 B-reps are used to compute the following
metrics, as used by [11]:

$\bullet$ Novel percentage of data that do not appear in training set.

$\bullet$ Unique percentage of data that appears only once in generation.

$\bullet$ Valid percentage of data that are water-tight solids.

###  6.2 Unconditional B-rep Generation

We compare with DeepCAD [11] and SolidGen [11] for unconditional generation of
mechanical parts. DeepCAD is evaluated based on the reconstructed B-reps from
its generated sketch and extrude sequences.

Quantitative Evaluation. Tab. 1 reports the averaged scores from 10 different
runs. BrepGen consistently outperforms both baselines with better COV, MMD
scores and a substantially lower JSD, demonstrating improvements in generation
quality and a closer match to the ground-truth distribution. Generated data
from BrepGen is also novel and different from the training set, as indicated
by the high novel and unique scores. Here, we consider two B-reps identical if
the topology connections are the same, and the shape geometry points from the
faces are equal after 4 bit quantization. The valid ratio of BrepGen is also
better compared to the SolidGen direct B-rep generation baseline. Our criteria
for validity enforces water-tight CAD models with no broken topology, and the
Novel and unique scores are calculated using the valid B-reps.

Figure 6: BrepGen generation results conditioned on the class label. Last row
shows the wireframe connected by the generated edges and vertices. The results
clearly show that BrepGen excels in generating well-structured CAD models with
free-form surfaces and curvatures.

Qualitative Evaluation. Fig. 5 illustrates qualitative results for
unconditional generation. We see that BrepGen generates appealing B-reps made
of a wide range of topologically connected faces, showing better generation
diversity than the baselines. The structural complexity is also higher with
fewer unbounded open regions or self-intersecting edges. Additional results
are in Fig. 13.

Table 1: Quantitative evaluations for DeepCAD unconditional generation based on the Coverage (COV) percentage, Minimum Matching Distance (MMD), Jensen-Shannon Divergence (JSD), and Unique, Novel, Valid ratio. Method | COV | MMD | JSD | Novel | Unique | Valid
---|---|---|---|---|---|---
|  % $\uparrow$ | $\downarrow$ | $\downarrow$ |  % $\uparrow$ |  % $\uparrow$ |  % $\uparrow$
DeepCAD | 65.46 | 1.29 | 1.67 | 87.4 | 89.3 | 46.1
SolidGen | 71.03 | 1.08 | 1.31 | 99.1 | 96.2 | 60.3
BrepGen | 71.26 | 1.04 | 0.09 | 99.8 | 99.7 | 62.9

###  6.3 Controllable B-rep Generation

We demonstrate BrepGen results on Furniture B-rep Dataset and show two
applications for CAD design. With no positional encoding and random shuffling
at training, BrepGen is permutation invariant with respect to the input
tokens, eliminating the need for a specific sequence order between existing
and to-be-generated components.

Conditional Generation. A class-conditioned BrepGen is trained on the
Furniture B-rep Dataset using classifier-free guidance [10]. Class embeddings
are added to every input token and an extra label indicates the unconditional
case, which occurs with a 10% probability at training. Fig. 6 shows the
generation results for all 10 categories. BrepGen generates the correct
topological connections across different components and outputs free-form
surfaces not possible with SolidGen. The generated edges form connected
wireframes with loops precisely trimming the faces into water-tight solids.
Additional results are provided in Fig. 14.

CAD Autocompletion. We use a pretrained BrepGen to autocomplete the full B-rep
from partial faces provided by the user. Motivated by RePaint [22], a random
subset of face tokens are replaced with the provided face geometry diffused to
that time step during face denoising. The replacement is discontinued in the
final 50 steps to allow minor adjustments to the provided faces. Subsequent
edges and vertices are also regenerated. Fig. 7 shows diverse autocompleted
results from different Gaussian noise. Even disjoint parts are connected and
transformed into water-tight solids with faces seamlessly generated between,
below, or above the given input. The inpainting process can make slight
adjustments to user input so that it connects with the rest of the geometry.
For example, the radius of the legs has been slightly increased for the middle
bed in the last row.

Figure 7: Given partial faces (gray), BrepGen auto-completes the geometry and
topology for a diverse set of CAD models (blue).

