Strominger–Yau–Zaslow

Strominger–Yau–Zaslow
Name	Strominger–Yau–Zaslow conjecture
Field	Differential geometry, Algebraic geometry, Mathematical physics
Proposer	Andrew Strominger, Shing-Tung Yau, Eric Zaslow
Year	1996

Strominger–Yau–Zaslow is a conjecture in differential geometry and mathematical physics proposing a geometric mechanism for mirror symmetry by relating Calabi–Yau manifolds to dual torus fibrations, invoking special Lagrangian submanifolds and dualities from string theory. It connects ideas from symplectic geometry, complex algebraic geometry, and topological string theory, and has stimulated work across research on the Calabi conjecture, homological mirror symmetry, and tropical geometry. The conjecture shaped subsequent advances involving the moduli of special Lagrangian cycles, the SYZ picture of T-duality, and constructions by Kontsevich and Fukaya.

Background and Motivation

The conjecture arose from efforts to explain the mirror symmetry phenomena observed in computations by Philip Candelas, Brian Greene, Paul Aspinwall, and Cumrun Vafa during studies of Calabi–Yau manifold compactifications in superstring theory, influenced by results such as the Calabi conjecture proved by Shing-Tung Yau and the development of T-duality in Type II string theory. It sought to give a geometric origin for the exchange of Hodge numbers noted in examples like the Quintic threefold calculations by Candelas et al. and to relate enumerative predictions from mirror symmetry to counts of holomorphic disks appearing in work by Kontsevich and Gromov–Witten theory. The proposal builds on structures from special Lagrangian submanifold theory studied by Harvey and Lawson and on ideas from the SYZ paper by its proposers.

Statement of the Conjecture

The conjecture asserts that a Calabi–Yau manifold admitting a Ricci-flat Kähler metric from the Calabi–Yau theorem should carry a special Lagrangian torus fibration whose generic fibers are n-dimensional tori, and that the mirror Calabi–Yau arises as the dual torus fibration via fiberwise T-duality or Fourier–Mukai type transform as envisioned by Strominger, Yau, and Zaslow. It predicts that mirror pairs such as those studied by Candelas, Green, and Hubsch correspond to dual integral affine manifolds with singularities studied in the context of affine geometry and tropical geometry. The statement links to categorical formulations like Homological mirror symmetry proposed by Maxim Kontsevich and to symplectic duality considerations by Paul Seidel and Kenji Fukaya.

Mathematical Framework and Definitions

Key definitions include Calabi–Yau manifold, Ricci curvature, Kähler manifold, special Lagrangian submanifold, Maslov index, and integral affine manifold. The framework uses the Calabi–Yau theorem of Yau for existence of Kähler–Einstein metrics, the construction of moduli via Torelli theorem analogues, and analytic methods developed by Siu, Arezzo–Pacard, and Donaldson. The conjecture employs the Fourier–Mukai transform from Derived category of coherent sheaves theory introduced by Muk90 and related to equivalences in Derived categories studied by Bondal and Orlov. The role of singular fibers invokes techniques from Morse theory and resolution methods linked to Gross–Siebert logarithmic geometry and deformation theory as developed by Deligne and Illusie.

Key Results and Developments

Developments include partial constructions of SYZ fibrations for toric Calabi–Yau varieties by Guillemin and Abreu inspired work, the Gross–Siebert program reconstructing mirrors via tropical and logarithmic techniques by Mark Gross and Bernd Siebert, and the use of non-archimedean geometry by Vladimir Berkovich and Maxim Kontsevich in wall-crossing and scattering diagrams. Work on special Lagrangian existence and singularity analysis by Neal Katzarkov, Tommaso Pacini, Dominic Joyce, and John Loftin advanced understanding of singular fibers and moduli. Homological formulations have been advanced by Kontsevich, Fukaya, Seidel, and Huybrechts, relating derived categories of coherent sheaves to Fukaya categories via Strominger–Yau–Zaslow philosophy. Results on SYZ in toric mirror symmetry were obtained by Fang, Liu, Zhou, and others using open Gromov–Witten invariants studied by Cho–Oh and Auroux.

Examples and Applications

Explicit SYZ-type constructions appear for elliptic K3 surfaces studied by Kodaira and in toric cases like mirrored pairs from Batyrev dual polytopes, with computations connecting to Gromov–Witten invariants and enumerative predictions first verified for the quintic threefold by Candelas. The Gross–Siebert reconstruction yields mirrors for degenerations considered by Friedman and Morrison–Plesser, while non-archimedean and tropical approaches produce examples related to Mikhalkin curves and amoebas used in Nishinou–Nohara–Ueda type analyses. Applications extend to studies of moduli spaces appearing in Donaldson–Thomas theory, wall-crossing formulae of Kontsevich–Soibelman, and developments in topological string theory pursued by Gopakumar–Vafa and Witten.

Proofs, Progress, and Open Problems

No general proof exists; progress divides into existence of special Lagrangian fibrations, reconstruction of mirrors from degenerations, and categorical equivalences implied by SYZ. Open problems include establishing existence and regularity of special Lagrangian torus fibrations for compact Calabi–Yau threefolds as sought by Joyce and analytic challenges tied to the Monge–Ampère equation and singularity formation studied by Yau and Siu. Further challenges involve matching instanton corrections within the Gross–Siebert program to enumerative invariants computed via Gromov–Witten theory and proving full equivalence with Homological mirror symmetry conjectured by Kontsevich. The SYZ conjecture continues to guide interactions among researchers at institutions like Institute for Advanced Study, Clay Mathematics Institute, Princeton University, and Harvard University in pursuits combining algebraic, symplectic, and physical perspectives.

Category:Conjectures in geometry