Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation

Technical Summary: Interpretable Analytic Calabi–Yau Metrics via Symbolic Distillation

Problem Statement
The paper addresses the challenge of obtaining compact, analytic descriptions of Ricci-flat Calabi–Yau (CY) metrics. While Yau's theorem guarantees the existence of such metrics, it does not provide closed-form formulas. Current numerical approaches, such as Donaldson's balanced-metric program, yield accurate approximations but result in large algebraic objects (e.g., $H$-matrices) rather than structural, interpretable formulas. The authors focus on a specific, geometrically meaningful scalar observable derived from the metric: the pointwise log-determinant-ratio, $R_\psi(z) = \log(\det g_{RF}(z; \psi) / \det g_{FS}(z))$. This observable measures the deviation of the Ricci-flat metric from the Fubini–Study baseline. The central question is whether this scalar field can be described compactly using a small number of projective invariants and whether such a description remains stable across complex-structure moduli.

Methodology
The authors employ a "symbolic distillation" strategy, treating a high-accuracy numerical solution as a teacher and compressing it into a sparse symbolic formula.

1. Algebraic Teacher Construction:

   • The study focuses on the Dwork quintic family, defined by $\sum z_i^5 - 5\psi \prod z_i = 0$.
   • Donaldson's balanced-metric algorithm is implemented at polynomial degree $k=10$ (basis size 875) to generate an algebraic teacher metric ($g_{alg}$).
   • The "Ricci-flatness indicator" $\sigma(\eta)$, the standard deviation of $\eta = \det(g) |\Omega|^2 / \det(g_{FS})$, is used to validate the teacher. At the Fermat point ($\psi=0$), $\sigma \approx 0.65\%$, indicating a high-quality approximation. At deformed points ($\psi \in [0.2, 0.8]$), $\sigma \approx 8-9\%$, serving as local exploratory teachers.
   • Data is generated by sampling $10^5$ points on the quintic via intersection with random complex lines, normalized to $\sum |z_i|^2 = 1$.

2. Restricted Feature Space:

   • Instead of using the full invariant algebra, the authors restrict the regression to permutation-symmetric functions of the coordinate moduli $|z_i|^2$.
   • The candidate feature set consists of power sums $p_k = \sum |z_i|^{2k}$ and elementary symmetric polynomials $e_k$.
   • The study specifically tests the sufficiency of the two lowest-order non-trivial generators: $p_2$ and $\sigma_3 = e_3$.

3. Symbolic Regression:

   • The PySR library is used to search for compact formulas mapping $(p_2, \sigma_3)$ to the target $R_\psi$.
   • The search space includes standard arithmetic operators, logarithms, and square roots.
   • Model selection uses a Pareto criterion balancing loss and complexity.

Key Contributions and Results

1. Near-Two-Dimensionality of the Observable:

   • Ablation Studies: Within the restricted symmetric feature class, the observable $R_\psi$ is almost entirely captured by just two coordinates, $p_2$ and $\sigma_3$.
   • Adding higher-order symmetric generators ($e_4, e_5, p_3, \dots$) to a degree-3 polynomial model increases the held-out test $R^2$ by less than $10^{-3}$ (from $0.9447$ to $0.9453$).
   • This suggests that within this specific feature class, the observable is effectively two-dimensional.

2. Rational Scaffold Discovery:

   • While low-degree polynomials saturate at $R^2 \approx 0.944$, symbolic regression identifies a five-term rational-polynomial scaffold that achieves a held-out test $R^2 = 0.998$ (RMSE = 0.0205).
   • The representative form is:
     $$\log\left(\frac{\det g_{alg}}{\det g_{FS}}\right) \approx c_0 + c_1 \frac{1}{p_2^2} + c_2 \frac{\sigma_3}{p_2^3} + c_3 p_2 + c_4 \sigma_3$$
   • The inverse-power terms ($1/p_2^n$) are critical; removing them degrades performance, and they appear in 100% of independent regression runs. These terms are bounded on the manifold ($p_2 \ge 1/5$) and likely encode extreme-coordinate behavior.

3. Moduli Stability and Refitting:

   • The functional form of the scaffold remains valid across the moduli range $\psi \in [0, 0.8]$.
   • By fixing the functional form discovered at $\psi=0$ and refitting only the five scalar coefficients via least-squares at each new $\psi$, the model maintains $R^2 \ge 0.947$ across the range.
   • The coefficients vary smoothly, with $c_1$ decreasing monotonically and $c_2$ changing sign between $\psi=0.6$ and $0.8$.

4. Robustness Checks:

   • Multi-seed validation: 10 independent regression runs consistently recover the $(p_2, \sigma_3)$ pair with a mean $R^2 = 0.997$.
   • Cross-$k$ validation: The scaffold trained on $k=10$ data generalizes to $k=6$ and $k=8$ teachers (after offset correction), suggesting the form reflects the observable's structure rather than finite-$k$ artifacts.
   • Permutation tests: Shuffling $p_2$ or $\sigma_3$ causes $R^2$ to drop to negative values, confirming their predictive necessity.

Significance and Scope
The paper claims a structural rather than solver-level contribution. It does not provide a closed-form formula for the full Ricci-flat metric or Kähler potential. Instead, it demonstrates that for the specific observable $R_\psi$, the complex geometry of the Dwork quintic can be compressed into a compact, interpretable rational expression involving only two symmetric invariants.

• Interpretability: The resulting formula is small enough to inspect term-by-term, offering insight into how the metric deviation scales with coordinate distribution ($p_2$) and three-way correlations ($\sigma_3$).
• Efficiency: Once the scaffold is established, evaluating the observable is $10,000\times$ faster than direct Donaldson $H$-matrix evaluation.
• Limitations: The results are bounded by the finite-$k$ teacher used for distillation. The authors explicitly state they do not claim the exact Ricci-flat metric is two-dimensional, nor do they validate the model against metric-sensitive physical quantities (like curvature integrals or physical Yukawa couplings) which require the full metric tensor. The holomorphic Yukawa coupling $\kappa_{111}=5$ is reproduced only as a normalization check, not as a metric-sensitive validation.

In summary, the work establishes that a specific metric-derived scalar observable on the Dwork family admits a highly compact, stable, and interpretable symbolic description within a restricted symmetric feature space, bridging the gap between numerical accuracy and structural understanding for that specific observable.