CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks

A. Chervov; F. Levkovich-Maslyuk; A. Smolensky; F. Khafizov; I. Kiselev; D. Melnikov; I. Koltsov; S. Kudashev

hep-thMar 23, 2026

CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks

Authors: A. Chervov, F. Levkovich-Maslyuk, A. Smolensky, F. Khafizov, I. Kiselev, D. Melnikov, I. Koltsov, S. Kudashev, D. Shiltsov, M. Obozov, S. Krymskii, V. Kirova, E. V. Konstantinova, A. Soibelman, S. Galkin, L. Grunwald, A. Kotov, A. Alexandrov, S. Lytkin, D. Fedoriaka, A. Chevychelov, Z. Kogan, A. Natyrova, L. Cheldieva, O. Nikitina, S. Fironov, A. Vakhrushev, A. Lukyanenko, V. Ilin, D. Gorodkov, N. Bogachev, I. Gaiur, M. Zaitsev, F. Petrov, L. Petrov, T. Gaintseva, A. Gavrilova, M. N. Smirnov, N. Kalinin, A. Khan, K. Jung, H. Mousset, H. Isambert, O. Debeaupuis

Abstract

This is the fourth paper in the CayleyPy project, which applies AI methods to the exploration of large graphs. In this work, we suggest the existence of a new discrete version of holographic string dualities for this setup, and discuss their relevance to AI systems and mathematics. Many modern AI tasks -- such as those addressed by GPT-style language models or RL systems -- can be viewed as direct analogues of predicting particle trajectories on graphs. We investigate this problem for a large family of Cayley graphs, for which we show that surprisingly it admits a dual description in terms of discrete strings. We hypothesize that such dualities may extend to a range of AI systems where they can lead to more efficient computational approaches. In particular, string holographic images of states are proposed as natural candidates for data embeddings, motivated by the "complexity = volume" principle in AdS/CFT. For Cayley graphs of the symmetric group S_n, our results indicate that the corresponding dual objects are flat, planar polygons. The diameter of the graph is equal to the number of integer points inside the polygon scaled by n. Vertices of the graph can be mapped holographically to paths inside the polygon, and the usual graph distances correspond to the area under the paths, thus directly realising the "complexity = volume" paradigm. We also find evidence for continuous CFTs and dual strings in the large n limit. We confirm this picture and other aspects of the duality in a large initial set of examples. We also present new datasets (obtained by a combination of ML and conventional tools) which should be instrumental in establishing the duality for more general cases.

Figures & tables

Figure 1. Left to right: the graph $Gr_{2,5}$ with marked vertices (red and green), and a polygon illustrating the corresponding paths (red and green).

Figure 2. Left to right: the graph $Gr_{2,6}$ with marked vertices (red and green), and a polygon illustrating the corresponding paths (red and green). Interactive widget (F.Khafizov) available at link . The distance between nodes on graph is equal to area between the paths - ”complexity = area” principle. Which in that case is equality of the Mann–Whitney statistic and the area under the ROC curve.

Figure 3. The Young tableau (upper left corner) provides a solution to the string equations of motion. It encodes the motion of the string as depicted on the right. In the right panel, each position of the string at times $t=1,\dots,6$ is shown in a different color, and the number in each box indicates the step at which the string passes through it. Each obtained string is a non-decreasing path, thus it belongs to the image of the holography map. In the holographically dual picture, the tableau encodes one of the shortest paths on the graph – indicated by wavy edges.

Figure 4. Lehmer-code representation of the permutations $(3,2,1,0)$ (cyan) and $(1,0,2,3)$ (magenta). Their Lehmer codes are $L(3,2,1,0)=(3,2,1,0)$ and $L(1,0,2,3)=(1,0,0,0)$ , respectively. On the left, the corresponding vertices of the permutohedron are marked in matching colors. On the right, each permutation is represented by a lattice path given by the upper boundary of its Lehmer diagram inside the staircase Young diagram. The complexities of these permutations are 1 and 6, respectively, which coincide with the areas under the corresponding paths.