Design Interpolation. Given a pair of CAD models, face tokens from the two
shapes are concatenated, diffused for 150 steps, repeatedly padded and passed
to the face denoiser. Edges and vertices are also regenerated. This is
different from blending the latent value as mixing two geometry tokens will
not lead to valid shapes. Fig. 8 shows the interpolated results. From left to
right, face tokens from the target are increasingly added to the source for
face generation, after which the source tokens are removed until only the
target tokens are left. The topology and geometry changes are smooth, with
interpolated design directly generated in B-rep format.

##  7 Limitations and Future Work

BrepGen enables the generation of 3D models in the Boundary representation but
has several limitations that warrant future work. BrepGen supports only water-
tight single body solids; more complicated CAD models with multiple assembled
bodies are left to future work. If edges or faces are too close to one another
they will be merged and identified as one after deduplication. In practice,
this implies a minimum threshold, determining how close two similar surfaces
or edges can be. We train only on data with a minimum threshold of $0.05$
after normalizing the CAD geometry to the $[-3,3]$ range. This limitation is
similar to other sketch-and-extrude generation approaches, e.g. [42], that
quantize geometry to a set bit range resulting in vertex merging. The chosen
threshold of $0.05$ is roughly equivalent to a 1 bit difference after 7 bit
quantization. Finally, while the heuristic post-processing used to generate
the final B-rep is simple, fast, and can handle complicated data, to achieve
better results future work on a learning-based post-processing module may
provide more robust handling of invalid shapes.

Figure 8: Design interpolation from source to target. Geometry and topology of
the generated B-rep undergo progressive changes.

##  8 Conclusion

We introduced BrepGen, a generative diffusion model for direct B-rep
generation. Extensive experiments showed that BrepGen surpasses existing
methods and establishes state-of-the-art results for B-rep generation. We
outlined our newly collected Furniture B-rep Dataset and results that
demonstrate the ability of BrepGen to generate complicated B-rep 3D models
with free-form and doubly-curved surfaces for the first time. We believe
BrepGen moves us one step closer towards a CAD foundation model capable of
directly generating B-reps to reduce the extensive manual labor required from
skilled designers using today’s professional CAD software.

## Acknowledgements

This research is partially supported by NSERC Discovery Grants with
Accelerator Supplements and DND/NSERC Discovery Grant Supplement, NSERC
Alliance Grants, and John R. Evans Leaders Fund (JELF).

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## APPENDIX

##  Appendix A Furniture B-rep Dataset

We provide additional statistics for the Furniture B-rep Dataset. Fig. 9 shows
the category names in the dataset along with the number of 3D models per
category. Fig. 10 shows the number of bodies per design in the dataset as a
distribution. Fig. 11 shows the number of faces per design in the dataset as a
distribution. Fig. 12 shows the number of edges per design in the dataset as a
distribution.

##  Appendix B Additional Generation Results

Additional generation results on DeepCAD and Furniture B-rep Dataset are
provided in Figs. 13 and 14. Figs. 15, 16 and 17 also visualize the faces,
edges, and vertices at different stages of the reverse denoising process. For
simpler visualization, we sync the generated geometry to the same time steps,
and transformed them to the world coordinate space based on the generated
global position. The geometry removed after association deduplication are not
shown. The duplicated shared edges and vertices are visualized to show how
they were denoised to similar values at the end.

Figure 9: Design count per category. Figure 10: Number of bodies per design in
the dataset shown as a distribution. Figure 11: Number of faces per design in
the dataset shown as a distribution. Figure 12: Number of edges per design in
the dataset shown as a distribution. Figure 13: Unconditionally generated
mechanical part B-reps from BrepGen. Figure 14: Conditionally generated
furniture B-reps from BrepGen, colored by category. Figure 15: Visualization
of the reverse denoising process from left to right. Each row shows the faces
(colored) or edges and vertices (black and red) at a particular time step.
Column represents different time steps from T=1000 to T=0.  Figure 16:
Visualization of the reverse denoising process from left to right. Each row
shows the faces (colored) or edges and vertices (black and red) at a
particular time step. Column represents different time steps from T=1000 to
T=0. Figure 17: Visualization of the reverse denoising process from left to
right. Each row shows the faces (colored) or edges and vertices (black and
red) at a particular time step. Column represents different time steps from
T=1000 to T=0.


