Abstract
The geometry of 4-string contact interaction of closed string field theory is characterized using machine learning. We obtain Strebel quadratic differentials on 4-punctured spheres as a neural network by performing unsupervised learning with a custom-built loss function. This allows us to solve for local coordinates and compute their associated mapping radii numerically. We also train a neural network distinguishing vertex from Feynman region. As a check, 4-tachyon contact term in the tachyon potential is computed and a good agreement with the results in the literature is observed. We argue that our algorithm is manifestly independent of number of punctures and scaling it to characterize the geometry of n-string contact interaction is feasible.
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References
B. Zwiebach, Closed string field theory: Quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
C. de Lacroix et al., Closed Superstring Field Theory and its Applications, Int. J. Mod. Phys. A 32 (2017) 1730021 [arXiv:1703.06410] [INSPIRE].
T. Erler, Four Lectures on Closed String Field Theory, Phys. Rept. 851 (2020) 1 [arXiv:1905.06785] [INSPIRE].
H. Erbin, String Field Theory: A Modern Introduction, Springer Cham (2021) [https://doi.org/10.1007/978-3-030-65321-7] [INSPIRE].
A. Belopolsky and B. Zwiebach, Off-shell closed string amplitudes: Towards a computation of the tachyon potential, Nucl. Phys. B 442 (1995) 494 [hep-th/9409015] [INSPIRE].
A. Belopolsky, Effective Tachyonic potential in closed string field theory, Nucl. Phys. B 448 (1995) 245 [hep-th/9412106] [INSPIRE].
Y. Okawa and B. Zwiebach, Twisted tachyon condensation in closed string field theory, JHEP 03 (2004) 056 [hep-th/0403051] [INSPIRE].
N. Moeller, Closed bosonic string field theory at quartic order, JHEP 11 (2004) 018 [hep-th/0408067] [INSPIRE].
H.-T. Yang and B. Zwiebach, Testing closed string field theory with marginal fields, JHEP 06 (2005) 038 [hep-th/0501142] [INSPIRE].
H. Yang and B. Zwiebach, Dilaton deformations in closed string field theory, JHEP 05 (2005) 032 [hep-th/0502161] [INSPIRE].
H. Yang and B. Zwiebach, Rolling closed string tachyons and the big crunch, JHEP 08 (2005) 046 [hep-th/0506076] [INSPIRE].
H. Yang and B. Zwiebach, A closed string tachyon vacuum?, JHEP 09 (2005) 054 [hep-th/0506077] [INSPIRE].
N. Moeller, Closed Bosonic String Field Theory at Quintic Order: Five-Tachyon Contact Term and Dilaton Theorem, JHEP 03 (2007) 043 [hep-th/0609209] [INSPIRE].
N. Moeller and H. Yang, The nonperturbative closed string tachyon vacuum to high level, JHEP 04 (2007) 009 [hep-th/0609208] [INSPIRE].
N. Moeller, Closed Bosonic String Field Theory at Quintic Order. II. Marginal Deformations and Effective Potential, JHEP 09 (2007) 118 [arXiv:0705.2102] [INSPIRE].
N. Moeller, A tachyon lump in closed string field theory, JHEP 09 (2008) 056 [arXiv:0804.0697] [INSPIRE].
L. Schlechter, Closed Bosonic String Tachyon Potential from the \(\mathcal{N}\) = 1 Point of View, arXiv:1905.09621 [INSPIRE].
S.F. Moosavian and R. Pius, Hyperbolic geometry and closed bosonic string field theory. Part I. The string vertices via hyperbolic Riemann surfaces, JHEP 08 (2019) 157 [arXiv:1706.07366] [INSPIRE].
S.F. Moosavian and R. Pius, Hyperbolic geometry and closed bosonic string field theory. Part II. The rules for evaluating the quantum BV master action, JHEP 08 (2019) 177 [arXiv:1708.04977] [INSPIRE].
