Cubics, Integrable Systems, and Calabi-Yau Threefolds

Eyal Markman; Ron Donagi

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Cubics, Integrable Systems, and Calabi-Yau Threefolds

Ron Donagi, Eyal Markman

alg-geom · Submitted 9 Aug 1994

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.06 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds.	contribution 0.99	0.95	0.78	asserted 1.00
s1	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.	contribution 0.82	0.78	0.09	asserted 1.00
s2	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.	result 0.94	0.81	0.19	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 1.00.

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The paper

Original 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.

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References

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
math.AG 2008-11 conditional novelty 9.0 of 10

The paper constructs motivic Donaldson-Thomas invariants for 3d Calabi-Yau categories with stability structures and proves a multiplicative wall-crossing formula, conditional on an integral identity.
Mirror Counts of Spectral Curves
math.SG 2026-09 conditional novelty 8.0 of 10

Rational nodal curves on toric surfaces are counted by rational rulings of a Legendrian link, unifying six independent definitions.
Geometric constraints on the space of N=2 SCFTs I: physical constraints on relevant deformations
hep-th 2015-05 conditional novelty 7.0 of 10

A systematic scan of rank-1 planar Coulomb branch geometries yields 28 physically consistent deformation patterns, 17 of them new, conditional on an unproven 'no dangerously irrelevant operators' conjecture.

How the literature uses it

9 arXiv papers cite this in 18 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 8 sentences without a confident label

Uses its method or tool 1 paper

We outline a proof following [ref] ; for the original proof see [ref] .

Aspects of Calabi-Yau Integrable and Hitchin Systems · Compact Calabi–Yau integrable systems · 0.80

Compares with it 1 paper

With these methods we construct Calabi–Yau integrable systems from complete families of compact Calabi–Yau threefolds in a different way than originally done by Donagi–Markman [ref] .

Aspects of Calabi-Yau Integrable and Hitchin Systems · Structure of the notes · 0.98

Cites it as background 6 papers

The low-energy theory around a generic point on a rank-$r$ CB is captured by a fibration of rank-$r$ abelian varieties [ref] .

Completely (iso-)split scale-invariant Coulomb branch geometries are isotrivial · Introduction · 0.88
The SK structure of $$ is captured by a holomorphic algebraically completely integrable Hamiltonian system [ref] whose complex phase space we call the total space, $X$, of the CB.

Flavor symmetries and the topology of special K\"ahler structures at rank 1 · Generalization to higher ranks · 0.89
Instead, the natural invariant object is the holomorphic 2-form [ref] [equation] defined on the total space of the elliptic fibration.

Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated · Seiberg Witten curve and form · 0.80

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