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##  Appendix A Chirally symmetric confining $\psi\eta$ model

An interesting possibility pointed out for this model is that no condensates
form, the system confines and the flavor symmetry remains unbroken [5]. The
candidate massless degrees of freedom in the IR are

$B^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad A,B=1,2,\ldots,N+4\;,$
(A.1)

(baryons), antisymmetric in $A\leftrightarrow B$. All of the $SU(N+4)_{\rm
f}\times U(1)$ anomaly triangles are saturated by $B^{[AB]}$, see Table 2.

| fields | $SU(N)_{\rm c}$ | $SU(N+4)$ | ${U_{\psi\eta}}(1)$
---|---|---|---|---
i UV | $\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $N+4$
| $\eta^{A}$ | $(N+4)\cdot{\bar{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-(N+2)$
IR | $B^{[AB]}$ | $\frac{(N+4)(N+3)}{2}\cdot(\cdot)$ | ${\yng(1,1)}$ | $-N$
Table 2: Chirally symmetric “confining” phase of the $\psi\eta$ model. As in
other Tables of the text, the multiplicity, charges and the representation are
shown for each set of fermions. $(\cdot)$ stands for a singlet representation.

##  Appendix B Higgs phase of the $\psi\eta$ model

Another possibility for the $\psi\eta$ model is that a color-flavor locked
phase appears [12, 18], with

$\langle\psi^{\\{ij\\}}\eta_{i}^{B}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{jB}\;,\qquad
j,B=1,2,\dots N\;,$ (B.1)

where the symmetry is reduced to

$\displaystyle SU(N)_{\rm cf}\times SU(4)_{\rm f}\times U(1)^{\prime}\,.$
(B.2)

A subset of the same baryons ($B^{[A_{1}B_{1}]}$ and $B^{[A_{1}B_{2}]}$ in the
notation of Table 3) saturate all of the triangles for (B.2), see Table 3. The
massless degrees of freedom are $\tfrac{N^{2}+7N}{2}$ massless baryons
$B^{[A_{1}B]}$ and $8N+1$ NG bosons.

To reproduce correctly the strong anomaly in the IR, however, another
condensate $\langle{B}{B}\rangle\sim\langle\psi\eta\eta\psi\eta\eta\rangle$
and another set of massless baryon $B^{[A_{2}B_{2}]}$ (see Table 3) are
needed. This does not alter neither the symmetry breaking pattern (B.2), nor
the anomaly matching. See Sec. 5.2 for more detail.

| fields | $SU(N)_{\rm cf}$ | $SU(4)_{\rm f}$ | $U^{\prime}(1)$
---|---|---|---|---
i UV | $\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $1$
| $\eta^{A_{1}}$ | ${\bar{\yng(2)}}\oplus{\bar{\yng(1,1)}}$ | $N^{2}\,\cdot\,(\cdot)$ | $-1$
| $\eta^{A_{2}}$ | $4\cdot{\bar{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-\frac{1}{2}$
IR | $B^{[A_{1}B_{1}]}$ | ${\bar{\yng(1,1)}}$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $-1$
| $B^{[A_{1}B_{2}]}$ | $4\cdot{\bar{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-\frac{1}{2}$
| $B^{[A_{2}B_{2}]}$ | $6\cdot(\cdot)$ | ${\yng(1,1)}$ | $0$
Table 3: Color-flavor locked phase in the $\psi\eta$ model. $A_{1}$ or $B_{1}$
stand for the first $N$ flavors ($A_{1},B_{1}=1,2,\ldots,N$), whereas $A_{2}$
or $B_{2}$ run over the rest of the flavor indices, $N+1,\ldots,N+4$. Another
set of potentially massless baryons $B^{[A_{2}B_{2}]}$ do not contribute to
$SU(N)_{\rm cf}\times SU(4)_{\rm f}\times U^{\prime}(1)$ anomalies.

