Learning to Trace Seiberg Dualities

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Learning to Trace Seiberg Dualities

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arXiv CS · Papers · License: Open Access · 2026

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machine learning, deep learning, neural networks

Learning to Trace Seiberg Dualities Jonathan J. Heckman,1,2 Shani Nadir Meynet,1 Alessandro Mininno,3 and Gary Shiu3 1

Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA 2

arXiv:2607.28628v1 [hep-th] 30 Jul 2026

Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA

Department of Physics, University of Wisconsin–Madison, 1150 University Avenue, Madison, WI 53706, USA [email protected], [email protected], [email protected], [email protected]

Abstract Dualities play an important role in establishing both microscopic and emergent phenomena in a wide range of physical systems. In practice, though, it can often be computationally challenging to establish when two systems are dual, even when all of the “rules of the game” are well-known. Said differently, when confronted with two systems, how can one efficiently establish that they are in fact dual? In this paper we use machine learning methods to address this question for Seiberg dualities of supersymmetric quiver gauge theories. Mathematically, this involves establishing mutations of quivers, which is in turn a variation on the theme of “learning to unknot”. On the one hand, this leads us to a practical tool for establishing the computational complexity of different dualities. On the other hand, it also allows us to study how different network architectures learn how to trace Seiberg dualities. We find that for quivers with a modest number of quiver nodes (of order 10), different network architectures consisting of transformers and multi-layer perceptrons tend to outperform deterministic algorithms. Supplementing the network by well-established pathfinder algorithms (essentially “Google Maps for quivers”) leads to an additional improvement in the efficiency and accuracy of the search strategy. We anticipate that this class of questions can serve as a useful benchmark for frontier AI models applied to theoretical physics.

Contents 1 Introduction

2 Tracing Seiberg Dualities 2.1 Mutating Quiver Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mutation Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Seed and Validation Theories . . . . . . . . . . . . . . . . . . . . . . . . . .

6 7 9 11

3 Dataset Generation 3.1 Breadth-First Search over Duality Trees . . . . . . . . . . . . . . . . . . . .

12 12

4 Graph Neural Networks for Seiberg Dualities 4.1 Distance Graph Neural Network Architecture . . . . . . . . . . . . . . . . . 4.2 Adviser Graph Neural Network Architecture . . . . . . . . . . . . . . . . . .

15 15 20

5 Pathfinder for Seiberg Dualities 5.1 Brute-Force Search Baseline and Branching Factor . . . . . . . . . . . . . . . 5.2 Pathfinders for Seiberg Dualities . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Complexity Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Pathfinders Evaluation and Efficiency Metrics . . . . . . . . . . . . . . . . .

25 25 27 32 33

6 Results 6.1 In-Distribution Theories Analysis . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Out-of-Distribution Theories Analysis . . . . . . . . . . . . . . . . . . . . . .

35 35 46

7 Failure Modes

8 Conclusions

A Seed Theories for Dataset Generation A.1 In-Distribution Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Out-of-Distribution Theories . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 62

B Overview of Graph Neural Networks B.1 Message Passing Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Graph Transformers and Global Context . . . . . . . . . . . . . . . . . . . .

64 64 64

C Overview of Search Algorithms C.1 A∗ Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Beam Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66

Introduction

Dualities provide important complementary perspectives in a wide range of physical systems. Celebrated examples include the Kramers–Wannier duality of the 2D Ising model, the electricmagnetic duality of Maxwell theory, and many modern examples ranging from interacting quantum field theories to string / M- / F-theory to holography.1 Indeed, dualities provide a window into the structure of many physical systems at strong coupling. More precisely, a “duality” can often connect more than two presentations of a theory. Indeed, in many well-known examples, successive dualities lead to a wild proliferation of different possible descriptions. Turning the question around, if one is confronted with two theories, how does one establish that the pair is connected to one another via a chain of dualities, and if so, how many simple duality operations does it take to connect the pair? In this paper, we study this question for Seiberg dualities [1] applied to quiver gauge theories, i.e., mutations of quivers in the sense of [2, 3]. In the original examples of supersymmetric QCD, this duality relates an “electric” non-abelian gauge theory with a “magnetic” dual non-abelian gauge theory. This can be extended to a wide range of supersymmetric gauge theories with multiple simple gauge group factors, where the basic operation of duality involves acting on one gauge group factor at a time. A celebrated example of precisely this sort is the duality cascade of reference [4], in which there is a pair of gauge groups that share matter in bifundamental representations. This example also prominently features in many string theory applications since it characterizes fractional branes near the warped throat specified by the conifold [4–6]. Physically, the flow of gauge couplings in these examples dictates how duality proceeds (i.e., which gauge group is running to strong coupling). In many cases, however, the structure of dualities is far more involved, and there can often be more than one prescribed answer depending on the parameters of the quiver. For example, [7] showed that the far more generic situation is that there is an infinite family of dual theories. The specific trajectory also exhibits a high level of sensitivity to the initial conditions of gauge couplings, resulting in a huge chaotic range of possible duality sequences. From this perspective, there is already a practical question: Given two supersymmetric quiver gauge theories, are they in fact related via a sequence of Seiberg dualities? If so, what are the most efficient/optimal simple duality operations available to realize these transformations? In physical terms, this helps to establish the level of computational complexity in reaching dual phases of quiver gauge theories. From a holographic standpoint, this amounts to understanding how far into the bulk a given flow proceeds down a throat.2 1

The literature is vast, so rather than providing a partial and biased list of references, we trust the reader to search the original literature for themselves. 2 It is also of interest in the context of statistical inference in the sense of [8–10]: what is the level of distinguishability between different string compactifications?

Treating this question in purely analytic terms is a daunting task, and so we shall instead seek a more practical, artificial intelligence/machine learning (AI/ML)-based approach.3 As far as we are aware, previous AI/ML approaches to Seiberg dualities have focused on the special case where the quivers are of finite mutation type, i.e., they return to their original form after a finite number of dualities [13].4 As already mentioned, however, the far more generic case is the unwieldy scenario in which each successive duality dramatically increases the combinatorial possibilities. To focus on the essential points, in this paper we suppress some of the physical data of a quiver gauge theory. Along these lines, we shall specify a “simplified quiver” by a directed graph, with each node of the graph decorated with a choice of a non-negative integer (i.e., the integer N of an SU(N ) gauge theory). The quivers are dictated by the condition that the gauge theory is free of anomalies; namely, there is a weighted balancing condition on the arrows of the graph. A Seiberg duality on a node corresponds to a transformation of the gauge theory that alters the dualized node and changes the local connectivity of arrows on the graph (see Section 2 for details). A sequence of dualities can thus be viewed as specifying an alphabet, with words formed by concatenating all of these mutations. The underlying mathematical structure thus shares some similarities with the process of determining whether a given knot is related to an unknot [15]. Much as in [15], our aim here will be to compare a few AI methodologies to understand which neural network (NN) architectures are most effective at learning how to trace Seiberg dualities.5 Our purpose here is twofold: On the one hand, we use these different architectures for the practical goal of establishing the existence of dualities and obtaining an estimate of the overall computational complexity of duality transformations. On the other hand, there is also the question of performance, namely how efficient and accurate different approaches are in establishing a path to dualization. Indeed, while we focus our analysis on Seiberg dualities in quiver gauge theories, the same sort of combinatorial methods shows up in a broad class of other computational problems, ranging, e.g., from the mathematics of cluster algebras [2, 3] to scattering amplitudes [17]. As such, we anticipate that the analysis we provide will have far broader applications. Summary of Methodology and Results Let us now turn to a brief summary of our methodology. Additional details are presented later on. As in most machine learning tasks, we first collect/generate a suitable batch of training 3

One way to narrow the search is to also look for simple numerical invariants of the quiver which could be different, e.g., the defect group associated with a quiver, or its matrix Hilbert series. See e.g., [11, 12] for some recent examples of invariants along these lines. 4 See also [14] for the use of AI to study quivers with a finite number of mutations. 5 See also [16] for work on the study of how NNs learn the structure of arithmetic and group laws.

data. With this in place, we then train different neural network architectures (detailed below). We then consider the post-training phase, comparing their relative performance against some pre-specified performance metrics. In our case, the quivers used to generate the training data consist of D-branes probing a toric Calabi–Yau (CY) threefold singularity. This leads to a rather rich class of quiver gauge theories that have been extensively studied in the literature (see e.g., [18–20] for reviews). Starting from these “canonical presentations” of a quiver, we proceed to generate our training data by Seiberg dualizing these quivers multiple times. We primarily focus on quivers with 13 or fewer nodes, with the training set built from up to 12 Seiberg dualities. We note that after dualizing, one often reaches a non-toric phase (i.e., after flops of the geometry). In practice, generating the full dataset was accomplished with rather modest computational resources on personal laptops and the UPenn cluster over two to three days, with 256 GB of RAM across 8 CPUs. We then used this data to train different neural network architectures. To keep a broad perspective, we considered a few different choices of graph neural networks (GNNs) with transformer layers [21]. The first NN we trained is a GNN that estimates the distance, i.e., the number of minimal mutations, between a pair of theories. We call this network a Distance GNN (DGNN) because, conceptually, it compares the two graphs and attempts to estimate their distance/difference. The accumulated difference is used to estimate how far apart the two quivers are. We show a schematic representation of the DGNN architecture in Figure 6a, while the detailed description of each layer is given in Section 4.1. The second NN we trained is another GNN that aims to estimate, in a pair of theories, which vertices are most likely to be dualized to transition from one theory to the other. We call this network an Adviser GNN (AGNN), where the GNN functions as a policy adviser, i.e., it takes the input quiver and provides a probability that will be used by the pathfinder algorithm to make the choice of which node to dualize at each step. We show a schematic representation of the AGNN architecture that we will describe in Section 4.2, in Figure 6b. We find it most effective to use the NN as a policy guide for heuristic-driven pathfinders that search over possible ways of dualizing a quiver gauge theory. We considered two kinds of pathfinders: one is a bidirectional A∗ search pathfinder [22], and the other is a beam search pathfinder. For both, we used various definitions of cost and heuristic functions6 which guides the search for a path connecting a pair of quivers. The output of the DGNN will be used as the heuristic function, while the AGNN acts as the cost function. A schematic representation 6

Search algorithms seek a path that minimizes specific criteria. These are formalized using a cost function that computes the accumulated penalty of the route taken so far and a heuristic function that estimates the remaining effort to reach the goal. In this sense, a beam search pathfinder is an A∗ search without a heuristic function and with a reduced search frontier.

of the A∗ search is shown in Figure 13a, with pseudocode in Algorithm 2, while a beam search [23] is shown in Figure 13b and described in Algorithm 3. This approach of mixing a GNN with a heuristic search algorithm has been proven to be quite efficient in the data science literature (see e.g. [24]), but as far as we are aware, the present work constitutes the first application to a problem in theoretical physics. Moreover, we designed a physics-informed pathfinder, which we call the Lowest Common Ancestor (LCA) pathfinder, that is also an A∗ search pathfinder, with the policy determined by attempting to find the common quiver between a pair of theories with minimal ranks. This LCA pathfinder is inspired by the way (human) physicists approach the problem of finding the path between two theories related by Seiberg dualities, and it is similar to how we generated the database to train our NNs. The last pathfinder we studied is essentially a hybrid approach that unifies all the previous policies into a single pathfinder, guided both by the output of the NNs and the LCA policies. The policies for the pathfinders we designed are summarized in Table 1. We tested the performance of these pathfinders by comparing them first with a BreadthFirst Search (BFS) algorithm. In this way, we defined the efficiency ratio as the average number of nodes expected to be explored by a BFS baseline over the number of nodes explored by the heuristic pathfinder. We also kept track of their success rate, namely the percentage of paths found before exhausting the maximum number of steps. Each pathfinder has its pros and cons based on maximizing either the efficiency ratios or their success rates. We analyzed the performance over the dataset used to train the NNs, i.e., the in-distribution (ID) dataset, and over out-of-distribution (OOD) theories. See Table 2a for a summary of these results. We then compared the performance of the pathfinders among themselves, using the LCA pathfinder as the baseline, as summarized in Table 2b.7 In that table, one can also find the performance of the Hybrid LCA pathfinder, which is the final pathfinder we considered, with both the Hybrid and LCA policies added to guide the A∗ search. This pathfinder has the best overall performance of all the pathfinders we considered if we insist on having a 100% success rate. The NNs (with our best checkpoints) and the pathfinders are available at our GitHub repository. In order to study the breaking point of the NN-guided pathfinders, we also intro7

A careful comparison of Tables 2a and 2b reveals an apparent paradox when comparing the ERs of the OOD theories: the average EERs of the LCA relative to the BFS are superior to those of the Hybrid model. However, when evaluating the ER normalized by the LCA baseline, the Hybrid model outperforms the LCA. The reason is that the BFS baseline grows exponentially with the distances, whereas the LCA is efficient in finding the paths. On the other hand, the LCA pathfinder becomes trapped when the distance is not too large, whereas the Hybrid pathfinder is more efficient. We will discuss these performance tests further in Sections 6.1.4 and 6.1.6.

duce the concept of “complexity” defined as C = D log10 K, where D is the distance between two quivers and K is the number of nodes of the quivers. The NNs for which we provide the checkpoints have been trained up to complexity ∼ 11.5, as we considered theories with up to 13 nodes and distances of up to 12 mutations. In the conclusion of the analysis of the breaking point of the NN-guided pathfinders, we estimate that the Hybrid or Hybrid LCA pathfinders will perform better than pure LCA up to a complexity ∼ 1.5/2× of the largest complexity seen during the training of the NNs. Structure of the Paper The rest of this paper is organized as follows. In Section 2, we discuss some combinatorial aspects of Seiberg duality and pose the general question: What is the computational complexity of establishing whether two quiver gauge theories are Seiberg dual? With this in place, in Section 3, we provide further details on our method for generating training data for our neural networks. Section 4 discusses in greater detail the different GNN architectures, and Section 5 discusses pathfinder algorithms and their hybrid implementation with GNNs. In Section 6, we summarize the results from the various tests. Moreover, in Section 7, we analyze the breaking point of the pathfinders. Finally, we present our conclusions and potential future directions in Section 8. Additional details of GNNs are reviewed in Appendix B; while details on the different pathfinder algorithms used in the main body are summarized in Appendix C. The seed theories used for our training dataset are given in Appendix A. Note Added: As this paper was being finalized, reference [25] appeared, which also considers the task of identifying Seiberg dual quiver gauge theories. Our approach is centered on understanding the physics, computational complexity, and inference underlying such duality operations, whereas [25] focuses on evaluating AI models, specifically testing their ability to use a verifier to debug incorrect duality claims.

Tracing Seiberg Dualities

In this section, we discuss in more detail the central physics goal of the present work: determining how many “simple” Seiberg duality operations are necessary to connect two quiver gauge theories. To frame the discussion to follow, we briefly review the relevant physics and some motivating questions.

