Learning Size and Shape of Calabi-Yau Spaces

Magdalena Larfors
Department of Physics and Astronomy
Uppsala University
SE-751 20 Uppsala
Swedenmagdalena.larfors@physics.uu.se
Andre Lukas
Rudolf Peierls Centre for Theoretical Physics
Oxford University
Parks Road, Oxford OX1 3PU
United Kingdom
andre.lukas@physics.ox.ac.uk
Fabian Ruehle
Department of Physics
Northeastern University
360 Huntington Avenue, Boston MA 02115
United Statesf.ruehle@northeastern.edu
Robin Schneider
Department of Physics and Astronomy
Uppsala University
SE-751 20 Uppsala
Swedenrobin.schneider@physics.uu.se

Abstract
We present a new machine learning library for computing metrics of string compactification spaces. We benchmark the performance on Monte-Carlo sampled integrals
against previous numerical approximations and find that our neural networks are
more sample- and computation-efficient. We are the first to provide the possibility
to compute these metrics for arbitrary, user-specified shape and size parameters of
the compact space and observe a linear relation between optimization of the partial
differential equation we are training against and vanishing Ricci curvature.

1

Introduction

String theory is the leading candidate for a unification of general relativity and the standard model ofparticle physics. Its mathematical consistency, however, requires the introduction of six additional
small and compact dimensions, usually described by Calabi-Yau (CY) manifolds. The shape, size,
and topology of these internal spaces determine many interesting physical quantities such as thegauge group and particle content of the theory. CY manifolds are not only interesting objects inphysics, but exhibit fascinating mathematical properties, such as mirror symmetry.
Fourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

CY spaces have been extensively studied, starting with [1, 2]. However, above four compact
dimensions, there exists no known closed form expression for their metric tensor. The metric is ofutmost importance to connect string theory with observable physics. It is a necessary ingredient in thecomputation of physical Yukawa couplings from string theory and can be used to test conjectures fromthe swampland program [3, 4] or regarding mirror symmetry [5]. These applications furthermore
require control over how the metric depends on the size and shape parameters, so-called moduli, of
the CY geometry.
The standard numerical approximation for CY metrics is the Donaldson algorithm [6], a fixed point
iteration methods which expands the metric in a monomial basis of degree k and converges to the
CY metric as k → ∞. This algorithm has been utilized in examples [7–12]. However, there arethree main drawbacks: first, its time and space complexity scales poorly (factorially) with k. Second,changing the moduli of the metric requires running the whole fixed point iteration from scratch. Lastly,
the algorithm is in fact providing a metric with different properties, which happens to coincide with
the CY metric in the limit k → ∞. However, better direct optimizations of the CY metric are possible
for any finite, fixed k. Energy functionals [13, 14] and holomorphic neural networks [15] have provenmore efficient. Alternatively, one may directly learn the CY metric tensor [16–18]. By modeling themetric as neural network, this has also allowed studies of the shape moduli dependence [17].
In this paper we advance this promising line of research by presenting an open source package,cymetric1 , to learn CY metrics with TensorFlow [19] on a wide class of CY manifolds. Given
the lack of analytic metric solutions, direct supervised training of the neural network is impossible.
We therefore proceed in an almost self-supervised manner and introduce five loss functions thatgovern the learning process. Our neural network reproduces the full moduli dependence of the CY
metric, thus filling a crucial knowledge gap which has so far impeded phenomenological studies
in string theory. The implementation improves sample and computation efficiency compared toprevious approximations. Moreover, experiments run with our package establish a direct linear
relation between optimizing a simple surrogate loss and the computationally much more expensive
vanishing Ricci curvature.
The results of our experiments have promising implications for utilizing neural networks in solving
differential equations [20, 21]. In particular, we demonstrate that by using physical domain knowledge,
it is possible to find solutions to non-linear second-order differential (Monge-Ampère) equations bycomputing only a single derivative in the training process. There are (besides computation time) nolimitations on generating points samples as input data, and each sample point comes with machine
precision. This opens up the possibility to study various scaling laws of neural networks [22].
The training data consists of Monte-Carlo sampled points on the CY. Using a powerful theoremof Shiffman and Zelditch [23], we do not need to perform MCMC, but can construct the measure
directly by embedding the manifold into a higher-dimensional ambient space with known measure.

