Generative AI for Brane Configurations and Coamoeba

TL;DR

The paper develops a conditional variational autoencoder (CVAE) to learn a world model for coamoeba projections of the mirror curve $Σ$ of toric Calabi–Yau 3-folds and their associated Type IIB brane configurations. By conditioning on complex structure moduli ${\bf c}$, the CVAE generates high-fidelity coamoebae on $T^2$ and supports continuous transitions between toric phases, enabling near-continuous phase diagrams and phase-paths that correspond to Seiberg duality between 4d ${\rm N}=1$ quiver theories. A central finding is that the latent space $L$ not only captures multimodality in coamoeba representations but also acts as a new moduli-like space that controls the sharpness of the mirror curve's projection and brane intersections. The approach is demonstrated on the cone over the zeroth Hirzebruch surface $F_0$, with prospects for generalization to other toric CY3-folds and broader toric/brane configurations, offering a scalable framework for exploring toric phases and dualities in string/gauge theory contexts.

Abstract

We introduce a generative AI model to obtain Type IIB brane configurations that realize toric phases of a family of 4d N=1 supersymmetric gauge theories. These 4d N=1 quiver gauge theories are worldvolume theories of a D3-brane probing a toric Calabi-Yau 3-fold. The Type IIB brane configurations are given by the coamoeba projection of the mirror curve associated with the toric Calabi-Yau 3-fold. The shape of the mirror curve and its coamoeba projection, as well as the corresponding Type IIB brane configuration and the toric phase of the 4d N=1 theory, all depend on the complex structure moduli parameterizing the mirror curve. We train a generative AI model, a conditional variational autoencoder (CVAE), that takes a choice of complex structure moduli as input and generates the corresponding coamoeba. This enables us not only to obtain a high-resolution representation of the entire phase space for a family of 4d N=1 theories corresponding to the same toric Calabi-Yau 3-fold, but also to continuously track the movements of the mirror curve and the branes wrapping the curve in the corresponding Type IIB brane configurations during phase transitions associated with Seiberg duality.

Figures (8)