##  Appendix C Symmetric confining phase for the $\chi\eta$ model

Let us first examine the possible confining vacua, with full unbroken global
symmetry [5]. The massless baryons needed for ’t Hooft’s anomaly matching are

$B^{\\{CD\\}}=\chi_{[ij]}\,{\eta}^{i\,C}{\eta}^{j\,D}\;,\qquad
C,D=1,2,\ldots(N-4)\;,$ (C.1)

symmetric in $C\leftrightarrow D$.

| fields | $SU(N)_{\rm c}$ | $SU(N-4)$ | $U(1)_{\chi\eta}$
---|---|---|---|---
UV | $\chi$ | ${\bar{\yng(1,1)}}$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $N-4$
| ${\eta}^{A}$ | $(N-4)\cdot{{\yng(1)}}$ | $N\,\cdot\,{\yng(1)}$ | $-(N-2)$
i IR | $B^{\\{AB\\}}$ | $\frac{(N-4)(N-3)}{2}\cdot(\cdot)$ | ${\yng(2)}$ | $-N$
Table 4: Confinement and unbroken symmetry in the $\chi\eta$ model

##  Appendix D Higgs phase of the $\chi\eta$ model

It was pointed out [12, 18] that this system may instead develop a color-
flavor locked condensate,

$\langle\chi_{[ij]}{\eta}^{B\,j}\rangle={\rm
const}\,\Lambda^{3}\delta_{i}^{B}\;,\qquad i,B=1,2,\ldots,N-4\;.$ (D.1)

The symmetry is broken to

$\displaystyle SU(N-4)_{\rm cf}\times U(1)^{{}^{\prime}}\times SU(4)_{\rm
c}\,.$ (D.2)

The massless baryons (C.1) saturate all the anomalies associated with
$SU(N-4)_{\rm cf}\times U(1)^{{}^{\prime}}$. There are $\chi_{i_{2}j_{2}}$
fermions which remain massless and strongly coupled to the $SU(4)_{\rm c}$. We
assume that $SU(4)_{\rm c}$ confines, and the condensate

$\langle\chi\chi\rangle\neq 0\;,$ (D.3)

forms and $\chi_{i_{2}j_{2}}$ acquire mass dynamically. Assume that the
massless baryons are:

$B^{\\{AB\\}}=\chi_{[ij]}\,{\eta}^{i\,A}{\eta}^{j\,B}\;,\qquad
A,B=1,2,\ldots(N-4)\;,$ (D.4)

the saturation of all of the triangles associated can be seen in Table 5.

| fields | $SU(N-4)_{\rm cf}$ | $U^{\prime}(1)$ | $SU(4)_{\rm c}$
---|---|---|---|---
UV | $\chi_{i_{1}j_{1}}$ | ${\bar{\yng(1,1)}}$ | $N$ | $\frac{(N-4)(N-5)}{2}\cdot(\cdot)$
| $\chi_{i_{1}j_{2}}$ | $4\cdot{\bar{\yng(1)}}$ | $\frac{N}{2}$ | $(N-4)\cdot{\bar{\yng(1)}}$
| $\chi_{i_{2}j_{2}}$ | $\frac{4\cdot 3}{2}\cdot(\cdot)$ | $0$ | ${\bar{\yng(1,1)}}$
| ${\eta}^{i_{1},A}$ | $\yng(2)\oplus\yng(1,1)$ | $-N$ | $(N-4)^{2}\cdot(\cdot)$
| ${\eta}^{i_{2},A}$ | $4\,\cdot\,{\yng(1)}$ | $-\frac{N}{2}$ | $(N-4)\cdot\yng(1)$
i IR | $B^{\\{AB\\}}$ | ${\yng(2)}$ | $-N$ | $\frac{(N-4)(N-3)}{2}\cdot(\cdot)$
Table 5: Color-flavor locking in the $\chi\eta$ model. The color index $i_{1}$
or $j_{1}$ runs up to $N-4$ and the rest is indicated by $i_{2}$ or $j_{2}$.

Complementarity [97] appears to be working here, in the sense that the
massless sector of the dynamical Higgs phase has the same $SU(N-4)\times U(1)$
symmetry as in the confining phase discussed in Appendix C. See however a
discussion in [96] on this point, which seems to point to the conclusion that
the apparent complementarity (which occurs only in the $\chi\eta$ model, but
in none of other BY and GG models) is just a coincidence.

