×

zbMATH — the first resource for mathematics

Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning. (English) Zbl 1466.83111
Summary: We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in \(\mathbb{P}^4\).
MSC:
83E30 String and superstring theories in gravitational theory
81T12 Effective quantum field theories
32Q25 Calabi-Yau theory (complex-analytic aspects)
68T05 Learning and adaptive systems in artificial intelligence
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Yau, ST, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Commun. Pure Appl. Math., 3, 339 (1978) · Zbl 0369.53059
[2] M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory Vol. 2: 25th Anniversary Edition, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2012), [DOI] [INSPIRE].
[3] Headrick, M.; Wiseman, T., Numerical Ricci-flat metrics on K3, Class. Quant. Grav., 22, 4931 (2005) · Zbl 1085.53035
[4] S.K. Donaldson, Some numerical results in complex differential geometry, Pure Appl. Math. Q.2 (2009) 571, [math/0512625]. · Zbl 1178.32018
[5] Douglas, MR; Karp, RL; Lukic, S.; Reinbacher, R., Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic, JHEP, 12, 083 (2007) · Zbl 1246.14052
[6] Douglas, MR; Karp, RL; Lukic, S.; Reinbacher, R., Numerical Calabi-Yau metrics, J. Math. Phys., 49, 032302 (2008) · Zbl 1153.81351
[7] Braun, V.; Brelidze, T.; Douglas, MR; Ovrut, BA, Calabi-Yau Metrics for Quotients and Complete Intersections, JHEP, 05, 080 (2008)
[8] Braun, V.; Brelidze, T.; Douglas, MR; Ovrut, BA, Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds, JHEP, 07, 120 (2008)
[9] Headrick, M.; Nassar, A., Energy functionals for Calabi-Yau metrics, Adv. Theor. Math. Phys., 17, 867 (2013) · Zbl 1304.14050
[10] Anderson, LB; Braun, V.; Karp, RL; Ovrut, BA, Numerical Hermitian Yang-Mills Connections and Vector Bundle Stability in Heterotic Theories, JHEP, 06, 107 (2010) · Zbl 1288.81086
[11] Anderson, LB; Braun, V.; Ovrut, BA, Numerical Hermitian Yang-Mills Connections and Kähler Cone Substructure, JHEP, 01, 014 (2012) · Zbl 1306.81181
[12] Ashmore, A.; He, Y-H; Ovrut, BA, Machine Learning Calabi-Yau Metrics, Fortsch. Phys., 68, 2000068 (2020)
[13] Cui, W.; Gray, J., Numerical Metrics, Curvature Expansions and Calabi-Yau Manifolds, JHEP, 05, 044 (2020) · Zbl 1437.83130
[14] S. Kachru, A. Tripathy and M. Zimet, K3 metrics from little string theory, arXiv:1810.10540 [INSPIRE].
[15] S. Kachru, A. Tripathy and M. Zimet, K3 metrics, arXiv:2006.02435 [INSPIRE].
[16] A. Tripathy and M. Zimet, A plethora of K3 metrics, arXiv:2010.12581 [INSPIRE].
[17] Ooguri, H.; Vafa, C., On the Geometry of the String Landscape and the Swampland, Nucl. Phys. B, 766, 21 (2007) · Zbl 1117.81117
[18] Blumenhagen, R.; Conlon, JP; Krippendorf, S.; Moster, S.; Quevedo, F., SUSY Breaking in Local String/F-Theory Models, JHEP, 09, 007 (2009)
[19] Ruehle, F., Data science applications to string theory, Phys. Rept., 839, 1 (2020) · Zbl 1452.81004
[20] Tian, G., On a set of polarized Kähler metrics on algebraic manifolds, J. Diff. Geom., 32, 99 (1990) · Zbl 0706.53036
[21] Larfors, M.; Lukas, A.; Ruehle, F., Calabi-Yau Manifolds and SU(3) Structure, JHEP, 01, 171 (2019) · Zbl 1409.83191
[22] M.R. Douglas, S. Lakshminarasimhan and Y. Qi, Numerical Calabi-Yau metrics from holomorphic networks, arXiv:2012.04797 [INSPIRE].
[23] E. Calabi, On kähler manifolds with vanishing canonical class, in Algebraic geometry and topology, a symposium in honor of S. Lefschetz, vol. 12 (1957), pp. 78-89 [DOI]. · Zbl 0080.15002
[24] Candelas, P.; Dale, AM; Lütken, CA; Schimmrigk, R., Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B, 298, 493 (1988)
[25] A. Paszke et al., Pytorch: An imperative style, high-performance deep learning library, in Advances in Neural Information Processing Systems 32, H. Wallach et al. eds., Curran Associates, Inc. (2019) [arXiv:1912.01703] [http://papers.neurips.cc/paper/9015-pytorch- an-imperative-style-high-performance-deep-learning-library.pdf].
[26] M. Abadi et al., TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems, arXiv:1603.04467 [INSPIRE].
[27] J. Bradbury, R. Frostig, P. Hawkins, M.J. Johnson, C. Leary, D. Maclaurin and S. Wanderman-Milne, JAX: composable transformations of Python+NumPy programs, http://github.com/google/jax.
[28] D.P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, arXiv:1412.6980v9 [INSPIRE].
[29] Strominger, A., Superstrings with Torsion, Nucl. Phys. B, 274, 253 (1986)
[30] Hull, CM, Compactifications of the Heterotic Superstring, Phys. Lett. B, 178, 357 (1986)
[31] Lopes Cardoso, G.; Curio, G.; Dall’Agata, G.; Lüst, D.; Manousselis, P.; Zoupanos, G., NonKähler string backgrounds and their five torsion classes, Nucl. Phys. B, 652, 5 (2003) · Zbl 1010.83063
[32] Atiyah, MF; Bott, R.; Garding, L., Lacunas for hyperbolic differential operators with constant coefficients II, Acta. Math., 131, 145 (1973) · Zbl 0266.35045
[33] Donaldson, SK, Scalar curvature and projective embeddings II, Quart. J. Math., 56, 345 (2005) · Zbl 1159.32012
[34] Wang, X., Canonical metrics on stable vector bundles, Comm. Anal. Geom., 13, 253 (2005) · Zbl 1093.32008
[35] Anderson, LB; Gray, J.; Lukas, A.; Ovrut, B., Stability Walls in Heterotic Theories, JHEP, 09, 026 (2009)
[36] Strominger, A.; Yau, S-T; Zaslow, E., Mirror symmetry is T duality, Nucl. Phys. B, 479, 243 (1996) · Zbl 0896.14024
[37] Meurer, A., Sympy: symbolic computing in python, PeerJ Comput. Sci., 3, e103 (2017)
[38] S. Krippendorf and M. Syvaeri, Detecting Symmetries with Neural Networks, arXiv:2003.13679 [INSPIRE].
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.