2.1

Mutating Quiver Gauge Theories

We shall be interested in 4D N = 1 quiver gauge theories with SU-type gauge groups.8 In physical terms, we introduce a collection of 4D N = 1 vector multiplets for the SU(Ni ) gauge groups, which we denote by circles. Flavor symmetry factors can also be included, being formally viewed as gauge symmetry nodes at zero coupling.9 In the quiver gauge theory literature, these are typically presented as squares. In what follows, we focus on quivers without such flavor symmetry factors. The “matter” of the theory is dictated by a choice of adjacency matrix A. While in topdown constructions A arises from a signed intersection pairing matrix, for our purposes it will suffice to consider a presentation in which all entries are non-negative i.e., Aij ∈ Z≥0 . This entry specifies the number of 4D N = 1 chiral multiplets in the bifundamental representation (Ni , Nj ) of SU(Ni ) × SU(Nj ). We draw this as a directed arrow that emanates from node i to node j. We also allow for arrows i → j as well as j → i, which in the physics literature are often referred to as “vector-like pairs”. We note that the adjacency matrix is different from its anti-symmetrization B = A − AT . This quiver data amounts to specifying a directed graph along with the decoration provided by rank assignments {Ni }i∈Nodes (the rank of each SU(Ni ) being Ni − 1). The full specification of the theory is supplemented by additional data such as gauge-invariant superpotential interaction terms, gauge couplings, and other (possibly irrelevant) interaction terms of the QFT. While this additional physical data is quite important in fully specifying a theory, in what follows we shall primarily focus on the defining data of the directed graph (i.e., the adjacency matrix A) and the rank assignments. We denote this data as (A, N) in the obvious notation. This should be viewed as necessary but not sufficient data in fully specifying the 4D QFT. Now, for 4D quiver gauge theories, there are a few basic consistency conditions we need to maintain. First of all, the absence of SU(Ni )3 gauge anomalies means that the rank assignments are partially constrained. Consistency requires that for each node i, the following condition holds:10 X X No SU(Ni )3 Anomalies ⇒ Aij Nj − Nj Aji = 0 . (2.1) j

We do not impose any further constraints aside from the requirement that any duality operation produces another theory with rank assignments Ni ≥ 0. In special cases such as 8

We will not be dealing with global properties of the gauge group, so we will conflate the usage of gauge group and gauge algebra since it affects nothing in our discussion. 9 One does not need to enforce 4D anomaly cancellation constraints on such nodes. 10 Recall that in our conventions, all entries of Aij are non-negative. Additionally, we are specifically excluding quivers with flavor symmetry factors. These can be included by extending the sum over nodes, but one does not need to impose an anomaly cancellation condition on the non-abelian flavor symmetry itself.

superconformal field theories, one often imposes additional constraints on the Ni . We do not impose such conditions here.11 Given a 4D N = 1 quiver gauge theory, we can consider a Seiberg dual presentation of the IR theory. This is an IR equivalence among theories, whose prototypical example is given by Supersymmetric Quantum Chromodynamics (SQCD), i.e., SU(Nc ) with Nf flavors, which is dual to SU(Nf − Nc ) with Nf flavors and a set of meson fields interacting via a cubic superpotential. This operation naturally extends to quiver gauge theories, where one considers the dualization of a specified node. We refer to this as a simple duality operation. Chaining together multiple such simple duality operations then relates many different quiver gauge theories. We refer to dualization of a quiver at node k by Dk (A, N). Generalizing the SQCD example above, the mutation maps Nk to Nfk − Nk , where Nfk is the number of flavors at node k, given by the sum of the ranks of all nodes j connecting to k, weighted by the number of incoming arrows Ajk . On the other hand, all the ranks of the other nodes in the quiver remain unchanged. In formulae, the dual theory has a rank vector  K X    Nj Ajk − Nk if i = k, Ni′ = j=1 (2.2)    Ni if i ̸= k . This operation is well-defined so long as the new rank Nk′ remains non-negative.12 Note also that duality preserves the condition of anomaly cancellation, i.e., we have an equal number of incoming and outgoing arrows (weighted by the ranks of the neighboring nodes).13 The mutation of the adjacency matrix modifies the matter content. The effects of dualizing node k include the reversal of the chiralities of the fundamental and antifundamental matter fields coupled to k, and the generation of composite meson fields corresponding to paths of length two that pass through k. The mesonic fields add new arrows between adjacent nodes i and j. We schematically show the action of Seiberg duality on node k in Figure 1. In addition to the changes to the directed graph and rank vectors, there is an accompa11

It is worth noting that even the condition of anomaly cancellation is a special feature of dealing with a 4D QFT. For example, in closely related 1D quiver quantum mechanics theories, no such condition needs to be imposed. 12 When the rank is too low compared to the number of flavors, a non-perturbative superpotential can be generated (see e.g, [26, 27]). This can often be diagnosed simply by dualizing to a theory with formally negative ranks, clearly a pathological situation. For our purposes, it suffices to work at the level of the directed graph and its ranks assignments. 13 We will also consider theories that are not necessarily anomaly free, but we will not use them as a training set for the neural networks; instead, they will only be used for our out-of-distribution tests. As already mentioned, such quivers also arise in physical systems such as quiver quantum mechanics. We will still impose that the resulting theory after dualization is given by a connected graph.

Nk Aik

Akj

l Nl Alk − Nk

Ajk

Aki Aik Akj

Figure 1: Schematic representation of Seiberg duality at the level of quiver theories. The labels on the arrows indicate the multiplicities of the fields in the various bifundamental representations. nying change to the superpotential, i.e., we add a new cubic interaction term that naturally generalizes the SQCD case. As already mentioned, we neglect such superpotential data in what follows, focusing exclusively on the data (A, N). The only way in which the physics of a superpotential enters is that any vector-like pairs generated by Seiberg duality/quiver mutation are automatically deleted. This naturally happens because vector-like pairs generated in this way also introduce a mass term that eliminates these contributions anyway. We note that this does not exclude the possibility of vector-like pairs in the original seed theory. Under this assumption, the Fomin–Zelevinsky mutation rule [2, 3] now becomes: ( Aji if i = k or j = k, A′ij = max(0, ∆ij ) otherwise,

(2.3)

where the net flow of arrows ∆ij from node i to node j is computed by taking the existing arrows Aij , adding the new mesonic arrows Aik Akj , and subtracting any oppositely oriented arrows: ∆ij = (Aij + Aik Akj ) − (Aji + Ajk Aki ) . (2.4)

2.2

Mutation Trees

Having specified the defining data for mutating a quiver gauge theory, we can now proceed to introduce a sequence of dualities, i.e., we pick a node of the quiver gauge theory Q = (A, N), mutate some node k to reach Dk Q, and then mutate again. Starting from a seed theory, we thus obtain a sequence of dualities, reaching a new quiver: Q(n) = Djn ◦ · · · ◦ Dj1 Q .

(2.5)

By inspection, each quiver obtained in this way amounts to a sequence of letters forming a mutation word. There can, in principle, be non-trivial relations between these words (i.e., a braid algebra), and in this sense, the structure of mutations shares many formal similarities with the problem of unknotting [15]. In a physical quiver gauge theory where we also include the data of the gauge couplings and superpotential couplings, the trajectory of an RG flow can often dictate a preferred sequence of dualities. That being said, there is also a great deal of sensitivity to these boundary 9

conditions, i.e., the RG flow trajectory is chaotic [7]. From this perspective, one can, in principle, entertain any sequence of mutations. The set of quiver gauge theories that can mutate into each other, therefore, forms a treelike structure. Indeed, at each stage, we can opt to mutate any of the nodes.14 Taking into account relations in the braid algebra, the number of distinct quivers reached in this way can be smaller. For example, finite-type quivers are precisely those that always return to the same starting point after a finite number of moves. On the other hand, the far more generic cas is that one can mutate endlessly. A simple ex(1,1,1) ample of this sort is the quiver obtained from N D3-branes probing the orbifold C3 /Z3 (see [7]), as schematically depicted in Figure 2. We also show the example of dualizing one (1,1,1) of the nodes of C3 /Z3 in Figure 3. Q(3)

Q(3) D3

Q(3)

Q(2)

Q(3) D3

Q(3)

Q(1) Q(2)

Q(3) D1

Q(1)

Q(2)

Q(3)

Q(0) D1

Q(1) D2

Q(2) D1

Q(3)

Q(2)

Q(3)

Figure 2: Seiberg Duality Tree for C3 /Z3 inspired by [32, Figure 2]. Each node in the tree is a gauge theory Q(n) with n being the number of non-trivial mutations Dj done from Q(0) . Generally, the action of Dj leads to different gauge theories. However, C3 /Z3 has a permutation symmetry such that each node of the same color corresponds to the same theory up to relabeling of the gauge nodes. 14

In the context of quiver gauge theories realized by D-branes probing a CY singularity, we comment that the braiding relations involve left- and right-mutation on an exceptional collection of sheaves [28–31].

1 (1,1,1)

Figure 3: Example of dualization of C3 /Z3

with respect to the red node (top quiver node).

In the context of string based constructions, RG flows/duality moves correspond to motion along a warped throat. Moving further into the bulk of a string compactification thus has a direct bearing on how energy scales impact the computational complexity of a given collection of quivers.

2.3

Seed and Validation Theories

A priori, there are an infinite number of possible quiver gauge theories one could introduce. To keep things concrete, we primarily focus on theories where the underlying quiver is dual to a well-known string theory construction, e.g., a D3-brane probing a toric Calabi–Yau singularity. At a small number of gauge group nodes, the entire list of such quiver gauge theories is completely known, and there are well-known algorithms for extracting more intricate quivers (see, e.g., [20] for an overview). In practice, then, we start with well-known toric Calabi–Yau geometries where the corresponding quiver gauge theory consists of up to 12 gauge group nodes. From this starting point, we then generate our training dataset by mutating many times. We comment that after mutation, the resulting geometry probed by the D3-brane may be in a non-toric phase. To test the efficacy of our method, we compare against a number of different kinds of quivers. First of all, we consider validation on Seiberg-dualized quivers derived from D3-branes probing toric and non-toric Calabi–Yau singularities not included in the original training set. Additionally, we also consider more abstract quivers which are not known to be realized by any D3-brane probing a Calabi–Yau singularity. This case is especially interesting to consider because it directly points to another implicit question: when does a quiver gauge theory arise from a D-brane probing a singular geometry? Having stated the physics motivation and rules, we now build a machine to check for Seiberg dualities.

Dataset Generation

In this section we describe the theories we have used to generate the database for the training of the NNs we considered in our work. We also explain the method used to generate the database via BFS over the duality tree of the theories in Appendix A.

3.1

Breadth-First Search over Duality Trees

We now explain how we generated the pairs of theories that we used to train and test the NNs and the pathfinders. We considered the theories described in Appendix A, which we will refer to as seed theories from this moment on. Let us call Q the set of quivers that can be generated by applying Seiberg duality to any of the seed theories Q(0) ∈ Q. Every quiver Q ∈ Q is defined by its rank vector N and adjacency matrix A. We then define Dk : Q → Q ∪ {∅} to be the map of a Seiberg duality transformation at node k. The operator maps a quiver to its dual, Dk Q(n) = Q(n+1) , according to the rules defined in Section 2. In the following, if mutating a node makes the rank of the dualized node non-positive, or if the quiver breaks into subquivers, we prune that branch; the operator returns ∅. We also introduce the concept of depth d of the duality tree, which is given by the number of non-trivial applications of Dk to Q(0) . In order to generate the pairs, we start from Q(0) at depth d = 0. We then make a BFS exploration, layer by layer, with Sd denoting the set of quivers Q discovered by mutating Q(0) d times. Moreover, we keep track of all visited quivers to avoid repetition in the database, and we call Vd the set of quivers found up to depth d.15 Starting from S0 = {Q(0) } ,

V0 = {Q(0) } ,

we proceed up to depth d, via the recursive definition: Sd = Q(d) = Dk Q(d−1) Q(d−1) ∈ Sd−1 , k ∈ {1, . . . , N }, Q(d) ̸= ∅, Q(d) ∈ / Vd−1 ,

(3.1)

Vd = Vd−1 ∪ Sd .

(3.2)

A schematic representation of the generation is shown in Figure 4 and the pseudocode is in Algorithm 1. Once we have generated the full duality tree for the quiver Q(0) up to dmax , we need to construct the pairs (QA , QB ) of theories divided by their distance d(QA , QB ), i.e., the minimal number of simple Seiberg duality operations that are necessary to go from QA to 15

Introducing Vd allows us to avoid artificially inflating the depth by applying Dk twice consecutively to the same node, which returns the original quiver Q. This ensures only “non-trivial” applications of Dk are considered.

S0 (Depth d = 0)

Q(0) D1 → drop D3

D1 D2

(1)

Qb D3

S2 (Depth d = 2)

∅

(1)

S1 (Depth d = 1)

(2)

Figure 4: Schematic representation of the BFS generated by the mutation Dk . QB . This can be obtained by considering the graph obtained by placing each mutant of the quiver on the nodes, with the edges representing the mutations relating them; we will denote this graph as Q∗ = (V ∗ , E ∗ ). We define the duality distance between (QA , QB ) as the length of the shortest sequence of edges connecting them in this graph: d(QA , QB ) = min {p : p number of edges between QA and QB in Q∗ } .

(3.3)

This distance is illustrated in Figure 5. The distance is computed during the BFS generation. Every discovered state Qi ∈ V ∗ is associated with a specific mutation path sequence Pi = (ki,1 , ki,2 , . . . , ki,di ) originating from Q(0) . Since all the quivers Q ∈ Q are generated up to depth dmax , the distance between any two quivers QA and QB is obtained from their path vectors. We define the LCA of the two theories as the state corresponding to the longest shared prefix between their path sequences: PLCA = PA ∩ PB .

(3.4)

The shortest path connecting QA to QB is constructed by taking the union of their path sequences and removing the shared intersection (the path from the root to the LCA). Expressed as a formula, this is given by: d(QA , QB ) = |PA | + |PB | − 2|PLCA | .

