Physics and Machine Learning¶
Learning to Trace Seiberg Dualities¶
We apply machine learning methods to identify Seiberg dualities [1] between supersymmetric quiver gauge theories.
Seiberg Dualities in Quiver Gauge Theories¶
We consider 4D \(\mathcal{N} = 1\) quiver gauge theories with \(\text{SU}(N_i)\) gauge groups. The nodes represent gauge factors, and the directed arrows specify bifundamental chiral multiplets. The adjacency matrix \(\mathbf{A}\) defines the matter content. Anomaly cancellation requires the rank assignments to satisfy:
Seiberg duality operates locally on a single node \(k\). The mutation alters the gauge group rank to \(N_k' = \sum_j N_j A_{jk} - N_k\). All other ranks \(N_i\) for \(i \neq k\) remain unchanged. Connected arrows reverse direction. Mesonic fields generate new arrows \(A_{ik}A_{kj}\) between adjacent nodes \(i\) and \(j\), while oppositely oriented arrows annihilate.
The mutation of the adjacency matrix follows the Fomin--Zelevinsky rule [2, 3]:
where the net flow of arrows \(\Delta_{ij}\) from node \(i\) to node \(j\) is computed by adding the new mesonic arrows and subtracting any oppositely oriented arrows:
Chaining these simple duality operations produces a sequence of mutations \(Q^{(n)} = D_{j_n} \dots D_{j_1} Q\). The set of theories connected by these operations forms a duality tree [4-9]. Finding the exact sequence of dualizations connecting two distinct quiver gauge theories requires navigating this tree. We formulate this as a pathfinding problem where the quiver forms a state in a graph and mutations represent the edges. We developed two Graph Neural Networks (GNNs) [10, 11] to guide search algorithms through this space:
Pathfinders for Duality Sequences¶
We implement five search algorithms to find paths between related theories.
DGNN Pathfinder: A bidirectional A* search using the DGNN distance output as the heuristic function. The evaluation functions for the forward and backward frontiers are:
The step cost is fixed at \(g(Q_{n+1}) = g(Q_n) + 1\). The heuristic occasionally exhibits non-monotonicity, forcing the priority queue to explore uninformative branches and delaying convergence.
AGNN Pathfinder: A unidirectional beam search (\(B=3\)). It uses the AGNN mutation probability distributions to compute the path cost. The trajectory cost accumulates as:
Highly probable mutations contribute minimally to the cost, while unlikely ones heavily penalize the branch. The fixed beam width discards alternative branches and lacks a backtracking mechanism.
Hybrid Pathfinder: A bidirectional A* search combining DGNN distance estimates and AGNN branch probabilities. The cumulative trajectory cost updates recursively via:
The AGNN acts as a soft pruning mechanism, penalizing unlikely branches and preventing the algorithm from stalling in flat regions of the DGNN landscape.
LCA Pathfinder: A bidirectional A* search without neural network guidance. The cost function \(g(D_k)\) penalizes mutations based on gauge ranks (\(c_{\text{dec}}\) if rank decreases, \(c_{\text{eq}}\) if rank is unchanged, and \(c_{\text{inc}}\) if rank increases):
It forces both search trees toward simpler quivers to locate a Lowest Common Ancestor.
Hybrid LCA Pathfinder: Integrates the LCA rank penalties into the Hybrid pathfinder. The heuristic function incorporates a penalty proportional to the total rank:
This secondary penalty biases the search toward smaller-rank theories when network confidence is low.
Results¶
We evaluate the pathfinder performances by averaging the metrics across all distances and node counts in each dataset. The Effective Efficiency Ratio (EER) multiplies the Success Rate (\(\overline{\text{SR}}\)) by the Efficiency Ratio (\(\overline{\text{ER}}\)). The first table compares the neural pathfinders against the uninformed Breadth-First Search (BFS) baseline. The second table compares them against the classical Lowest Common Ancestor (LCA) heuristic.
