homological mirror symmetry

homological mirror symmetry
Name	Homological mirror symmetry
Field	String theory, Algebraic geometry, Symplectic geometry
Introduced	1994
Introduced by	Maxim Kontsevich
Notable for	Bridge between Calabi–Yau manifold categories and Fukaya category

homological mirror symmetry

Homological mirror symmetry (HMS) is a conjectural relationship proposing an equivalence between categories arising in Symplectic geometry and Algebraic geometry. Formulated in 1994 by Maxim Kontsevich for Calabi–Yau manifold pairs motivated by mirror symmetry, HMS connects objects from the Fukaya category on one side to objects in the derived category of coherent sheaves on the mirror, tying together methods from String theory, Derived category theory, Floer homology, and Category theory.

Introduction

HMS arose from observations in mirror symmetry relating counts of Gromov–Witten invariants on Calabi–Yau manifolds to period integrals in Hodge theory on mirror manifolds, with early influence from Philip Candelas, Xenia de la Ossa, Paul S. Green, and Linda Parkes. The conjecture reframes these numerical predictions as an equivalence between the Fukaya category (symplectic side) and the bounded derived category of coherent sheaves on the mirror (algebraic side), linking techniques developed by Andrei Floer, Mikhail Gromov, Kyoji Saito, and David Mumford.

Background and mathematical context

HMS synthesizes structures from Symplectic geometry, Algebraic geometry, and Mathematical physics. The symplectic side invokes constructions of Fukaya category by contributors such as Kenji Fukaya, Paul Seidel, and Dominic Joyce, relying on Lagrangian submanifolds and Floer homology methods stemming from Andrei Floer and Yasha Eliashberg. The algebraic side uses derived categories of coherent sheaves developed through work by Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, and later formalized by Bernhard Keller and Jonathan L. Smith. The conjecture also interacts with Kontsevich–Soibelman wall-crossing phenomena, Stability conditions on triangulated categories by Tom Bridgeland, and moduli problems studied by David Mumford and Nicolaas H. Kuiper.

Statement of the conjecture

Kontsevich proposed that for a mirror pair (X, X^vee), there is an equivalence between the derived Fukaya category D^πFuk(X) and the bounded derived category D^bCoh(X^vee). The equivalence is expected to be compatible with additional structures: A∞-category structures from Stasheff polytope ideas, Hochschild cohomology identifications as in work of Maxim Kontsevich and Yuri Manin, and the identification of Gromov–Witten invariants with Period integrals connected to Variation of Hodge structure by researchers like Claire Voisin and Phillip Griffiths. Subsequent formulations relate HMS to T-duality studied by C. M. Hull and Andrew Strominger.

Key examples and computations

Early verifications include the case of elliptic curves treated by Alexander Polishchuk and Eric Zaslow, and the two-torus examples associated with work of Paul Seidel and Kontsevich himself. HMS has been established for certain K3 surfaces via contributions by Maxim Kontsevich, Paul Seidel, Ivan Smith, and Yukinobu Toda, and for Toric variety mirrors following work by Denis Auroux, Mark Gross, Paul Hacking, and Bernd Siebert. Computations using Matrix factorizations by Dmitry Orlov and Michael Khovanov have confirmed HMS in Landau–Ginzburg contexts studied by Jim Stasheff and Kentaro Hori. Important explicit examples include work on Fano variety mirrors by Dmitry Auroux and enumerative checks related to predictions by Philip Candelas and collaborators.

Techniques and proofs

Proof techniques mix analytic, algebraic, and categorical tools: virtual fundamental cycles and Kuranishi structures developed by Kenji Fukaya and collaborators; pseudoholomorphic curve methods from Gromov; obstruction theory from Kontsevich–Soibelman frameworks; and homological algebra methods stemming from Bernhard Keller and Amnon Neeman. Gluing techniques and deformation arguments often reference work by Michael Atiyah, Raoul Bott, and Jean-Michel Bismut. Mirror constructions use the SYZ conjecture of Andrew Strominger, Shing-Tung Yau, and Eric Zaslow, tropical geometry methods by Grigory Mikhalkin and Mark Gross, and techniques from Noncommutative geometry influenced by Alain Connes. Categorical equivalences frequently use enhancements by A∞-categories and model category approaches championed by Daniel Quillen.

Consequences and applications

HMS has reshaped research programs across String theory, Algebraic geometry, and Symplectic topology. It led to new invariants in low-dimensional topology influenced by Floer homology and to advances in enumerative geometry affecting research by Catherine Taubes and Richard Thomas. HMS informs mirror constructions for Toric varietys and Fano varietys, impacts derived autoequivalence studies linked to Paul Seidel and Tom Bridgeland, and suggests links to Enumerative combinatorics via tropical methods by Grigory Mikhalkin. Applications extend to mathematical aspects of Topological quantum field theory explored by Edward Witten and categorical interpretations relevant to Geometric Langlands program work by Edward Frenkel and Alexander Beilinson.

Open problems and developments

Major open problems include proving HMS for broad classes of Calabi–Yau manifolds beyond known examples, clarifying the role of singularities and Landau–Ginzburg models in mirror symmetry as pursued by Maxim Kontsevich and Denis Auroux, and establishing analytic foundations for virtual perturbation methods advocated by Kenji Fukaya and Paul Seidel. Further development connects HMS to the Geometric Langlands program initiatives by Edward Frenkel and to categorified invariants studied by Mikhail Khovanov and Jacob Lurie. Emerging directions explore derived symplectic geometry influenced by Dmitry Roytenberg and Tony Pantev, and interactions with computational approaches championed by David Eisenbud and Bernd Sturmfels.

Category:Mathematical conjectures