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| Kontsevich's homological mirror symmetry conjecture | |
|---|---|
| Name | Maxim Kontsevich |
| Birth date | 1954 |
| Nationality | Soviet Union |
| Field | Mathematics |
Kontsevich's homological mirror symmetry conjecture Kontsevich's homological mirror symmetry conjecture proposes a deep equivalence between categories arising from complex algebraic geometry and symplectic geometry, connecting ideas from Calabi–Yau manifolds, String theory, and Mirror symmetry. Formulated by Maxim Kontsevich in 1994 at the International Congress of Mathematicians in Zurich and influenced by work of Philip Candelas, Paul Aspinwall, Brian Greene, and Edward Witten, the conjecture reshaped research across Algebraic geometry, Symplectic geometry, Category theory, and Mathematical physics.
The conjecture arose from observations in Superstring theory and computations by groups including Philip Candelas, Xenia de la Ossa, Paul S. Green, Thomas Hubsch and later numerical tests by Borisov and Morrison. Kontsevich synthesized insights from Mikhail Gromov's pseudoholomorphic curve techniques, Maxwell Kontsevich's own work in Deformation quantization, and categorical notions popularized in Alexander Grothendieck's program, building on foundations by André Weil, David Mumford, and Jean-Pierre Serre. The motivation linked enumerative predictions used by Candelas et al. to categorical dualities envisioned by Kontsevich and anticipated applications to conjectures of Homological algebra origin like those of Pierre Deligne and Grothendieck–Riemann–Roch.
Kontsevich posited that for a mirror pair consisting of a complex Calabi–Yau manifold X and a mirror Y, there is an equivalence between the derived category of coherent sheaves on X, denoted D^b(Coh(X)), and the derived Fukaya category Fuk(Y); symmetrically, D^b(Coh(Y)) should correspond to Fuk(X). This equivalence ties objects such as Coherent sheafs and Lagrangian submanifolds, and morphisms computed by Ext groups correspond to Floer cohomology groups defined via methods of Andreas Floer and Yakov Eliashberg. The statement synthesizes categorical dualities explored by Bernhard Keller and the notion of A-model/B-model duality from Cecotti–Vafa frameworks in Topological string theory.
Central objects include D^b(Coh(X)) from derived categories developed by Alexander Grothendieck's students and Fukaya categories introduced by Kenji Fukaya. The Fukaya category uses Lagrangian intersection Floer homology techniques of Andreas Floer, Paul Seidel extended these ideas through monodromy and mapping class group actions studied by William Thurston and Maxim Kontsevich. The mirror correspondence employs Variation of Hodge structures studied by Phillip Griffiths and period integrals related to work by Bernard Malgrange and Pierre Deligne. Deformation theory inputs come from Maurice Auslander-type frameworks and Gerstenhaber algebras used in Deformation quantization by Kontsevich himself, while stability conditions on triangulated categories were axiomatized by Tom Bridgeland.
Early evidence came from enumerative matches in the work of Candelas on counting rational curves on the quintic threefold and subsequent mathematical confirmations by Gromov–Witten theory pioneers including Maxim Kontsevich and Yuri Manin. Provable instances include homological mirror symmetry for elliptic curves established by Paul Seidel and Alexander Polishchuk, for higher-genus curves by Dmitry Kaledin and collaborators, for toric varieties and Fano toric mirrors by Kenji Fukaya, Mohammed Abouzaid, and Borisov–Hori–Vafa patterns, and for certain K3 surfaces through work by Paul Aspinwall, Shigeru Mukai, and Dolgachev. Results on equivalences of categories and computations of Hochschild cohomology have been advanced by Maxim Kontsevich, Bernhard Keller, Keller & Lefèvre-Hasegawa, and Mitya Boyarchenko.
Key examples include the quintic threefold and its mirror used in Candelas's calculations, mirror symmetry for elliptic curves exhibited by Polishchuk and Zaslow, homological mirror symmetry for the two-torus by Seidel and Zaslow, toric Fano varieties via the SYZ conjecture perspective by Strominger–Yau–Zaslow, and mirror dualities for K3 surfaces informed by lattice-theoretic methods of Shigeru Mukai and V. V. Nikulin. Explicit computations of Fukaya categories for punctured spheres and pair-of-pants decompositions were developed by Abouzaid and Seidel, while mirror descriptions of Landau–Ginzburg models relate to work by Hori and Vafa.
Approaches include categorical deformation theory from Gerstenhaber and Schlessinger-style frameworks, analytic pseudoholomorphic curve techniques of Gromov and Hofer in symplectic topology, and sheaf-theoretic methods from Grothendieck and Serre. Homological algebra tools used involve DG categories and A-infinity categories formalized by Jim Stasheff and developed by Bernhard Keller. Other methods use perverse sheaves and microlocal sheaf theory advanced by Masaki Kashiwara and Pierre Schapira, tropical and nonarchimedean techniques related to Vladimir Berkovich and Grigory Mikhalkin, and string-theoretic intuition from Edward Witten and Cumrun Vafa to guide conjectural correspondences.
Extensions include the SYZ conjecture by Andrew Strominger, Shing-Tung Yau, and Zaslow relating torus fibrations to mirror duality, homological mirror symmetry for Fano varieties and Landau–Ginzburg models explored by Auroux and Hori–Iqbal–Vafa influences, and categorical formalisms connecting to the Geometric Langlands program attributed to Edward Witten and Alexander Beilinson. Refinements involve stability conditions by Tom Bridgeland, noncommutative mirror symmetry studied by Ginzburg and Van den Bergh, and relations to Donaldson–Thomas invariants by Richard Thomas and Kontsevich–Soibelman developments. Ongoing links with Mathematical physics communities, including researchers at Institute for Advanced Study and Simons Foundation workshops, continue to expand applications across Algebraic geometry, Symplectic geometry, and beyond.