K. Costello and B. Zwiebach, Hyperbolic string vertices, JHEP 02 (2022) 002 [arXiv:1909.00033] [INSPIRE].
M. Cho, Open-closed Hyperbolic String Vertices, JHEP 05 (2020) 046 [arXiv:1912.00030] [INSPIRE].
A.H. Fırat, Hyperbolic three-string vertex, JHEP 08 (2021) 035 [arXiv:2102.03936] [INSPIRE].
P. Wang, H. Wu and H. Yang, Connections between reflected entropies and hyperbolic string vertices, JHEP 05 (2022) 127 [arXiv:2112.09503] [INSPIRE].
N. Ishibashi, The Fokker-Planck formalism for closed bosonic strings, PTEP 2023 (2023) 023B05 [arXiv:2210.04134] [INSPIRE].
A. Sen, Tachyon condensation on the brane anti-brane system, JHEP 08 (1998) 012 [hep-th/9805170] [INSPIRE].
A. Sen, Universality of the tachyon potential, JHEP 12 (1999) 027 [hep-th/9911116] [INSPIRE].
A. Sen and B. Zwiebach, Tachyon condensation in string field theory, JHEP 03 (2000) 002 [hep-th/9912249] [INSPIRE].
N. Moeller and W. Taylor, Level truncation and the tachyon in open bosonic string field theory, Nucl. Phys. B 583 (2000) 105 [hep-th/0002237] [INSPIRE].
N. Berkovits, A. Sen and B. Zwiebach, Tachyon condensation in superstring field theory, Nucl. Phys. B 587 (2000) 147 [hep-th/0002211] [INSPIRE].
N. Moeller, A. Sen and B. Zwiebach, D-branes as tachyon lumps in string field theory, JHEP 08 (2000) 039 [hep-th/0005036] [INSPIRE].
A. Sen and B. Zwiebach, Large marginal deformations in string field theory, JHEP 10 (2000) 009 [hep-th/0007153] [INSPIRE].
L. Rastelli, A. Sen and B. Zwiebach, Classical solutions in string field theory around the tachyon vacuum, Adv. Theor. Math. Phys. 5 (2002) 393 [hep-th/0102112] [INSPIRE].
W. Taylor, A perturbative analysis of tachyon condensation, JHEP 03 (2003) 029 [hep-th/0208149] [INSPIRE].
D. Gaiotto and L. Rastelli, Experimental string field theory, JHEP 08 (2003) 048 [hep-th/0211012] [INSPIRE].
K. Strebel, Quadratic Differentials, in Quadratic Differentials, Springer, Berlin Heidelberg (1984), p. 16–26 [https://doi.org/10.1007/978-3-662-02414-0_2].
F. Ruehle, Evolving neural networks with genetic algorithms to study the String Landscape, JHEP 08 (2017) 038 [arXiv:1706.07024] [INSPIRE].
J. Halverson, B. Nelson and F. Ruehle, Branes with Brains: Exploring String Vacua with Deep Reinforcement Learning, JHEP 06 (2019) 003 [arXiv:1903.11616] [INSPIRE].
M. Larfors and R. Schneider, Explore and Exploit with Heterotic Line Bundle Models, Fortsch. Phys. 68 (2020) 2000034 [arXiv:2003.04817] [INSPIRE].
A. Ashmore and F. Ruehle, Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold, Phys. Rev. D 103 (2021) 106028 [arXiv:2103.07472] [INSPIRE].
M.R. Douglas, S. Lakshminarasimhan and Y. Qi, Numerical Calabi-Yau metrics from holomorphic networks, arXiv:2012.04797 [INSPIRE].
L.B. Anderson et al., Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning, JHEP 05 (2021) 013 [arXiv:2012.04656] [INSPIRE].
H. Erbin and R. Finotello, Inception neural network for complete intersection Calabi-Yau 3-folds, Mach. Learn. Sci. Tech. 2 (2021) 02LT03 [arXiv:2007.13379] [INSPIRE].
H. Erbin and R. Finotello, Machine learning for complete intersection Calabi-Yau manifolds: a methodological study, Phys. Rev. D 103 (2021) 126014 [arXiv:2007.15706] [INSPIRE].