2

Methodology

Calabi-Yau manifolds A CY manifold, X, is a compact, complex, Ricci-flat Kähler manifold. To
describe it, we recall a few items from differential, complex and Kähler geometry. A CY geometryis specified by two nowhere-vanishing differential forms: a real Kähler 2-form J, associated to a
Kähler metric g, and a holomorphic n-form Ω, which specifies how n holomorphic coordinates are
selected among the 2n real ones. Given X, the Kähler form is not unique; however, within the same
cohomology class of any closed J 0 , there is unique representative J whose corresponding metricg has vanishing Ricci curvature by the celebrated Calabi-Yau theorem [24, 25]. On any Kähler
manifold, the metric is encoded in a Kähler potential K, and the Ricci curvature simplifies to¯ ,g = ∂ ∂K

R = ∂ ∂¯ log det g .

(1)

The unique volume form of a CY three-fold can be expressed in terms of either J or Ω. Combining
this with the cohomological equivalence of J and J 0 , we get the Monge-Ampère equation,
J ∧ J ∧ J = κ Ω ∧ Ω̄ = κ dVolCYwith¯
J = J 0 + ∂ ∂φ(2)

for a real scalar function φ, where κ is some complex constant. This second order complex non-linear
partial differential equation is satisfied iff g is the unique Ricci-flat metric on X [24, 25].
1https://github.com/pythoncymetric/cymetric2a) MA - Trainingb) MA - Validationc) transition - Validationd) dJ - Validation

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Figure 1: Fermat Quintic experiments: a) Monge-Ampère loss on training data; b)+c)+d) MongeAmpère, transition and Kähler loss on validation data; e) σ-measure f) volume and g) R-measure on
test data; h) the linear relationship between improvement in σ-measure and R-measure.
CY manifolds are commonly constructed as (intersections of) hypersurfaces, i.e. the vanishing loci ofpolynomial equations, in a higher-dimensional ambient space, A. In this paper we consider complete
intersection CY manifolds in A = Πri=1 Pni , where Pni are complex projective spaces [1, 2]. Our
TensorFlow models also work for the numerous CY hypersurfaces in toric varieties specified in
the Kreuzer-Skarke list [26] and for CY manifolds in complex dimension other than three. The
performance of our model is comparable for projective and toric ambient spaces, but is reduced for
CY manifolds that are either defined by multiple hypersurfaces, or have many shape moduli.
Point Sampling The training data consists of points on the CY X, which must be sampled uniformly with respect to a known measure. Rejection sampling is not feasible due to the high dimensionality of our problem. MCMC methods can deal with high dimensionality [27], but they require
rejecting many points, and so are not optimal. We therefore implemented a more efficient point
sampling method in NumPy [28], which employs theorems from complex geometry [8, 23], andapplies to CY spaces constructed from intersecting hypersurfaces. This results in points sampledaccording to a measure dA, which is computed as an anti-symmetric product of certain ambient spacemeasures pulled back to the CY. A function f may then be Monte-Carlo integrated over the CY using
Z
Z
dVolCY
1 X
dVolCYdVolCY f =dA f =
wi f |pi with wi =
|pi ,
(3)
dA
N
dA
X
X
i
where the measure dA and weights wi are given by the outlined procedure. Finally, the performanceof numeric metric approximations is evaluated by integrating equations (1) and (2) over X using (3).
Neural Networks The metric tensor g is a (3, 3) hermitian complex tensor. The obvious ansatz
is to let a neuralP
network (NN) freely predict nine real numbers, gpr = gNN , such that we have amapping f : R2 ni +1 → R9 . As we will show in the next section, this is not the most efficient
way to approach this problem, and one should modify the prediction of the NN with physicalintuition. Equation (2) shows that the unique Ricci-flat metric on X is given by an exact correction
to some reference Kähler metric gFS , which we may construct by pullback from the ambient space
A = Πri=1 Pni .2 We then define NNs that learn corrections to gFS such that the resulting metric gpr
is Ricci-flat: addition gpr = gFS + gNN , element-wise multiplication gpr = gFS + gFS gNN , matrix¯
product gpr = gFS + gFS · gNN , and Monge-Ampère type gpr = gFS + ∂ ∂φ.
A admits a metric, g̃FS , known as a Fubini-Study metric. The pullback gFS = Iai (g̃FS )ij̄ Ib̄j̄ (where Iai =∂zi /∂xa , and zi and xa are coordinates on A and X) yields a globally well-defined Kähler metric on X.
2