(3.5)

In generating the dataset, we keep track of the distance as well as the specific path we used to compute it. However, as is clear from Figure 5, this path is, in general, not unique. Moreover, if the quiver has permutation symmetries, the distance (3.5) becomes an upper 13

bound. As we will explain in Section 4.1.1, we trained the NNs using (3.5) as the target distance to be learned, without identifying quivers related by symmetries. This allows us to validate the training using the mean absolute error (MAE) between the predicted distance and (3.5). However, as we will see later, the DGNN loses accuracy at large distances, predicting shorter distances compared to the database distance (3.5). Once the permutations are taken into account, the accuracy of the network improves, meaning that the model was already accounting for these symmetries.

km ki

Figure 5: Schematic representation of the duality distance d(QA , QB ) = 3, defined as the shortest path (red) corresponding to the dualizations Dkm ◦ Dkj ◦ Dki QA . Algorithm 1 BFS over Duality Trees 1: Input: Initial quiver Q(0) , maximum depth dmax 2: Output: Set of visited quivers V 3: S0 ← {Q(0) } 4: V ← {Q(0) } 5: for d ← 1 to dmax do 6: Sd ← ∅ 7: for Q(d−1) ∈ Sd−1 do 8: for k ← 1 to K do 9: Q(d) ← Dk Q(d−1) 10: if Q(d) ̸= ∅ and Q(d) ∈ / V then (d) 11: Sd ← Sd ∪ {Q } 12: V ← V ∪ {Q(d) } 13: end if 14: end for 15: end for 16: if Sd = ∅ then 17: break 18: end if 19: end for 20: return V

▷ Iterate over all nodes ▷ Check validity and uniqueness

▷ Terminate if no new states are generated

Graph Neural Networks for Seiberg Dualities

In this section, we describe the specific Graph Neural Networks (GNNs) we designed for this work. See Appendix B for a review of terminology as well as a general introduction to the Graph Neural Network paradigm. We primarily consider two types of GNN architectures, though they will serve two different purposes. The first one, called the Distance GNN (DGNN), is trained to estimate the distance d(QA , QB ) between two theories QA and QB related by Seiberg duality. The second GNN, called the Adviser GNN (AGNN), predicts the first node to mutate in the sequence of Seiberg dualities that brings QA to QB . Both our architectures share the building blocks of standard MPNN [33] and Graph Transformers (similar to GraphGPS models, where GPS stands for General, Powerful, and Scalable [34]), but we adapt them to evaluate duality operations rather than classification tasks. Specifically, they use a message-passing tokenizer followed by a Transformer encoder. The tokenizer captures the structure of each node in the quiver, e.g., the rank of its gauge group or the number of edges connecting to it, while the global attention mechanism allows the network to process the totality of the graph. The differences lie in how the architectures handle the outputs. Graph QA xA,i , wA,ij

GNN Tokenizer (3 Layers)

Graph QB xB,i , wB,ij

Transformer (2 Layers)

Shared Weights

GNN Tokenizer (3 Layers)

Transformer (2 Layers)

Embeddings hA,i ∈ RH

Global Mean zA Node-wise |hA,i − hB,i | ↓ Mean (dlocal )

Embeddings hB,i ∈ RH

Concat [zA ⊕ zB ⊕ dlocal ]

MLP Head (ReLU → Drop)

Softplus Output dˆ

Global Mean zB

(a) DGNN Architecture. Graph QA yA,i , LPEA

Graph QB yB,i , LPEB

Hybrid GPS Encoder

Dense Nodes hA,i

Shared Weights

Node-wise Diff ∆deep = hB,i − hA,i i ∆raw and ∆Ai i

Hybrid GPS Encoder

Dense Nodes hB,i

Concat Zi [hA ⊕ hB ⊕ ∆]

MLP Classifier (Node-Level)

Action Masking M

Softmax P (Di |QA , QB )

(b) AGNN Architecture.

Figure 6: Schematic representations of DGNN and AGNN architectures.

4.1

Distance Graph Neural Network Architecture

We now explain how the DGNN architecture is constructed. Its purpose is to estimate the distance d(QA , QB ) between two quivers QA and QB related by mutations. The DGNN ar15

chitecture is depicted in Figure 6a, showing how it processes a pair (QA , QB ) through weightsharing encoding branches to map them into a latent space. Processing the latent space leads to the prediction of a continuous distance dˆ ∈ R+ . First of all, every quiver Q is represented by its ranks and adjacency matrix (N, A). These inputs are element-wise normalized as: xi = log(1 + Ni ) ,

wij = log(1 + Aij ) .

(4.1)

The normalized inputs are then given to an encoder consisting of a 3-layer GNN tokenizer. The primary role of this module is to construct locally-aware “tokens”, so that starting from the first layer encoding the information about the ranks of the nodes xi , the NN can process more and more information about the edges connecting each node. (ℓ)

Let hi ∈ RH represent the hidden feature vector of node i at layer ℓ,16 where the (0) initial input is the 1-dimensional normalized rank hi ≡ xi ∈ R1 . The convolution layers are (ℓ) (ℓ) parametrized by two distinct learnable weight matrices each: Wself for self-loops and Wmsg for neighbor messages.17 At each step, the unactivated update rule aggregates information from the immediate neighborhood: X (ℓ) (ℓ) (ℓ) (ℓ+1) (ℓ) (4.2) Wmsg hj · wji . = Wself hi + mi j∈N (i) (ℓ)

(ℓ)

(ℓ+1)

The weights Wself and Wmsg are shared between QA and QB . These vectors mi are normalized via LayerNorm and passed through a LeakyReLU activation function with a negative slope α = 0.2.18 The fully activated node representation is therefore: (ℓ+1) (ℓ+1) h̃i = LeakyReLU LayerNorm mi . (4.3) Crucially, residual skip connections are enforced for all layers ℓ > 0 such that: (ℓ+1)

(ℓ)

(ℓ+1)

= hi + h̃i

(4.4)

Standard convolutional GNNs replace the node state at each step. This architecture differs by (ℓ+1) using residual connections to accumulate the outputs, adding the new features h̃i directly (ℓ) to the prior state hi . We accumulate these features to ensure that the network retains some of the initial information of the quiver extracted in previous layers. Consequently, the final (3) token representation hi is a sum of the information gathered by all the convolutional layers, and it will be the token passed through the transformer we describe next. 16

In our case, H = 64. For the first layer ℓ = 0, the weight matrices project from the 1-dimensional input to the hidden dimension (0) (0) (ℓ) (ℓ) H, i.e., Wself , Wmsg ∈ RH×1 . For subsequent layers ℓ > 0, the matrices are square, Wself , Wmsg ∈ RH×H . 18 The precise definitions of LayerNorm and LeakyReLU are discussed in Appendix B.1. 17

While the GNN tokenizer has captured the local information of each node and its surroundings up to ℓ = 3, when comparing (QA , QB ), we need a way to understand the global (3) changes in the quiver that distinguish them after dualization. To achieve this, we pad hi into X ∈ RK×H and process it using a 2-layer Transformer encoder via the Scaled Dot-Product Self-Attention mechanism [35]. We use Multi-Head Attention (MHA) with h = 4 independent heads and a projection dimension dk = H/h = 16, followed by a position-wise Feed-Forward Network (FFN). The progression of the node state X through a single encoder layer uses standard residual connections and LayerNorm: X̃ = LayerNorm(X + MHA(X)) , Xout = LayerNorm(X̃ + FFN(X̃)) .

(4.5)

After passing through the 2 distinct Transformer layers, the model yields hA,i and hB,i for the paired quivers QA and QB , respectively. From these vectors, we define a node-wise difference vector dlocal ∈ RH : K 1 X dlocal = |hA,i − hB,i | , (4.6) K i=1 which explicitly records the local discrepancies caused by the sequence of Seiberg dualities. Finally, we define the average of hA,i and hB,i over the nodes of the quiver: K

1 X zA = hA,i , K i=1

1 X zB = hB,i , K i=1

(4.7)

which provide global representations of each theory to the distance regressor. The averaging operation smooths out the individual node features, giving the macroscopic difference dglobal = |zA − zB |. For our purposes, we need the combined information of dglobal and dlocal . Indeed, unlike standard GNNs that typically pool node features to process the graph data (e.g., for classification purposes), the DGNN computes the absolute difference between corresponding nodes of two graphs, and it needs to retain finer information. If this discrepancy were evaluated solely after global pooling (yielding only dglobal ), localized node variations would be smoothed out. Incorporating the pre-pooling difference dlocal ensures that the regressor retains fine-grained structures that global averaging alone would discard. After the various encoding layers, the continuous distance is processed through a MultiLayer Perceptron (MLP) head, with [zA ⊕zB ⊕dlocal ] ∈ R3H as input. The purpose of this part of the NN is to reduce the 3H-dimensional vector down to a single scalar. To achieve this, we use standard linear projection with a ReLU activation function and a Dropout regularization layer (p = 0.2) to mitigate overfitting. The output of this projection is passed through a Softplus function, i.e. dˆ = Softplus(s) = log(1 + es ) , (4.8) ˆ leading to the predicted distance d. 17

4.1.1

Training Evaluation 2

MAE

Loss

6 4

1.5

2 0

100 150 Epoch Train Loss

200

250

100 150 Epoch

Val Loss

200

250

Val MAE

(a) Training and Validation MSE Loss

(b) Validation MAE

Figure 7: Training logs for the DGNN model The network parameters are optimized by minimizing the Mean Squared Error (MSE) between the prediction dˆ and the path distance d computed at the generation of the database. For a given batch of size B, the loss function is defined as: B

LMSE =

1 X ˆ (dk − dk )2 . B k=1

(4.9)

While MSE is utilized for gradient descent to penalize outlier predictions, the accuracy of the model is tracked using the Mean Absolute Error (MAE): B

1 X ˆ MAE = |dk − dk | . B k=1

(4.10)

In the context of distance regression, MAE provides a direct measure of the average step deviation. The DGNN was trained for approximately 250 epochs to estimate the distance between quivers. Figure 7 displays the training trajectory. Checkpoints for the final evaluation were selected based on the epoch minimizing the MAE. Specifically, the checkpoint used for the pathfinder was reached at epoch 247 with a final MSE loss of 1.137. There can be multiple reasons why the MAE could not go lower. One possibility is that whenever the distance between two quivers is larger than the number of nodes, the graph difference taken by the DGNN of the hidden layers of the NN becomes less reliable, leading to an inability to estimate larger distances than the number of nodes of the quivers. This is 18

0 1

6 7 8 Distance

9 10 11 12

(a) MAE by Distance

1.224

1.229

1.155

1.280

1.132

1.476

1.551

2.232

2.680

6 3.785

MAE

6.547

corroborated by looking at the MAE by the number of nodes in Figure 8, where it is clear that the NN struggled to predict larger distances for smaller quivers while being extremely accurate for larger quivers.

7 8 9 10 11 12 13 Nodes

(b) MAE by Nodes

25k

Median: −0.954

Pairs

20k 15k 10k 5k 0 −12 −10 −8 −6 −4 −2

Error

Figure 8: Benchmark results for DGNN. Because we apply the DGNN as the heuristic function of an A∗ search, as we will explain in Section 5.2, we have decided to perform further benchmark tests on the kind of heuristic function that the DGNN can provide. One test is to check if the heuristic function is monotonic, so that we reduce the possibility that the A∗ search requires processing the same node multiple times. However, as we show in Figure 9, the DGNN output exhibits non-monotonicity. More precisely, we note that the DGNN is non-monotonic approximately 32.07% of the time. Consequently, during a pure A∗ search, the priority queue can be delayed by repeatedly exploring uninformative branches. We will see that, despite the high success of the A∗ search guided by the DGNN for both in distribution (ID) and out of distribution (OOD) theories, it is precisely the absence of monotonicity in the DGNN that causes the algorithm’s failure. 19

Monotonicity Violation

300k

250k

Violation: 32.07%

200k Pairs

40 30

150k 100k

50k 10 2

6 8 Distance

(a) Monotonicity Violations by Distance

3 ∆h

(b) Jump Distribution

Figure 9: Evaluation of DGNN monotonicity. The non-monotonicity of the heuristic results in excessive node expansions and timeouts.

4.2

Adviser Graph Neural Network Architecture

We now proceed to describe the AGNN architecture. The objective of the AGNN is to predict the most probable node that should be mutated first in the sequence of Seiberg dualities from quiver QA to quiver QB . Unlike the DGNN, which predicts a distance, the AGNN acts as a localized policy function, yielding a probability distribution over the K nodes of QA . The AGNN architecture is described in Figure 6b. First of all, much as in the DGNN case, we want to encode the information of the rank and adjacency matrix of the quiver via a suitable “tokenization”. The initial feature vector is 5-dimensional: out in out yi = log(1 + Ni ), log(1 + Ain i ), log(1 + Ai ), log(1 + Ni ), log(1 + Ni )

(4.11)

Aij Nj .

(4.12)

where Ain i =

K X j=1

Aji ,

Aout = i

K X

Aij ,

Niin =

j=1

X j

Aji Nj ,

Niout =

X j

To allow the network to distinguish nodes, we evaluate the Laplacian Positional Encodings (LPE).19 Let us denote as LPEi ∈ RdLPE the Laplacian eigenvectors. These LPEi vectors and the features xi are projected into the hidden layer of dimension H, summed, and normalized 19

Let A be the adjacency matrix of the directed quiver. We define the symmetric adjacency matrix of the underlying undirected graph as Asym = A + AT . The degree di of node i is, then, the total number of edges P connected to the node, i.e. di = j (Asym )ij . The corresponding graph Laplacian is L = D − Asym , where D is the diagonal degree matrix with entries Dii = di and Di̸=j = 0.

(0)

to form the initial embeddings hA,i and hB,i : (0)

hi = LayerNorm (Wnode yi + WLPE LPEi ) ,

(4.13)

where Wnode ∈ RH×5 and WLPE ∈ RH×dLPE are learnable projection matrices, shared by both quivers QA and QB . These initial embeddings are then passed through a Hybrid GPS Encoder, identical in structure to the combined GNN Tokenizer and Transformer used in the DGNN, from which we extract hA,i and hB,i , as before. However, despite what one might expect, we find that these more refined inputs empirically under-perform when used in the training of the DGNN. Indeed, the introduction of the LPEi vectors as an input degraded training performance, most likely because it made it harder to extract macroscopic properties of the graph, resulting in higher error rates when predicting distances. Similarly to the DGNN case, we also compute node-wise differences: (0)

= hB,i − hA,i , ∆deep i

(0)

∆raw = hB,i − hA,i , i

as well as the element-wise absolute difference of the adjacency matrices: X ∆Ai = |AB,ij − AA,ij | .

(4.14)

(4.15)

In this way, we keep track of all possible local differences between the two quivers. The dense nodes and these local differences are concatenated into Zi ∈ R4H+6 for each node: h i raw ⊕ ∆ Zi = hA,i ⊕ hB,i ⊕ ∆deep ⊕ y ⊕ ∆A (4.16) A,i i . i i The concatenated vector Zi is then processed by an MLP classification head, which projects the features down to a single scalar value for each node i. This scalar, called logit, represents the network’s preliminary preference for mutating node i. However, not all nodes can, in general, be dualized since, for instance, mutating certain nodes might yield negative gauge ranks or disconnect the quivers. We generate a binary action mask M prior to evaluation, which takes the value Mi = 1 if the mutation at node i P is permissible, and Mi = 0 otherwise. In practice, the network checks if Ni′ = j Aji Nj − Ni is positive. Before converting the logits into probabilities, we apply this mask by manually overriding the scores of all invalid nodes to −∞. Finally, we apply a Softmax function across the N nodes. Because the exponential of −∞ is exactly zero, this masking guarantees that the network assigns a 0% probability to any mutation that is not possible, forcing the remaining valid nodes to absorb the entirety of the probability distribution. The final policy is thus given by: ( yi if Mi = 1 , P (Di |QA , QB ) = Softmax(ỹ)i , where ỹi = (4.17) −∞ if Mi = 0 , 21

where yi denotes the logit output of the MLP. Once this valid probability distribution is obtained, we sort the probabilities P (Di |QA , QB ) in descending order and extract the top-k nodes. 4.2.1

Training Evaluation

Accuracy (%)

Loss

1.2

0.8

40 0

20 30 Epoch Train Loss

Val Loss

Top-1

(a) Training and Validation Cross-Entropy Loss

20 30 Epoch

Top-2

Top-3

(b) Next-Step Action Accuracy

Figure 10: Training logs for the Autoregressive model. The AGNN is optimized by minimizing the Cross Entropy (CE) loss between the predicted probability distribution and the exact first-step mutation extracted from the pre-computed shortest path sequence. For a given mini-batch of size B, let tq ∈ {1, . . . , K} denote the index of the node that must be dualized in the q-th quiver QA,q . The loss function penalizes the network based on the predicted log-probability assigned to these actions and is defined as: B

1 X LCE = − log P (Dtq |QA,q , QB,q ) . B q=1

(4.18)

During training, we check the prediction accuracy of the top-k nodes, which monitor whether the node we predict to be the one to dualize, i.e., tq , appears within the highest probability choices predicted by the network. However, it is crucial to note that this metric does not perfectly reflect the true success rate of the AGNN. Due to quiver symmetries and the nonuniqueness of shortest paths in the duality tree, multiple nodes may represent equally optimal first-step mutations. The Cross Entropy loss penalizes the network for deviating from tq in our dataset, even if the AGNN can, in principle, propose a perfectly valid alternative node. Consequently, the top-k accuracy serves as a lower bound on the true success rate of the policy.

