A Calabi-Yau-to-Curve Correspondence for Feynman Integrals

TL;DR

The paper introduces a Calabi–Yau–to–curve correspondence that reinterprets the maximal cut of the equal‑mass four‑loop banana integral as a period of a genus‑two curve. It develops a holomorphic realignment using polarised holomorphic intermediate Jacobians to construct a genus‑two curve whose period matrix reproduces the same Picard–Fuchs operator acting on the CY periods, thereby unifying CY and curve perspectives for Feynman integrals. The construction relies on the Hulek–Verrill CY family and mirror symmetry to establish integral periods, then uses Weil and Griffiths Jacobians to derive a holomorphic genus‑two curve via a polarised holomorphic intermediate Jacobian, with detailed explicit steps, theta‑function representations, Igusa invariants, and transcendentality arguments. The results not only provide a concrete example of CY‑to‑curve correspondence but also outline criteria and potential generalizations for applying the method to other one‑ and two‑parameter CY families in perturbative quantum field theory. The approach offers a motivic viewpoint and practical benefits by relating CY data to well‑studied higher‑genus curve moduli and their modular properties, enabling new avenues for analytic and numerical control of Feynman integrals.

Abstract

It has long been known that the maximal cut of the equal-mass four-loop banana integral is a period of a family of Calabi-Yau threefolds that depends on the kinematic variable $z=m^2/p^2$. We show that it can also be interpreted as a period of a family of genus-two curves. We do this by introducing a general Calabi-Yau-to-curve correspondence, which in this case locally relates the original period of the family of Calabi-Yau threefolds to a period of a family of genus-two curves that varies holomorphically with the kinematic variable $z$. In addition to working out the concrete details of this correspondence for the equal-mass four-loop banana integral, we outline when we expect a correspondence of this type to hold.

Figures (7)

• Figure 1: A flow chart visualising the construction of the Calabi--Yau-to-Curve correspondence presented in this paper.
• Figure 2: The four-loop banana graph.
• Figure 3: A diagram summarising, for $m=1,2$ the construction of the map $\Phi$ realising the Calabi--Yau-to-curve correspondence. The map $\pi^{-1}$ gives a Calabi--Yau manifold $Y_z$ given a point $z$ in the local moduli space $\Delta$. Here $J_2^W$ denotes both the map that, given a Calabi--Yau manifold $Y_z$, gives the Weil intermediate Jacobian $J_2^W(Y_z)$, but also the composition $J_2^W \circ \pi^{-1} : \Delta \to \mathcal{S}_{m+1}$. $J_1$ denotes the map that maps a stable curve of genus-$(m{+}1)$ to its intermediate Jacobian.
• Figure 4: The Hodge numbers $h^{p,q}$ for the Calabi--Yau manifolds $Y$ and $Y^{\mathrm{mirror}}$.
• Figure 5: Plots showing the smallest eigenvalue $\lambda_{\text{min}}$ of $\text{Im}(\bm{H}(z))$, visualising the range of values of $z$, for which the polarised holomorphic intermediate Jacobian is positive definite and therefore in $\mathcal{H}_2$. Left: Contour plot over the complex $z$-plane. The red points mark the physical singularities $z=0$ and $z=\frac{1}{25}$, and the hatching corresponds to the region $\lambda_{\text{min}}>1$. The red lines denote branch cuts of $\bm{H}(z)$. In the uncoloured region we have $\lambda_{\text{min}}< 0$ and $\text{Im}(\bm{H}(z))$ is indefinite. The coloured area thus shows the region $U$ where the second polarised holomorphic intermediate Jacobian $J_2^{\Delta_{\mathbb{R}}}(Y_z)$ is defined. Right: Values of $\lambda_{\text{min}}$ along the real $z$-axis. At the physical singularity $z=\tfrac{1}{25}$, the eigenvalue $\lambda_{\text{min}}$ vanishes, corresponding to the boundary of $\mathcal{H}_2$.