##  Appendix E Confining symmetric phase of the BY models

The candidate massless composite fermions for the Bars-Yankielowicz models
are:

${({B}_{1})}^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{({B}_{2})}^{a}_{A}=\bar{\psi}_{ij}\bar{\eta}^{i}_{A}\xi^{j,a}\;,\qquad{({B}_{3})}_{\\{ab\\}}=\psi^{ij}{\bar{\xi}}_{i,a}{\bar{\xi}}_{j,b}\;,$
(E.1)

the first is anti-symmetric in $A\leftrightarrow B$ and the third is symmetric
in $a\leftrightarrow b$, see Table 6.

| $SU(N)_{\rm c}$ | $SU(N+4+p)$ | $SU(p)$ | ${U}(1)_{\psi\eta}$ | ${U}(1)_{\psi\xi}$
---|---|---|---|---|---
${{B}_{1}}$ | $\frac{(N+4+p)(N+3+p)}{2}\cdot(\cdot)$ | ${\yng(1,1)}$ | $\frac{(N+4+p)(N+3+p)}{2}\cdot(\cdot)$ | $-N+p$ | $p$
${{B}_{2}}$ | $(N+4+p)p\cdot(\cdot)$ | $p\cdot\bar{\yng(1)}$ | $(N+4+p)\cdot{\yng(1)}$ | $-(p+2)$ | $-(N+p+2)$
${{B}_{3}}$ | $\frac{p(p+1)}{2}\cdot(\cdot)$ | $\frac{p(p+1)}{2}\cdot(\cdot)$ | $\bar{\yng(2)}$ | $N+4+p$ | $2N+4+p$
Table 6: Chirally symmetric phase of the BY model.

Explicit anomaly-matching checks can be found in e.g., [71].

##  Appendix F Higgs phase in the BY models

Something nontrivial happens in the dynamical Higgs phase for general BY
models. i.e., with $p>0$. There are now two symmetry breaking channels,
$\psi\eta$ and $\xi\eta$. We assume that both condensates occur as:

$\displaystyle\langle\psi^{ij}\eta_{i}^{B}\rangle=\,c_{\psi\eta}\,\Lambda^{3}\delta^{jB}\neq
0\;,\qquad j,B=1,\dots,N\;,$
$\displaystyle\langle\xi^{i,a}\eta_{i}^{A}\rangle=\,c_{\eta\xi}\,\Lambda^{3}\delta^{aA}\neq
0\;,\qquad a=1,\dots,N\;,\quad A=N+1,\dots,N+p\;,$ (F.1)

where $\Lambda$ is the renormailization-invariant scale dynamically generated
by the gauge interactions and $c_{\eta\xi},c_{\psi\eta}$ are coefficients both
of order one. The resulting symmetry breaking pattern s

$\displaystyle SU(N)_{\mathrm{c}}\times SU(N+4+p)_{\eta}\times
SU(p)_{\xi}\times U(1)_{\psi\eta}\times U(1)_{\psi\xi}$
$\displaystyle\xrightarrow{\langle\xi\eta\rangle,\langle\psi\eta\rangle}SU(N)_{{\rm
cf}_{\eta}}\times SU(4)_{\eta}\times SU(p)_{\eta\xi}\times
U(1)_{\psi\eta}^{\prime}\times U(1)_{\psi\xi}^{\prime}\;.$ (F.2)

The color gauge symmetry is completely (dynamically) broken, leaving color-
flavor diagonal $SU(N)_{{\rm cf}_{\eta}}$ symmetry. $U(1)_{\psi\eta}^{\prime}$
and $U(1)_{\psi\xi}^{\prime}$ are combinations respectively of
$U(1)_{\psi\eta}$ (4.6) and $U(1)_{\xi\eta}$ (4.7) with the element of
$SU(N+4+p)_{\eta}$ generated by

$t_{SU(N+4+p)_{\eta}}=\left(\begin{array}[]{c|c|c}(-\alpha(p+2)-p\beta){\bf
1}_{N\times N}&&\\\ \hline\cr&\frac{\alpha(N-p)-\beta p}{2}{\bf 1}_{4\times
4}&\\\ \hline\cr&&(\alpha+\beta)(N+2){\bf 1}_{p\times p}\\\
\end{array}\right)\;.$ (F.3)

Making the decomposition of the fields one gets Table 7.

