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Level Crossings, Attractor Points and Complex Multiplication
High Energy Physics - Theory · Submitted 31 Mar 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication.
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The abstract, sentence by sentence
We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau $n$-folds in P^{n+1}. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the the distance from degeneration points in complex structure moduli space.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau $n$-folds in P^{n+1}. | contribution 0.96 | 0.90 | 0.04 | asserted 0.93 |
| s1 | It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. | background 1.00 | 0.04 | 0.03 | asserted 0.99 |
| s2 | We identify the cause for this behavior for the torus. | contribution 0.78 | 0.93 | 0.06 | asserted 1.00 |
| s3 | We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. | result 0.99 | 0.93 | 0.35 | asserted 1.00 |
| s4 | We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. | interpretation 0.66 | 0.21 | 0.27 | speculative 1.00 |
| s5 | To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. | method 0.68 | 0.96 | 0.14 | asserted 0.97 |
| s6 | To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the the distance from degeneration points in complex structure moduli space. | method 0.97 | 0.95 | 0.02 | asserted 0.96 |
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The paper
Original abstract
We study the complex structure moduli dependence of the scalar Laplacian eigenmodes for one-parameter families of Calabi-Yau $n$-folds in P^{n+1}. It was previously observed that some eigenmodes get lighter while others get heavier as a function of these moduli, which leads to eigenvalue crossing. We identify the cause for this behavior for the torus. We then show that at points in a sublocus of complex structure moduli space where Laplacian eigenmodes cross, the torus has complex multiplication. We speculate that the generalization to arbitrary Calabi-Yau manifolds could be that level crossing is related to rank one attractor points. To test this, we compute the eigenmodes numerically for the quartic K3 and the quintic threefold, and match crossings to CM and attractor points in these varieties. To quantify the error of our numerical methods, we also study the dependence of the numerical spectrum on the quality of the Calabi-Yau metric approximation, the number of points sampled from the Calabi-Yau variety, the truncation of the eigenbasis, and the the distance from degeneration points in complex structure moduli space.
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References
How the literature uses it
10 arXiv papers cite this in 18 sentences
Papers citing it · 11 sentences without a confident label
Cites it as support 2 papers
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This reinforces the observation of [ref] which relates Laplacian eigenmodes and attractor points.
Attractors, Geodesics, and the Geometry of Moduli Spaces · Abelian varieties · 0.97
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This is consistent with the broader numerical lesson that spectra and metric dependent observables are sensitive to the quality and distribution of the point cloud [ref] .
What to do with a Ricci-flat Calabi--Yau metric? · Improvements from point selection · 0.96
Cites it as background 4 papers
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This is perhaps the simplest example for a bona fide CY threefold and has served as a prime example in the context of geodesics [ref] , attractors [ref] , and marked moduli spaces [ref] .
Attractors, Geodesics, and the Geometry of Moduli Spaces · The one-parameter family of Quintics · 0.97
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Beyond many excellent programs of work seeking to numerically construct the elusive Ricci-flat Calabi-Yau metrics [ref] enabling further steps in string phenomenology [ref] , there has been a variety of papers finding real efficacy of machine learning in predicting Calabi-Yau topological properties.
Calabi-Yau Four/Five/Six-folds as $\mathbb{P}^n_\textbf{w}$ Hypersurfaces: Machine Learning, Approximation, and Generation · Introduction · 0.84
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More details can be found in, for example, [ref] .
Numerical spectra of the Laplacian for line bundles on Calabi-Yau hypersurfaces · Numerical results · 1.00
What it cites · 26 sentences without a confident label
Builds on 11 papers
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Building on earlier work [ref] , neural networks were developed to approximate the Ricci-flat metric on CY manifolds [ref] .
Numerical Calabi-Yau metrics · Introduction · 0.81
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Building on earlier work [ref] , neural networks were developed to approximate the Ricci-flat metric on CY manifolds [ref] .
Some numerical results in complex differential geometry · Introduction · 0.81
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Building on earlier work [ref] , neural networks were developed to approximate the Ricci-flat metric on CY manifolds [ref] .
Neural Network Approximations for Calabi-Yau Metrics · Introduction · 0.81
Uses the method or tool of 5 papers
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To generate points on the Calabi-Yau which are distributed according to the pullback of the Fubini-Study (FS) metric on the ambient $P^n$, we use the techniques detailed in [ref] and implemented in the cymetric package [ref] .
Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning · Point generation · 1.00
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To generate points on the Calabi-Yau which are distributed according to the pullback of the Fubini-Study (FS) metric on the ambient $P^n$, we use the techniques detailed in [ref] and implemented in the cymetric package [ref] .
Learning Size and Shape of Calabi-Yau Spaces · Point generation · 1.00
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To generate points on the Calabi-Yau which are distributed according to the pullback of the Fubini-Study (FS) metric on the ambient $P^n$, we use the techniques detailed in [ref] and implemented in the cymetric package [ref] .
Numerical Metrics for Complete Intersection and Kreuzer-Skarke Calabi-Yau Manifolds · Point generation · 1.00
Uses data or values from 1 paper
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For the value of the crossings, we simply take some of the ones observed in [ref] , which appear at around [equation] cf. Figure [ref].
Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold · Crossing and attractor points for the quintic · 0.99
Compares with 1 paper
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It is very suggestive that a similar effect explains this behavior observed in [ref] for the quintic as well: There, $ψ$ was varied along a geodesic with $ψ∈R$, which connects the conifold point to the large complex structure point.
Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold · Eigenvalue analysis · 0.90
Cites as support 1 paper
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Our motivation comes explaining explain the behavior of the eigenmodes of the Laplace operator on the quintic observed in [ref] : some eigenvalues become heavier and others lighter as we vary the complex structure, such that eigenmodes cross along special loci in complex structure moduli space.
Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold · Conclusion and Outlook · 0.99
Cites as background 7 papers
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In his seminal work [ref] , Moore established a connection between Calabi-Yau compactifications and number theory.
Arithmetic and Attractors · Introduction · 0.99
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On the other hand, the swampland distance conjecture (SDC) [ref] , which has received a lot of attention recently, states that the mass gap of the eigenspectrum closes exponentially fast as a function of the geodesic distance between two points in moduli space.
On the Geometry of the String Landscape and the Swampland · Introduction · 0.94
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Complex Multiplication for higher-dimensional Calabi-Yau manifolds have been studied in [ref] .
Rational Conformal Field Theories and Complex Multiplication · Generalization of CM for Calabi–Yau n-folds · 0.97
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