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Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning

Anderson, Lara B., Gerdes, Mathis, Gray, James, Krippendorf, Sven, Raghuram, Nikhil, Ruehle, Fabian·2021·Journal of High Energy Physics
DOI 10.1007/jhep05(2021)013
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Abstract

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 ℙ 4.

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Figures

Image 1
CC-BY · From the PDF
Figure 2 ( Left ): errors σ for Donaldson's algorithm as a function of ψ for random sampling. ( Middle ): values of ψ used in the training and evaluation set. ( Right ): comparison of σ for different approximation approaches.
Figure 2 ( Left ): errors σ for Donaldson's algorithm as a function of ψ for random sampling. ( Middle ): values of ψ used in the training and evaluation set. ( Right ): comparison of σ for different approximation approaches.CC-BY · From the PDF
Figure 3 ( Left ): errors σ for Donaldson's algorithm as a function of ψ for sparse sampling. ( Middle ): values of ψ used in the training and evaluation set. ( Right ): comparison of σ for different approximation approaches.
Figure 3 ( Left ): errors σ for Donaldson's algorithm as a function of ψ for sparse sampling. ( Middle ): values of ψ used in the training and evaluation set. ( Right ): comparison of σ for different approximation approaches.CC-BY · From the PDF
Figure 4 σ accuracies at k = 6 achieved by the dense network with one and two hidden layers. The shaded area indicates the range of | ψ | that was not used during training, and thus shows the extrapolation behavior of the networks. For reference, the σ accuracy achieved by Donaldson's algorithm for the same range of | ψ | is shown. The dashed line corresponds to the extrapolation of using Donaldson's balanced metric at ψ = 100 over real values of ψ . The error band in each case corresponds to the maximal and minimal value obtained respectively when evaluating the σ accuracy at different angles.
Figure 4 σ accuracies at k = 6 achieved by the dense network with one and two hidden layers. The shaded area indicates the range of | ψ | that was not used during training, and thus shows the extrapolation behavior of the networks. For reference, the σ accuracy achieved by Donaldson's algorithm for the same range of | ψ | is shown. The dashed line corresponds to the extrapolation of using Donaldson's balanced metric at ψ = 100 over real values of ψ . The error band in each case corresponds to the maximal and minimal value obtained respectively when evaluating the σ accuracy at different angles.CC-BY · From the PDF
Figure 5 Evolution of the training loss during training. ( Left ): optimizing the NN with all three losses. ( Middle ): optimizing the NN without Kähler loss (i.e. λ 2 = 0 ). ( Right ): optimizing the NN without overlap loss (i.e. λ 3 = 0 ).
Figure 5 Evolution of the training loss during training. ( Left ): optimizing the NN with all three losses. ( Middle ): optimizing the NN without Kähler loss (i.e. λ 2 = 0 ). ( Right ): optimizing the NN without overlap loss (i.e. λ 3 = 0 ).CC-BY · From the PDF
Figure 6 ( Left ): evolution of the validation loss components for our network trained on the range 0 < | ψ | < 10 . The loss components are averaged over the respective patch-networks. ( Middle ): σ accuracy of our network in comparison with the induced Fubini-Study accuracy. The data points indicate the mean value of the performance, and the error bars show the minimal and maximal accuracy on different patches. (We have multiple networks which are trained here.) This average is also over four different angles θ for ψ = re iθ , namely 0 , π/ 2 , π , and 3 π/ 2 . The networks were trained on values of | ψ | < 10 , and the performance in the range 10 < | ψ | < 20 is obtained from extrapolation beyond the training set. ( Right ): deviation from the induced Fubini study metric during training, averaged over each patch-network and across the ψ values used for training.
Figure 6 ( Left ): evolution of the validation loss components for our network trained on the range 0 < | ψ | < 10 . The loss components are averaged over the respective patch-networks. ( Middle ): σ accuracy of our network in comparison with the induced Fubini-Study accuracy. The data points indicate the mean value of the performance, and the error bars show the minimal and maximal accuracy on different patches. (We have multiple networks which are trained here.) This average is also over four different angles θ for ψ = re iθ , namely 0 , π/ 2 , π , and 3 π/ 2 . The networks were trained on values of | ψ | < 10 , and the performance in the range 10 < | ψ | < 20 is obtained from extrapolation beyond the training set. ( Right ): deviation from the induced Fubini study metric during training, averaged over each patch-network and across the ψ values used for training.CC-BY · From the PDF