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Figure 2: Bicubic experiments: a) σ-measure b) R-measure and c) the linear relationship between
improvement in σ-measure and R-measure and d) mean slope (with one standard deviation as error)
computations for five line bundles on test data.
Loss contributions We define five loss functions that ensure that the numerical metric satisfies
the Monge-Ampère equation (2), is Kähler, i.e. dJ = 0, is well defined over patch transitions, hasvanishing Ricci curvature (1), and results in the overall volume as the reference metric gFS (this is anecessary condition to preserve the Kähler class). The total training loss is thus
L = α1 LMA + α2 LdJ + α3 Ltransition + α4 LRicci + α5 Lvol-K(4)

where αi are hyperparameters (by default set to αi = 1.0), and
LMA = 1 −LdJ =X

1 det gprκ Ω ∧ Ω̄,

LRicci = ||R||n = ∂ ∂¯ ln det gprnn||Re cijk ||n + ||Im cijk ||n ,with cijk = gij̄,k − gkj̄,i,(5)

and gij̄,k = ∂k gij̄

(6)

ijk

1 X (t)†(s)
Ltransition =gpr − T(s,t) · gpr· T(s,t)
d
(s,t)

Z
n

and Lvol-K =Z
det g̃FS −det gpr. (7)nHere the subscript n denote the Ln norms (default is n = 1 for all but LdJ which has n = 2),
T(s,t) = ∂~x(s) /∂~x(t) denote transition matrices between patches s and t, d is the number of patch
transitions, and the integral is taken over mini-batches. By virtue of the Calabi-Yau theorem, LMA
is a surrogate loss (which does not involve costly derivatives) for the Ricci-loss LRicci ; hence thelatter is disabled by default. Moreover, the φ-model is by construction Kähler with fixed size moduli,
and thus comes with a disabled LdJ and Lvol-K . It does requires two derivatives with respect to the
input parameters instead, which makes it comparable in speed to the other NNs. The derivatives withrespect to the input coordinates are computed with TensorFlow’s automatic differentiation, whichworks reliable even when training the φ−model against the Ricci-scalar with a total of five nested
gradient tapes.

3

Experiments

Fermat Quintic To illustrate our package and benchmark it against previous results we firstP4
consider the Fermat Quintic, which is defined as the zero locus of Q = i=0 zi5 in A = P4 . Figure 1
shows the results for each of the five model ansätze introduced in the previous section averaged over
five runs. Training is done with an Adam optimizer on a fully connected feed-forward network withthree hidden layers of 64 units each and GELU activation function for 100 epochs. Since we only
want to learn a correction to the known Fubini-Study metric, we initialize the NN weights with small
Gaussian variance N (0, 0.01) (N (0, 1) for the free network). The training set contains 198.000
points with a 0.1 validation split. We performed hyperparameter tuning via box searches for thenumber of hidden layers and activation function.
As seen in Figure 1, the two multiplication networks and the φ-model consistently arrive at the
Ricci-flat metric. We evaluate the performance of the NN on the established benchmarks [10, 11] ofsigma and Ricci measures, (σ, R), with an additional test set of 22.000 points in subplots e) and g).
The φ-models with lowest validation loss reach, within a few hours on a laptop, a mean accuracy of
σ = 0.0086 and R = 0.076. This is on par with k = 20 in Donaldson algorithm [11] with a training4

time of 35 years on 4.6E8 points [16]. We are the first to systematically study R, and demonstrate,
in subplot h), a linear relation, σ ≈ 0.06R, between optimization of the surrogate Monge-Ampère
equation (2) and decrease in Ricci measure R. Finally, subplot f) shows that the normalized volumeof the manifold is constant, and hence that we do not change the Kähler class during training.
Bicubic As a second example we consider the bicubic manifold X given by a homogeneous degree(3,3) polynomial in A = P2 × P2 and size moduli, t1 = 1 = t2 . For phenomenological reasons,and also to improve performance, we choose a member of this family which is invariant under a Z3
symmetry in its shape moduli. Figure 2 shows the averaged results of five experiments using thepreviously described experimental setup. As for the quintic, the three “multiplicative” NNs that arelearning corrections relative to the value of the reference metric gFS outperform the free and additive
NNs, whose output is not scaled with the reference metric and varies over several orders of magnitude.
The φ-model is particularly successful and the linear relation persists, σ ≈ 1.4R − 0.1.
For CYs with multiple size moduli ti , an important consistencyof the ti -dependence of theR √ checkab̄
NN is given by the so-called slope, which is defined as µ =gg Fab̄ for a vector√
bundle V withcurvature F . Clearly, this slope depends not only on the CY volume, encoded in g, but on thefull metric tensor that is approximated by the NN. Crucially, µ may also be computed exactly using
topological data of (X, V ), and can in fact be shown to vanish for stable, holomorphic V [29, 30].
Subtable d) of Figure 2 shows that the slope computed using the φ-model for various line bundles isin excellent agreement with the exact, topological results, both for stable and non-stable bundles.