Figure 5. Example of the adjacent transposition $(0,1)$ . Top: $\sigma=(0,3,1,2)$ , $\sigma^{\prime}=(3,0,1,2)$ ; one inversion is added. Bottom: $\sigma=(2,0,1,3)$ , $\sigma^{\prime}=(0,2,1,3)$ ; one inversion is removed. Only the components $k_{0}$ and $k_{1}$ of the Lehmer code change. All other components remain unchanged.

Figure 6. Roots of the $H$ -polynomials in the complex plane. In the left panel, all non-trivial roots have modulus equal to one, so the “Riemann conjecture” holds, whereas in the right panel it is violated. Both panels correspond to $k$ -consecutive cycle generators with $k=6$ : the left uses an inverse-closed generating set, while the right uses generators without inverses. These correspond to Schreier coset graphs for the subgroup $S_{\lfloor n/2\rfloor}\times S_{n-\lfloor n/2\rfloor}$ . Whether the property that all roots have modulus one persists for larger $k$ remains unclear.

Figure 7. Farey graph on the upper halfplane. From [ CamiloMelnikovNovaesPrudenziati2019 ] . The dashed curves are “holography images” of some elements of

SL(2,\mathbb{Z})

areas under them correspond to complexities of these elements. “Complexity = area” holds approximately.

Figure 8. Multiset $\{0^{4}\,1^{3}\,2^{5}\,3^{2}\}$ , i.e. $(n_{0},n_{1},n_{2},n_{3})=(4,3,5,2)$ ; $N=4+3+5+2=14$ . The rectangles from left to right are $R_{0}$ (size $2\times 12$ ), $R_{1}$ ( $5\times 7$ ), and $R_{2}$ ( $3\times 4$ ). $E=[0,0,0,0,1,1,1,2,2,2,2,2,3,3]$ , $P=[2,1,3,2,2,0,0,2,0,1,2,1,3,0]$ , and $d(P,E)=A_{0}+A_{1}+A_{2}=11+25+6=42$ .

Panel of Figure 9 — Figure 9. Encoding the words $012$ and $210$ by the lattice paths. The distance between the words coincides with the oriented area $S$ between the associated paths.

Panel of Figure 10 — Figure 10. Encoding the words by the lattice paths.

Panel of Figure 11 — Figure 11. Encoding of the words $00112$ (blue) and $21100$ (red) by lattice paths.

Panel of Figure 12 — Figure 12. Encoding of the words $001122$ (blue) and $221100$ (red) by lattice paths.

Figure 13. Box-Ball System Illustration. Example for $K=4$ balls and $N=20$ boxes.

Figure 14. Limit shape for ROC curves (rectangular Young diagrams, or equivalently Dyck paths), defined as the average shape of the curve when all configurations are weighted by $q^{\mathrm{area}}$ . (Here $q$ is a parameter; in the plot it is chosen to be $0.95$ .) The solid red line represents the theoretical limit curve obtained by A. M. Vershik [ vershik1996statistical ] . An interactive simulation widget developed by L. Petrov is available at link .

Figure 15. Limit shape for the ROC curves (Dyck paths) with fixed area under curve. That is - average of all ROC curves with fixed area under them.

Figure 16. Monte Carlo simulation results for starting sequence $0^{k}1^{p-k})$ : left, the averaged occupancy heatmap $u(x,s)$ with fitted green boundary $s(x)\approx 0.259+2.846x-2.574x^{2}$ ; right, multi-time comparison of exact Burgers profiles and MC data with per-slice $L^{2}$ errors. An interactive simulation is available at link .

Figure 17. The Schreier graph $S_{8}/(S_{3}\times S_{5})$ with Coxeter generators, similarity to lattice can be clearly seen. Widget is available link . Other graph examples available by the link.

Figure 18. Fitted numerical data shows quite good coincidence of $c_{0}$ with Mahler measure which is around 1.67338. Leading coefficient $c_{0}$ does not depend much on the degree of the polynomial approximation. Notebook

n	d=2	d=3	d=4	d=5	d=6	d=7
2
3	1	2
4	2	3	4
5	2	4	5	6
6	3	5	7	8	9
7	3	6	8	10	11	12

Table 1. Experimental data (

d

different, for

2

-cycles)

Figure 29. The Schreier graph as a quarter Aztec diamond for $n=5$ . Note that $(x,y)$ coordinates are indices of 1s. E.g., for $A=(00011)$ $(x_{A},y_{A})=(4,5)$ or $B=(00110)$ $(x_{B},y_{B})=(3,4)$ , etc.

