[Paper Review] Cubics, Integrable Systems, and Calabi-Yau Threefolds

Ron Donagi, Eyal Markman|ArXiv.org|Aug 9, 1994

Algebraic Geometry and Number TheoryMathematics19 references70 citations

TL;DR

This paper constructs an analytically completely integrable Hamiltonian system (ACIHS) canonically associated to any family of Calabi-Yau threefolds, where the base is a moduli space of gauged Calabi-Yaus and fibers are Deligne cohomology groups (intermediate Jacobians). The key result is that the Yukawa cubic—encoding the infinitesimal variation of Hodge structure—is equivalent to the symplectic form on the total space, and normal functions (Abel-Jacobi images of curves) are Lagrangian, linking mirror symmetry to integrable systems via Hodge theory.

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.

Motivation & Objective

• To establish a canonical analytically completely integrable Hamiltonian system (ACIHS) for families of Calabi-Yau threefolds.
• To identify the conditions under which a family of abelian varieties (e.g., intermediate Jacobians of Calabi-Yau threefolds) admits a symplectic structure making fibers Lagrangian.
• To explore the role of the Yukawa cubic in encoding the symplectic structure of the total space of the integrable system.
• To investigate whether the infinitesimal invariants of normal functions (derived from Abel-Jacobi maps of curves) contain enough information to reconstruct curve counts in mirror symmetry.
• To propose a decomposition of the mirror conjecture into a formal part (mirror transform on integrable systems with Lagrangian multisections) and a geometric Torelli-type part.

Proposed method

• Define a symplectic form σ on the total space of a family of Calabi-Yau threefolds such that fibers (intermediate Jacobians) are Lagrangian submanifolds.
• Establish that the existence of such a symplectic structure is equivalent to a 'cubic condition' on the period map: the differential of the period map is given by contraction with a field of cubics on the tangent bundle of the base.
• Show that for Calabi-Yau threefolds, this cubic field is precisely the Yukawa cubic, arising from the infinitesimal variation of Hodge structure.
• Use action variables from symplectic geometry to recover the natural affine coordinates used in mirror symmetry.
• Characterize the infinitesimal invariant δν of a normal function ν as an element of the Jacobian ring of the Yukawa cubic, specifically in the second graded piece of the Koszul complex associated to the cubic.
• Propose that the weight function wν for curve counting should depend only on δν, not on the actual curves, and conjecture a formula wν = ∑ Nν,g λ^{g−1} such that ∑ wν = ∑ Nk,g q^k λ^{g−1}.

Experimental results

Research questions

• RQ1 Under what conditions does a family of abelian varieties admit a symplectic structure for which the fibers are Lagrangian submanifolds?
• RQ2 How is the Yukawa cubic—encoding the infinitesimal variation of Hodge structure—related to the symplectic form on the total space of the integrable system?
• RQ3 Can the partition functions of the A- and B-models in mirror symmetry be reconstructed directly from the integrable system and its Lagrangian multisections?
• RQ4 Is the infinitesimal invariant δν of a normal function sufficient to determine the curve counts (via weight functions wν) on a generic Calabi-Yau threefold?
• RQ5 To what extent can the geometry of a Calabi-Yau threefold be recovered from its associated integrable system and the associated data of Lagrangian multisections?

Key findings

• The symplectic structure on the total space of the integrable system is equivalent to the Yukawa cubic, establishing a deep link between Hodge theory and symplectic geometry in Calabi-Yau families.
• Normal functions arising from curves on a generic Calabi-Yau threefold are always Lagrangian with respect to the constructed symplectic form.
• The action variables of the integrable system correspond exactly to the natural affine coordinates on the moduli space used in the mirror correspondence.
• The infinitesimal invariant δν of a normal function ν lies in the second graded piece of the Jacobian ring of the Yukawa cubic, specifically in R^2 = Sym^2(H^{2,1})^* modulo the Jacobian ideal generated by partial derivatives of the cubic.
• The weight function wν for curve contributions is conjectured to depend only on δν, suggesting that curve counts may be encoded in the Hodge-theoretic data of normal functions.
• The problem of reconstructing curve counts from δν resembles a variational Torelli problem, and while partial results exist, the full reconstruction remains open.