4

Conclusion

We have presented a computational library that learns Ricci-flat metrics for a wide range of CY
manifolds. The package comprises five conceptually different NN architectures, and we compare
their efficacy in learning the metric. Two novel features compared to previous studies, are that thenumerical metrics can be computed for any value of the CY shape and size moduli, and the package
allows for direct studies of the Ricci scalar. Our package has a significantly lower entry barrier thanprevious methods. Moreover, using domain knowledge in point sampling and during training greatly
improves the performance. We emphasize that our results can be further improved by more careful
hyperparameter optimization, more sample points, or simply longer training time.

Broader Impact
We have more efficient algorithms than before, which reduces the carbon footprint of numericalcomputations for phenomenological and mathematical studies of string theory. We democratize thelearning of Calabi-Yau metrics by creating an open source package with low entry barrier, to thebenefit of researchers that have smaller computational budgets. This may in part offset the positiveenvironmental effect just mentioned, as energy consumption will grow with the number of users of
the package. The package can be used to test swampland conjectures which are actively discussed in
the string theory community. This might open up resources for other applications down the line.

Acknowledgments and Disclosure of Funding
RS and ML are funded in part by the Swedish Research Council (VR) under grant number 202003230. The work of FR is supported by startup funding from Northeastern University. Computations
were in part enabled by resources provided by the Swedish National Infrastructure for Computing(SNIC) at the HPC cluster Tetralith, partially funded by the Swedish Research Council through grant
agreement no. 2018-05973, the hydra cluster of the Physics Department at the University of Oxford,
and the computing cluster at CERN.

References[1] P. Candelas, A. M. Dale, C. A. Lutken, and R. Schimmrigk. Complete Intersection Calabi-Yau
Manifolds. Nucl. Phys., B298:493, 1988. doi:10.1016/0550-3213(88)90352-5.
5[2] Paul S. Green, Tristan Hubsch, and Carsten A. Lutken. All Hodge Numbers of All Complete
Intersection Calabi-Yau Manifolds. Class. Quant. Grav., 6:105–124, 1989. doi:10.1088/0264-9381/6/2/006.
[3] Cumrun Vafa. The String landscape and the swampland. 9 2005. URL: https://arxiv.org/
abs/hep-th/0509212, arXiv:hep-th/0509212.
[4] Anthony Ashmore and Fabian Ruehle. Moduli-dependent KK towers and the swamplanddistance conjecture on the quintic Calabi-Yau manifold. Phys. Rev. D, 103(10):106028, 2021.
arXiv:2103.07472, doi:10.1103/PhysRevD.103.106028.
[5] Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T duality. Nucl.
Phys. B, 479:243–259, 1996. arXiv:hep-th/9606040, doi:10.1016/0550-3213(96)00434-8.
[6] S. K. Donaldson. Some numerical results in complex differential geometry. arXiv Mathematics
e-prints, page math/0512625, December 2005. arXiv:math/0512625.
[7] Michael R. Douglas, Robert L. Karp, Sergio Lukic, and Rene Reinbacher. Numerical CalabiYau metrics. J. Math. Phys., 49:032302, 2008. arXiv:hep-th/0612075, doi:10.1063/1.
2888403.
[8] Volker Braun, Tamaz Brelidze, Michael R. Douglas, and Burt A. Ovrut. Calabi-Yau Metrics
for Quotients and Complete Intersections. JHEP, 05:080, 2008. arXiv:0712.3563, doi:
10.1088/1126-6708/2008/05/080.
[9] Volker Braun, Tamaz Brelidze, Michael R. Douglas, and Burt A. Ovrut. Eigenvalues andEigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds. JHEP, 07:120, 2008.
arXiv:0805.3689, doi:10.1088/1126-6708/2008/07/120.
[10] Michael R. Douglas, Robert L. Karp, Sergio Lukic, and Rene Reinbacher. Numerical solution
to the hermitian Yang-Mills equation on the Fermat quintic. JHEP, 12:083, 2007. arXiv:
hep-th/0606261, doi:10.1088/1126-6708/2007/12/083.
[11] Lara B. Anderson, Volker Braun, Robert L. Karp, and Burt A. Ovrut. Numerical HermitianYang-Mills Connections and Vector Bundle Stability in Heterotic Theories. JHEP, 06:107,
2010. arXiv:1004.4399, doi:10.1007/JHEP06(2010)107.
[12] Lara B. Anderson, Volker Braun, and Burt A. Ovrut. Numerical Hermitian Yang-Mills
Connections and Kahler Cone Substructure. JHEP, 01:014, 2012. arXiv:1103.3041,
doi:10.1007/JHEP01(2012)014.
[13] Matthew Headrick and Ali Nassar. Energy functionals for Calabi-Yau metrics. Adv. Theor. Math.
Phys., 17(5):867–902, 2013. arXiv:0908.2635, doi:10.4310/ATMP.2013.v17.n5.a1.
[14] Wei Cui and James Gray. Numerical Metrics, Curvature Expansions and Calabi-Yau Manifolds.
JHEP, 05:044, 2020. arXiv:1912.11068, doi:10.1007/JHEP05(2020)044.
[15] Michael R. Douglas, Subramanian Lakshminarasimhan, and Yidi Qi. Numerical Calabi-Yau
metrics from holomorphic networks. 12 2020. URL: https://arxiv.org/abs/2012.04797,
arXiv:2012.04797.
[16] Anthony Ashmore, Yang-Hui He, and Burt A. Ovrut. Machine Learning Calabi–Yau Metrics.
Fortsch. Phys., 68(9):2000068, 2020. arXiv:1910.08605, doi:10.1002/prop.202000068.
[17] Lara B. Anderson, Mathis Gerdes, James Gray, Sven Krippendorf, Nikhil Raghuram, and Fabian
Ruehle. Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning.
JHEP, 05:013, 2021. arXiv:2012.04656, doi:10.1007/JHEP05(2021)013.
[18] Vishnu Jejjala, Damian Kaloni Mayorga Pena, and Challenger Mishra. Neural Network Approximations for Calabi-Yau Metrics. 12 2020. URL: https://arxiv.org/abs/2012.15821,
arXiv:2012.15821.
6

