Abstract
This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope.
Highlights
In 1991, a group of physicists astounded the algebraic geometry community with conjectures for the number of rational curves of degree d lying on the famous quintic threefold, a variety of dimension three and degree five in 4-dimensional projective space
The key tool used in the paper [1] is mirror symmetry, which is an isomorphism of quantum field theories that turns a difficult counting problem on the quintic threefold into a much simpler problem on the so-called quintic mirror
All of this seems far removed from the polar duality of polytopes, but as we will see in this article, there is a surprisingly deep connection based on the reflexive polytopes introduced by Batyrev [2] in the context of toric geometry
Summary
In 1991, a group of physicists astounded the algebraic geometry community with conjectures for the number of rational curves of degree d lying on the famous quintic threefold, a variety of dimension three and degree five in 4-dimensional projective space. The key tool used in the paper [1] is mirror symmetry, which is an isomorphism of quantum field theories that turns a difficult counting problem on the quintic threefold into a much simpler problem on the so-called quintic mirror. All of this seems far removed from the polar duality of polytopes, but as we will see in this article, there is a surprisingly deep connection based on the reflexive polytopes introduced by Batyrev [2] in the context of toric geometry
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