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In mathematics, a Manin triple consists of a Lie algebra with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras and such that is the direct sum of and as a vector space. A closely related concept is the (classical) Drinfeld double, which is an even dimensional Lie algebra which admits a Manin decomposition.
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 to integrals from a family of varieties. In short, this means there is a relation between the number of genus algebraic curves of degree on a Calabi-Yau variety and integrals on a dual variety . These relations were original discovered by Candelas, de la Ossa, Green, and Parkes in a paper studying a generic quintic threefold in as the variety and a construction from the quintic Dwork family giving . Shortly after, Sheldon Katz wrote a summary paper outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.
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References
- Iskovskih, V. A.; Manin, Ju. I. (1971), "Three-dimensional quartics and counterexamples to the Lüroth problem", Matematicheskii Sbornik , Novaya Seriya, 86: 140–166, doi:10.1070/SM1971v015n01ABEH001536, MR 0291172
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