• Figure 1: (a) The coamoeba for the cone over the zeroth Hirzebruch surface $F_0$ are shown at 3 different values of the complex structure moduli $(c_{11}, c_{12}, c_{21}, c_{22})$. The white regions are occupied by the D5-brane, whereas the black regions are occupied by the NS5-brane wrapping $\Sigma$. (b)-(c) The skeleton graph of the coamoeba is a bipartite periodic graph on $T^2$ known as a dimer model or a brane tiling. This work concentrates on the coamoeba projection itself and its role in locating the D5- and NS5-branes in the Type IIB brane configuration as shown in (a). In (c), the polygonal faces in the bipartite graph are labelled by integers $i=1, \dots, G$, where $G$ is the number of $U(N)_i$ gauge groups in the corresponding $4d$$\mathcal{N}=1$ gauge theory. These faces are due to the bipartite nature of the graph even-sided and have therefore even number of boundary edges and nodes. Each face corresponds to a $U(N)_i$ gauge group of the $4d$$\mathcal{N}=1$ theory. Edges connecting white and black nodes of the dimer correspond to bifundamental chiral fields $X_{ij}$, where the orientation around white nodes is clockwise and around black nodes is counter-clockwise. These orientations also give the correct gauge-invariant combinations of chiral fields around white and black nodes that are associated respectively to positive and negative terms in the superpotential $W$ of the $4d$$\mathcal{N}=1$ theory. (d)-(e) The dual graph of the brane tiling forms what we refer to as a periodic quiver on $T^2$. The associated superpotentials are shown in (f). We note here that the continuous change of complex structure moduli changes the shape of the coamoeba on $T^2$, which in turn modifies the associated brane tilings and corresponding $4d$$\mathcal{N}=1$ theories. These $4d$$\mathcal{N}=1$ theories corresponding to the same toric Calabi-Yau 3-fold, here the cone over $F_0$, are related by Seiberg duality and represent toric phases.
• Figure 2: (a) The $m_{\text{PCA}}=2$ dimensional phase diagram for $F_0$ obtained in Seong:2023njx from $N=2304$ different coamoeba with complex structure moduli components $c_{ab} \in \{0, \pm 3, \pm 6, \pm 9\}$. Each point in the phase diagram represents a coamoeba with a unique choice of complex structure moduli. Using our trained CVA model, we are now able to generate many more coamoeba for $F_0$ within the same range of complex structure moduli. In (b), we obtain the $m_{\text{PCA}}=2$ dimensional phase diagram for $F_0$ based on $N=389376$ different coamoeba generated by our CVAE model with complex structure moduli components $c_{ab} \in \{0, \pm 0.5, \pm 1.0, \dots, \pm 5.5 ,\pm 6.0\}$. The resulting phase diagram is such that points representing different coamoeba are much more closely placed by the PCA, giving a near-continuous phase diagram for $F_0$.
• Figure 3: (a) We have a selection of 5 coamoeba for $F_0$ with all having components of complex structure moduli $(c_{12}, c_{21}, c_{22})=(-9.0, -3.0, -3.0)$, whereas $-6.0 \leq c_{11} \leq +6.0$ is varied at coarse steps of $\Delta c_{11} = 3.0$. This sample of the training dataset is used to train the CVAE model, which after training is capable to generate (b) a selection of 25 coamoeba with $(c_{12}, c_{21}, c_{22})=(-9.0, -3.0, -3.0)$ and $-6.0 \leq c_{11} \leq +6.0$, where now $c_{11}$ can be varied at much smaller intervals $\Delta c_{11} = 0.5$. This includes coamoeba that were not part of the original training dataset and allows us to track infinitesimal changes in the coamoeba and its associated Type IIB brane configurations.
• Figure 4: (a) The coamoeba given by $\mathbf{x}$ and the associated complex structure moduli $\mathbf{c}$ are combined into a tensor $\overline{\mathbf{x}}$ of dimension $5\times 64\times 64$, which is the input tensor for the conditional variational autoencoder (CVAE) model illustrated here. (b) The encoder network of the CVAE model consists of a fully-connected convolutional neural network with 4 layers $f_e^{(u)}(\theta_e)$, where the output layer is then connected to a fully-connected linear layer that outputs the mean $\mu_k(\theta_e)$ and logarithmic variance $\log \sigma_k^2(\theta_e)$ of the approximated posterior distribution $q_{\theta_e}(\mathbf{z}|\mathbf{x}, \mathbf{c})$, where $k=1, \dots, m$ and $m$ is the dimension of the latent space $L$. (c) The decoder network of the CVAE model also consists of a fully-connected convolutional neural network with 4 layers $f_d^{(v)}(\theta_d)$, which outputs the probability distribution for the likelihood in terms of $q_{\theta_d}(x_{ij}=1 | \mathbf{z}, \mathbf{c})$. This measures the probability of the coamoeba occupying grid point $(i,j)$ in the discretized unit cell of $T^2$. (d) The decoder takes as an input the reparameterized latent space vector $\mathbf{z}$ and (e) outputs what we refer to as the generated coamoeba $\hat{\mathbf{x}}$ corresponding to $\mathbf{z}$ and a choice of complex structure moduli $\mathbf{c}$.
• Figure 5: (a)-(d) Certain choices of latent space vectors $\mathbf{z}$ are less optimal than other choices for coamoeba generated by the CVAE model. The choice in (e) is what we use throughout this work as the optimal choice for the latent space vector $\mathbf{z}$. The optimal choice for $\mathbf{z}$ appears to produce coamoeba on $T^2$ with sharp boundaries. Given that the CVAE model outputs a probability map on $T^2$ given in terms of $q_{\theta_d}(x_{ij}=1|\mathbf{z}, \mathbf{c})$, the less optimal choices for $\mathbf{z}$ appear to generate probability distributions that are more spread out, reflecting an increased level of uncertainty in the locality of the coamoeba on $T^2$ and the branes in the corresponding Type IIB brane configuration.