Machine-learning dessins d’enfants: explorations via modular and Seiberg-Witten curves. (English) Zbl 1519.14058
Summary: We apply machine-learning to the study of dessins d’enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg-Witten (SW) curves and extremal elliptic \(K3\) surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications reached 0.92 accuracy with 0.03 standard error relatively quickly. The SW curves for those with rational coefficients are also tabulated.
MSC:
| 14Q05 | Computational aspects of algebraic curves |
| 14H57 | Dessins d’enfants theory |
| 11G32 | Arithmetic aspects of dessins d’enfants, Belyĭ theory |
| 14H81 | Relationships between algebraic curves and physics |
| 68T07 | Artificial neural networks and deep learning |
Software:
TensorFlowReferences:
| [1] | Belyǐ G V 1980 On Galois extensions of a maximal cyclotomic field Math. USSR-Izvestiya14 247 · Zbl 0429.12004 · doi:10.1070/im1980v014n02abeh001096 |
| [2] | Grothendieck A 1984 Esquisse d’un programme available from: https://webusers.imj-prg.fr/leila.schneps/grothendieckcircle/EsquisseFr.pdf · Zbl 0901.14001 |
| [3] | Klein F 1879 Ueber die transformation elfter Ordnung der elliptischen functionen Math. Ann.15 533-55 · JFM 11.0299.06 · doi:10.1007/bf02086276 |
| [4] | Girondo E and González-Diez G 2011 Introduction to Compact Riemann Surfaces and Dessins d’Enfants(London Mathematical Society Student Texts) (Cambridge: Cambridge University Press) · doi:10.1017/CBO9781139048910 |
| [5] | Guillot P 2014 An elementary approach to dessin d’enfants and the Grothendieck-Teichmüller group (arXiv:1309.1968 [math]) · Zbl 1364.11126 |
| [6] | Lando S K, Zvonkin A K, Gamkrelidze R V and Vassiliev V A (ed) 2004 Graphs on Surfaces and Their Applications(Encyclopaedia of Mathematical Sciences vol 141) (Heidelberg: Springer) · Zbl 1040.05001 · doi:10.1007/978-3-540-38361-1 |
| [7] | Zapponi L 2003 What is a…dessin d’enfant Not. AMS50 788-9 available from: https://ams.org/notices/200307/what-is.pdf · Zbl 1211.14001 |
| [8] | Ashok S K, Cachazo F and dell’Aquila E 2007 Children’s drawings from Seiberg-Witten curves Commun. Numer. Theor. Phys.1 237-305 · Zbl 1230.14050 · doi:10.4310/cntp.2007.v1.n2.a1 |
| [9] | Seiberg N and Witten E 1994 Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory Nucl. Phys. B 426 19-52 · Zbl 0996.81510 · doi:10.1016/0550-3213(94)90124-4 |
| [10] | Jejjala V, Ramgoolam S and Rodriguez-Gomez D 2011 Toric CFTs, permutation triples and Belyi pairs J. High Energy Phys. JHEP03(2011)065 · Zbl 1301.81141 · doi:10.1007/jhep03(2011)065 |
| [11] | Hanany A, He Y, Jejjala V, Pasukonis J, Ramgoolam S and Rodriguez-Gomez D 2011 The beta ansatz: a tale of two complex structures J. High Energy Phys. JHEP06(2011)056 · Zbl 1298.81293 · doi:10.1007/jhep06(2011)056 |
| [12] | He Y-H, Hu Z, Probst M and Read J 2018 Yang-Mills theory and the ABC conjecture Int. J. Mod. Phys. A 33 1850053 · Zbl 1388.81328 · doi:10.1142/s0217751x18500537 |
| [13] | Franco S, Hanany A, Martelli D, Sparks J, Vegh D and Wecht B 2006 Gauge theories from toric geometry and brane tilings J. High Energy Phys. JHEP01(2006)128 · doi:10.1088/1126-6708/2006/01/128 |
