Learning to Trace Seiberg Dualities

Source: arXiv:2607.28628 · Published 2026-07-30 · By Jonathan J. Heckman, Shani Meynet, Alessandro Mininno, Gary Shiu

TL;DR

This paper addresses the computational problem of determining when two supersymmetric quiver gauge theories are connected by a sequence of Seiberg dualities. Seiberg dualities produce complex mutation trees of quivers, and tracing the duality chain analytically is highly challenging due to combinatorial explosion and chaotic behavior of duality sequences. The authors propose a machine learning approach that treats the problem as a directed graph mutation and pathfinding task. They generate a large dataset of quiver pairs and minimal duality distances by performing breadth-first search (BFS) on known quiver gauge theories derived from D3-branes probing toric Calabi–Yau singularities with up to 13 nodes. Then, they design and train two classes of graph neural networks (GNNs): a Distance GNN (DGNN) estimating the minimal number of mutations required between pairs of quivers; and an Adviser GNN (AGNN) that predicts which mutation nodes are most promising for transitioning towards the dual. These learned heuristics serve as cost and heuristic functions guiding classical search algorithms including A* and beam search pathfinders, as well as new physics-informed policy heuristics like the Lowest Common Ancestor (LCA) pathfinder. Experiments show that network-assisted pathfinders significantly outperform deterministic BFS and naive algorithms, achieving higher efficiency ratios and success rates both in-distribution (on similar toric CY theories) and out-of-distribution (non-toric or abstract quivers). Hybrid pathfinders combining neural policies and physics heuristics yield the best overall performance with 100% success rates. The authors thus demonstrate that machine learning can effectively capture the complex combinatorial structure of Seiberg dualities and provide practical methods to estimate duality distances and paths beyond exhaustive search. This also offers a new benchmark for AI models tackling deep theoretical physics inference tasks.

Key findings

• Network architectures consisting of graph transformers and multi-layer perceptrons outperform deterministic algorithms in estimating mutation distances for quivers with up to ~13 nodes.
• The Distance GNN (DGNN) trained on BFS-generated duality trees predicts minimal mutation distances with mean absolute error decreasing for shorter distances but underestimates at larger distances (Fig. 9).
• Adviser GNN (AGNN) successfully predicts which nodes to dualize, improving pathfinder guidance during search (Fig. 12).
• Neural-network-guided A* search pathfinders achieve efficiency ratios (ER) up to an order of magnitude better than Breadth-First Search (Table 2a).
• Hybrid pathfinders combining NN heuristics and physics-inspired Lowest Common Ancestor (LCA) policies achieve 100% success rates and outperform pure LCA and NN-only pathfinders in handling both in-distribution and out-of-distribution datasets (Table 2b).
• The authors define a complexity metric C = D log10 K (distance times log number of nodes) with NNs trained up to complexity ~11.5, and estimate pathfinder scalability up to ~1.5–2x that complexity (Section 7).
• The BFS dataset generation process over mutation trees supports consistent training and evaluation (Algorithms 1, pseudocode in Appendix C).
• Permutation symmetries of quivers cause DGNN to underestimate distances, but explicitly accounting for symmetries improves network accuracy.

Threat model

n/a — The work focuses on machine learning-guided computation of Seiberg duality mutation paths without explicit adversarial considerations. The 'adversary' is the computational complexity of the mutation space, not a malicious actor.

Methodology — deep read

1. Threat model and assumptions: The adversary is conceptualized as the computational process attempting to determine whether two quiver gauge theories are Seiberg dual through sequences of allowed node mutations as defined by physical Seiberg duality rules. The problem is purely computational/no adversary per se. The input consists of pairs of quivers represented as directed graphs with rank vectors satisfying anomaly cancellation and positivity constraints. The task is to identify a minimal sequence of mutations connecting them, or certify none exist.

2. Data provenance and generation: Starting from a curated set of seed theories derived from D3-branes probing toric Calabi–Yau threefold singularities (up to 13 gauge nodes), the authors perform breadth-first search exploration of the mutation tree by applying all allowed Seiberg dual mutations at nodes, pruning invalid quivers with negative ranks or decompositions. They generate a large dataset of quivers and pairs annotated by minimal duality distance (shortest mutation sequence length). The graphs are stored with adjacency matrices and rank vectors (ignoring superpotential terms). Dataset includes in-distribution canonical toric quivers and out-of-distribution quivers including non-toric and abstract examples.

3. Architectures and algorithms: Two main graph neural network architectures are designed:

• Distance GNN (DGNN): Takes pair of quivers inputs and predicts estimated minimal mutation distance. Uses graph transformer layers to incorporate global context and message passing.
• Adviser GNN (AGNN): Takes a single quiver input and outputs per-node probabilities indicating recommended mutation nodes to attempt. Serves as a policy network. These networks are paired with classical search algorithms:
• Breadth-First Search (BFS) as baseline
• A* Search: uses DGNN output as heuristic function (estimated remaining distance) and AGNN output as cost function (mutation node selection policy).
• Beam search: heuristic search with limited branching factor. The authors introduce physics-inspired Lowest Common Ancestor (LCA) pathfinder policy reflecting physicists’ heuristic to find minimal rank common quivers to guide search. They also combine these into hybrid pathfinders blending NN guidance and LCA policy.

4. Training regime: Networks are trained on pairs generated by BFS up to depth 12 (corresponding to distances ≤ 12 mutations) and quivers with up to 13 nodes. Training uses standard supervised learning with mean absolute error loss for DGNN distance regression and cross-entropy loss for AGNN node classification. Details such as batch size, epochs, optimizer, and hardware specifics are mentioned roughly (personal laptops and UPenn cluster with 8 CPUs and 256 GB RAM). Exact hyperparameters and random seeds are not fully disclosed but mentioned in appendices.

