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The homological mirror symmetry conjecture of Maxim Kontsevich states that the derived category of coherent sheaves on one Calabi–Yau manifold is equivalent in a certain sense to the Fukaya category of its mirror. [54] This equivalence provides a precise mathematical formulation of mirror symmetry in topological string theory.
The homological mirror symmetry conjecture of Maxim Kontsevich predicts an equality between the Lagrangian Floer homology of Lagrangians in a Calabi–Yau manifold and the Ext groups of coherent sheaves on the mirror Calabi–Yau manifold. In this situation, one should not focus on the Floer homology groups but on the Floer chain groups.
In the classical bosonic sector of a supersymmetric field theory, the Bogomol'nyi–Prasad–Sommerfield (BPS) bound (named after Evgeny Bogomolny, M.K. Prasad, and Charles Sommerfield [1] [2]) provides a lower limit on the energy of static field configurations, depending on their topological charges or boundary conditions at spatial infinity.
In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromov–Witten invariants) to integrals from a family of varieties (encoded as period integrals on a variation of Hodge structures).
Mirror symmetry not only replaces the homological dimensions but also the symplectic structure and complex structure on the mirror pairs. That is the origin of homological mirror symmetry. In 1990-1991, Candelas et al. 1991 had a major impact not only on enumerative algebraic geometry but on the whole mathematics and motivated Kontsevich (1994).
In 1991 the paper [5] about mirror symmetry on the quintic threefold in from the string theoretical viewpoint gives numbers of degree d rational curves on for all >. Prior to this, algebraic geometers could calculate these numbers only for d ≤ 5 {\displaystyle d\leq 5} .
This equation is completely coordinate- and metric-independent and says that the electromagnetic flux through a closed two-dimensional surface in space–time is topological, more precisely, depends only on its homology class (a generalization of the integral form of Gauss law and Maxwell–Faraday equation, as the homology class in Minkowski ...
Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality [note 1] or S-duality according to current terminology. [note 2] It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magnetic monopoles, which are usually viewed as emergent quasiparticles that are "composite" (i.e. they are solitons or topological ...