Performance vs BFS
| Data | Pathfinder | \(\overline{\text{SR}}\) (%) | \(\overline{\text{ER}}\) | \(\overline{\text{EER}}\) |
|---|---|---|---|---|
| ID | DGNN | \(99.94\) | \(430.25\) | \(429.99\) |
| AGNN | \(76.12\) | \(928.59\) | \(706.84\) | |
| Hybrid | \(100.00\) | \(790.08\) | \(790.07\) | |
| LCA | \(100.00\) | \(763.66\) | \(763.66\) | |
| OOD 1 | DGNN | \(99.51\) | \(134.38\) | \(133.72\) |
| AGNN | \(50.91\) | \(305.15\) | \(155.35\) | |
| Hybrid | \(99.84\) | \(221.99\) | \(221.63\) | |
| LCA | \(99.97\) | \(176.13\) | \(176.08\) | |
| OOD 2 | DGNN | \(99.34\) | \(714.42\) | \(709.74\) |
| AGNN | \(78.64\) | \(1175.18\) | \(924.21\) | |
| Hybrid | \(99.92\) | \(881.67\) | \(880.96\) | |
| LCA | \(100.00\) | \(568.25\) | \(568.25\) |
Performance vs LCA
| Data | Pathfinder | \(\overline{\text{SR}}\) (%) | \(\overline{\text{ER}}\) | \(\overline{\text{EER}}\) |
|---|---|---|---|---|
| ID | DGNN | \(99.94\) | \(0.83\) | \(0.83\) |
| AGNN | \(76.12\) | \(2.04\) | \(1.55\) | |
| Hybrid | \(100.00\) | \(1.10\) | \(1.10\) | |
| Hybrid LCA | \(100.00\) | \(1.14\) | \(1.14\) | |
| OOD 1 | DGNN | \(99.51\) | \(0.54\) | \(0.53\) |
| AGNN | \(50.91\) | \(1.03\) | \(0.52\) | |
| Hybrid | \(99.84\) | \(0.83\) | \(0.83\) | |
| Hybrid LCA | \(100.00\) | \(0.98\) | \(0.98\) | |
| OOD 2 | DGNN | \(99.34\) | \(0.83\) | \(0.83\) |
| AGNN | \(78.64\) | \(1.91\) | \(1.50\) | |
| Hybrid | \(99.92\) | \(1.16\) | \(1.16\) | |
| Hybrid LCA | \(100.00\) | \(1.18\) | \(1.18\) |
Overview of Search Algorithms¶
A* Search: An informed search algorithm that selects paths by minimizing the evaluation function \(f(n) = g(n) + h(n)\). The cost \(g(n)\) accumulates along the path, while the heuristic \(h(n)\) estimates the remaining distance. The algorithm maintains unexplored branches in a priority queue. This structure enables backtracking when a path reaches a dead end. Bidirectional A* accelerates the search by expanding forward from the initial theory and backward from the target theory simultaneously.
Beam Search: A greedy algorithm that limits memory overhead. It sorts generated candidate paths by cost and retains only the top \(B\) quivers at each depth step. The algorithm permanently discards all other branches. This reduces execution time but eliminates the ability to backtrack.
Graph Neural Network Architectures¶
We base our models on a Message Passing paradigm [10] combined with Graph Transformers [11, 12]. Nodes update their features by aggregating messages from their immediate neighborhood. We supplement this with global attention mechanisms to extract long-range topological data and prevent oversmoothing [13] in deep layers.