H. Erbin, R. Finotello, R. Schneider and M. Tamaazousti, Deep multi-task mining Calabi-Yau four-folds, Mach. Learn. Sci. Tech. 3 (2022) 015006 [arXiv:2108.02221] [INSPIRE].
A. Ashmore, L. Calmon, Y.-H. He and B.A. Ovrut, Calabi-Yau Metrics, Energy Functionals and Machine-Learning, International Journal of Data Science in the Mathematical Sciences 1 (2023) 49 [arXiv:2112.10872] [INSPIRE].
M. Larfors, A. Lukas, F. Ruehle and R. Schneider, Learning Size and Shape of Calabi-Yau Spaces, arXiv:2111.01436 [INSPIRE].
M. Larfors, A. Lukas, F. Ruehle and R. Schneider, Numerical metrics for complete intersection and Kreuzer-Skarke Calabi-Yau manifolds, Mach. Learn. Sci. Tech. 3 (2022) 035014 [arXiv:2205.13408] [INSPIRE].
V. Jejjala, W. Taylor and A. Turner, Identifying equivalent Calabi-Yau topologies: A discrete challenge from math and physics for machine learning, in the proceedings of the Nankai Symposium on Mathematical Dialogues: In celebration of S.S.Chern’s 110th anniversary, China, August 02–13 (2021) [arXiv:2202.07590] [INSPIRE].
J. Halverson, A. Maiti and K. Stoner, Neural Networks and Quantum Field Theory, Mach. Learn. Sci. Tech. 2 (2021) 035002 [arXiv:2008.08601] [INSPIRE].
J. Halverson, Building Quantum Field Theories Out of Neurons, arXiv:2112.04527 [INSPIRE].
H. Erbin, V. Lahoche and D.O. Samary, Non-perturbative renormalization for the neural network-QFT correspondence, Mach. Learn. Sci. Tech. 3 (2022) 015027 [arXiv:2108.01403] [INSPIRE].
F. Ruehle, Data science applications to string theory, Phys. Rept. 839 (2020) 1 [INSPIRE].
J. Schmidt-Hieber, Nonparametric regression using deep neural networks with ReLU activation function, arXiv:1708.06633 [https://doi.org/10.1214/19-AOS1875].
K. Zhang and Y.-X. Wang, Deep Learning meets Nonparametric Regression: Are Weight-Decayed DNNs Locally Adaptive?, arXiv:2204.09664.
R. Balestriero, J. Pesenti and Y. LeCun, Learning in High Dimension Always Amounts to Extrapolation, arXiv:2110.09485.
K. Xu et al., How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks, arXiv:2009.11848.
H. Erbin and A.H. Fırat, Characterizing n-string contact interactions using machine learning, to appear.
B. Zwiebach, Consistency of Closed String Polyhedra From Minimal Area, Phys. Lett. B 241 (1990) 343 [INSPIRE].
B. Zwiebach, How covariant closed string theory solves a minimal area problem, Commun. Math. Phys. 136 (1991) 83 [INSPIRE].
M. Saadi and B. Zwiebach, Closed String Field Theory from Polyhedra, Annals Phys. 192 (1989) 213 [INSPIRE].
T. Kugo, H. Kunitomo and K. Suehiro, Nonpolynomial Closed String Field Theory, Phys. Lett. B 226 (1989) 48 [INSPIRE].
E. Witten, Noncommutative Geometry and String Field Theory, Nucl. Phys. B 268 (1986) 253 [INSPIRE].
H. Sonoda and B. Zwiebach, Closed String Field Theory Loops With Symmetric Factorizable Quadratic Differentials, Nucl. Phys. B 331 (1990) 592 [INSPIRE].
R. Gopakumar, From free fields to AdS: III, Phys. Rev. D 72 (2005) 066008 [hep-th/0504229] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar, B. Knighton and P. Maity, From symmetric product CFTs to AdS3, JHEP 05 (2021) 073 [arXiv:2011.10038] [INSPIRE].
