Mapping kinematic parameters to complex-structure moduli in banana integrals

Determine the explicit relation between the kinematic parameters (internal masses and external momenta) of multi-loop banana Feynman integrals and the complex-structure moduli of the associated Calabi–Yau manifolds, including a general characterization beyond special cases such as elliptic curves.

Background

In the banana integral family, the maximal cut defines a family of Calabi–Yau manifolds whose moduli are functions of kinematic parameters. Understanding this map is crucial for relating physical inputs to geometric structures and for deriving differential equations.

The authors stress that, even in the simplest elliptic case, this relation involves modular functions and is in general unknown, hindering systematic geometric analysis of these integrals.