Cubics, Integrable Systems, and Calabi-Yau Threefolds

TL;DR

This paper constructs a completely integrable Hamiltonian system associated with families of Calabi-Yau threefolds, linking geometric moduli, cohomology, and mirror symmetry through novel symplectic and Lagrangian structures.

Contribution

It introduces a new integrable system framework connecting Calabi-Yau moduli, cohomology, and mirror symmetry, with properties relating Abel-Jacobi images and Yukawa cubic.

Findings

Multivalued sections are Lagrangian.

Affine coordinates serve as action variables.

Yukawa cubic relates to the symplectic structure.

Abstract

In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds. The base of this system is a moduli space of gauged Calabi-Yaus in the family, and the fibers are Deligne cohomology groups (or intermediate Jacobians) of the threefolds. This system has several interesting properties: the multivalued sections obtained as Abel-Jacobi images, or ``normal functions'', of a family of curves on the generic variety of the family, are always Lagrangian; the natural affine coordinates on the base, which are used in the mirror correspondence, arise as action variables for the integrable system; and the Yukawa cubic, expressing the infinitesimal variation of Hodge structure in the family, is essentially equivalent to the symplectic structure on the total space.