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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.
Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985) , after Eugenio Calabi ( 1954 , 1957 ), who first conjectured that such surfaces might exist, and ...
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).
One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial = + + + + Computing the partial derivatives of gives the four polynomials = = = = = Since the only points where they vanish is given by the coordinate axes in , the vanishing locus is empty since [::::] is not a point in .
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2-dimensional space, there is a line/axis of symmetry, in 3-dimensional space, there is a plane of symmetry
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).
By applying mirror symmetry, mathematicians have translated this problem into an equivalent problem for the mirror Calabi–Yau, which turns out to be easier to solve. [12] In physics, mirror symmetry is justified on physical grounds. [13] However, mathematicians generally require rigorous proofs that do not require an appeal to physical intuition.
These two conjectures encode the predictions of mirror symmetry in different ways: homological mirror symmetry in an algebraic way, and the SYZ conjecture in a geometric way. [ 6 ] There should be a relationship between these three interpretations of mirror symmetry, but it is not yet known whether they should be equivalent or one proposal is ...