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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.
The non-chiral Su–Schrieffer–Heeger model (=), can be associated with symmetry class BDI with an integer topological invariant due to gauge invariance. [6] [7] The problem is similar to the integer quantum Hall effect and the quantum anomalous Hall effect (both in =) which are A class, with integer Chern number.
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).
(The class of a Kähler metric is the cohomology class of its associated 2-form.) Calabi showed such a metric is unique. There are many other inequivalent definitions of Calabi–Yau manifolds that are sometimes used, which differ in the following ways (among others): The first Chern class may vanish as an integral class or as a real class.
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).
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]
The cohomology ring of real projective space RP n with Z/2 coefficients is Z/2[x]/(x n+1), with x in degree 1. [9] Here x is the class of a hyperplane RP n−1 in RP n; this makes sense even though RP j is not orientable for j even and positive, because Poincaré duality with Z/2 coefficients works for arbitrary manifolds.