The AGNN was trained to predict the next-step Seiberg duality operation, with the NN outputting the probabilities over the nodes. Figure 10 shows the cross-entropy loss and top-1 next-step accuracy across the 49 training epochs, at which point the training was stopped to prevent overfitting the data, since the training loss kept decreasing while the validation loss started increasing. However, the accuracy converges rapidly, with the optimal validation loss achieved at epoch 39. Due to the symmetry of the quivers, the top-1 accuracy does not exceed 70%, meaning that for a better policy, we need to rely on more than one probability. In fact, top-3 accuracy is, on average, around 96%, resulting in the correct node being dualized most of the time. This can be seen from Figure 11. We note that the accuracy actually increases with distance, resulting in a better oracle whenever two quivers are far apart, while the accuracy drops for smaller distances. However, unlike the DGNN, there is not much difference in accuracy for the different theories used for training. The AGNN will also be utilized to guide a pathfinding search. Initially, we evaluate it as a unidirectional pathfinder, where the trajectory is constructed by iteratively selecting the actions with the top-k probabilities predicted by the AGNN. To benchmark the reliability of the AGNN in this guiding role, we analyze the policy margin, defined as the difference between the probability assigned to the correct node (Pcorrect ) and the highest probability assigned to an incorrect node (Pbest incorrect ).20 A negative policy margin indicates an inversion, meaning the network prefers an incorrect operation over the correct one. As shown in Figure 12, the model exhibits such inversions 20% to 40% of the time. However, despite this inversion rate, the average policy margin remains positive across all target distances; this occurs because the network is highly confident when it is correct (yielding large positive margins), easily outweighing the margins of the incorrect predictions, which tend to be only slightly negative. From the discussion above, we can conclude that, considering the three most probable nodes as branching, the AGNN is correct over 90% of the time. This insight is critical: while the AGNN is not an infallible oracle and cannot navigate the duality tree on its own, it suppresses the vast majority of incorrect branches. This confidence motivates the last kind of pathfinder that will use the AGNN policy to restrict the branching factors, allowing the DGNN to evaluate only the most promising paths. Furthermore, analyzing the tail of the margin distribution in Figure 12 reveals that for a substantial subset of states (approximately 5.7 × 104 theories), the network exhibits an extreme policy margin of Pcorrect − Pbest incorrect ≈ 1.0. This signifies that the AGNN model is highly polarized: it appears to be uncertain between two options for only a small subset of nodes. Large regions of the duality tree are instead considered trivial and are solved with high 20

Because multiple valid paths can connect two theories, we define an ‘incorrect’ node as any node that does not belong to the specific shortest path recorded during our dataset generation.

4 0.23 0.77 0.81 0.79 0.76 0.78 0.75 0.80 0.82 0.88 0.91 0.94 5 0.40 0.86 0.82 0.74 0.69 0.68 0.68 0.69 0.70 0.77 0.80 0.83

0.8

Nodes

6 0.59 0.88 0.79 0.71 0.64 0.62 0.59 0.63 0.62 0.68 0.67 0.78 7 0.76 0.88 0.83 0.77 0.70 0.66 0.60 0.63 0.65 0.68 0.69 0.68

0.8

0.6

Accuracy

3 0.11 0.66 0.58 0.82 0.84 0.87 0.91 0.86 0.97 1.00 1.00 1.00

0.6 0.4

8 0.83 0.83 0.78 0.71 0.63 0.62 0.58 0.60 0.57 0.62 0.60 0.67 9 0.53 0.79 0.71 0.74 0.73 0.67 0.65 0.66 0.64 0.63 0.63 0.61

0.4

0.2

10 0.39 0.68 0.61 0.67 0.62 0.61 0.57 0.56 0.54 0.57 0.55 0.61 11 0.32 0.66 0.60 0.70 0.61 0.58 0.59 0.57 0.56 0.59 0.54 0.61 12 0.35 0.57 0.53 0.61 0.58 0.56 0.54 0.52 0.51 0.53 0.51 0.54 13 0.36 0.57 0.56 0.65 0.62 0.58 0.60 0.54 0.55 0.54 0.55 0.49 1

6 8 Distance

Top-1

Top-2

Top-3

Distance

(a) Top-1 Accuracy by Nodes and Distance

Cross-Entropy Loss

(b) Top-k Accuracy Decay over Distance

1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 2

6 8 Distance

Figure 11: Benchmark results for the AGNN. While next-step accuracy degrades at larger duality distances, the validity rate remains high, indicating that the network has implicitly learned the rules of Seiberg dualities.

30k 25k Pairs

Pcorrect − Pbest incorrect

1 0.75 0.5 0.25 0 −0.25 −0.5 −0.75 −1

20k 15k 10k 5k

Top-1

6 8 Distance

Top-2

Top-3

−1

−0.5

0.5

Pcorrect − Pbest incorrect Top-1

(a) Policy Margin Decay by Distance

Top-2

Top-3

Inversion

(b) Top-k Policy Margin Distributions

Figure 12: Policy Margin analysis. While Top-1 inversions cause some paths to fail, the high Top-3 margins enable pruning of a large portion of the duality tree. confidence. From an algorithmic perspective, this is beneficial, as a guided search experiences a zero effective branching factor in these states. However, this same polarization means that when the network does make an error (inversions where the margin is < 0), it also often does so with high confidence. This extreme confidence in both correct and incorrect directions further highlights why a pure greedy search with AGNN is brittle, necessitating the safety net of the DGNN to backtrack out of confident dead-ends.

Pathfinder for Seiberg Dualities

In this section, we discuss the other component of our inference model, namely the guided pathfinders that are used to determine the chain of dualities that connect two theories QA and QB . The pathfinders are of two kinds: a Bidirectional A∗ Search Pathfinder and a Beam Search Pathfinder, as generally reviewed in Appendix C. For the latter kind, the beam is guided by the output of the AGNN model. On the other hand, we propose different kinds of cost and heuristic functions for the A∗ Search Pathfinder, as summarized in Table 1.

5.1

Brute-Force Search Baseline and Branching Factor

To benchmark the efficiency of guided search, we first formalize the complexity of unguided exploration. In standard graph search algorithms, the branching factor b characterizes the number of direct children a node possesses. In the present framework, the branching factor of a specific quiver Q = (N, A) corresponds to the number of valid Seiberg duality transformations it admits. Let K be the total number of nodes in the quiver. A priori, one might attempt 25

to dualize any of the K nodes, implying a maximum branching factor of K. However, as established in Section 3, a mutation Dk Q is admissible if and only if it satisfies two conditions: 1. Rank Positivity: The mutated gauge rank must remain positive. 2. Graph Connectivity: The resulting adjacency matrix A′ still describes a connected quiver. Therefore, the valid branching factor bQ for any state Q is defined as the number of valid mutations: bQ = |{k ∈ {1, . . . , K} | Dk Q ̸= ∅}| ≤ K . (5.1) For a bidirectional search between QA and QB , bA and bB designate the valid branching factors: • bA ≡ bQA : The number of valid dualizations originating from QA . • bB ≡ bQB : The number of valid dualizations originating from QB . To establish a baseline for the expected number of mutations performed by an unguided bidirectional search (NBFS ), we observe the following fact: since Seiberg duality is an involution, i.e. Dk ◦ Dk Q = Q,21 the branching factor can be estimated to be, at depth i ≥ 1, at most bavg. (bavg. − 1)i−1 . This means that starting from Q, we reach distance d with Ndir (d) dualities, which can be expressed as22 Ndir (d) = 1 + bQ

d X

(bQ − 1)i−1 = 1 + bQ

i=1

(bQ − 1)d − 1 . (bQ − 1) − 1

(5.2)

If we start a bidirectional search between two quivers at distance d, to guarantee finding an intersection, the unguided pathfinder that explores must reach depth d/2 from each side. To be precise, one of the two sides will reach ⌊d/2⌋ depth, and the other ⌈d/2⌉. The expectation is the sum of these two expansions: bavg. NBFS ≈ Ndir (⌊d/2⌋) + Ndir (⌈d/2⌉) = 2 + (bavg. − 1)⌊d/2⌋ + (bavg. − 1)⌈d/2⌉ − 2 . (5.3) bavg. − 2 This formulation implies asymmetric growth. When d changes from an even to an odd distance, one tree must expand by an entire extra depth level, causing a spike in the total number of explored nodes. Conversely, when d goes from odd to even, the second tree catches up to the same depth, resulting in a smaller multiplicative increase. The exponential growth of NBFS therefore oscillates depending on the parity of d, an artifact that will be visible in the efficiency metrics defined in Section 6. 21

Strictly speaking, this requires a left- and right-mutation on the exceptional collection of sheaves rather than two mutations of the same sort (see e.g., [36]). Such distinctions will not matter in the present analysis but would be interesting to study further. 22 If bQ = 2, Ndir (d) = 1 + 2d.

5.2

Pathfinders for Seiberg Dualities

We now proceed to present the NN-informed A∗ pathfinder we developed in detail. A summary of the algorithms is given in Appendix C, while their evaluations, costs, and objective functions are given in Table 1. Since quivers are defined up to node permutations, at each step of the pathfinder, we compare a canonicalized version of the quivers via a Weisfeiler–Lehman (WL) hashing of the graphs.23 Once a pair is identified, the search terminates, and the duality sequence is reconstructed. QA

Pruned

Top 2 Top 1

gfwd

Vfwd hbwd (Qm )

Qn′

hfwd (Qn ) gbwd

Vbwd

(a) Bidirectional A∗ search pathfinder.

(b) Beam search pathfinder

Figure 13: Schematic representations of the two kinds of pathfinders we consider in this work. In contrast to the unguided expansion NBFS , the total computational cost of the guided pathfinder is the sum of the unique quivers explored: Nexp = |Vfwd | + |Vbwd | . 23

This avoids missing the pair because of a “wrong” ordering of the nodes.

(5.4)

If the pathfinder is unidirectional, as will be the case for the AGNN pathfinder, then Vbwd = ∅. If the search exceeds a predefined threshold of steps without finding an intersection, execution halts. This outcome indicates either that the two theories are not related by Seiberg dualities, or that their separation exceeds the maximum search depth. Pathfinder

Cost g(.)

Heuristic h(.)

DGNN

Bidir. A∗

DGNN(Qn , Qtarget )

AGNN

B = 3 Beam

− log P (Di |Qn , Qtarget )

Hybrid

Bidir. A∗

1 − λAR log P (Di |Qn , Qtarget )

λDGNN DGNN(Qn , Qtarget )

LCA

Bidir. A∗

c(Di )

Hybrid LCA

Bidir. A∗

(1 − λdet cost ) + λdet cost cdet (Dk ) −λAR log P (Dk |Qn , Qtarget )

0 λDGNN DGNN(Q n , Qtarget ) P rank(Q n) +λLCA P rank(Qroot )

Table 1: Summary of Pathfinders and their cost and heuristic functions. BFS Pathfinder: As a fundamental baseline for evaluating our neural network-guided models, we implement an unguided Breadth-First Search (BFS) pathfinder. This algorithm operates by systematically exploring the Seiberg duality graph level by level, generating all admissible mutations at depth t before evaluating any transition at depth t+1. Structurally, it represents an unguided A∗ search where the heuristic function is identically zero (h(Q) ≡ 0) and the cumulative path cost simply tracks the traversal depth (g(Qn+1 ) = g(Qn ) + 1), assigning an equal step cost to every edge. Consequently, the BFS pathfinder is topologically agnostic, treating every valid mutation as equally promising without any predictive physical intuition. To optimize execution and match the structural framework of the DGNN and Hybrid models, the BFS pathfinder is implemented as a bidirectional search originating simultaneously from both the root theory QA and the target theory QB . DGNN Pathfinder: It consists of a bidirectional A∗ search algorithm with a heuristic function h(·) given by the output of the DGNN. This means that for any intermediate quiver states Qn and Qm originating from mutating QA and QB , respectively, the evaluation functions for the forward and backward frontiers are defined as: ffwd (Qn ) = gfwd (Qn ) + DGNN(Qn , QB ) fbwd (Qm ) = gbwd (Qm ) + DGNN(Qm , QA ) ,

(5.5)

with gfwd (QA ) = gbwd (QB ) = 0. The dequeued element at each step is determined by which evaluation function is smaller. This represents the first of a set of pathfinders in which the NN prediction provides the policy to guide the pathfinder. As we will discuss in Section 6, despite being quite efficient and successful, the main problem with using the DGNN as a heuristic 28

is its non-monotonicity; i.e., the distance between the two quivers is not monotonically decreasing. In fact, sometimes the heuristic delays the search queue, eventually degrading to a BFS algorithm. AGNN Pathfinder: Unlike the distance-guided approach, this implementation utilizes the probability distributions output by the Adviser GNN (AGNN) to parameterize the cost function g(Q) of a beam search with beam width B.24 Given an initial theory QA and a target theory QB , the algorithm explores the space of valid Seiberg dualities starting from QA . Following our empirical evaluation of the network’s classification accuracy in Section 4.2, we set the fixed beam width to B = 3. At each expansion step, the algorithm processes all quivers currently residing in the beam. For a given state Qn ∈ Bt , the AGNN predicts the mutation probability distribution P (Dk |Qn , QB ) across all admissible nodes. The algorithm selects the B actions with the highest predicted probabilities and generates the corresponding mutant quivers Qn+1 = Dk Qn . To formulate the trajectory evaluation as a cost minimization problem, each valid successor state Qn+1 is assigned a cumulative trajectory cost g(Qn+1 ), defined as the accumulated negative log-probability of the mutation sequence: g(Qn+1 ) = g(Qn ) − log P (Dk |Qn , QB ) ,

(5.6)

where the root theory is initialized at g(QA ) = 0. Since log P ≤ 0, highly probable mutations contribute minimally to the accumulated cost, while unlikely ones heavily increase the cost. To prevent cyclic exploration, the algorithm checks the best candidates against the WL hashes of all previously evaluated quivers. If the candidate’s structure has already been visited, the candidate is discarded.25 Once possible loop branches have been pruned, the algorithm collects all surviving candidates generated across the current beam, sorts them in ascending order based on their cumulative trajectory cost g(Qn+1 ), and retains only the top B quivers to form the new beam Bt+1 . The search procedure terminates successfully the moment a generated candidate is confirmed to be isomorphic to the target theory QB . Conversely, the search returns a failure status if the exploration reaches a predefined maximum depth or if the beam becomes empty due to dead-end mutations.26 24

Because beam search permanently discards alternative branches without a backtracking mechanism, the probability that a forward beam originating from QA and a backward beam from QB intersect in the vast intermediate mutation space is vanishingly small. Therefore, unlike the A∗ pathfinder, beam search is strictly implemented as a unidirectional search originating from QA . 25 Recall that the beam search algorithm simultaneously tracks up to B active trajectories at each step. In this sense, it can be conceptualized as an A∗ search with a truncated frontier, trading the guarantee of completeness for memory efficiency. 26 A dead-end occurs when all active trajectories in the beam are pruned. This scenario arises if the AGNN

Hybrid Pathfinder: To overcome the limitations of the DGNN A∗ search and the AGNN beam search, we developed a Hybrid Pathfinder. This approach combines the distance estimates of the DGNN with the branch probabilities of the AGNN. Specifically, the AGNN acts as a soft pruning mechanism, discouraging exploration along improbable branches and preventing the algorithm from stalling in flat or non-monotonic regions of the DGNN landscape. Retaining the A∗ priority queue provides a mechanism to backtrack if the AGNN predictions diverge. The Hybrid Pathfinder inherits the bidirectional exploration structure of the DGNN A∗ search but modifies the step-cost accumulation using the policy output of the AGNN. For any valid mutation Qn+1 = Dk (Qn ), the evaluation function f (Qn+1 ) is defined again as: f (Qn+1 ) = g(Qn+1 ) + λDGNN DGNN(Qn+1 ) ,

(5.7)

where λDGNN is a scaling hyperparameter balancing the heuristic weight, and the cumulative trajectory cost g(Qn+1 ) updates recursively via: g(Qn+1 ) = g(Qn ) + 1 − λAR log P (Dk | Qn , Qtarget ) .

