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

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.