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The DNA of Calabi-Yau Hypersurfaces

Nate MacFadden, Andreas Schachner, Elijah Sheridan

TL;DR

We address the challenge of exploring the enormous Calabi–Yau landscape arising from the Kreuzer–Skarke database by introducing a DNA encoding of non-two-face-equivalent FRSTs (NTFE FRSTs) and applying Bayesian-optimized Genetic Algorithms to optimize observables in Type II string compactifications. The method maps two-face triangulations to full FRSTs via an extension procedure, enabling efficient search while avoiding trivial equivalences implied by Wall's theorem. Across polytopes with $h^{1,1}=23,60,491$, the GA outperforms random sampling, MCMC, and Simulated Annealing in maximizing Calabi–Yau volume and tuning axion-related observables, with hyperparameters tuned by Bayesian optimization. This approach reduces redundancies in the triangulation-to-CY map and demonstrates tractability for large KS polytopes, suggesting future work to co-optimize polytopes and moduli spaces for phenomenology.

Abstract

We implement Genetic Algorithms for triangulations of four-dimensional reflexive polytopes which induce Calabi-Yau threefold hypersurfaces via Batyrev's construction. We demonstrate that such algorithms efficiently optimize physical observables such as axion decay constants or axion-photon couplings in string theory compactifications. For our implementation, we choose a parameterization of triangulations that yields homotopy inequivalent Calabi-Yau threefolds by extending fine, regular triangulations of two-faces, thereby eliminating exponentially large redundancy factors in the map from polytope triangulations to Calabi-Yau hypersurfaces. In particular, we discuss how this encoding renders the entire Kreuzer-Skarke list amenable to a variety of optimization strategies, including but not limited to Genetic Algorithms. To achieve optimal performance, we tune the hyperparameters of our Genetic Algorithm using Bayesian optimization. We find that our implementation vastly outperforms other sampling and optimization strategies like Markov Chain Monte Carlo or Simulated Annealing. Finally, we showcase that our Genetic Algorithm efficiently performs optimization even for the maximal polytope with Hodge numbers $h^{1,1} = 491$, where we use it to maximize axion-photon couplings.

The DNA of Calabi-Yau Hypersurfaces

TL;DR

We address the challenge of exploring the enormous Calabi–Yau landscape arising from the Kreuzer–Skarke database by introducing a DNA encoding of non-two-face-equivalent FRSTs (NTFE FRSTs) and applying Bayesian-optimized Genetic Algorithms to optimize observables in Type II string compactifications. The method maps two-face triangulations to full FRSTs via an extension procedure, enabling efficient search while avoiding trivial equivalences implied by Wall's theorem. Across polytopes with , the GA outperforms random sampling, MCMC, and Simulated Annealing in maximizing Calabi–Yau volume and tuning axion-related observables, with hyperparameters tuned by Bayesian optimization. This approach reduces redundancies in the triangulation-to-CY map and demonstrates tractability for large KS polytopes, suggesting future work to co-optimize polytopes and moduli spaces for phenomenology.

Abstract

We implement Genetic Algorithms for triangulations of four-dimensional reflexive polytopes which induce Calabi-Yau threefold hypersurfaces via Batyrev's construction. We demonstrate that such algorithms efficiently optimize physical observables such as axion decay constants or axion-photon couplings in string theory compactifications. For our implementation, we choose a parameterization of triangulations that yields homotopy inequivalent Calabi-Yau threefolds by extending fine, regular triangulations of two-faces, thereby eliminating exponentially large redundancy factors in the map from polytope triangulations to Calabi-Yau hypersurfaces. In particular, we discuss how this encoding renders the entire Kreuzer-Skarke list amenable to a variety of optimization strategies, including but not limited to Genetic Algorithms. To achieve optimal performance, we tune the hyperparameters of our Genetic Algorithm using Bayesian optimization. We find that our implementation vastly outperforms other sampling and optimization strategies like Markov Chain Monte Carlo or Simulated Annealing. Finally, we showcase that our Genetic Algorithm efficiently performs optimization even for the maximal polytope with Hodge numbers , where we use it to maximize axion-photon couplings.
Paper Structure (13 sections, 18 equations, 9 figures, 1 table)

This paper contains 13 sections, 18 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Calabi-Yau Hypersurfaces from Reflexive Polytopes
  3. Generating two-face inequivalent triangulations
  4. Equivalences beyond two-face triangulations
  5. Genetic Algorithms for Polytope Triangulations
  6. Genetic Algorithms
  7. Implementation for polytope triangulations
  8. Hyperparameter tuning from Bayesian optimization
  9. Applications to String Compactifications
  10. Maximizing the volume --- $h^{1,1}=23$
  11. Optimizing axion decay constants --- $h^{1,1}=60$
  12. Maximizing axion-photon couplings --- $h^{1,1}=491$
  13. Conclusions

Figures (9)

  • Figure 1: Left: Total number of NTFE FRSTs, FRST classes, Calabi-Yau (CY) classes and FRSTs for all polytopes at $h^{1,1}\leq 7$. Right: Ratio of different equivalence classes of FRSTs over the number of FRSTs.
  • Figure 2: Distance Comparison. Left: Mean and maximum Calabi-Yau volume as a function of flip (top) and Hamming (bottom) distance. The fact that the mean target function value correlates with distance supports the idea that the search space of DNA exhibits a funnel-like topography near the global maximum, motivating the use of a GA. Right: Calabi-Yau volume as a function of flip distance for fixed Hamming distance. We see that both forms of distance experience correlation with the average behavior of target function when the other is held fixed, suggesting that the target function experience ensemble-level continuity with both Hamming and flip distance independently.
  • Figure 3: Distribution of values for each generation averaged over $25$ runs for a population size of 100 for the same choice of optimized hyperparameters, but random initial populations. The dashed line shows the full distribution of $\mathcal{V}$ for all NTFE FRSTs of $\Delta^\circ$.
  • Figure 4: Performance comparison between different sampling and optimization algorithms when maximizing the Calabi-Yau volume. In particular, we plot the average best encountered target $\log_{10}(\mathcal{V})$ value as a function of unique encountered DNA. We see the DNA encoding of CYs enables many canonical optimization methods to noticeably outperform brute force search, with the GA and Best-First Search performing best. Moreover, we witness the non-trivial effect of Bayesian Optimization of hyperparameters.
  • Figure 5: Comparison between number of unique DNA encountered before one of the two maximal values of $\log_{10}(\mathcal{V})$ is achieved for the two best optimization methods --- genetic algorithms and best-first search --- along with random sampling of DNA, for comparison purposes.
  • ...and 4 more figures