[19] Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro,
Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow,
Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser,
Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek
Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal
Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete
Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Largescale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.
URL: https://www.tensorflow.org/.
[20] Henry Jin, Marios Mattheakis, and Pavlos Protopapas. Unsupervised neural networks forquantum eigenvalue problems. NeurIPS Machine Learning for Physical Sciences, 2020. URL:
https://arxiv.org/abs/2010.05075, arXiv:2010.05075.
[21] Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks:
A deep learning framework for solving forward and inverse problems involving nonlinear partial
differential equations. Journal of Computational Physics, 378:686–707, 2019.
[22] Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B. Brown, Benjamin Chess, Rewon
Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neurallanguage models. CoRR, abs/2001.08361, 2020. URL: https://arxiv.org/abs/2001.
08361, arXiv:2001.08361.
[23] Bernard Shiffman and Steve Zelditch. Distribution of zeros of random and quantum chaotic
sections of positive line bundles. Communications in Mathematical Physics, 200(3):661–683,
January 1999. arXiv:math/9803052, doi:10.1007/s002200050544.
[24] Eugenio Calabi. On Kähler Manifolds with Vanishing Canonical Class:, pages 78–89.
Princeton University Press, 2015. URL: https://doi.org/10.1515/9781400879915-006,
doi:doi:10.1515/9781400879915-006.
[25] Shing-Tung Yau. On the ricci curvature of a compact kähler manifold and the complex mongeampére equation, I. Commun. Pure Appl. Math., 31(3):339–411, 1978. doi:10.1002/cpa.
3160310304.
[26] Maximilian Kreuzer and Harald Skarke. Complete classification of reflexive polyhedra in
four-dimensions. Adv. Theor. Math. Phys., 4:1209–1230, 2002. arXiv:hep-th/0002240,
doi:10.4310/ATMP.2000.v4.n6.a2.
[27] Matthew D Hoffman, Andrew Gelman, et al. The no-u-turn sampler: adaptively setting path
lengths in hamiltonian monte carlo. J. Mach. Learn. Res., 15(1):1593–1623, 2014.
[28] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith,Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan
Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant,
Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, andTravis E. Oliphant. Array programming with NumPy. Nature, 585(7825):357–362, 2020.
doi:10.1038/s41586-020-2649-2.
[29] Simon K Donaldson. Anti self-dual yang-mills connections over complex algebraic surfaces
and stable vector bundles. Proceedings of the London Mathematical Society, 3(1):1–26, 1985.
[30] Karen Uhlenbeck and Shing-Tung Yau. On the existence of hermitian-yang-mills connections
in stable vector bundles. Communications on Pure and Applied Mathematics, 39(S1), 1986.

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8