Mirror Symmetry and Polar Duality of Polytopes

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.

Anomalous Mirror Symmetry Generated by Optical Illusion

“…In Figure 6 we show the intersection of the scene by a plane x = t parallel to the yz plane. As shown in this figure, the space curve c 1 (t) that appears a 1 (t) when it is seen in the view direction v 1 , and that appears b 1 (t) when it is seen in the view direction v 2 , can be represented as…”

“…Hence b 1 (t) = (t, b 1 (t), 0) and b 2 (t) = (t, b 2 (t), 0) constitute another closed curve on the xy plane. We want to construct a closed space curve that appears to be the closed curve composed of a 1 (t) and a 2 (t) when it is seen in the view direction v 1 , and that appears to be the closed curve composed of b 1 (t) and b 2 (t) when it is seen in the view direction v 2 , and it generates anomalous symmetry when we place a mirror parallel to the xz plane.…”

“…Although the concept of the mirror symmetry has been generalized in many directions in mathematics and physics [1,2], the original mirror symmetry with respect to a planar mirror is simple and well understood from a mathematical point of view.…”

Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

“…This "key reduction" formula (14) is instrumental in separating the p-adic terms in a way convenient to our future analysis. We will use the following local analytic properties of the p-adic Gamma function (see for example [37]).…”

Batyrev Mirror Symmetry

“…The material for this contribution is mostly taken from Cox's expository paper "Mirror Symmetry and Polar Duality of Polytopes" [Cox15], and parts of Cox-Katz, "Mirror Symmetry and Algebraic Geometry" [CK99] (in particular Sections 4.1 and 4.2).…”