(5.8)

Here, the base step cost of 1 ensures admissibility by guaranteeing strictly increasing path costs, while the parameter λAR ≥ 0 controls the penalty imposed on actions deviating from the optimal AGNN policy. When λAR = 0, the expansion reverts to the standard DGNN A∗ search; conversely, as λAR → ∞ with λDGNN = 0, the trajectory cost dominates, recovering the behavior of the AGNN pathfinder. LCA Pathfinder: This algorithm works as an A∗ search without a heuristic function, where at each iteration, the pathfinder expands the active frontier in the direction that has the minimum path cost determined by hard-coded rules, not learned from training. At every state Qn , the cost of mutating it with Dk is based on the change in the rank of the dualized node k. The cost function g(Dk ) is determined by three weights, defined as cdec : ceq : cinc ∼ 1 : 10 : 100, such that the cost of every step is    cdec if rank(k) decreases , (5.9) g(Dk ) = ceq if rank(k) is unchanged ,   c if rank(k) increases . inc

Because we penalize the increase of rank for both frontiers, the search does not attempt to find the direct path between two theories. Instead, the cost function acts as a “gravity well” that funnels both search trees towards the model that has the lowest rank. policy repeatedly favors mutations that lead to previously visited states. Because the algorithm systematically filters out historical structures to prevent infinite loops, an overconfident policy trapped in a cycle will cause all candidate branches to be discarded, leaving Bt+1 = ∅.

The motivation behind the LCA is to construct an algorithm that mimics the process that (human) physicists follow when guessing if two theories are related by Seiberg dualities: one tries to obtain a “simpler theory” from both quivers, reducing ranks and arrows between nodes, hoping to reach a common ancestor that generates the pair of quivers. This heuristic serves as a more robust baseline than standard BFS for benchmarking our neural pathfinders. By explicitly incorporating the structural “complexity” of the quiver, ranks, and total number of arrows, this algorithm effectively prioritizes structural simplification. This contrasts with a standard BFS, which is topologically agnostic and blindly treats every valid mutation as equally promising, regardless of its impact on the theory’s complexity. Hybrid LCA Pathfinder: Finally, a natural possibility is the integration of the LCA and Hybrid pathfinders. The idea is to further help the Hybrid pathfinder overcome the nonmonotonicity and inversion problems: if the guidance of the NNs is unsure, the LCA can move the search towards the minimal quiver. The LCA cost function enters into modifying both the cost and heuristic functions of the Hybrid pathfinder. To this end, we modify the total edge cost for the priority queue, which accumulates as: g(Qn+1 ) = g(Qn ) + (1 − λdet cost ) + λdet cost cdet (Dk ) − λAR log P (Dk |Qn , Qtarget ) , where cdet (Dk ) is defined similarly to (5.9), i.e.    cdet dec if rank(k) decreases , cdet (Dk ) = cdet eq if rank(k) is unchanged ,   c if rank(k) increases .

(5.10)

(5.11)

det inc

Similarly, the heuristic incorporates a penalty proportional to the total rank of the quiver to encourage moving along quivers with smaller ranks. The modified function is defined as: P rank(Qn ) h(Qn ) = λDGNN DGNN(Qn , Qtarget ) + λLCA P , (5.12) rank(Qroot ) where Qroot denotes the initial graph QA or QB depending on the active frontier. By adjusting the weights λdet cost , λAR , λDGNN , and λLCA , this pathfinder averages between a purely NN-guided framework and the LCA pathfinder. However, this integration can cause conflicting search signals between the two pathfinders. The Hybrid pathfinder uses the guidance of DGNN and AGNN, which have learned to find the shortest path between two theories. This means that such a path might necessitate moving along trajectories where the quivers increase their ranks, completely avoiding the LCA. On the other hand, the LCA pathfinder works like an effective potential, moving precisely to find the theories with lower 31

ranks. If the LCA is integrated into the Hybrid with the same hierarchy of weights defined for the LCA pathfinder, the resulting algorithm would simply reduce to the LCA pathfinder without any guidance given by the NNs. To balance the two guiding policies of the pathfinder, we introduce hyperparameters designed to maximize the efficiency ratio while maintaining a 100% success rate. The optimizer converged to cdet dec = 0.3, cdet eq = 2.7, and cdet inc = 3.1, alongside a global weight λdet cost = 1.3. Substituting these into the definition of cbase yields effective base step costs of cbase dec = 0.09, cbase eq = 3.21, and cbase inc = 3.73 for the priority queue. This calibration imposes a penalty for rank-increasing mutations that is of the same order of magnitude as the neural network coefficients while contributing negligibly to the cost of rank-reducing mutations. Consequently, the LCA penalty does not dominate the Hybrid cost function. Instead, it biases search toward smaller-rank theories when network confidence is low while allowing high-probability shortcuts to proceed uninterrupted.

5.3

Complexity Comparison

We, now, compare the computational complexity of the NN-guided pathfinders against the unguided bidirectional BFS baseline established in Section 5.1. In an unguided expansion, d/2 the number of explored states scales exponentially as O bavg. , where bavg. is the average valid branching factor and d is the minimum mutation distance between the theories. In the ideal limit of a monotonic heuristic, an A∗ search expands only the states along the optimal trajectory, achieving linear complexity O(d). However, as shown in Section 4.1.1, the distance estimates generated by the DGNN exhibit non-monotonic fluctuations across mutation sequences, violating the formal conditions for algorithmic consistency. Because A∗ maintains an exhaustive frontier of unexplored branches, a locally misleading heuristic can attract the search into non-optimal trajectories. Consequently, in the worst-case scenario, the computational cost not only degrades to the exponential scaling of the BFS baseline but can even exceed it, exhausting the step budget before recovering the true path. The same worst-case bound applies to the LCA baseline, which operates as an A∗ search guided purely by a deterministic topological cost function. Because the LCA heuristic strictly penalizes increases in gauge ranks and arrow counts, the algorithm can easily become trapped in a local minimum of topological “complexity”. In this case, the search might stagnate in minimal-rank configurations, eventually exhausting its computational budget. Conversely, the AGNN beam search strictly confines the expansion tree to a maximum of B candidate quivers per level. The number of evaluated states is strictly bounded by O(B · bavg. · d), thereby guaranteeing predictable linear scaling with respect to distance at the cost of a decrease in NN accuracy. Indeed, this massive reduction in memory and execution 32

time sacrifices algorithmic completeness: as discussed in Section 4.2.1, if an incorrect AGNN prediction prunes the true path from the active beam, the algorithm permanently loses the ability to backtrack and fails to find a solution. The Hybrid Pathfinder overcomes the limitations of both by integrating localized pruning policies with a backtracking memory. First, by applying a Top-k filter based on the AGNN policy distributions, the effective branching factor is bounded by beff. ≤ B ≪ bavg. . Second, embedding the negative log-likelihood penalty − log P (Dk |Qn , Qtarget ) into the step-cost g(Q) counteracts the non-monotonicity of the DGNN heuristic. Furthermore, because P ≤ 1, the modified step cost is bounded from below by g(Qn+1 ) − g(Qn ) ≥ 1. This positivity guarantees that, even under conditions where both NNs produce inaccurate predictions, the Hybrid Pathfinder will degrade to a BFS-like exploration. Finally, the Hybrid LCA variant preserves these theoretical complexity bounds while introducing a bias that favors rank-reducing mutations when the NN-guidance is uncertain.

5.4

Pathfinders Evaluation and Efficiency Metrics

To quantify the performance and efficiency of the pathfinders, we compare them to baseline algorithms, namely the BFS search and the LCA pathfinder. Although the AGNN pathfinder performs a unidirectional search, we nevertheless decided to compare it with the bidirectional baselines to facilitate comparisons with the other models. Success Rate (SR): We define, for each pathfinder, the success rate as the fraction of test pairs for which the model successfully finds a path, i.e., Nsucc. , out of the total number of test pairs evaluated, i.e., Ntot. , obtaining SR =

Nsucc. . Ntot.

(5.13)

Efficiency Ratio (ER): This indicates how successful the guidance coming from the cost and heuristic functions is at pruning the search space compared to a given baseline search. An ER > 1 indicates an active reduction in search complexity. We define two different baselines: the BFS search, representing an uninformed baseline, and the LCA search, serving as a physics-informed baseline. We thus compute the ERs against either the BFS explored nodes NBFS or the LCA ones NLCA . The ratio is evaluated against the baseline expectation: ERbidir =

Nbaseline , Nexp

(5.14)

where Nexp is the total number of nodes explored by search algorithms as in (5.4), and Nbaseline corresponds to NBFS, bidir or NLCA depending on the comparison.27 Note that when Nbaseline = 27

For the LCA pathfinder, the comparison is obviously done over the BFS baseline.

NBFS , the ratio inherits the oscillations originating from the asymmetric bidirectional BFS expansion. These spikes at odd distances are an artifact of the unguided baseline and do not reflect the underlying behavior of the pathfinders, as confirmed by the absence of such oscillations when evaluating the efficiency against the LCA pathfinder. Guidance Efficiency (GE): This metric quantifies the precision of the neural network policy by measuring the search relative to the theoretical minimum number of steps required to discover a path of distance d. Because it is defined as a ratio of observed evaluations to the ideal minimum, it is strictly bounded by GE ≥ 1, where values approaching 1 indicate a near-perfect trajectory with minimal branching redundancy. An optimal search would never expand branches outside the shortest path. Accounting for the immediate neighbors evaluated to confirm each step along the trajectory, the theoretical minimum number of explored states is bavg. at the root level and bavg. −1 for the remaining d−1 transition steps. The bidirectional guidance efficiency is thus defined as: GEbidir =

Nexp . bavg. + (d − 1)(bavg. − 1)

(5.15)

For the AGNN pathfinder, however, the search is performed by only keeping a beam width B. Under perfect guidance, the AGNN evaluates only the single root quiver at depth 0 and at most min(B, bavg. ) candidate branches at each subsequent depth level. The unidirectional guidance efficiency is therefore defined as GEunidir =

Nexp . 1 + (d − 1) min(B, bavg. )

(5.16)

Effective Efficiency Ratio (EER): Because the evaluated models exhibit varying SR across different duality distances and quiver complexities, and since the underlying ERs are formulated differently for A∗ and beam search architectures, a unified comparison requires a composite metric. We, therefore, define the Effective Efficiency Ratio as the direct product of efficiency and accuracy: EER = ER × SR . (5.17) This formulation naturally balances the trade-off between search speed and algorithmic completeness, strictly penalizing pathfinders that fail to converge by assigning zero effective efficiency to unresolved pairs. The benchmarking evaluations described above are initially conducted on an in-distribution (ID) test set drawn from the theory families detailed in Appendix A.1. This assesses pathfinder performance when the neural networks operate within familiar topological domains encountered during training. Furthermore, because the search algorithms actively transform theory structures through sequential Seiberg dualities, it is important to evaluate their robustness 34

against unfamiliar architectures. Consequently, we extend our benchmarks to the out-ofdistribution (OOD) theories cataloged in Appendix A.2 to test the generalization capabilities of the learned policies.

Results

In this section, we evaluate the computational efficiency and routing performance of the GNN pathfinders introduced in Section 5. We begin our analysis by benchmarking the models on the in-distribution (ID) test set detailed in Appendix A.1, evaluating their search capabilities within familiar domains. In Section 6.2, we investigate their generalization properties when applied to the out-of-distribution (OOD) architectures listed in Appendix A.2. To ensure statistical consistency across all evaluations, each experimental benchmark is performed on a randomized sample of 500 quiver pairs, systematically categorized by node count and Seiberg duality distance.

6.1

In-Distribution Theories Analysis

6.1.1

DGNN Pathfinder Performance

We first examine the baseline performance of the A∗ search, guided purely by the DGNN distance heuristic, as formulated in Section 5.2. As illustrated in Figure 14, the DGNN pathfinder demonstrates exceptional reliability across the ID evaluation set, achieving an overall SR of 99.94%. The algorithm successfully resolves the vast majority of duality trajectories, exhibiting only minor performance degradation when traversing large duality distances. As illustrated in Figure 15a, the search efficiency of the DGNN pathfinder remains stable and does not degrade significantly at large distances. However, the guidance of the DGNN model in Figure 15b reduces when considering pairs of quivers separated by larger distances. This means that the pathfinder is slowly degrading to an unguided pathfinder. Despite this, the EER does not show signs of decreasing with distance; hence, it still outperforms a pure BFS algorithm. However, if we had the possibility to improve the guidance of the NN, the performance of the pathfinder could, in principle, improve. One of the reasons why the DGNN pathfinder becomes unguided at large distances is due to the non-monotonicity of the heuristic function given by the output of the DGNN. One way to compensate for this is to try to modify the cost function so that the pathfinder can prioritize certain directions when exploring the possible nodes to dualize. The AGNN pathfinder, discussed below, addresses this issue by modifying the cost function.

(a) Performance against BFS Baseline

Dataset

ID dataset

OOD dataset in Figure 37

OOD dataset in Figure 38

Pathfinder

SR (%)

EER

DGNN AGNN Hybrid LCA

99.94 76.12 100.00 100.00

430.25 928.59 790.08 763.66

429.99 706.84 790.07 763.66

DGNN AGNN Hybrid LCA

99.51 50.91 99.84 99.97

134.38 305.15 221.99 176.13

133.72 155.35 221.63 176.08

DGNN AGNN Hybrid LCA

99.34 78.64 99.92 100.00

714.42 1175.18 881.67 568.25

709.74 924.21 880.96 568.25

(b) Performance against LCA Baseline

Dataset

ID dataset

OOD dataset in Figure 37

OOD dataset in Figure 38

Pathfinder

SR (%)

EER

DGNN AGNN Hybrid Hybrid LCA

99.94 76.12 100.00 100.00

0.83 2.04 1.10 1.14

0.83 1.55 1.10 1.14

DGNN AGNN Hybrid Hybrid LCA

99.51 50.91 99.84 100.00

0.54 1.03 0.83 0.98

0.53 0.52 0.83 0.98

DGNN AGNN Hybrid Hybrid LCA

99.34 78.64 99.92 100.00

0.83 1.91 1.16 1.18

0.83 1.50 1.16 1.18

Table 2: Summary of pathfinder performances evaluated as the average across all distances and node counts in each dataset. The EER is the effective efficiency ratio obtained by multiplying the success rate by the corresponding efficiency ratio as in (5.17). Table 2a shows the comparison of the pathfinders against the BFS baseline, while Table 2b shows the comparison of the pathfinders against the LCA baseline.