Distance GNN (DGNN): Estimates the scalar mutation distance between two quivers. The architecture uses a 3-layer GNN tokenizer with residual connections, followed by a 2-layer Transformer encoder. The fully activated node representation incorporates LayerNorm [14] and LeakyReLU [15] activations: \(\tilde{h}_i^{(\ell+1)} = \text{LeakyReLU}( \text{LayerNorm}( m_i^{(\ell+1)} ) )\). The network computes the element-wise absolute difference between the encoded nodes of both quivers, \(d_{\text{local}} = \frac{1}{K} \sum_{i=1}^K | h_{A,i} - h_{B,i} |\), concatenates this with global graph averages, and projects the result to a positive distance via a Multi-Layer Perceptron. We optimize the network using Mean Squared Error.
Adviser GNN (AGNN): Predicts the optimal first-step mutation. The initial feature vector encodes node ranks, incoming/outgoing arrows, and total adjacent ranks. The network adds Laplacian Positional Encodings (LPE) to distinguish structurally similar nodes. A Hybrid GPS encoder [11] yields dense nodes \(h_{A,i}\) and \(h_{B,i}\). An MLP classifier maps concatenated differences into unnormalized mutation scores (logits). The network applies a binary action mask \(\mathcal{M}\) to set scores for invalid mutations (e.g., those producing negative ranks) to negative infinity. A Softmax layer then normalizes the remaining scores into a valid probability distribution: \(P(D_i | Q_A, Q_B) = \text{Softmax}(\tilde{y})_i\).
Path Generation Script¶
The script pathfinders/find_path.py automates the search for Seiberg duality sequences between an initial and target quiver gauge theory. The output JSON structures the path details and search metrics.
The algorithm parses the starting and target graph data (ranks and adjacency matrices) and executes the selected search strategies.
Search Execution
The script runs multiple pathfinders sequentially based on command-line flags.
- --dgnn, --agnn, --hybrid, --lca, and --hybrid_lca trigger the respective search algorithms.
Path Output
The output records the specific mutation sequence.
- path contains the exact array of node indices mutated to connect the initial and target graphs.
- status confirms if the algorithm succeeded or encountered a dead end.
Search Metrics
The script records the number of explored nodes and outputs the search metrics.
- visited_states counts the total number of unique graph configurations generated and evaluated.
- meeting_depth_fwd and meeting_depth_bwd record the specific search depths where the two frontiers intersected in bidirectional algorithms.
References
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[9] P. S. Aspinwall and I. V. Melnikov, "D-Branes on Vanishing del Pezzo Surfaces", JHEP 12, 042 (2004), arXiv:hep-th/0405134.
[10] J. Gilmer, S. S. Schoenholz, P. F. Riley, O. Vinyals, and G. E. Dahl, "Neural Message Passing for Quantum Chemistry", (2017), arXiv:1704.01212.
[11] L. Ramp{\'a}{\v{s}}ek, M. Galkin, V. P. Dwivedi, A. T. Luu, G. Wolf, and D. Beaini, "Recipe for a General, Powerful, Scalable Graph Transformer", (2022), arXiv:2205.12454.
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[15] A. L. Maas, "Rectifier Nonlinearities Improve Neural Network Acoustic Models", (2013).
Exploring Line Bundle Standard Models with Transformers¶
Heterotic Compactifications and Line Bundle Standard Models¶
We focus on finding \(\text{SU}(5)\) Grand Unified Theory (GUT) models from \(E_8 \times E_8\) heterotic string theory compactified on a Calabi-Yau threefold \(\mathbf{X}_3\) [1-7]. We associate a trivial line bundle to the hidden sector and require an equivariant vector bundle \(V\) that breaks the ten-dimensional gauge group into a split structure group \(H = \text{S}(\text{U}(1)^5)\) [8-10]. This breaks the observable \(E_8\) group down to \(G \times H = \text{SU}(5) \times \text{S}(\text{U}(1)^5)\).
We restrict \(V\) to be a direct sum of line bundles:
The integer vectors \(\mathbf{k}_a\) encode the first Chern classes expanded in the Kähler cone basis of \(\mathbf{X}_3\). To yield a consistent, supersymmetric vacuum, the bundle \(V\) must satisfy specific topological and physical constraints.
