Mirror symmetry conjecture

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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). In short, this means there is a relation between the number of genus ${\displaystyle g}$ algebraic curves of degree ${\displaystyle d}$ on a Calabi-Yau variety ${\displaystyle X}$ and integrals on a dual variety ${\displaystyle {\check {X}}}$. These relations were original discovered by Candelas, de la Ossa, Green, and Parkes^[1] in a paper studying a generic quintic threefold in ${\displaystyle \mathbb {P} ^{4}}$ as the variety ${\displaystyle X}$ and a construction^[2] from the quintic Dwork family ${\displaystyle X_{\psi }}$ giving ${\displaystyle {\check {X}}={\tilde {X}}_{\psi }}$. Shortly after, Sheldon Katz wrote a summary paper^[3] outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.