1 3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.8

5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99

0.6

Nodes

7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.95 9 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99

0.4

10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 11 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.2

12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 13 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1

Distance

Figure 14: Success Heatmap

104

150

103 GE

EER

100 102

50 101 0

100 2

K=3 K=5 K=7 K=9 K=11 K=13

Distance

Distance K=4 K=6 K=8 K=10 K=12 BFS

K=3 K=5 K=7 K=9 K=11 K=13

(a) Efficiency Ratio

K=4 K=6 K=8 K=10 K=12 Opt

(b) Guidance Efficiency

Figure 15: Global performance of the DGNN pathfinder tested on theories in the testing set of DGNN.

6.1.2

AGNN Pathfinder Performance

We next evaluate the unidirectional beam search guided by the AGNN policy. The main difference of the beam search compared to the A∗ search is the limited frontier that, at each step of the path, is maintained. However, limiting the frontier, if the policy is not perfect (as anticipated in Section 4.2.1), may result in the pathfinder being unable to find a path at all. For this reason, the AGNN pathfinder will have a success rate lower than that of the DGNN A∗ pathfinder. As a benchmark, we evaluate the AGNN pathfinder across the same test set used in the previous section. As illustrated in Figure 16, the overall SR for the AGNN pathfinder converges to approximately 76.12%. Nevertheless, the AGNN pathfinder achieves high search efficiency. 1 3 1.00 1.00 0.66 0.55 0.59 0.46 0.43 0.38 0.35 0.34 0.30 0.33 4 0.90 0.90 0.88 0.84 0.77 0.71 0.63 0.60 0.51 0.47 0.38 0.39 5 0.92 0.92 0.93 0.89 0.86 0.80 0.71 0.66 0.57 0.51 0.38 0.35

0.8

6 0.96 0.95 0.96 0.94 0.89 0.81 0.70 0.64 0.53 0.48 0.34 0.35 Nodes

7 0.96 0.96 0.95 0.95 0.91 0.86 0.77 0.72 0.60 0.58 0.43 0.38

0.6

8 0.97 0.96 0.97 0.94 0.91 0.85 0.77 0.69 0.52 0.51 0.35 0.33 9 0.93 0.91 0.87 0.85 0.83 0.79 0.78 0.74 0.70 0.68 0.60 0.56

0.4

10 0.95 0.96 0.92 0.89 0.87 0.86 0.79 0.71 0.69 0.65 0.44 0.38 11 0.98 0.96 0.95 0.93 0.95 0.93 0.94 0.92 0.94 0.87

0.2

12 0.96 0.93 0.90 0.86 0.86 0.82 0.80 0.73 0.69 0.65 13 0.99 0.97 0.97 0.94 0.95 0.93 0.93 0.92 0.91 0.87 1

Distance

Figure 16: Success Heatmap The AGNN pathfinder is the only pathfinder in this work that is unidirectional. However, for consistency in the analysis with the other pathfinders, we still compare it with the bidirectional BFS baseline. Additionally, the beam search we performed set the beam width to B = 3. This is the reason why, in Figure 17a, for quivers with K = 3 and small distances, the performance of the AGNN pathfinder is worse than that of the BFS baseline since, by construction, the AGNN pathfinder explores all the nodes of the quiver at each step. However, for larger quivers, the AGNN pathfinder is an order of magnitude better than the bidirectional BFS baseline, despite its reduced SR. Furthermore, the efficiency of the pathfinder grows with the distance, even for small quivers. This sustained efficiency is directly explained by the GE in Figure 17b, which remains 38

stable across all distances. The AGNN pathfinder resolves the large-distance breakdown seen in DGNN pathfinder for large distances because the pathfinder has a more definite criterion for choosing the direction of the path. However, this gain in efficiency comes at the expense of success rate 104 3

EER

103 102

101 1

100 2

Distance K=3 K=5 K=7 K=9 K=11 K=13

Distance K=4 K=6 K=8 K=10 K=12 BFS

K=3 K=5 K=7 K=9 K=11 K=13

(a) Effective Efficiency Ratio

K=4 K=6 K=8 K=10 K=12 Opt

(b) Guidance Efficiency

Figure 17: Global performance of the AGNN pathfinder evaluated on theories in the testing set of AGNN. Combining the AGNN and DGNN policies into a single pathfinder aims to recover the success rate while maintaining high GE.28 6.1.3

Hybrid Pathfinder Performance

We now evaluate the Hybrid pathfinder, designed to integrate the complementary guidance policies of the AGNN and DGNN to address their limitations. The evaluation is once again performed on the same test set. The most immediate result is the recovery of a 100% SR across all tested duality distances, as shown in Figure 18. Figure 19a confirms that the ER outperforms the unguided BFS baseline across all distances, becoming more efficient also for small quivers. We also see that the GE in Figure 19b is closer, in order of magnitude, to the one observed for the AGNN pathfinder, starting to diverge only at large distances. The reason for this behavior is that the Hybrid pathfinder keeps the whole frontier of nodes to dualize, as does the DGNN pathfinder, but each node 28

Recall that, by definition in (5.15), the GE is better the closer it is to unity.

1 3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.8

6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Nodes

7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.6

8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 9 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.4

10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 11 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.2

12 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 13 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1

Distance

Figure 18: Success Heatmap has a cost function given by the probability distribution predicted by the AGNN model.29 For ID theories, the Hybrid pathfinder resolves the guidance inefficiency of DGNN and the success-rate drop of the AGNN pathfinder, giving the so-far most efficient NN-guided pathfinder with a 100% SR in finding a path (if the path exists). 6.1.4

LCA Pathfinder Performance

In this section, we discuss the performance of the LCA pathfinder over the BFS baseline. Despite the fact that this pathfinder does not have a NN-guided policy, we test it on the same sample of theories that we used for the previous pathfinders to be able to compare its performance. As shown in Figure 20, the LCA pathfinder reaches an SR near 100%, so we can use the ER as a ratio of efficiency when we compare it with the BFS baseline. We see from Figure 21a that the LCA pathfinder reaches levels of performance comparable to the Hybrid pathfinder in Figure 19a. However, the way in which each pathfinder is designed is different. The LCA pathfinder finds a common ancestor with lower ranks, while the Hybrid pathfinder has two NN-guided policies that try to reduce the distance between the quivers and predict the nodes to be dualized at each step. It is true that the NNs have been trained to minimize the differences between the quivers; however, the distances predicted or the probability distributions are generated on the resulting graphs processed by the Transformers. 29

In Figure 19b, we can see that the GE can become smaller than 1. This happens when, given a starting quiver QA , multiple nodes exist that can be dualized to reach QB , while (5.15) assumes that there is only one path connecting each pair.

104 2

EER

103 102

1.5

101

100 2

Distance K=3 K=5 K=7 K=9 K=11 K=13

Distance K=4 K=6 K=8 K=10 K=12 BFS

K=3 K=5 K=7 K=9 K=11 K=13

(a) Efficiency Ratio

K=4 K=6 K=8 K=10 K=12 Opt

(b) Guidance Efficiency

Figure 19: Global performance of the Hybrid pathfinder tested on theories in the testing set of DGNN and AGNN.

1 3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 5 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

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Distance

Figure 20: Success Heatmap

The fact that both pathfinders perform similarly is, in some sense, surprising because it seems that the NNs have learned some pattern to guide the path towards a common quiver. This raises the question of how much the Hybrid pathfinder improves upon the LCA baseline. 1.6

104

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Distance K=4 K=6 K=8 K=10 K=12 BFS

K=3 K=5 K=7 K=9 K=11 K=13

(a) Efficiency Ratio

K=4 K=6 K=8 K=10 K=12 Opt

(b) Guidance Efficiency

Figure 21: The global performance of the LCA pathfinder tested on the ID dataset. 6.1.5

Comparing Pathfinders with LCA Baseline

Although the BFS baseline is the only objective comparison to judge the efficiency of a pathfinder, since it represents the blind search one would perform when given two graphs, the LCA pathfinder represents the algorithm that a physicist would write for the task of finding the chain of dualities connecting two theories; this should be, from now on, the baseline against which to compare our NN-guided pathfinders. In summary, DGNN achieves a high SR with degrading GE, whereas AGNN yields a lower SR with superior EER and GE. The Hybrid pathfinder reaches a 100% SR, good performance, and stable GE. In Figure 22, we show how those same EERs compare when evaluated against the LCA pathfinder as the baseline. The first observation in Figure 22 is that the scale of the EER is now linear instead of logarithmic. Unlike the BFS comparisons, no model outperforms the LCA pathfinder by orders of magnitude; only the AGNN pathfinder in Figure 22b has a ∼ 10× speedup for small distances. The Hybrid pathfinder achieves a global average speedup of 1.09× over the LCA baseline, but its performance depends on the quiver size and the distance: 42

8 1 EER

EER

6 4

0.5 2 0 2

Distance

(a) DGNN Pathfinder

(b) AGNN Pathfinder

1.4

K=3 K=5 K=7 K=9 K=11 K=13

1.2 EER

K=4 K=6 K=8 K=10 K=12 LCA

0.8 0.6 2

Distance

Figure 22: EER for the three pathfinders using LCA baseline in the ID dataset.

• For small quivers (K < 5), the LCA pathfinder is more efficient, while Figure 23b shows that the Hybrid pathfinder outperforms LCA on larger graphs (K ≥ 5), with its average EER scaling from 1.05× up to 1.18× for K ≥ 10. • Figure 23c shows an immediate advantage for the Hybrid model across distances d ∈ [2, 8], peaking at ∼ 1.21× for d ∈ [4, 5]. The EER drops below unity only for small quivers at very high distances, d > 10. This is corroborated by the absolute node savings (∆Nexp = NLCA − NHyb ) in Figure 23a, confirming that NN-guidance becomes important as soon as the size of the quiver increases. K=3 K=5 K=7 K=9 K=11 K=13

∆Nexp

K=4 K=6 K=8 K=10 K=12

−20 2

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(a) Difference of the nodes explored 1.2

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1.1 EER

EER

1.1

0.9

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Nodes EER

Avg (1.09×)

EER

(b) Average EER vs nodes

Avg (1.09×)

Figure 23: Nodes explored and average EER per nodes and distances of the Hybrid pathfinder compared with the LCA pathfinder in the ID dataset We must determine whether the Hybrid model’s advantage over LCA stems from finding shorter paths or simply exploring fewer nodes. It could then be that the better performance (within limits) of the Hybrid pathfinder over the LCA that we are seeing in Figure 23 is only due to the pathfinder being able to find shortcuts instead of finding the path to the 44

common ancestor. For this reason, in Figure 24, we computed the percentage of times that the Hybrid pathfinder found a shorter or longer path than the LCA pathfinder.30 On average, in the ID dataset, the Hybrid pathfinder found a shorter path only ∼ 3.4% of the time and found a longer path (due to the confusion of the NN policies) ∼ 12.6% of the time. This means that ∼ 84% of the time, the Hybrid pathfinder found a path of the same length as the LCA pathfinder, and its speedup is purely due to the exploration of fewer nodes during each step, as shown in Figure 23a. Therefore, despite the different logic of guidance between the pathfinders, the Hybrid pathfinder behaves similarly to LCA 84% of the time, while it is also able to find shortcuts when possible.31 30

Percentage (%)

25 20 15 10 5 0

8 9 Nodes

Shorter

10 11 12 13

Longer

Figure 24: Total percentage of path deviations (Shorter and Longer) found in ID dataset by Hybrid pathfinder compared to the LCA pathfinder.

6.1.6

Hybrid LCA Pathfinder Performance

The purpose of the Hybrid LCA pathfinder is to find a balance between the physics-informed policy defined by the LCA and the NN-guided policies of the Hybrid pathfinder. In this way, similarly to when we discussed the Hybrid pathfinder, we can try to improve the performance of the pathfinders: when the Hybrid pathfinder is struggling, i.e., at larger distances or smaller 30

By the length of a path, we mean the number of mutations connecting the pair of theories. The presence of longer paths can be due to symmetries of the quivers that allow for multiple paths connecting the pair or the non-perfect guidance of the NN policies. Nevertheless, it is interesting to note that despite the path being longer, the number of nodes explored by the Hybrid pathfinder is generally smaller than those explored by the LCA pathfinder. 31

quivers, the LCA pathfinder can dominate the contribution to the heuristic. On the other hand, when the LCA pathfinder explores too many nodes in order to find the common ancestor, the NN can select a shorter and better path to improve overall efficiency. Figure 25 shows that the Hybrid LCA pathfinder improves upon, with the parameters discussed in the previous section, is an improvement compared to the corresponding performance of the Hybrid (DGNN–AGNN only) pathfinder, which we can see in Figures 22c, 30c and 31c. K=3 K=5 K=7 K=9 K=11 K=13

EER

1.2

K=4 K=6 K=8 K=10 K=12 LCA

Distance

Figure 25: EER for the Hybrid LCA pathfinder on the ID dataset By inspection of Figure 26, the curves now show a positive trend in average EER for both nodes and distances, surpassing the pure LCA pathfinder as soon as the pairs of quivers are large and sufficiently separated. Despite the average EER being only slightly improved compared to what is shown in Figure 23, the trends seem to have a slower decay, guaranteeing that it is better than pure LCA for longer distances than the Hybrid pathfinder was. Finally, testing reveals that there is no significant competition between the Hybrid and LCA policies in the search for a path, leading to the preference for finding a path over finding the shortest path. We see in Figure 27 that there is no significant increase in longer paths over the pure Hybrid pathfinder statistics shown in Figure 24. We believe that this is one of the signals that the optimization of the parameters has been successful and that the two policies are assisting each other in finding the path (rather than interfering with each other).