\(E_8\) Embedding: The line bundle sum must embed into \(E_8\), which imposes \(c_1(V) = \sum_{a=1}^5 \mathbf{k}_a = 0\). To prevent accidental isomorphisms that yield a non-unitary structure group and break the \(\text{SU}(5)\) GUT embedding, all proper subsets \(S \subset \{1,\ldots, 5\}\) must also satisfy \(\sum_{a \in S} \mathbf{k}_a \neq 0\) [11].
Anomaly Cancellation: We assume sufficient NS5-branes wrap effective curves to cancel the anomaly via the Green-Schwarz mechanism. This requires the curve \(c_2(T\mathbf{X}_3) - c_2(V)\) to be effective, imposing the bound:
Poly-Stability: The resulting effective field theory must be supersymmetric. The gauge connection on \(V\) must satisfy the Hermitian Yang-Mills equations at zero slope [12, 13]. For a direct sum of line bundles, this requires the slope \(\mu(L_a)\) of each individual line bundle to vanish simultaneously somewhere in the interior of the Kähler cone parameterized by the moduli \(t^i\):
The tensor \(\kappa_{ijk}\) denotes the intersection numbers of the Calabi-Yau threefold \(\mathbf{X}_3\).
Particle Spectrum: The spectrum must be chiral and contain exactly three families of multiplets in the \(\mathbf{10}-\overline{\mathbf{10}}\) representation. To achieve three chiral families upon quotienting the manifold by a freely acting discrete symmetry group \(\Gamma\), the index on the upstream covering space must scale with the group order [11, 14, 15]:
To prevent anti-generations and avoid Higgs color triplets that cannot be projected out by Wilson lines, the individual and pairwise indices must satisfy strict bounds [11, 14, 15]:
| Conditions | |
|---|---|
| \(E_8\) Embedding | \(c_1(V) = 0\,, \sum_{a\in S} c_1(L_a) \neq 0\) |
| Anomaly Cancellation | \(c_2(T\mathbf{X}_3) - c_2(V) \geq 0\) |
| Poly-Stability | \(\mu(L_a) = 0\) |
| Three Chiral Families | \(\text{ind}(V) = -3\vert\Gamma\vert\) |
| No Exotic Representations | \(-3\vert\Gamma\vert \leq \text{ind}(L_a) \leq 0\,, -3\vert\Gamma\vert \leq \text{ind}(L_a \otimes L_b) \leq 0\) |
Network Architecture and Reinforcement Learning¶
The search for vector line bundles is formulated as an autoregressive sequence generation task and solved using Proximal Policy Optimization (PPO) [16]. The objective is to construct the \(h^{1,1}(\mathbf{X}_3) \times 5\) matrix \(\mathbf{K}\) token-by-token. The first Chern class condition, \(c_1(V) = 0\), is imposed by construction by requiring that:
The remaining entries of \(\mathbf{K}\) are flattened in row-major order into a token sequence of length \(S = h^{1,1}(\mathbf{X}_3) \times 4\). The action space consists of integers \(a_t \in [0, 2k_{\text{max}}]\), which map to the sequence values via \(s_t = a_t - k_{\text{max}}\).
Actor-Critic Architecture: The policy and value functions are parameterized by a shared, decoder-only Transformer network [17]. Discrete input tokens map to dense vectors via a learnable embedding matrix, and we introduce a learnable positional encoding to break the permutation equivariance of the attention mechanism.
The input sequence is processed by \(L=4\) identical transformer blocks. Each layer applies a Masked Multi-Head Self-Attention (MHA) operation with \(H=8\) independent attention heads. The attention mechanism enforces causality via an upper-triangular mask. We apply a residual connection followed by Layer Normalization [18], and a Feed-Forward Network [19] with ReLU activation expands the dimensionality before projecting back down.