| [14] | He Y-H, McKay J and Read J 2013 Modular subgroups, dessins d’enfants and elliptic K3 surfaces J. Comput. Math.16 271-318 · Zbl 1298.11058 · doi:10.1112/s1461157013000119 |
| [15] | He Y-H and McKay J 2013 N = 2 gauge theories: congruence subgroups, coset graphs, and modular surfaces J. Math. Phys.54 012301 · Zbl 1290.81067 · doi:10.1063/1.4772976 |
| [16] | Gaiotto D 2012 N = 2 dualities J. High Energy Phys. JHEP08(2012)034 · Zbl 1397.81362 · doi:10.1007/jhep08(2012)034 |
| [17] | Vidunas R and He Y 2017 Genus one Belyi maps by quadratic correspondences Int. J. Math.31 2050034 (arXiv:1706.04258 [math.AG]) · Zbl 1458.11104 · doi:10.1142/S0129167X20500342 |
| [18] | He Y 2017 Deep-learning the landscape (arXiv:1706.02714 [hep-th]) |
| [19] | He Y-H 2017 Machine-learning the string landscape Phys. Lett. B 774 564-8 · doi:10.1016/j.physletb.2017.10.024 |
| [20] | Krefl D and Seong R K 2017 Machine learning of Calabi-Yau volumes Phys. Rev. D 96 066014 · doi:10.1103/physrevd.96.066014 |
| [21] | Ruehle F 2017 Evolving neural networks with genetic algorithms to study the string landscape J. High Energy Phys. JHEP08(2017)038 · Zbl 1381.83128 · doi:10.1007/jhep08(2017)038 |
| [22] | Carifio J, Halverson J, Krioukov D and Nelson B D 2017 Machine learning in the string landscape J. High Energy Phys. JHEP09(2017)157 · Zbl 1382.81155 · doi:10.1007/jhep09(2017)157 |
| [23] | He Y 2018 The Calabi-Yau landscape: from geometry, to physics, to machine-learning (arXiv:1812.02893 [hep-th]) |
| [24] | Ruehle F 2020 Data science applications to string theory Phys. Rep.839 1-117 · Zbl 1452.81004 · doi:10.1016/j.physrep.2019.09.005 |
| [25] | Bao J, He Y, Hirst E and Pietromonaco S 2020 Lectures on the Calabi-Yau landscape (arXiv:2001.01212 [hep-th]) |
| [26] | He Y 2019 Machine-learning mathematical structures Oxford ML Physics Seminar Series https://youtube.com/watch?v=nMP2f14gYzc |
| [27] | Ashmore A, He Y and Ovrut B A 2019 Machine learning Calabi-Yau metrics (arXiv:1910.08605 [hep-th]) |
| [28] | Bull K, He Y-H, Jejjala V and Mishra C 2018 Machine learning CICY threefolds Phys. Lett. B 785 65-72 · doi:10.1016/j.physletb.2018.08.008 |
| [29] | Brodie C R, Constantin A, Deen R and Lukas A 2020 Machine learning line bundle cohomology Fortschr. Phys.68 1900087 · Zbl 1535.32024 · doi:10.1002/prop.201900087 |
| [30] | He Y and Kim M 2019 Learning algebraic structures: preliminary investigations (arXiv:1905.02263 [cs.LG]) |
| [31] | Bao J, Franco S, He Y, Hirst E, Musiker G and Xiao Y 2020 Phys. Rev. D.102 086013 Quiver Mutations, Seiberg Duality and Machine Learning · doi:10.1103/PhysRevD.102.086013 |
| [32] | He Y and Yau S T 2020 Graph Laplacians, Riemannian manifolds and their machine-learning (arXiv:2006.16619) |
| [33] | Alessandretti L, Baronchelli A and He Y 2019 Machine learning meets number theory: the data science of Birch-Swinnerton-Dyer (arXiv:1911.02008 [math.NT]) |
| [34] | Miranda R and Persson U 1989 Configurations of in fibers on elliptic K3-surfaces Math. Z.201 339-61 · Zbl 0694.14019 · doi:10.1007/bf01214900 |
| [35] | Beukers F and Montanus H 2008 Explicit calculation of elliptic fibrations of K3-surfaces and their Belyi-maps Number Theory and Polynomials(LMS Lecture Note Series vol 352) (Cambridge: Cambridge University Press) pp 33-51 Dessins coming from the Miranda-Persson table. Available from: http://staff.science.uu.nl/beuke106/MirandaPersson_table/Dessins.html · Zbl 1266.11078 · doi:10.1017/CBO9780511721274.005 |