| $SU(N)_{{\rm cf}_{\eta}}$ | $SU(4)_{\eta}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\psi\eta}^{\prime}$ | $U(1)_{\psi\xi}^{\prime}$
---|---|---|---|---|---
$\psi$ | ${\yng(2)}$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $\frac{N(N+1)}{2}\cdot(\cdot)$ | $N+4+p$ | $p$
$\eta_{1}$ | ${\bar{\yng(2)}}\oplus{\bar{\yng(1,1)}}$ | $N^{2}\cdot(\cdot)$ | $N^{2}\cdot(\cdot)$ | $-(N+4+p)$ | $-p$
$\eta_{2}$ | $4\cdot{\bar{\yng(1)}}$ | $N\cdot{\yng(1)}$ | $4N\cdot(\cdot)$ | $-\frac{N+p+4}{2}$ | $-\frac{p}{2}$
$\eta_{3}$ | $p\cdot{\bar{\yng(1)}}$ | $Np\cdot(\cdot)$ | $N\cdot\bar{\yng(1)}$ | $0$ | $N+2$
$\xi$ | $p\cdot{{\yng(1)}}$ | $Np\cdot(\cdot)$ | $N\cdot{{\yng(1)}}$ | $0$ | $-(N+2)$
Table 7: UV fieds in the BY model, decomposed as a direct sum of the representations of the unbroken group of Eq. (F.2).  | $SU(N)_{{\rm cf}_{\eta}}$ | $SU(4)_{\eta}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\psi\eta}^{\prime}$ | $U(1)_{\psi\xi}^{\prime}$
---|---|---|---|---|---
${B}_{1}$ | ${\bar{\yng(1,1)}}$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $\frac{N(N-1)}{2}\cdot(\cdot)$ | $-(N+4+p)$ | $-p$
${B}_{2}$ | $4\cdot{\bar{\yng(1)}}$ | $N\cdot{\yng(1)}$ | $4N\cdot(\cdot)$ | $-\frac{N+p+4}{2}$ | $-\frac{p}{2}$
Table 8: IR fieds in the BY model, the massless subset of the baryons in Tab.
6 in the Higgs phase.

The composite massless baryons are subset of those in (E.1):

$\displaystyle{B}_{1}^{[AB]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{B}_{2}^{[AC]}=\psi^{ij}\eta_{i}^{A}\eta_{j}^{\rm
c}\;,$ $\displaystyle A,B=1,\dots,N\;,\quad C=N+1,\dots,N+4\;.$ (F.4)

##  Appendix G Confining symmetric phase of the GG models

The candidate massless composite fermions for the generalized Georgi-Glashow
models are:

${({B}_{1})}^{\\{AB\\}}=\chi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad{({B}_{2})}^{a}_{A}=\bar{\chi}_{ij}\bar{\eta}^{i}_{A}\xi^{j,a}\;,\qquad{({B}_{3})}_{[ab]}=\chi^{ij}{\bar{\xi}}_{i,a}{\bar{\xi}}_{j,b}\;,$
(G.1)

the first symmetric in $A\leftrightarrow B$ and the third anti-symmetric in
$a\leftrightarrow b$. All anomaly triangles are saturated by these candidate
massless composite fermions (Table 9 vs Tab. LABEL:sua). For explicit matching
equations, see e.g., [71].

| $SU(N)_{\rm c}$ | $SU(N-4+p)$ | $SU(p)$ | ${U}(1)_{\chi\eta}$ | ${U}(1)_{\chi\xi}$
---|---|---|---|---|---
${B}_{1}$ | $\frac{(N-4+p)(N-3+p)}{2}\cdot(\cdot)$ | ${\yng(2)}$ | $\frac{(N-4+p)(N-3+p)}{2}\cdot(\cdot)$ | $-N+p$ | $p$
${B}_{2}$ | $(N-4+p)p\cdot(\cdot)$ | $p\cdot\bar{\yng(1)}$ | $(N-4+p)\cdot{\yng(1)}$ | $-(p-2)$ | $-(N+p-2)$
${B}_{3}$ | $\frac{p(p-1)}{2}\cdot(\cdot)$ | $\frac{p(p-1)}{2}\cdot(\cdot)$ | $\bar{\yng(1,1)}$ | $N-4+p$ | $2N-4+p$
Table 9: IR massless fermions in the chirally symmetric phase of the GG model.

