# Homological mirror symmetry

﻿
Homological mirror symmetry

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.

In an address to the 1994 International Congress of Mathematicians in Zurich, Kontsevich speculated that mirror symmetry for a pair of Calabi-Yau manifolds "X" and "Y" could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of "X" and another triangulated category constructed from the symplectic geometry of "Y".

Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B models. These models concern maps from Riemann surfaces into a fixed target - usually a Calabi-Yau manifold. Most of the mathematical predictions of mirror symmetry are embedded in the physical equivalence of the A-model on "Y" with the B-model on its mirror "X". When the Riemann surfaces have empty boundary, they represent the worldsheets of closed strings. To cover the case of open strings, one must introduce boundary conditions to preserve the supersymmetry. In the A-model, these boundary conditions come in the form of Lagrangian submanifolds of "Y" with some additional structure (often called a brane structure). In the B-model, the boundary conditions come in the form of holomorphic (or algebraic) submanifolds of "X" with holomorphic (or algebraic) vector bundles on them. These are the objects one uses to build the relevant categories. They are often called A and B branes respectively. Morphisms in the categories are given by the massless spectrum of open strings stretching between two branes.

The closed string A and B models only capture the so-called topological sector - a small portion of the full string theory. Similarly, the branes in these models are only topological approximations to the full dynamical objects that are D-branes. Even so, the mathematics resulting from this small piece of string theory has been both deep and difficult.

Only in a few examples have mathematicians been able to verify the conjecture. In his seminal address, Kontsevich commented that the conjecture could be proved in the case of elliptic curves using theta functions. Following this route, Alexander Polishchuk and Eric Zaslow provided a proof of a version of the conjecture for elliptic curves. Kenji Fukaya was able to establish elements of the conjecture for abelian varieties. Later, Kontsevich and Yan Soibelman provided a proof of the majority of the conjecture for nonsingular torus bundles over affine manifolds using ideas from the SYZ conjecture. Recently, Paul Seidel proved the conjecture in the case of the quartic surface.

ee also

*Topological quantum field theory

*Category theory

*Floer homology

*Fukaya category

*Derived category

References

* Kontsevich, Maxim. "Homological algebra of mirror symmetry." [http://front.math.ucdavis.edu/alg-geom/9411018]

* Seidel, Paul. "Homological mirror symmetry for the quartic surface." [http://front.math.ucdavis.edu/math.SG/0310414]

* Kontsevich, Maxim, Soibelman Yan. "Homological Mirror Symmetry and torus fibrations." [http://front.math.ucdavis.edu/math.SG/0011041]

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Mirror symmetry — may refer to: Mirror symmetry (string theory), a relation between two Calabi Yau manifolds in string theory Homological mirror symmetry, a mathematical conjecture about Calabi Yau manifolds made by Maxim Kontsevich Reflection symmetry, a… …   Wikipedia

• Mirror symmetry (string theory) — In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi Yau manifolds. It happens, usually for two such six dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they… …   Wikipedia

• SYZ conjecture — The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled Mirror Symmetry is T duality …   Wikipedia

• Floer homology — is a mathematical tool used in the study of symplectic geometry and low dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an… …   Wikipedia

• List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

• Timeline of category theory and related mathematics — This is a timeline of category theory and related mathematics. By related mathematics is meant first hand * Homological algebra * Homotopical algebra * Topology using categories, especially algebraic topology * Categorical logic * Foundations of… …   Wikipedia

• Maxim Kontsevich — Born 25 August 1964 (1964 08 25) (age 47) Khimk …   Wikipedia

• Alexander Givental — ist ein russischstämmiger US amerikanischer Mathematiker, der sich mit symplektischer Topologie, Singularitätentheorie und algebraischer Geometrie mit Wechselwirkungen zur Stringtheorie beschäftigt. Givental ist ein Schüler von Wladimir Arnold,… …   Deutsch Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• Hodge structure — In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. A mixed Hodge… …   Wikipedia