Calabi-Yau Metrics with Kähler Moduli Dependence

Abstract

We present a method to construct approximate analytic expressions for Ricci-flat Kähler metrics on Calabi-Yau threefolds with explicit dependence on the Kähler moduli. Our strategy combines numerical data obtained from machine learning with an explicit analytic Ansatz for the Kähler potential and symbolic regression methods. Specifically, we use neural networks to learn the Kähler potential at selected points in Kähler moduli space, fit this data to analytic expressions with Kähler moduli-dependent parameters, and determine an analytic form of these coefficients as functions of the Kähler moduli using symbolic regression. In this way, we reconstruct closed-form approximations to the Ricci-flat metric that retain explicit Kähler-moduli dependence. We apply this method to two Calabi-Yau threefolds with $h^{1,1}=2$, namely a bicubic hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$ and a bi-degree $(2,4)$ hypersurface in $\mathbb{P}^1 \times \mathbb{P}^3$, both of which admit nontrivial discrete symmetry groups that simplify the structure of the metric. In both cases, the resulting analytic expressions reproduce the numerically learned Kähler potentials with percent-level accuracy and respect the discrete symmetry of the underlying manifold. Our results represent a concrete bridge between purely numerical results for Calabi-Yau metrics and analytic constructions, opening the door to a systematic study of their dependence on Kähler moduli.

Figures (6)

• Figure 1: Left: final $\sigma$-measure achieved by the neural network as a function of the moduli ratio $t_1/t_2$. Right: average percentage deviation $\langle |K_N-K|/K_N\rangle$ as a function of $t_1/t_2$, comparing different ansätze for the Kähler potential. The blue points correspond to the pure Fubini--Study potential ($\phi=0$), while the orange and green points show fits obtained from the Ansatz in Eq. \ref{['eq:phi']} with $(k_1,k_2)=(1,1)$ and $(k_1,k_2)=(2,2)$, respectively. Increasing the truncation order leads to a significant improvement in agreement with the numerically learned Kähler potential. Averaged over $t_1/t_2$, the deviations are $17.1\%$, $5.3\%$, and $1.8\%$ for the three cases, respectively.
• Figure 2: Best-fit coefficients as functions of $t_1/t_2$ for the $(k_1,k_2)=(2,2)$ truncation. Out of the $36\times 36$ entries of the Hermitian matrix $\alpha_{IJ}$, only $36$, the diagonal entries, are non-vanishing within numerical accuracy; these naturally organise into eight groups, which are displayed in different colours. Coefficients corresponding to monomials that belong to the same $H$-singlet are shown with the same colour and labelled as $\alpha_0,\ldots,\alpha_7$. The left panel displays coefficients associated with singlets invariant under $x \leftrightarrow y$, while the remaining panels show pairs of coefficients associated with singlets that are exchanged under $x \leftrightarrow y$, exhibiting the expected crossing behaviour at $t_1=t_2$.
• Figure 3: Best-fit numerical values of the coefficients $\alpha_i$ in the Ansatz \ref{['eq:phiGinv']} as functions of the moduli ratio $t_1/t_2$, obtained for $(k_1,k_2)=(2,2)$.
• Figure 4: Left: final $\sigma$-measure as a function of the moduli ratio $t_1/t_2$. Right: average relative deviation $\langle |K_N-K|/K_N\rangle$ between the analytic Kähler potential $K$ and the numerically learned potential $K_N$, shown as a function of $t_1/t_2$ for different truncations of the Ansatz. The blue points correspond to the pure Fubini--Study potential ($\phi=0$), while the remaining points correspond to increasing values of $(k_1,k_2)$ as indicated in the legend. Averaged over all sampled values of $t_1/t_2$, the deviations are $18\%$, $9\%$, $8.5\%$, and $1.9\%$, in the order given in the legend.
• Figure 5: Monomial coefficients $\alpha_i$ as functions of $t_1/t_2$ in the $(k_1,k_2)=(2,2)$ truncation. The left panel displays the diagonal coefficients, while the other two panels present the leading off-diagonal ones. The black lines corresponds to the analytic expression obtained using symbolic regression.