6.2

Out-of-Distribution Theories Analysis

To probe the generalization capabilities of our NNs, we test them on the out-of-distribution (OOD) quiver families detailed in Appendix A.2, comparing our neural pathfinders directly against the BFS and LCA baselines. This OOD dataset encompasses both finite- and infinitemutation type quivers, i.e., quivers with finite or infinitely many mutants, as well as quivers that violate gauge anomaly cancellation conditions. Although such anomalous configurations lack a physical realization as 4D N = 1 supersymmetric gauge theories, mutations are fundamentally well-defined combinatorial operations on directed graphs. Consequently, these 46

K=3 K=5 K=7 K=9 K=11 K=13

∆Nexp

K=4 K=6 K=8 K=10 K=12

0 4

Distance

(a) Difference of the nodes explored 1.2 1.2

EER

1.15 1.1

1.1

1.05 1

1 4

Distance

Nodes EER

Avg (1.13×)

EER

(b) Average EER per nodes

Avg (1.13×)

Figure 26: Nodes explored and average EER per nodes and distances of the Hybrid LCA pathfinder compared with the LCA pathfinder in the ID dataset

Percentage (%)

25 20 15 10 5 0

8 9 Nodes

Shorter

10 11 12 13

Longer

Figure 27: Comparison of paths found in ID dataset by Hybrid LCA pathfinder when compared to the LCA pathfinder. “unphysical” examples provide a rigorous mathematical stress-test to evaluate the topological robustness of our pathfinder algorithms beyond familiar physical constraints.32 6.2.1

Success Rates on OOD Theories

We now analyze the performance of the various NN-guided pathfinders when tested on OOD theories. The heatmaps for the SRs for the theories in Figure 37 are shown in Figure 28. We see that the DGNN pathfinder has an SR over 97% in Figure 28a, proving the ability of DGNN to guide the pathfinder regardless of whether that theory was part of the training set. This, however, does not apply to the AGNN pathfinder, which once again struggles to keep the SR up for large distances. For special sorts of OOD theories, we see in Figure 28b that the AGNN pathfinder fails completely due to the special nature of theories such as Q1, Q2, Q3, Q13, and Q15 in Figure 37 that either do not admit any non-anomalous realizations or admit only a finite number of mutations. The struggle of the AGNN model to guide the pathfinder is also reflected at the level of the Hybrid pathfinder, which failed to find the path for quivers with a large number of nodes, despite the DGNN pathfinder managing to 32

Although these anomalous theories are inconsistent when interpreted as 4D gauge theories, they are consistent as quiver quantum mechanics theory, and appear, for example, as the BPS quivers for various 4D N = 2 QFTs. In that context, mutations correspond to moving from one BPS chamber to another (see e.g., [37]).

do so alone. The success rate of 100% is, however, achieved when the Hybrid pathfinder is augmented with the LCA policy, as shown in Figure 28d. 3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

3 1.00 1.00 0.67 0.61 0.48 0.47 0.40 0.41 0.32 0.32 0.28 0.29

4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

4 0.83 0.82 0.67 0.61 0.54 0.49 0.39 0.35 0.31 0.28 0.19 0.19 0.8

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7 0.83 0.79 0.75 0.66 0.64 0.62 0.63 0.62 0.49 0.39 0.32 0.28

9 0.17 0.00 0.00 0.00 0.00 0.00

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0.67 0.00 0.00 0.00 0.00 0.00

(a) DGNN Pathfinder

(b) AGNN Pathfinder

3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99

4 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.8

5 1.00 1.00

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1.00 1.00 1.00 1.00 0.00 0.00

1.00 1.00 1.00 1.00 1.00 1.00

(d) Hybrid LCA Pathfinder

Figure 28: Success Heatmap for Theories in Figure 37 (OOD dataset). The blank white regions are a consequence of these theories being of finite mutation type, i.e., there is an upper bound on the distance between any pair of theories. The OOD theories in Figure 38 admit infinitely many mutations, so they are closer to the theories considered during the training, although they were never seen by the NNs during the training. However, we see that the pathfinders behave similarly to the analysis done for each pathfinder in Section 6.1: the DGNN pathfinder has a SR of over 90%, while the AGNN pathfinder struggles to find a path at large distances, as we see in Figures 29a and 29b. The Hybrid pathfinder has almost perfect SR due to the AGNN policy failing at large distances, but the perfect score is obtained by the Hybric LCA pathfinder, as we show in Figures 29c and 29d.

7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.98 0.92 0.77

7 0.85 0.89 0.82 0.81 0.79 0.71 0.62 0.54 0.41 0.33 0.26 0.23 0.8

0.8 8 0.99 0.99 0.99 0.98 0.96 0.91 0.83 0.77 0.66 0.59

0.6 9 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.4

Nodes

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10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99

0.6 9 0.99 0.99 0.96 0.94 0.91 0.90 0.84 0.80 0.75 0.76 0.4 10 0.99 0.98 0.95 0.91 0.78 0.73 0.65 0.57 0.49 0.47

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(a) DGNN Pathfinder

(b) AGNN Pathfinder 1

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0.6 9 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.4 10 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

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0.2 11 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0 1

Distance

(d) Hybrid LCA Pathfinder

Figure 29: Success Heatmap for Theories in Figure 38 (OOD dataset)

6.2.2

Efficiency on OOD Theories

The efficiencies of the NN-guided pathfinders on the OOD datasets, when compared to the LCA baseline, show interesting results. We can see in Figure 30 that when the number of mutations of the quivers is finite or the theories are anomalous, all pathfinders perform worse than the LCA pathfinder in general ground. However, the worse performance of the Hybrid LCA is around the LCA performance. We believe that this is a sign that if the pairs of theories are not known to admit infinitely many mutations, the use of the Hybrid LCA pathfinder could still be more efficient, since it will only be outperformed by the LCA when the theories are anomalous. Moreover, beating the LCA pathfinder on the datasets we generated, as mentioned already in the previous sections, is the most difficult test for our pathfinders because the LCA pathfinder follows the same criterion (but opposite) that we used to generate the dataset in the first place, as explained in Section 3.1. Therefore, the datasets, in particular when the number of allowed mutations is finite, are biased towards pairs that share a “lowest common ancestor”, by construction. If, instead, we consider theories that admit an infinite number of mutations, such as those in Figure 38, the NN-guided pathfinders behave similarly to how they performed on the ID datasets. We see in Figure 31 that, in particular, the Hybrid LCA pathfinder has an LCA baseline that is usually below the efficiency of the NN-guided pathfinders. The overall performance is also consistent with the results found in the ID dataset, with a peak efficiency around 1.6× faster than the LCA pathfinder, showing symptoms of degradation for large distances. Overall, however, we can conclude that the NNs guidance is not specific to the theories that were in the dataset; instead, it is based solely on the number of nodes and distances that were observed during the training of the NNs. For this reason, in Section 7, we study the limitations of the pathfinders, i.e., where they start to exhibit failure modes, to understand how long we can achieve better performance than the LCA pathfinder given a certain training of the NNs.

Failure Modes

So far, we have tested the performance of the different pathfinders against each other, analyzing the pros and cons of each strategy and trying to maximize efficiency while maintaining the highest possible success rate. In general, we see that all pathfinders, even those that are NN-guided, do not distinguish much between ID and OOD theories, maintaining similar efficiencies when compared to a BFS or LCA baseline. From the discussion in Section 6, the Hybrid LCA pathfinder appears to be the most consistent, outperforming pure LCA every time when the theories admit an infinite number of mutations. However, even for these theories, the EER starts to decrease when small quivers are very far apart in terms of the number of mutations. The purpose of this section is to understand what the common failure 51

4 1

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(d) Hybrid LCA Pathfinder K=3 K=8 K=13

K=4 K=9 K=14

K=5 K=10 K=15

K=6 K=11 K=16

Figure 30: EER for the pathfinders using LCA baseline in the OOD dataset of Figure 37.

EER

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(a) DGNN Pathfinder

(b) AGNN Pathfinder

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K=7

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K=9

K=10

K=11

Figure 31: EER for the pathfinders using LCA baseline in the OOD dataset of Figure 38.

modes and limitations (i.e., “the breaking point”) of the pathfinders we propose are, given the training of the NNs over theories with at most Ktrain nodes and at a distance Dtrain from each other. In particular, we want to find the turning point when the efficiency ratio decreases and the maximum complexity after which the pathfinders degrade worse than either BFS or LCA baselines. For this analysis, we considered only theories with an infinite number of mutations, as found in the ID dataset and Figure 38 of the OOD dataset. We introduce two new parameters: Nexp Nexp 1 C = dtrue log10 K , W = EW = . (7.1) dtrue + 1 dtrue + 1 SR The first is the logarithm of the complexity K dtrue for a pair of quivers with K nodes at a distance dtrue from each other, as saved from the database using BFS. The distance dtrue is not the distance found by the pathfinders; rather, it is the reference distance we save when generating the database, as explained in Section 3.1. The parameter W (resp. EW) is the “wandering” (resp. “effective wandering”), which measures how many nodes the model actually explored compared to the minimum required if it made zero mistakes. Throughout this work, and with the checkpoints available on the GitHub repository, the NNs have been trained up to distance 12 for theories in the ID dataset with up to 10 nodes and up to distance 10 for theories in the ID dataset with up to 13 nodes. The complexities associated with these extremal values of training are C1 = 12 log10 10 = 12 ,

C2 = 10 log10 13 = 11.1394 .

(7.2)

Therefore, despite being very close, we consider C1 = Ctrain as the reference complexity associated to the pathfinders for the moment. We can then plot in Figure 32 how the SR, EER (with respect to LCA baseline), and EW differ across complexity. Interestingly, we see in Figure 32a that the model, which started degrading even at small complexities, is the AGNN pathfinder. On the other hand, Figure 32b presents it as the best model. This is consistent with the discussion that the efficiency of the AGNN is unmatched, but the lack of backtracking makes it unreliable for large complexities. However, the most interesting plot is Figure 32c, where we see that the pathfinder that “wandered” more is actually the DGNN, while all other models remained consistent, as shown in Figure 32d.33 In fact, the EER and EW of all models except for DGNN are better than the LCA baseline. The models show no signs of degradation even at the largest complexities used for training, i.e., Ctrain . In order to find the breaking point at which the NNs’ guidance causes the pathfinder to degrade and become worse than an unguided LCA, we decided to artificially generate a 33

By definition, W and EW are “better” the closer they are to 1.

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100 0

(a) Success Rate

(b) Effective Efficiency Ratio

101

102

101

100.5

DGNN

(d) Effective Wandering without DGNN Hybrid

LCA

Hybrid LCA

Ctrain

Figure 32: Success Rate, Effective Efficiency Ratio (with respect to LCA baseline) and Effective Wandering for theories when considering NNs trained up to distance 12 and 13 nodes.

situation where we could push the pathfinders to large complexities while training the NN on smaller ones. We therefore re-trained the NNs to a maximum complexity Ctrain = 6 log10 9 ≃ 5.73 and tested the pathfinders up to a complexity C = 13.37, obtained by considering theories in the ID dataset with K = 13 at distance 12.34 The results are shown in Figure 33. 1

EER

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(b) Effective Efficiency Ratio

105 102

103

104

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101 100

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(d) Effective Wandering without DGNN Hybrid

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Figure 33: Success Rate, Effective Efficiency Ratio (with respect to LCA baseline) and Effective Wandering for theories when considering the NNs trained up to distance 6 on theories with up to K = 9. This time, we clearly see that the pathfinders start to degrade when the Ctrain threshold is passed. In fact, by focusing on the EER and EW for each model, we can predict the complexity value after which the LCA pathfinder will perform better than the NN-guided 34

We considered theories in the ID dataset with up to K = 14 and up to distance 10, i.e., with complexity C = 11.46.

pathfinders: Cpeak (SRpeak )

Cpred (SRpred )

Cpred /Ctrain

EER

DGNN

4.00 (1.00)

5.90 (1.00)

0.70

1.03

AGNN

6.48 (0.78)

10.79 (0.33)

48.02 (0.00)

18.61 (0.10)

8.39

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6.00 (1.00)

8.00 (1.00)

8.27 (1.00)

8.92 (1.00)

1.44

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6.00 (1.00)

8.00 (1.00)

11.25 (1.00)

10.93 (1.00)

1.97

1.91

(7.3)

We see that the DGNN pathfinder is the one degrading even before reaching Ctrain , while the AGNN pathfinder appears to be the most efficient for larger complexities, but the predicted SR is rapidly dropping to zero. The most interesting pathfinders are the Hybrid and Hybrid LCA pathfinders, which are able to perform better than LCA for complexities 1.46 to 1.97× larger than the training complexity, respectively. Also, note that the breaking point of the Hybrid and Hybrid LCA occurs before their success rates drop below 100%, meaning that they will be reliable for the entire duration they perform better than LCA. We believe this to be a good estimate of how long the NN-guided pathfinders can outperform the LCA pathfinder.

Conclusions

Dualities between quantum field theories provide important access to deep non-perturbative phenomena. It is therefore natural to ask: given two QFTs, could it be that they are actually dual? If so, what is the sequence of simple duality operations that connects the two theories? In practice, this is a challenging question and is amenable to machine learning techniques. Given the encouraging success of Transformers in tackling complex problems in string theory [38–41] and conformal field theories [42], it is natural to ask whether this architecture can be similarly incorporated into the duality pathfinder we aim to develop. In this paper, we have constructed a dataset of dual QFTs with a seed theory given by a D3-brane probing a toric singularity. We used this dataset to train different graph neural networks. Combining this with pathfinder algorithms, we have studied the computational complexity of connecting Seiberg dual theories and established that a hybrid approach tends to be both more efficient and more accurate. In the remainder of this section, we discuss some potential avenues for future investigation. In this work, we have investigated the computational complexity of dualities between quiver gauge theories. From a holographic standpoint, such sequences of dualities prominently appear in warped throats of a string compactification, where motion down the throat triggers a specific sequence of duality moves. It would be interesting to investigate whether this notion 57

of computational complexity matches more precisely with other bulk proposals of complexity (see, e.g., [43–45]).35 An ambitious aim for the future would be to classify all physically distinct quiver gauge theories. To accomplish this, one would also need to have a sharp understanding of how many quiver gauge theories are in fact dual (i.e., to identify the equivalence classes). We anticipate using this work to tackle this classification problem. We have primarily focused on 4D quiver gauge theories and their Seiberg duals, but we anticipate that the same techniques apply more broadly. It would be interesting to study the computational complexity of dualities in other well-motivated settings, including dualities of string, M-, or F-theory compactifications, as well as quantum field theories in other spacetime dimensions. At a practical level, it would be interesting to increase both the size of the training dataset and the complexity of the network architectures used to establish Seiberg dual pairs. In particular, studying possible scaling laws in the degree of inference as a function of dataset and network size would be instructive. Given recent advancements in large language models (LLMs), it is natural to ask whether the custom GNN architecture used here is really necessary. Said differently, can one simply provide two dualized quivers and ask an LLM to find the most efficient and accurate path between these theories? This provides a natural class of benchmark problems for studying the capabilities of state of the art inference models, but at the same time, it is difficult to provide a completely “fair comparison” since it can be challenging to properly compare the computational resources of an LLM with the specific (quite modest) computational budget used in the present work.36 Acknowledgements We thank F. Carta, M. Danese, S. De, C. Lawrie, H. Lee, and D.S. Park for helpful comments and discussions. The work of JJH and SNM is supported by DOE (HEP) Award DE-SC0013528, BSF grant 2022100, and a University Research Foundation grant at the University of Pennsylvania. AM and GS are supported in part by DOE (HEP) Awards DESC0017647 and DE-SC0023719. This work was performed in part while JJH, AM, and GS were at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. JJH and GS especially thank the organizers of the Aspen workshop “Theoretical Physics for Artificial Intelligence” for providing a stimulating atmosphere that directly led to the start of this project. SNM and AM thank the Simons Center for Geometry 35

It would also be interesting to study various generalizations of complexity distance measures of the sort introduced in [46]. 36 Can one truly put a price tag on finding all Seiberg dual quiver gauge theories? We think not.

and Physics for the hospitality during the “23rd Simons Physics Summer Workshop: Theory, Experiment and the Emerging New Physics”. The code produced for this work was written with the help of Gemini Pro 3.1.

Seed Theories for Dataset Generation

In this appendix, we describe in more detail the seed theories used for dataset generation. We focus primarily on D3-branes probing toric CY singularities. See [20] and references therein for additional discussion of the construction and study of these theories.