To predict the next action \(a_t\) and evaluate the current state, the model extracts the final hidden state vector \(h_t\) corresponding to the last processed token. This vector routes through two independent linear layers:
- Actor Head (Policy): Maps \(h_t\) to logits across the action space, converting them via a softmax activation into the probability distribution \(\pi_\theta(a_t | \mathbf{S}_t)\) for the next token.
- Critic Head (State-Value Estimate): Maps \(h_t\) to a scalar \(V_\theta(\mathbf{S}_t)\) predicting the expected cumulative reward for the sequence prefix.
Reward Shaping: The environment assigns a dense reward \(r_S\) only at the terminal step when the full vector bundle configuration is specified, evaluating the constraints using a composite scoring system:
- Triviality Masking: A binary mask zeros the final score if the bundle is degenerate.
-
Continuous Scoring: A continuous score is computed for each constraint using a log-barrier function:
\[ \begin{equation} S_{i, \text{cont}} = \frac{1}{1 + w_i \ln(1 + e_i)}\,. \label{eq:log_barrier_score} \end{equation} \]This range is constrained to \((0,1]\), preserving a gradient signal via the logarithm even for large errors that would otherwise suppress the reward. We define a continuous base score \(B = \sum_i w_i S_{i, \text{cont}}\) using uniform weights.
-
Perfection Bonus: A valid matrix satisfying all constraints receives an additive spike (\(+5\) bonus).
- Novelty Penalty: To avoid mode collapse into a single known solution, a rolling First-In-First-Out (FIFO) novelty buffer penalizes duplicates (under \(S_5\) permutation symmetry) by scaling down their reward by a penalty factor \(p_{\text{pen}} = 0.5\).
Synthesizing the triviality masking, the base score \(B\), the perfection bonus, and the novelty penalty, the terminal reward \(r_S\) assigned by the environment at step \(t=S\) for matrix \(\mathbf{K}\) is:
Because intermediate steps receive no immediate feedback, \(r_S\) is propagated backwards using Generalized Advantage Estimation (GAE) [20].
Complete Intersection Calabi-Yaus¶
Complete intersection Calabi-Yau (CICY) threefolds are algebraic varieties constructed as intersections of the zero loci of \(k\) homogeneous polynomials \(p_j\) in an ambient space \(\mathcal{A}\) [21]. This space is a product of projective spaces \(\mathbb{P}^{n_1} \times \dots \times \mathbb{P}^{n_m}\). The complex dimension of the threefold \(\mathbf{X}_3\) equals the dimension of \(\mathcal{A}\) minus \(k\). This yields the constraint:
A multi-degree \(q_j^i\) characterizes each polynomial \(p_j\), specifying the degrees in the homogeneous coordinates of each \(\mathbb{P}^{n_i}\). This information populates a configuration matrix:
The manifold \(\mathbf{X}_3\) is Calabi-Yau if the first Chern class of its tangent bundle \(c_1(T\mathbf{X}_3)\) vanishes. This imposes a condition on the polynomial degrees:
Each homogeneous polynomial \(p_j\) acts as a holomorphic section of a line bundle \(\mathcal{O}_{\mathcal{A}}(q_j)\) over the ambient space \(\mathcal{A}\). We define the normal bundle \(\mathcal{N}_{\mathbf{X}_3/\mathcal{A}}\) to \(\mathbf{X}_3\) as the direct sum of these \(k\) line bundles.
We restrict our analysis to favorable CICYs [22, 23]. In favorable geometries, the second cohomology group descends entirely from the ambient space. This means the restriction map \(\iota^* : H^2(\mathcal{A}, \mathbb{Z}) \to H^2(\mathbf{X}_3, \mathbb{Z})\) is surjective, leading to \(h^{1,1}(\mathbf{X}_3) = h^{1,1}(\mathcal{A}) = m\).