##  Appendix H Higgs phase in the GG models

The Higgs phase of the GG model may be described by either of the two possible
bifermion channels $\chi\eta$ and $\xi\eta$. We assume that both occur

$\displaystyle\langle\chi^{ij}\eta_{i}^{A}\rangle=\,c_{\chi\eta}\,\Lambda^{3}\delta^{jA}\neq
0\;,\qquad j=1,\dots,N-4\;,\quad A=1,\dots,N-4\;,$
$\displaystyle\langle\xi^{i,a}\eta_{i}^{B}\rangle=\,c_{\eta\xi}\,\Lambda^{3}\delta^{aB}\neq
0\;,\qquad a=1,\dots,p\;,\quad B=N-4+1,\dots,N-4+p\;.$

The symmetry-breaking pattern is

$\displaystyle SU(N)_{\mathrm{c}}\times SU(N-4+p)_{\eta}\times
SU(p)_{\xi}\times U(1)_{\chi\eta}\times U(1)_{\chi\xi}$
$\displaystyle\xrightarrow{\langle\xi\eta\rangle,\langle\chi\eta\rangle}SU(4)_{{\rm
c}}\times SU(N-4)_{{\rm cf}_{\eta}}\times SU(p)_{\eta\xi}\times
U(1)_{\chi\eta}^{\prime}\times U(1)_{\chi\xi}^{\prime}\;.$ (H.2)

The color gauge symmetry is partially (dynamically) broken, leaving color-
flavor diagonal global $SU(N-4)_{{\rm cf}_{\eta}}$ symmetry and an
$SU(4)_{{\rm c}}$ gauge symmetry. $U(1)_{\chi\eta}^{\prime}$ and
$U(1)_{\chi\xi}^{\prime}$ are a combinations respectively of $U(1)_{\chi\eta}$
(LABEL:uce) and $U(1)_{\chi\xi}$ (LABEL:ucx) with the elements of
$SU(N)_{\mathrm{c}}$ and $SU(N-4+p)_{\eta}$ generated by:

$\displaystyle\qquad t_{SU(N)_{\rm
c}}=\left(\begin{array}[]{c|c}2\frac{\alpha(N-4+p)+\beta p}{N-4}{\bf
1}_{(N-4)\times(N-4)}&\\\ \hline\cr&-\frac{\alpha(N-4+p)+\beta p}{2}{\bf
1}_{4\times 4}\\\ \end{array}\right)\ ,$ (H.5) $\displaystyle
t_{SU(N-4+p)_{\eta}}=\left(\begin{array}[]{c|c}-\frac{p(\alpha+\beta)(N-2)}{N-4}{\bf
1}_{(N-4)\times(N-4)}&\\\ \hline\cr&(\alpha+\beta)(N-2){\bf 1}_{p\times p}\\\
\end{array}\right)\;.$ (H.8)

Making the decomposition of the fields in the direct-sum representations in
the subgroup one arrives at Table 10.