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	True	False

Table 27

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	False	False

Table 29

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	True	False

Table 31

Figure 38. Consecutive 6-cycles [0..01..1]

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	False	False

Table 36

Figure 39. Consecutive 3-cycles +inv [0..01..1]

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	False	False

Table 38

Figure 40. Consecutive 4-cycles +inv [0..01..1]

Figure 41. Consecutive 5-cycles +inv [0..01..1]

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	False	False

Table 41

Nonnegative	Symmetric	Unimodal	Weakly log concave
True	False	False	False

Table 42

Figure 42. Consecutive 6-cycles +inv [0..01..1]

$n$	k=2			k=3			k=4			k=5
	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$
4	4	2	1	2	1	1	2	1	0
5	6	3	0	3	2	-1	3	1	-1	2	1	0
6	9	3	1	5	1	1	4	0	2	3	1	0
7	12	4	0	6	2	0	4	2	-1	4	1	-1
8	16	4	1	8	2	1	6	1	1	5	0	2

Table 2. Diameters

D_{k}(n)

and increments, consecutive

k

-cycle, inverse closed

Figure 43. Consecutive k=3 +inv (coincide 3)

Figure 44. Consecutive k=3 +inv (coincide 4)

Figure 45. Consecutive k=4 +inv (coincide 3)

Figure 46. Consecutive k=4 +inv (coincide 4)

k	n	no coincide D	coincide 2 D	coincide 3 D	coincide 4 D
3	4	3	3	2	–
3	5	5	5	4	2
3	6	7	7	6	5
3	7	10	10	9	8
3	8	14	14	13	11
3	9	18	18	17	15

Table 3. God’s numbers for consecutive cycles with inverses coset: central state

(0,1,2,...n-D-1,n-D,n-D,...,n-D)

(

D

coincide at the end)

k	n	no c. D	D = 2	D = 3	D = 4	D = 5	D = 6
2	3	3	2	–	–	–	–
2	4	6	5	3	–	–	–
2	5	10	9	7	4	–	–
2	6	15	14	12	9	5	–
2	7	21	20	18	15	11	6
2	8	28	27	25	22	18	13

Table 4. God’s numbers for consecutive cycles without inverses coset: central state

(0,1,2,\dots,n-D-1,n-D,n-D,\dots,n-D)

(

D

coincide at the end)

$k$	$N_{k}$	Offset ( $c_{k}$ )	$d_{k}(n)$
2	3	$-3$	$2n-3$
3	4	$-1$	$n-1$
4	5	$1$	$\big\lfloor\frac{2n+1}{3}\big\rfloor$
5	6	$4$	$\big\lfloor\frac{2n+4}{4}\big\rfloor$
6	8	$10$	$\left\lfloor\frac{2n+10}{5}\right\rfloor$
7	8	$14$	$\left\lfloor\frac{2n+14}{6}\right\rfloor$

Table 5. 3-Different diameter conjectures for

2\leqslant k\leqslant 20

Figure 47. Last layer size for k=6…10. Empirical data coincides with the proposed formula for large enough $n$ .

Figure 48. Last layer size for k=16…20. Empirical data coincides with the proposed formula for large enough $n$ .

Figure 49. Difference to naive estimation $\lceil l(n-l)/2(k-1)\rceil$ wrapped 4 cycles inverse closed. Bi-variable quasi-polynomials pattern.

Figure 50. Difference to naive estimation $\lceil l(n-l)/2(k-1)\rceil$ wrapped 3 cycles inverse closed. Bi-variable quasi-polynomials pattern.

Figure 51. Difference to naive estimation $\lceil l(n-l)/2(k-1)\rceil$ wrapped 5 cycles inverse closed. Bi-variable quasi-polynomials pattern.