The basis of divisors \(J_i\) on \(\mathbf{X}_3\) corresponds to the pullback of the ambient hyperplanes. We expand the K\"ahler form on \(\mathbf{X}_3\) as \(J = t^i J_i\), with positive K\"ahler moduli \(t^i\). We define the triple intersection numbers \(\kappa_{abc}\) as:
The integral factorizes over the \(\mathbb{P}^{n_i}\) components of \(\mathcal{A}\). It evaluates to a non-zero value only when the integrand contains the exact volume form of \(\mathbb{P}^{n_i}\) for every factor.
Using the divisors \(J_i\), we calculate the total Chern class of \(\mathbf{X}_3\) via adjunction:
The adjunction formula allows us to compute the Euler characteristic of \(\mathbf{X}_3\) and extract the dimension \(h^{2,1}(\mathbf{X}_3)\) for favorable configurations.
Results¶
LB-Explorer evaluated ~50 favorable CICYs with \(h^{1,1}(\mathbf{X}_3) \in [4,15]\) and \(|\Gamma| \in [2,4]\). The pipeline ran across five seeds for 10 million episodes per geometry. The system filters the final solution sets to impose equivariant structure.
Staged Discovery¶
The neural network learns constraints sequentially. It prioritizes anomaly cancellation and poly-stability in early training episodes. The network discovers valid solutions only after satisfying the chirality condition. Additional spectrum constraints fine-tune the search process.
Transfer Learning¶
Pre-trained models accelerate discovery times on new geometries. Transferring from a source geometry with a small symmetry group \(|G_{\mathbf{X}_3}|\) to a target with a large \(|G_{\mathbf{X}_3}|\) maintains broad exploration capabilities and yields diverse solution sets.
Transferring from large to small symmetry groups traps the policy in local minima. The network finds the first solution rapidly but fails to accumulate unique line bundle sums efficiently. Similar directional asymmetries occur for shifts in \(h^{1,1}(\mathbf{X}_3)\) and \(|\Gamma|\).
Optimal transfer directions include:
- Small \(|G_{\mathbf{X}_3}|\) to large \(|G_{\mathbf{X}_3}|\)
- Large \(|\Gamma|\) to small \(|\Gamma|\)
- Large \(h^{1,1}(\mathbf{X}_3)\) to small \(h^{1,1}(\mathbf{X}_3)\)
Running the LB-Explorer Pipeline¶
The codebase handles geometry parsing, transformer training, and physical verification.
Geometry Parsing¶
Convert the master CICY database into the specific format required by the explorer.
python scripts/create_LB-Explorer_inputs.py \
--db_path databases/full_cicy_database.json \
--output_dir cy_geometry_exports
Training the RL Agent¶
Train the PPO agent to search for line bundle matrices. Execution requires specifying the \(h^{1,1}\) dimension, the database index, the symmetry group order \(\Gamma\), and the maximum integer bounds for the line bundle charges.
python LB-Explorer.py \
--h11 5 \
--cy_index 7447 \
--gamma 2 \
--m_bound 8 \
--stability_range 2 \
--use_minibatches \
--run_id h11_5_g2__cy7447__s42
CP-SAT Hybrid Closure¶
For highly constrained geometries, the reinforcement learning agent can search for partial matrices by disabling the exact chiral index sum penalty (--sum_coef 0). Feed these partial solutions into a Constraint Programming SAT solver to systematically find the remaining columns required to close the matrix and satisfy all constraints.
python CPSAT-closure.py \
--solutions sol_runs_h11_5_g2__cy7447__s42/solutions_gpu_*.jsonl \
--geometry cy_geometry_exports/all_geometry_h11_5.json \
--cy_index 7447 \
--gamma 2
Physics Verification¶
Process the raw generated matrices through standalone scripts to verify equivariance, exact poly-stability (via SciPy optimization), and compute the explicit bundle cohomologies (via pyCICY).
python scripts/check_validity_solutions.py --rank 5 --input_dir Sol_Runs
python scripts/check_equivariance.py --workers 8
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