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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.
The R-symmetry group of a 2-dimensional N = (2,2) field theory is U(1) × U(1), twists by the two different factors lead to the A and B models respectively. The topological twisted construction of topological string theories was introduced by Edward Witten in his 1988 paper. [1]
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 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).
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 ...
The existence of a topological defect can be demonstrated whenever the boundary conditions entail the existence of homotopically distinct solutions. Typically, this occurs because the boundary on which the conditions are specified has a non-trivial homotopy group which is preserved in differential equations; the solutions to the differential equations are then topologically distinct, and are ...