Panel of Figure 52 — Figure 52. Zeros of $H$ -polynomials for wrapped $k$ -cycle Schreier graphs $S_{n}/\bigl(S_{\lfloor n/2\rfloor}\times S_{n-\lfloor n/2\rfloor}\bigr)$ (not inverse-closed) for $k=2,3,4$ .

$n$	k=2			k=3			k=4			k=5			k=6
	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$
3	1	1	0
4	2	1	1	2	0	1
5	3	2	-1	2	1	0	3	0	0
6	5	1	1	3	1	0	3	0	1	4	0	0
7	6	2	0	4	1	0	3	1	0	4	0	0	5	0	0

Table 6. Diameters

D_{k}(n)

and their increments and second increments for wrapped

k

-cycle coset

S_{n}/\bigl(S_{\lfloor n/2\rfloor}\times S_{n-\lfloor n/2\rfloor}\bigr)

(not inverse-closed). In that example second increments allow to see periodic structure, i.e. fit quasi-polynomials.

$n$	k=2 inv			k=3 inv			k=4 inv			k=5 inv			k=6 inv
	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$
3	1	1	0
4	2	1	1	2	0	1
5	3	2	-1	2	1	-1	2	1	-1
6	5	1	1	3	0	1	3	0	1	3	0	1
7	6	2	0	3	1	0	3	1	-1	3	1	-1	3	1	-1

Table 7. Diameters

D_{k}(n)

and their increments, wrapped k-cycle, inverse closed,

S_{n}/\bigl(S_{\lfloor n/2\rfloor}\times S_{n-\lfloor n/2\rfloor}\bigr)

residue ( $r$ )	$a_{r}$	$b_{r}$	polynomial $p_{r}(n)$
$(0)$	$0$	$0$	$\displaystyle\frac{n^{2}}{16}$
$(1)$	$\tfrac{1}{8}$	$-\tfrac{3}{16}$	$\displaystyle\frac{n^{2}}{16}+\frac{n}{8}-\frac{3}{16}$
$(2)$	$0$	$\tfrac{12}{16}=\tfrac{3}{4}$	$\displaystyle\frac{n^{2}}{16}+\tfrac{3}{4}$
$(3)$	$\tfrac{1}{8}$	$\tfrac{1}{16}$	$\displaystyle\frac{n^{2}}{16}+\frac{n}{8}+\frac{1}{16}$

Table 61

residue $r$	$\mathbf{a_{r}}$	$\mathbf{b_{r}}$	polynomial $p_{r}(n)$
0	$\tfrac{1}{8}$	$0$	$\displaystyle\frac{n^{2}}{32}+\frac{n}{8}$
1	$\tfrac{3}{16}$	$-\tfrac{7}{32}$	$\displaystyle\frac{n^{2}}{32}+\frac{3n}{16}-\frac{7}{32}$
2	$\tfrac{1}{8}$	$\tfrac{5}{8}$	$\displaystyle\frac{n^{2}}{32}+\frac{n}{8}+\frac{5}{8}$
3	$\tfrac{3}{16}$	$\tfrac{5}{32}$	$\displaystyle\frac{n^{2}}{32}+\frac{3n}{16}+\frac{5}{32}$
4	$\tfrac{1}{8}$	$0$	$\displaystyle\frac{n^{2}}{32}+\frac{n}{8}$
5	$\tfrac{3}{16}$	$\tfrac{9}{32}$	$\displaystyle\frac{n^{2}}{32}+\frac{3n}{16}+\frac{9}{32}$

Table 62

residue ( $r$ )	$\mathbf{a_{r}}$	$\mathbf{b_{r}}$	polynomial $p_{r}(n)$
0	$\tfrac{1}{6}$	$0$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{6}$
1	$\tfrac{5}{24}$	$-\tfrac{11}{48}$	$\displaystyle\frac{n^{2}}{48}+\frac{5n}{24}-\frac{11}{48}$
2	$\tfrac{1}{6}$	$\tfrac{7}{12}$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{6}+\frac{7}{12}$
3	$\tfrac{5}{24}$	$\tfrac{3}{16}$	$\displaystyle\frac{n^{2}}{48}+\frac{5n}{24}+\frac{3}{16}$
4	$\tfrac{1}{6}$	$0$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{6}$
5	$\tfrac{5}{24}$	$\tfrac{7}{16}$	$\displaystyle\frac{n^{2}}{48}+\frac{5n}{24}+\frac{7}{16}$