| $SU(N-4)_{{\rm cf}_{\eta}}$ | $SU(4)_{{\rm c}}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\chi\eta}^{\prime}$ | $U(1)_{\chi\xi}^{\prime}$
---|---|---|---|---|---
$\chi_{1}$ | ${\yng(1,1)}$ | $\frac{(N-4)(N-5)}{2}\cdot(\cdot)$ | $\frac{(N-4)(N-5)}{2}\cdot(\cdot)$ | $\frac{(N-4+p)N}{(N-4)}$ | $p\frac{N}{N-4}$
$\chi_{2}$ | $4\cdot{\yng(1)}$ | $(N-4)\cdot{\yng(1)}$ | $4(N-4)\cdot(\cdot)$ | $\frac{(N-4+p)N}{2(N-4)}$ | $\frac{pN}{2(N-4)}$
$\chi_{3}$ | $6\cdot(\cdot)$ | ${\yng(1,1)}$ | $6\cdot(\cdot)$ | $0$ | $0$
$\eta_{1}$ | ${\bar{\yng(2)}}\oplus{\bar{\yng(1,1)}}$ | $(N-4)^{2}\cdot(\cdot)$ | $(N-4)^{2}\cdot(\cdot)$ | $-\frac{(N-4+p)N}{(N-4)}$ | $-\frac{pN}{N-4}$
$\eta_{2}$ | $p\cdot\bar{\yng(1)}$ | $p(N-4)\cdot(\cdot)$ | $(N-4)\cdot\bar{\yng(1)}$ | $-2-2\frac{p}{N-4}$ | $N-2-\frac{2p}{N-4}$
$\eta_{3}$ | $4\cdot{\bar{\yng(1)}}$ | $(N-4)\cdot{\bar{\yng(1)}}$ | $4(N-4)\cdot(\cdot)$ | $-\frac{(N-4+p)N}{2(N-4)}$ | $-\frac{pN}{2(N-4)}$
$\eta_{4}$ | $4p\cdot(\cdot)$ | $p\cdot{\bar{\yng(1)}}$ | $4\cdot\bar{\yng(1)}$ | $\frac{N-4+p}{2}$ | $N-2+\frac{p}{2}$
$\xi_{1}$ | $p\cdot{{\yng(1)}}$ | $p(N-4)\cdot(\cdot)$ | $(N-4)\cdot{{\yng(1)}}$ | $2+2\frac{p}{N-4}$ | $-(N-2)+\frac{2p}{N-4}$
$\xi_{2}$ | $4p\cdot(\cdot)$ | $p\cdot{{\yng(1)}}$ | $4\cdot{{\yng(1)}}$ | $-\frac{N-4+p}{2}$ | $-(N-2)-\frac{p}{2}$
Table 10: UV fieds in the GG model, decomposed as a direct sum of the representations of the unbroken group of Eq. (H.2).  | $SU(N-4)_{{\rm cf}_{\eta}}$ | $SU(p)_{\eta\xi}$ | $U(1)_{\chi\eta}^{\prime}$ | $U(1)_{\chi\xi}^{\prime}$
---|---|---|---|---
${B}$ | ${\bar{\yng(2)}}$ | $\frac{(N-4)(N-3)}{2}\ \cdot(\cdot)$ | $-\frac{(N-4+p)N}{(N-4)}$ | $-\frac{pN}{N-4}$
Table 11: IR fied in the GG model in the dynamical Higgs phase.

The composite massless baryons are subset of those in (G.1):

$\displaystyle{B}^{\\{AB\\}}=\chi^{ij}\eta_{i}^{A}\eta_{j}^{B}\;,\qquad
A,B=1,\dots,N-4\;.$ (H.9)

In the IR these fermions saturate all the anomalies of the unbroken chiral
symmetry.

There is a novel feature in the GG models, which is not shared by the BY
models. There is an unbroken strong gauge symmetry $SU(4)_{{\rm c}}$, with a
set of fermions,

$\chi_{3}\;,\quad\chi_{2}\;,\quad\eta_{3}\;,\quad\eta_{4}\;,\quad\xi_{2}\;,$
(H.10)

charged with respect to it. The pairs $\\{\chi_{2}\;,\,\eta_{3}\\}$ and
$\\{\eta_{4}\;,\,\xi_{2}\\}$ can form massive Dirac fermions and decouple.
These are vectorlike with respect to the surviving infrared symmetry, (H.2),
thus irrelevant to the anomalies. The fermion $\chi_{3}$ can condense

$\langle\chi_{3}\chi_{3}\rangle\ $ (H.11)

forming massive composite mesons, $\sim\chi_{3}\chi_{3}$, which also decouple.
It is again neutral with respect to the unbroken symmetry. To summarize,
$SU(4)_{c}$ is invisible - confines - in the IR, and only the unpaired part of
the $\eta_{1}$ fermion $\big{(}{\bar{\yng(2)}}\big{)}$ remains massless. It’s
anomalies are reproduced by the composite fermions, (H.9).

The massive mesons $\chi_{2}\,\eta_{3}$, $\eta_{4}\,\xi_{2}$,
$\chi_{3}\,\chi_{3}$ are not charged with the flavor symmetries surviving in
the infrared. It is tempting to consider them as a sort of “dark matter”, as
contrasted to the fermions ${B}^{AB}$ which constitute the ordinary, “visible”
sector, in a toy-model interprepretation.