Figure 7 Change in loss during training for the SU(3) -structure example.
Figure 7 Change in loss during training for the SU(3) -structure example.CC-BY · From the PDF
Figure 8 Illustration of rejection sampling ( left ) and how the induced measure for sampling spherical coordinates with a flat prior can be non-flat ( right ) using the example of the unit disk.
Figure 8 Illustration of rejection sampling ( left ) and how the induced measure for sampling spherical coordinates with a flat prior can be non-flat ( right ) using the example of the unit disk.CC-BY · From the PDF
Figure 9 Scatter-plot comparing real and imaginary parts of the affine coordinates of the variety in P 2 defined by z 3 0 + z 3 1 + z 3 2 +10 z 0 z 1 z 2 = 0 , generated using either sampling algorithm introduced in section A. The values all lie in the affine patch ˆ U 0 defined by | z 1 | , | z 2 | ≤ | z 0 | = 1 .
Figure 9 Scatter-plot comparing real and imaginary parts of the affine coordinates of the variety in P 2 defined by z 3 0 + z 3 1 + z 3 2 +10 z 0 z 1 z 2 = 0 , generated using either sampling algorithm introduced in section A. The values all lie in the affine patch ˆ U 0 defined by | z 1 | , | z 2 | ≤ | z 0 | = 1 .CC-BY · From the PDF
Figure 10 Clustering of elements in H for ψ = 10 . ( Top row ): orange clusters correspond to vanishing elements, blue clusters correspond to non-vanishing values. ( Bottom row ): closeup view of the blue clusters. The number of vertical lines is in close relation to the number of equivariant components.
Figure 10 Clustering of elements in H for ψ = 10 . ( Top row ): orange clusters correspond to vanishing elements, blue clusters correspond to non-vanishing values. ( Bottom row ): closeup view of the blue clusters. The number of vertical lines is in close relation to the number of equivariant components.CC-BY · From the PDF
Figure 11 σ -accuracies achieved by an optimization of the H matrix using gradient descent with respect to the Monge-Ampère loss on the left, and the Ricci loss of equation (C.1) on the right.
Figure 11 σ -accuracies achieved by an optimization of the H matrix using gradient descent with respect to the Monge-Ampère loss on the left, and the Ricci loss of equation (C.1) on the right.CC-BY · From the PDF
Figure 12 Accuracies achieved by two network architectures detailed in section C.5 trained over uniform values | ψ | < 10 using the MA loss. The shaded area points out the extrapolation of the networks. As a comparison, the σ accuracies achieved by Donaldson's algorithm over real values of ψ are shown. The error band indicates the minimal and maximal values measured over multiple complex angles.
Figure 12 Accuracies achieved by two network architectures detailed in section C.5 trained over uniform values | ψ | < 10 using the MA loss. The shaded area points out the extrapolation of the networks. As a comparison, the σ accuracies achieved by Donaldson's algorithm over real values of ψ are shown. The error band indicates the minimal and maximal values measured over multiple complex angles.CC-BY · From the PDF

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Research works printed but not linked: 1

  1. S. Kachru, A. Tripathy and M. Zimet, K3 metrics, arXiv:2006.02435 [INSPIRE]. · arXiv

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Metadata

Authors
Anderson, L. B., Gerdes, M., Gray, J., Krippendorf, S., Raghuram, N., & Ruehle, F.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2021-05
Volume
2021
Issue
5
ISSN
1029-8479
DOI
10.1007/jhep05(2021)013
Cited by
50
References
40
Language
en
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CoChat. (2026). Citation check: Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning. https://app.cochat.ai/papers/10.1007/jhep05(2021)013
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Anderson, L. B., Gerdes, M., Gray, J., Krippendorf, S., Raghuram, N., & Ruehle, F. (2021). Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning. Journal of High Energy Physics. https://doi.org/10.1007/jhep05(2021)013

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