Table 63

residue $r$	$\mathbf{a_{r}}$	$\mathbf{b_{r}}$	polynomial $p_{r}(n)$
0	$0$	$2$	$\displaystyle\frac{n^{2}}{48}+2$
1	$\tfrac{1}{24}$	$\tfrac{31}{16}$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{24}+\frac{31}{16}$
2	$\tfrac{1}{12}$	$\tfrac{3}{4}$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{12}+\frac{3}{4}$
3	$\tfrac{1}{8}$	$\tfrac{7}{16}$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{8}+\frac{7}{16}$
4	$\tfrac{1}{12}$	$\tfrac{1}{3}$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{12}+\frac{1}{3}$
5	$\tfrac{1}{8}$	$\tfrac{41}{48}$	$\displaystyle\frac{n^{2}}{48}+\frac{n}{8}+\frac{41}{48}$

Table 64

$n$	k=2			k=3			k=4			k=5
	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$
3	1	1	-1
4	2	0	1	2	0	1
5	2	1	0	2	1	-1	2	1	-1
6	3	1	-1	3	0	1	3	0	0	3	0	0
7	4	0	2	3	1	0	3	0	1	3	0	1

Table 8. eccentricities and increments for

[0,1]^{\lfloor n/2\rfloor}+[0]^{\,n-2\lfloor n/2\rfloor}

) , wrapped k-cycle, not inverse-closed

$n$	k=2 inv			k=3 inv			k=4 inv			k=5 inv
	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$	$d$	$\Delta d$	$\Delta^{2}d$
3	1	1	-1
4	2	0	1	2	0	1
5	2	1	0	2	1	-1	2	1	-2
6	3	1	-1	3	0	1	3	-1	2	3	-1	2
7	4	0	2	3	1	0	2	1	0	2	1	0

Table 9. eccentricities and increments for

[0,1]^{\lfloor n/2\rfloor}+[0]^{\,n-2\lfloor n/2\rfloor}

) , wrapped k-cycle, inverse-closed

k	n	no c. d	d = 2	d = 3	d = 4	d = 5	d = 6
4	5	3	3	2	1
4	6	4	4	3	2	1
4	7	6	5	5	4	3	1
4	8	7	6	6	5	4	3
4	9	7	7	6	6	5	4
4	10	10	8	8	7	7	6

Table 10. Experimental data (

d

coincide)

$k$	$N_{k}$	period	$r_{k}(n)$
3	$4$	$4$	$(0,0,0,0)$
4	$5$	$6$	$(0,0,0,0,0,1)$
5	$6$	$8$	$(0,0,0,0,0,0,0,0)$
6	$7$	$10$	$(0,0,0,0,0,0,-1,0,0,1)$
7	$9$	$12$	$(-1,-1,0,0,0,0,0,0,0,0)$
8	$10$	$14$	$(0,-1,0,0,0,0,0,-1,-1,-1,0,0,1,0)$

Table 11.

r_{k}(n)

period

$k$	$N_{k}$	period	$l_{k}(n)$
3	$8$	$4$	$(16,4,2,30)$
4	$11$	$6$	$(2,29,2,34,114,34)$
5	$22$	$8$	$(62,124,10,84,2,6,87,6)$
6	$37$	$10$	$(258,5,26,168,10,28,48,203,18,117)$
7	$44$	$12$	$(37,14,98,278,34,96,225,218,1,34,235,2)$
8	$76$	$14$	$(1,14,61,118,137,190,3,20,225,2,4,2,9,82)$

Table 12. Last layer size period for wrapped consecutive coset 4-different.

Jul 30, 2026

Learning to Trace Seiberg Dualities

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.