| [36] | Sebbar A 2001 Classification of torsion-free genus zero congruence groups Proc. Am. Math. Soc.129 2517-27 · Zbl 0981.20038 · doi:10.1090/s0002-9939-01-06176-7 |
| [37] | McKay J and Sebbar A 2001 J-invariants of arithmetic semistable elliptic surfaces and graphs Proc. on Moonshine and Related Topics (Providence, RI: American Mathematical Society) 119-30 · Zbl 1191.11010 · doi:10.1090/crmp/030/11 |
| [38] | He Y and Read J 2015 Dessins d’enfants in N=2 generalised quiver theories J. High Energy Phys. JHEP08(2015)085 · Zbl 1388.81830 · doi:10.1007/jhep08(2015)085 |
| [39] | Beukers F (n.d.) montanuslist.txt available from: https://staff.science.uu.nl/beuke106/mirandapers-son/montanuslist.txt |
| [40] | Edwards H M 1984 A note on Galois theory Arch. Hist. Exact Sci.41 163-9 · Zbl 0764.01006 · doi:10.1007/bf00411863 |
| [41] | Cox D A 2012 Galois Theory (New York: Wiley) · Zbl 1247.12006 · doi:10.1002/9781118218457 |
| [42] | Shimura G 1971 Introduction to the Arithmetic Theory of Automorphic Functions (Princeton, NJ: Princeton University Press) · Zbl 0221.10029 |
| [43] | Kimura Y 2017 Gauge symmetries and matter fields in F-theory models without section - compactifications on double cover and Fermat quartic K3 constructions times K3 (arXiv:1603.03212 [hep-th]) |
| [44] | Kimura Y 2019 Nongeometric heterotic strings and dual F-theory with enhanced gauge groups (arXiv:1810.07657 [hep-th]) · Zbl 1411.83125 |
| [45] | Hanany A, He Y-H, Jejjala V, Pasukonis J, Ramgoolam S and Rodriguez-Gomez D 2012 Invariants of toric Seiberg duality Int. J. Mod. Phys. A 27 1250002 · Zbl 1247.81380 · doi:10.1142/s0217751x12500029 |
| [46] | Tachikawa Y 2014 N (Berlin: Springer) available from: https://springer.com/gp/book/9783319088211 |
| [47] | Cachazo F, Seiberg N and Witten E 2003 Phases of N = 1 supersymmetric gauge theories and matrices (arXiv:hep-th/0301006) |
| [48] | Sun C, Shrivastava A, Singh S and Gupta A 2017 Revisiting unreasonable effectiveness of data in deep learning era (arXiv:1707.02968 [cs]) |
| [49] | Banko M and Brill E 2001 Mitigating the paucity-of-data problem: exploring the effect of training corpus size on classifier performance for natural language processing Proc. of the 1st Int. Conf. on Human Language Technology Research · doi:10.3115/1072133.1072204 |
| [50] | Abadi M et al 2015 TensorFlow: large-scale machine-learning on heterogeneoussystems software available from: tensorflow.org |
| [51] | Buduma N and Locascio N 2017 Fundamentals of Deep Learning: Designing Next-Generation Machine Intelligence Algorithms (Sebastopol, CA: O’Reilly & Associates) |
| [52] | Ketkar N 2017 Deep learning with Python: a hands-on introduction |
| [53] | Ruder S 2016 An overview of gradient descent optimization algorithms (arXiv:1609.04747 [cs]) |
| [54] | Boughorbel S, Jarray F and El-Anbari M 2017 Optimal classifier for imbalanced data using Matthews correlation coefficient metric PloS One12 e0177678 · doi:10.1371/journal.pone.0177678 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
