Mirror symmetry

Mirror symmetry
Dbc334 · Public domain · source
Name	Mirror symmetry
Field	Algebraic geometry, Symplectic geometry, Theoretical physics
Introduced	1990s
Notable people	Philip Candelas, Xenia de la Ossa, Paul Aspinwall, Maxim Kontsevich, Edward Witten, Andrew Strominger, Cumrun Vafa, Shing-Tung Yau, David Morrison, Paul Seidel, Anton Kapustin, Kentaro Hori, Cumrun Vafa, Edward Frenkel, Richard Thomas, Mark Gross, Bernd Siebert, Dmitri Orlov, Alexander Polishchuk, Jacob Lurie, Mikhail Kapranov, Kontsevich–Soibelman, Kenji Fukaya, Kenji Ueda, Yongbin Ruan, Aleksey Zinger, Robin Hartshorne, Gérard Laumon, Arend Bayer, Tom Bridgeland, Akira Ishii, Paul Hacking, David Treumann, Luca Migliorini, Kathy M. O'Grady, Denis Auroux, Eric Zaslow, David Nadler, Paul Borodzik, Klaus Hulek, Willem de Smit

Mirror symmetry is a duality that relates pairs of compact Calabi–Yau manifolds and connects techniques from Algebraic geometry, Symplectic geometry, and String theory. First observed in enumerative predictions for quintic threefolds, it matured into precise conjectures linking categories, moduli, and physical quantities across collaborations among researchers at institutions such as Princeton University, Harvard University, and Institut des Hautes Études Scientifiques. The subject influences active research in Hodge theory, Gromov–Witten theory, Derived categories, and the mathematical formalism of Topological quantum field theory.

Introduction

Mirror symmetry arose when computations by a team including Philip Candelas, Xenia de la Ossa, Paul S. Green, and Linda Parkes compared enumerative numbers for a quintic threefold with predictions from a dual model, sparking connections with work by Shing-Tung Yau on the Calabi conjecture. The phenomenon connects complex-analytic invariants of a complex manifold to symplectic invariants of a different manifold; contemporaneous developments included insights from Edward Witten on topological twists and from Andrew Strominger, Shing-Tung Yau, and Eric Zaslow on geometric transitions. Subsequent mathematical frameworks involve constructions and conjectures by Maxim Kontsevich and algebraic formalizations developed by teams at University of California, Berkeley and Massachusetts Institute of Technology.

Mathematical Foundations

Foundations draw on classical and modern tools: Hodge theory for variations of Hodge structure, Gromov–Witten theory for curve counting, Quantum cohomology for deformed cup product structures, and Toric geometry for explicit models. Foundational results include the predictions of instanton numbers for quintic threefolds from periods of Picard–Fuchs equations studied by groups at University of Oxford and Max Planck Institute for Mathematics, and formal statements about mirror pairs framed via Batyrev construction for reflexive polytopes and via degenerations studied by Mark Gross and Bernd Siebert. Analytic tools such as the study of Variation of Hodge structure and monodromy around large complex structure limits connect to lattice-theoretic work by researchers at University of Chicago and Institut des Hautes Études Scientifiques.

Homological Mirror Symmetry

The Homological Mirror Symmetry conjecture by Maxim Kontsevich posits an equivalence between the derived category of coherent sheaves on a complex manifold and the Fukaya category of its mirror, a statement that situates mirror symmetry within Derived categories and A∞-categories. Major progress includes constructions by Kenji Fukaya, Paul Seidel, and Ivan Smith for particular classes of manifolds, and algebraic approaches by Alexander Polishchuk, Dmitri Orlov, and Tom Bridgeland. Techniques involve stability conditions linked to work by Bridgeland at University of Glasgow, deformation theory methods informed by Kontsevich–Soibelman wall-crossing, and categorical mirror constructions used by groups at Stanford University and Courant Institute.

Examples and Constructions

Concrete examples arise in Toric variety settings via the Batyrev construction and in non-toric contexts via Strominger–Yau–Zaslow-type fibrations proposed by Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Key explicit models include the mirror of the quintic threefold studied by Philip Candelas et al., mirrors for K3 surfaces analyzed by work at University of Cambridge and Institute for Advanced Study, and local mirror symmetry examples for neighborhoods of curves investigated by Cumrun Vafa and collaborators. Techniques for producing mirrors employ tropical geometry developed by Grigory Mikhalkin and Gross–Siebert reconstruction programs, with computational implementations influenced by researchers at University of Toronto and Imperial College London.

Physical Origins and String Theory Context

In physics, mirror symmetry emerges from dualities in Superstring theory compactifications on Calabi–Yau manifolds and from topological string theories formulated by Edward Witten, Cumrun Vafa, and Nathan Seiberg. The relationship links A-model topological strings computing Gromov–Witten invariants to B-model computations of complex moduli and period integrals; early physical perspectives appeared in seminars at CERN and workshops at Aspen Center for Physics. Mirror symmetry interacts with dualities such as T-duality and plays a role in understanding phenomena studied in M-theory and in gauge/string correspondences explored by groups at Perimeter Institute and SLAC National Accelerator Laboratory.

Applications and Consequences

Applications span enumerative predictions for curve counting pursued by teams at Princeton University and Brown University, advances in understanding of Derived categories with implications for birational geometry studied by Caucher Birkar-adjacent research groups, and interactions with Tropical geometry informing moduli compactifications considered at ETH Zurich. Mirror symmetry influences computations in Topological recursion and in the study of stability conditions with consequences for representation theory work at IHÉS and École Normale Supérieure. Broader mathematical impacts include new perspectives on degenerations analyzed by Kollar and Mori, algorithmic mirror computations implemented by research groups at University of California, Santa Barbara and Max Planck Institute for Mathematics and continuing cross-disciplinary collaborations between mathematicians and physicists at institutions including Princeton University and Institute for Advanced Study.

Category:Algebraic geometry