Jonathan J. Heckman, Shani Meynet, Alessandro Mininno +1

Department of Physics and Astronomy, 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

Transformer Scientific ML Learning-Augmented Algorithms

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Mar 5, 2026

Solving an Open Problem in Theoretical Physics using AI-Assisted Discovery

This paper demonstrates that artificial intelligence can accelerate mathematical discovery by autonomously solving an open problem in theoretical physics. We present a neuro-symbolic system, combining the Gemini Deep Think large language model with a systematic Tree Search (TS) framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings. Specifically, the agent evaluated the core integral

I(N,α)

for arbitrary loop geometries, directly improving upon recent AI-assisted attempts \cite{BCE+25} that only yielded partial asymptotic solutions. To substantiate our methodological claims regarding AI-accelerated discovery and to ensure transparency, we detail system prompts, search constraints, and intermittent feedback loops that guided the model. The agent identified a suite of 6 different analytical methods, the most elegant of which expands the kernel in Gegenbauer polynomials

C_l^{(3/2)}

to naturally absorb the integrand's singularities. The methods lead to an asymptotic result for

I(N,α)

at large

N

that both agrees with numerical results and also connects to the continuous Feynman parameterization of Quantum Field Theory. We detail both the algorithmic methodology that enabled this discovery and the resulting mathematical derivations.

Michael P. Brenner, Vincent Cohen-Addad, David Woodruff

Google Research · School of Engineering and Applied Sciences, Harvard University · School of Computer Science, Carnegie Mellon University and Google Research

Large Language Model-Guided Scientific Discovery Automated Algorithm Discovery

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May 26, 2026

Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

We introduce

dualGNN', an autoregressive message-passing GNN for sampling fine, regular triangulations of lattice polytopes. dualGNN operates on a generalization of the dual graph of a triangulation, with edges labeled by

signed circuits' -- combinatorial invariants from the theory of oriented matroids. We show that these circuits are necessary and sufficient to determine a triangulation's regularity from the graph, provided certain magnitude information is retained. The model is independent of the polytope's point count and invariant under its orientation-preserving symmetries (

\mathrm{SL}(d,\mathbb{Z}) \ltimes \mathbb{Z}^d

), and our masking procedure further guarantees that every rollout produces a fine triangulation (in 2D). On unseen polygons with

N_\mathrm{pts} \leq 40

, dualGNN is the only sampler we tested that is consistent with uniform sampling across all our diagnostics (KL divergence from uniformity, collision counts, and sample autocorrelation). The model is small (

\sim92

k parameters) and trains in

\sim7.5

hours on a single consumer GPU. We apply dualGNN to string theory, sampling Calabi-Yau threefolds uniformly at

h^{1,1}=86

; we also sample CYs at

h^{1,1}=128

, observing no deviations from uniformity, but our diagnostics are weaker here. Code, training scripts, and pretrained models are available at https://github.com/natemacfadden/dualGNN (pip install dualgnn), and dualGNN is integrated into CYTools.

Nate MacFadden

Department of Physics, Cornell University, Ithaca, NY 14853 USA

Graph Neural Networks Generative Modeling

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k	n	no c. D	D = 2	D = 3	D = 4	D = 5	D = 6
2	3	3	2	–	–	–	–
2	4	6	5	3	–	–	–
2	5	10	9	7	4	–	–
2	6	15	14	12	9	5	–
2	7	21	20	18	15	11	6
2	8	28	27	25	22	18	13

k	n	no c. D	D = 2	D = 3	D = 4	D = 5	D = 6
2	3	3	2	–	–	–	–
2	4	6	5	3	–	–	–
2	5	10	9	7	4	–	–
2	6	15	14	12	9	5	–
2	7	21	20	18	15	11	6
2	8	28	27	25	22	18	13

Abstract

Explore similar work

Learning to Trace Seiberg Dualities

Solving an Open Problem in Theoretical Physics using AI-Assisted Discovery

Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs

k	n	no c. D	D = 2	D = 3	D = 4	D = 5	D = 6
2	3	3	2	–	–	–	–
2	4	6	5	3	–	–	–
2	5	10	9	7	4	–	–
2	6	15	14	12	9	5	–
2	7	21	20	18	15	11	6
2	8	28	27	25	22	18	13