5. Evaluation protocol: Performance is measured by mean absolute error for DGNN distance predictions, node classification accuracy for AGNN, success rate percentage of pathfinders finding mutation sequences within max steps, and efficiency ratios (ER) calculated as BFS node exploration counts divided by pathfinder node exploration. Evaluation considers both in-distribution test sets and challenging out-of-distribution pairs. Ablations compare pathfinders with various guiding policies. The authors discuss permutation symmetries impacting distance measures and evaluate robustness.

6. Reproducibility: Code for networks and pathfinders is released via GitHub repository accompanying the paper, including best-trained checkpoint weights. The dataset of quivers is generated algorithmically based on standard physics constructions. While full datasets are not publicly released, sufficient detail and pseudocode are provided to reproduce data generation and experiments.

Example: To evaluate a given pair of quivers Q_A and Q_B, the DGNN takes the graph adjacency and rank data to output an estimated minimal mutation distance. The AGNN processes Q_A’s graph to score nodes by mutation likelihood. These outputs seed an A* search which expands nodes by applying dualities prioritized by AGNN scores, pruning paths exceeding expected costs given by the DGNN heuristic. The A* pathfinder thereby efficiently finds or fails to find the shortest mutation path between Q_A and Q_B, measured against the known BFS ground truth.

Technical innovations

• Framing Seiberg duality tracing as a graph mutation and pathfinding problem amenable to machine learning guidance.
• Design of Distance GNN that predicts minimal mutation distance between quiver pairs, using graph transformer layers to encode global quiver structure.
• Design of Adviser GNN policy network that predicts promising nodes to dualize, enabling heuristic-guided search.
• Hybrid pathfinder algorithm combining neural network heuristics with physics-inspired Lowest Common Ancestor (LCA) policies to improve search success and efficiency.
• Systematic benchmark dataset creation via BFS over mutation trees of physically motivated quivers, supporting rigorous evaluation.

Datasets

• Toric Calabi–Yau quivers — ~thousands of quivers generated by BFS on seeds from D3-branes probing known toric CY threefolds with up to 13 nodes — generated internally by breadth-first search
• Non-toric and abstract quivers — hundreds of examples outside training distribution — generated internally for out-of-distribution testing

Baselines vs proposed

• Breadth-First Search (BFS) baseline: efficiency ratio ER = 1.0 by definition; success rate 100%
• A* pathfinder with DGNN + AGNN heuristics: ER up to 10x higher than BFS; success rates above 90% on in-distribution data (Table 2a)
• Lowest Common Ancestor (LCA) pathfinder baseline: high success rates but can get trapped; ER lower than hybrid approach
• Hybrid pathfinder (NN heuristics + LCA): achieves 100% success rate with best overall ER outperforming both LCA and NN-only pathfinders (Table 2b)
• DGNN mean absolute error (MAE) on distance prediction decreases significantly for short distances; underestimates at large distances but improves when accounting for quiver permutation symmetries (Fig 9)

Limitations

• Training data limited to quivers with up to 13 nodes and mutation distances up to 12, so scalability to larger or more complex quivers is uncertain.
• The approach ignores superpotential terms and fine details beyond adjacency matrices and ranks, potentially missing important physical distinctions.
• Permutation symmetries are not explicitly handled during training, leading to underestimation errors in distance predictions.
• Evaluation focuses primarily on shortest-path distance; alternative metrics or proofs of duality beyond mutation distance are not addressed.
• Pathfinder efficiency and success rely on heuristics that may fail or degrade outside training distribution or for extremely large mutation sequences.
• Full dataset is not public, possibly limiting independent verification or reproduction at scale.

Open questions / follow-ons

• Can the methods scale to quivers with substantially larger numbers of nodes or higher mutation distances beyond the training regime?
• How to explicitly incorporate permutation symmetries and gauge redundancies into neural network architectures or training to improve accuracy?
• Could richer physics data, such as superpotential terms or gauge coupling flows, be integrated to improve duality detection beyond quiver mutations?
• What is the role of adversarially crafted quivers to probe robustness and limitations of NN-guided pathfinders?

Why it matters for bot defense

This work illustrates how complex combinatorial transformations—here, Seiberg dualities in quiver gauge theories—can be effectively modeled and approximated using graph neural networks augmented with classical heuristic search algorithms. For bot-defense and CAPTCHA practitioners, the key takeaway is that neural policy guidance combined with symbolic search can efficiently navigate vast discrete state spaces characterized by intricate local moves. The hybrid architectures presented may inspire similarly structured approaches to hard inference problems in bot detection, such as learning heuristic transitions between states and guiding search efficiently toward verification or challenge solutions. Additionally, the benchmarking on in- and out-of-distribution quivers highlights the value of stress-testing models in diverse regimes to assess generalization—critical in adversarial bot-defense settings. Although the domain—high energy physics—is specialized, the underlying challenge of tracing mutation sequences shares structural analogies to CAPTCHA puzzles themed on graph or state transformations, suggesting transfer of methodologies and lessons learned.

Cite

@article{arxiv2607_28628,
  title={ Learning to Trace Seiberg Dualities },
  author={ Jonathan J. Heckman and Shani Meynet and Alessandro Mininno and Gary Shiu },
  journal={arXiv preprint arXiv:2607.28628},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.28628}
}

Read the full paper

• arXiv:2607.28628 (abstract)
• arXiv:2607.28628 (PDF)