F. Bhat, R. Gopakumar, P. Maity and B. Radhakrishnan, Twistor coverings and Feynman diagrams, JHEP 05 (2022) 150 [arXiv:2112.05115] [INSPIRE].
B. Knighton, Classical geometry from the tensionless string, JHEP 05 (2023) 005 [arXiv:2207.01293] [INSPIRE].
E.W. Dijkstra, A Note on Two Problems in Connexion with Graphs, in Edsger Wybe Dijkstra, ACM (2022), p. 287–290 [https://doi.org/10.1145/3544585.3544600].
G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control Signals Syst. 2 (1989) 303 [INSPIRE].
K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989) 359 [INSPIRE].
K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Networks 4 (1991) 251 [INSPIRE].
M. Leshno, V.Y. Lin, A. Pinkus and S. Schocken, Multilayer feedforward networks with a nonpolynomial activation function can approximate any function, Neural Networks 6 (1993) 861.
B.C. Csáji, Approximation with Artificial Neural Networks, Ph.D. Thesis, Eötvös Loránd University, H-1053 Budapest, Hungary (2001).
C. Trabelsi et al., Deep Complex Networks, arXiv:1705.09792 (2018).
S. Scardapane, S.V. Vaerenbergh, A. Hussain and A. Uncini, Complex-valued Neural Networks with Non-parametric Activation Functions, arXiv:1802.08026.
J. Bassey, L. Qian and X. Li, A Survey of Complex-Valued Neural Networks, arXiv:2101.12249.
J. Bradbury et al., JAX: composable transformations of Python+NumPy programs, (2018).
E.J. Michaud, Z. Liu and M. Tegmark, Precision Machine Learning, arXiv:2210.13447 [https://doi.org/10.3390/e25010175].
T.S. Cohen and M. Welling, Group Equivariant Convolutional Networks, arXiv:1602.07576 [INSPIRE].
T. Cohen, M. Geiger and M. Weiler, A General Theory of Equivariant CNNs on Homogeneous Spaces, arXiv:1811.02017.
J.E. Gerken et al., Geometric Deep Learning and Equivariant Neural Networks, arXiv:2105.13926 [INSPIRE].
J. Zhou et al., Graph Neural Networks: A Review of Methods and Applications, arXiv:1812.08434 [INSPIRE].
M. Headrick and B. Zwiebach, Minimal-area metrics on the Swiss cross and punctured torus, Commun. Math. Phys. 377 (2020) 2287 [arXiv:1806.00450] [INSPIRE].
M. Headrick and B. Zwiebach, Convex programs for minimal-area problems, Commun. Math. Phys. 377 (2020) 2217 [arXiv:1806.00449] [INSPIRE].
U. Naseer and B. Zwiebach, Extremal isosystolic metrics with multiple bands of crossing geodesics, Adv. Theor. Math. Phys. 26 (2022) 1273 [arXiv:1903.11755] [INSPIRE].
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, US Government printing office 55 (1964).
S. Falkner, A. Klein and F. Hutter, BOHB: Robust and efficient hyperparameter optimization at scale, in Proceedings of the 35th International Conference on Machine Learning, Stockholmsmässan, Stockholm Sweden, July 10–15 (2018) [J. Dy and A. Krause eds., PMLR 80 (2018), p. 1437–1446].
Acknowledgments
We would like to thank Barton Zwiebach for endless discussions and introducing us to the world of quadratic differentials. We also would like to thank Riccardo Finotello, Manki Kim, Fabian Ruehle, and Siddharth Mishra-Sharma for discussions. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics of U.S. Department of Energy under grant Contract Number DE-SC0012567. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 891169.
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Erbin, H., Fırat, A.H. Characterizing 4-string contact interaction using machine learning. J. High Energ. Phys. 2024, 16 (2024). https://doi.org/10.1007/JHEP04(2024)016
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DOI: https://doi.org/10.1007/JHEP04(2024)016