A.1

In-Distribution Theories

We trained our NNs on five families of 4d N = 1 SCFTs that arise from D3-branes probing toric CY threefolds: (1,1,−2)

1. The 4d N = 1 SCFTs on D3-branes probing C3 /Zn , with n being odd and orbifold 3 action (1, 1, −2) on the C coordinates. These are orbifolds of N = 4 SU(N ) SYM, usually characterized by three chiral fields X,Y ,Z in the adjoint of SU(N ), with a superpotential term W = Tr (X[Y, Z]) , (A.1) where the trace is over color indices. The action of the orbifold leads to a theory with n gauge nodes SU(Ni ), with chiral fields in the bifundamental representation of SU(Ni ) × SU(Nj ). Using the same notation of the parent theory, they can be grouped in Xi|i−2 mod n , Yi|i+1 mod n , Zi|i+1 mod n , (A.2) while the superpotential becomes n X WC3 /Zn = Tr Xi|i−2 Yi−2|i−1 Zi−1|i − Xi|i−2 Zi−2|i−1 Yi−1|i .

(A.3)

i=1

Figure 34 shows representative C3 /Zn quivers. For NN training, we used odd orbifolds with 3 ≤ n ≤ 13. 2. The 4d N = 1 SCFTs on D3-branes probing C(dP1 ), where by C(.) we mean the complex cone over the dP1 surface. This theory is characterized by four gauge groups and bifundamental fields X14 , Y14 , X21 , X24 , X31 , X32 , Y32 , X43 , Y43 , Z43 ,

(A.4)

subjected to the superpotential WC(dP1 ) = X14 Y43 Y32 X21 − Y14 Y43 X32 X21 + X24 Z43 X32 − X24 X43 Y32 + X31 Y14 X43 − X31 X14 Z43 . 59

(A.5)

2 2

(a) C3 /Z3

(b) C3 /Z5

Figure 34: Examples of C3 /Zn quivers theories. The corresponding superpotential can be derived from (A.3). The quiver for this theory is shown in Figure 35a. 3. The 4d N = 1 SCFTs on D3-branes probing C(dP2 ). This theory is characterized by five gauge groups and bifundamental fields X15 , Y15 , X21 , X25 , X31 , X32 , X42 , X43 , X53 , X54 , Y54 ,

(A.6)

subjected to the superpotential WC(dP2 ) = X15 X53 X31 − X25 X53 X32 + X25 X54 X42 − X31 Y15 X54 X43 − X15 Y54 X42 X21 + X21 Y15 Y54 X43 X32 .

(A.7)

The quiver for this theory is shown in Figure 35b. 4. The 4d N = 1 SCFTs on D3-branes probing C(dP3 ). This theory is characterized by six gauge groups and bifundamental fields X15 , X16 , X21 , X26 , X31 , X32 , X42 , X43 , X53 , X54 , X64 , X65 ,

(A.8)

subjected to the superpotential WC(dP3 ) = X15 X53 X31 − X15 X54 X42 X21 + X26 X64 X42 − X16 X64 X43 X31 − X26 X65 X53 X32 + X16 X65 X54 X43 X32 X21 .

(A.9)

The quiver for this theory is shown in Figure 35c. 5. The 4d N = 1 SCFTs on D3-branes probing the C(Y p,q ) singularities. This family of theories is characterized by 2p gauge groups and bifundamental fields that can be categorized into SU(2) doublets U α , V α , and singlets Z, Y . With gauge node indices understood modulo 2p, they can be grouped as α U2k−1|2k

α (k = 1 . . . p) , V2k|2k+1

(k = 1 . . . q) , 60

Z2k|2k+1

(k = q+1 . . . p) ,

(A.10)

2 2

(a) C(dP1 )

(b) C(dP2 )

Figure 35: Quivers of C(dPn ) theories. The corresponding superpotentials are in Eqs. (A.5), (A.7) and (A.9). Y2k+1|2k−1 , Y2k+2|2k

(k = 1 . . . q) ,

Y2k+2|2k−1

(k = q + 1 . . . p) ,

(A.11)

where α = 1, 2 is the doublet index. These fields enter to a superpotential as WC(Y p,q ) =

q X

β β α α ϵαβ U2k−1|2k V2k|2k+1 Y2k+1|2k−1 − V2k|2k+1 U2k+1|2k+2 Y2k+2|2k

k=1 p X

(A.12) k−q−1

(−1)

β α ϵαβ U2k−1|2k Z2k|2k+1 U2k+1|2k+2 Y2k+2|2k−1 .

k=q+1

Some examples of C(Y p,q ) quivers we considered in our database are shown in Figure 36. There are some known identities for the C(Y p,q ) theories, such as C(Y 2,1 ) ≃ C(dP1 ), or C(Y p,0 ) ≃ C/Zp , where C is the conifold theory. Moreover, C(Y p,p ) ≃ C3 /Z2p . Since we want to avoid 2-loops in the quiver, we exclude the latter theories, and C(Y 1,0 ) ≃ C. For the training of the NNs we considered 2 ≤ p ≤ 6 and 1 ≤ q ≤ 6 with q ̸= p, so that the total number of nodes in the quivers was up to 12. 2

(a) C(Y 3,1 )

(b) C(Y 3,2 )

Figure 36: Examples of C(Y p,q ) quivers theories. The corresponding superpotential can be derived from (A.12).

A.2

Out-of-Distribution Theories

To evaluate the neural network’s generalization to unseen theories, we tested it on the family of quivers Q1–Q15 from [13] (Figure 37), using non-anomalous rank assignments when possible. We also evaluated quivers from D3-branes probing non-toric CY threefolds [47, 48], shown in Figure 38.

··· n−2

··· n−4

(a) Q1

(b) Q2

(d) Q4 or C(F0 ) ≃ C(Y 2,0 )

(e) Q5

(f) Q6

3 3

(g) Q7

(h) Q8

(i) Q9 or C3 /Z3 .

(j) Q10

(k) Q11

(l) Q12

2 2

(m) Q13

(n) Q14

(o) Q15

Figure 37: Quiver families considered in [13]. We comment that some of these quivers are intrinsically anomalous when inerpreted as 4D N = 1 QFTs (i.e., inconsistent), but are perfectly consistent when viewed as the BPS quivers of 4D N = 2 QFTs. As such, they provide an interesting testing ground for studying mutations of quivers.

2 5 2

2 2

(a) C(dP4 )

(b) C(dP5 )

2 4

(d) C(dP7 ) 3

3 3

(e) C3 /∆27

Figure 38: Further quiver theories used to test the NN. Nodes with a rank other than one are specified whenever it is necessary to make the theory non-anomalous.

Overview of Graph Neural Networks

Graph Neural Networks (GNNs) process data natively structured as graphs by assigning features to each node and iteratively updating these representations over successive layers. This refinement is regulated by a localized neighborhood aggregation mechanism known as message passing [33]. In the following, we provide a brief review of this technique.

B.1

Message Passing Paradigm

In a standard Message Passing Neural Network (MPNN) [33], the network updates the hidden representation hi of node i at layer ℓ by aggregating messages from its neighbors j ∈ N (i). The unactivated update rule commonly takes the form: X (ℓ+1) (ℓ) (ℓ) (ℓ) (ℓ) mi = Wself hi + Wmsg hj , (B.1) j∈N (i)

where Wself and Wmsg are learnable weight matrices. After aggregating the messages, the network normalizes the features. A frequent choice is Layer Normalization [49], which standardizes a vector mi ∈ RH using its mean µi and variance σi2 , and shifts it with learnable parameters γ, β: mi − µi LayerNorm(mi ) = γ ⊙ p 2 +β, (B.2) σi + ϵ where ⊙ is the Hadamard product, i.e., element-wise multiplication. The normalized vector is then passed through an activation function. Since Layer Normalization can shift feature values to be negative, the Leaky Rectified Linear Unit (LeakyReLU) [50] is often used to prevent “dead” neurons: ( x if x ≥ 0 , LeakyReLU(x) = (B.3) αx if x < 0 . The parameter α is a small positive constant. Stacking several message passing layers allows a node to gather information from nodes multiple hops away in the graph.

B.2

Graph Transformers and Global Context

While standard MPNNs effectively capture local graph topology, their ability to gather longrange information is limited. In particular, the phenomenon of oversmoothing [51], where node representations become indistinguishable after too many layers, remains one of the primary architectural bottlenecks of deep MPNNs. To solve this, architectures like GraphGPS [34] and Graph Transformers combine local message passing with global attention mechanisms [21]. 64

In a Graph Transformer layer, the sequence of node embeddings X ∈ RK×H is projected into Queries (Q), Keys (K), and Values (V ) using learnable linear transformations: Q = XW Q ,

K = XW K ,

V = XW V .

(B.4)

The attention weights determine how much focus each node places on every other node in the graph: QK T Attention(Q, K, V ) = softmax √ V. (B.5) dk To evaluate different types of relationships simultaneously, the network splits the embeddings into multiple independent heads using Multi-Head Attention (MHA). The outputs are concatenated and projected back to the original dimension: MHA(X) = Concat(head1 , . . . , headh )W O .

(B.6)

A single Transformer encoder layer typically wraps this operation with position-wise FeedForward Networks (FFN) and residual connections: X̃ = LayerNorm(X + MHA(X)) , Xout = LayerNorm(X̃ + FFN(X̃)) .

(B.7)

Overview of Search Algorithms

In this Appendix, we review some generalities about A∗ Search and Beam Search pathfinders.

C.1

A∗ Search Algorithms

A∗ is an informed search algorithm optimized for weighted graphs: starting from a specific node, the algorithm seeks a path to a given final node by minimizing a specified cost criterion. The algorithm identifies the trajectory by maintaining a tree of paths originating from the starting point, extending them until the target node is reached. In contrast to uninformed search strategies like Dijkstra’s algorithm, A∗ determines which path to extend based on an evaluation function f (n) that it minimizes. This function is defined as f (n) = g(n) + h(n) ,

(C.1)

where n designates the next node on the path, g(n) denotes the accumulated cost of the path from the starting node to n, and h(n) represents the heuristic function estimating the cost of the cheapest remaining path from n to the goal. Typically, the A∗ implementation relies on a priority queue known as the frontier. At each iteration, the node with the lowest f (·) value is selected (dequeued) from the frontier, compared against the target, and, if it differs, 65

all its neighbors are added to the queue. Crucially, these new states supplement rather than overwrite the existing frontier. Preserving previously explored branches is essential to allow the algorithm to backtrack if a promising path reaches a dead-end. This cycle of selecting the lowest-cost nodes and expanding the frontier repeats continuously until the dequeued node coincides with the target. We now see how bidirectional A∗ search (illustrated schematically in Figure 13a and detailed in Algorithm 2) is implemented to explore the space of quiver mutations in Section 5.1. The search expands two frontiers simultaneously: a forward frontier (Qfwd ) originating from QA and a backward frontier (Qbwd ) originating from QB . For any intermediate state Qn reached from QA or Qm reached from QB , the forward and backward evaluation functions are defined as: ffwd (Qn ) = gfwd (Qn ) + hfwd (Qn ) ,

fbwd (Qm ) = gbwd (Qm ) + hbwd (Qm ) ,

(C.2)

where g(·) and h(·) depend on the specific pathfinder considered in Section 5.2. At each iteration, the algorithm prioritizes the expansion of whichever frontier currently holds the minimum estimated path cost min(f ). When a quiver is selected for expansion, its valid Seiberg mutants are generated and compared against the states already visited by the opposing frontier.

C.2

Beam Search Algorithms

While A∗ employs a selection strategy, prioritizing the state with the lowest f -score at each iteration, it preserves completeness by maintaining the entire frontier of unexplored branches. This memory allows the algorithm to backtrack from dead-ends, guaranteeing that a valid path will be found in a finite number of steps if one exists. In contrast, we now introduce a Beam Search pathfinder, which is a purely greedy pathfinder that retains only a subset of candidates at each step, discarding all alternative branches. While this approach reduces memory overhead and execution time, the absence of a backtracking mechanism makes it vulnerable to local minima: a single misleading evaluation by the heuristic can permanently trap the search in a dead-end. Formally, for a K-node quiver, we fix a “beam width” B ≤ K and define the “beam” Bt as the set of candidate quivers retained in memory at depth t, with |Bt | ≤ B. At each exploration step, the algorithm generates the pool of all valid Seiberg mutants originating from the current beam: [ Ct+1 = {Dk Q | Dk Q ̸= ∅} . (C.3) Q∈Bt

Algorithm 2 Bidirectional A∗ Search for Seiberg Dualities 1: Input: Initial quiver QA , target quiver QB 2: Output: Shortest path between QA and QB , or Failure 3: Initialize Qfwd and Qbwd as empty priority queues 4: Vfwd ← {QA : 0}, Vbwd ← {QB : 0} ▷ Track states and accumulated costs 5: Push (gfwd (QA ) + hfwd (QA ), QA ) to Qfwd 6: Push (gbwd (QB ) + hbwd (QB ), QB ) to Qbwd 7: while Qfwd is not empty and Qbwd is not empty do 8: if min(ffwd ) ≤ min(fbwd ) then 9: Pop Qn from Qfwd 10: for k ← 1 to K do 11: Qn′ ← Dk Qn 12: if Qn′ ̸= ∅ then 13: if ∃ Qm ∈ Vbwd s.t. Qn′ ∼ = Qm then 14: return ReconstructPath(Vfwd , Vbwd , Qn′ , Qm ) ▷ Intersection found up to isomorphism 15: end if 16: gnew ← Vfwd [Qn ] + step cost 17: if Qn′ ∈ / Vfwd or gnew < Vfwd [Qn′ ] then 18: Vfwd [Qn′ ] ← gnew 19: Push (gnew + hfwd (Qn′ ), Qn′ ) to Qfwd 20: end if 21: end if 22: end for 23: else 24: // Perform expansion for the backward frontier (Qbwd ) 25: end if 26: end while 27: return Failure

Rather than maintaining an exhaustive frontier, the new beam Bt+1 is formed by selecting the B quivers from Ct+1 that minimize a guiding cost function g(Q): Bt+1 = {Q ∈ Ct+1 | |{Q′ ∈ Ct+1 \ {Q} | g(Q′ ) < g(Q)}| < B} ,

(C.4)

where ties at the boundary |Bt+1 | = B are broken deterministically by graph lexicographical ordering. Morally, the Beam Search is therefore an A∗ Search with f (n) = g(n) and h(n) = 0, and a limited frontier. In Section 5.2, g(·) is estimated using the output of the AGNN model. During expansion, every generated candidate Q′ is checked against the target QB . A schematic representation of the beam search is shown in Figure 13b, with the corresponding pseudocode provided in Algorithm 3. Algorithm 3 Beam Search Algorithm 1: Input: Initial quiver QA , target quiver QB , beam width B, max depth dmax 2: Output: Path to target state QB , or Failure 3: B0 ← {QA } 4: V ← {QA } 5: for t ← 0 to dmax − 1 do 6: Ct+1 ← ∅ ▷ Candidate set 7: for Q ∈ Bt do 8: for k ← 1 to K do 9: Q′ ← Dk Q 10: if Q′ ∼ = QB then 11: return Path(QA → · · · → Q′ ) ▷ Success up to isomorphism 12: end if 13: if Q′ ̸= ∅ and Q′ ∈ / V then 14: Compute cost g(Q′ ) ← g(Q) + cost function(Q′ ) 15: Ct+1 ← Ct+1 ∪ {(Q′ , g(Q′ ))} 16: end if 17: end for 18: end for 19: if Ct+1 = ∅ then 20: return Failure ▷ dead-end 21: end if 22: Bt+1 ← arg min-B g(s) ▷ Retain top B candidates s∈Ct+1

23: V ← V ∪ Bt+1 24: end for 25: return Failure

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