# Mirror Symmetry and Polar Duality of Polytopes

## Abstract

**:**

## 1. Introduction

## 2. Reflexive Polytopes

**Figure 1.**A cube $P\subseteq {\mathbb{R}}^{3}$ and its dual octahedron ${P}^{\circ}$. Reprinted from [4] (p. 81) with permission of the American Mathematical Society.

**Example 2.1.**Consider the Standard 4-Simplex

- P has 125 lattice points, i.e., $|P\cap {\mathbb{Z}}^{4}|=125$.
- ${P}^{\circ}$ has 6 lattice points, i.e., $|{P}^{\circ}\cap {\mathbb{Z}}^{4}|=6$.

**Definition 2.1.**A lattice polytope P is

**reflexive**if $0\in \mathrm{Int}\left(P\right)$ and ${P}^{\circ}$ is also a lattice polytope.

**Figure 2.**A reflexive polytope in ${\mathbb{R}}^{3}$ with 14 vertices. Reprinted from [6] (p. 180) with permission of the American Mathematical Society.

**Figure 3.**The 16 classes of reflexive lattice polygons in ${\mathbb{R}}^{2}$. The open circles represent the origin and the labels record the number of boundary lattice points. Reprinted from [4] (p. 382) with permission of the American Mathematical Society.

## 3. Mirror Symmetry

**Example 3.1.**The simplest Calabi–Yau threefold is the quintic threefold. We start with ${\mathbb{P}}^{4}$, the 4-dimensional projective space over the complex numbers. Points in ${\mathbb{P}}^{4}$ have homogeneous coordinates$({x}_{0},{x}_{1},{x}_{2},{x}_{3},{x}_{4})$, where the coordinates never vanish simultaneously and two sets of coordinates give the same point if and only if they differ by a nonzero scalar multiple. A homogeneous equation

- Kähler moduli parameters that encode the metric on V and control the A-model.
- Complex moduli parameters that encode the complex structure of V and control the B-model.

**Example 3.2.**To construct the quintic mirror, we begin with the threefold in $W\subseteq {\mathbb{P}}^{4}$ defined by

- Rational curves on V of various degrees are important in enumerative algebraic geometry. These can be encoded into Gromov–Witten invariants that are intimately related to Kähler moduli and the A-model of V.
- By mirror symmetry and the mirror map, we can switch to the B-model of ${V}^{\circ}$, where the complex moduli and B-model can be studied by the differential equations that arise in the variation of Hodge structure on ${V}^{\circ}$. This is straightforward to study since ${h}^{21}\left({V}^{\circ}\right)=1$.
- The result is an explicit formula for all of the Gromov–Witten invariants! A careful description of the formula is appears in ([10] Chapter 2).

## 4. Mirror Symmetry and Reflexive Polytopes

**Example 4.1.**When we apply this process to the polytope $P=5{\Delta}_{4}-(1,1,1,1)$ from Example 2.1, we get ${X}_{P}={\mathbb{P}}^{4}$, and the hypersurface is our friend the quintic threefold $V\subseteq {\mathbb{P}}^{4}$.

- A 4-dimensional reflexive polytope P gives a hypersurface in ${X}_{P}$ that becomes a Calabi–Yau threefold V after a suitable resolution of singularities.
- The dual polytope ${P}^{\circ}$ gives a Calabi–Yau threefold ${V}^{\circ}$, and the Hodge numbers of V and ${V}^{\circ}$ are related by$${h}^{11}\left({V}^{\circ}\right)={h}^{21}\left(V\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{h}^{21}\left({V}^{\circ}\right)={h}^{11}\left(V\right),$$

- Not all Calabi–Yau threefolds arise from Batyrev’s construction. We will learn more about this later in the paper.
- It is still an open question in physics as to whether V and ${V}^{\circ}$ give isomorphic SCFTs when P is an arbitrary 4-dimensional reflexive polytope. The evidence is compelling, but an actual isomorphism is only known for certain special cases, such as the quintic threefold and its mirror.

## 5. Duality and Symmetry in Mirror Symmetry

**Figure 4.**The Hodge diamond of an arbitrary smooth projective threefold (

**a**); the Hodge diamond of a Calabi–Yau threefold (

**b**).

**Figure 5.**$\chi =2({h}^{11}-{h}^{21})$ (horizontal) versus ${h}^{11}+{h}^{21}$ (vertical) for Calabi–Yau threefolds coming from 4-dimensional reflexive polytopes. Reprinted from [14] (p. 432) with permission of International Press of Boston.

## 6. CICYs and Duality of Cones

- A generic intersection of two cubic hypersurfaces in ${\mathbb{P}}^{5}$ is a Calabi–Yau threefold.
- A generic intersection of two quadric hypersurfaces and a cubic hypersurface in ${\mathbb{P}}^{6}$ is a Calabi–Yau threefold.

- The Minkowski sum of polytopes ${P}_{1},{P}_{2}\subseteq {\mathbb{R}}^{d}$ is defined by$${P}_{1}+{P}_{2}=\{u+v\mid u\in {P}_{1},v\in {P}_{2}\}.$$
- Points ${m}_{1},\cdots ,{m}_{s}\in {\mathbb{Z}}^{d}$ generate the rational convex polyhedral cone$$\sigma =\mathrm{Cone}({m}_{1},\cdots ,{m}_{s})=\{{\lambda}_{1}{m}_{1}+\cdots +{\lambda}_{s}{m}_{s}\mid {\lambda}_{i}\ge 0\}\subseteq {\mathbb{R}}^{d}.$$
- Given such a cone σ, its dual is$${\sigma}^{\vee}=\{u\in {\mathbb{R}}^{d}\mid u\xb7m\ge 0\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4pt}{0ex}}m\in \sigma \}.$$

- ${Q}_{1},\cdots ,{Q}_{r}$ are lattice polytopes containing the origin.
- $Q={Q}_{1}+\cdots +{Q}_{r}$ is a reflexive polytope of dimension $r+3$.

**Example 6.1.**Consider the reflexive polygon $P=3{\Delta}_{2}-(1,1)\subseteq {\mathbb{R}}^{2}$. This is the polygon labeled “9” in Figure 3. Note that P is the 2-dimensional analog of the polytope $5{\Delta}_{4}-(1,1,1,1)\subseteq {\mathbb{R}}^{4}$ that gives the quintic threefold.

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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Cox, D.A.
Mirror Symmetry and Polar Duality of Polytopes. *Symmetry* **2015**, *7*, 1633-1645.
https://doi.org/10.3390/sym7031633

**AMA Style**

Cox DA.
Mirror Symmetry and Polar Duality of Polytopes. *Symmetry*. 2015; 7(3):1633-1645.
https://doi.org/10.3390/sym7031633

**Chicago/Turabian Style**

Cox, David A.
2015. "Mirror Symmetry and Polar Duality of Polytopes" *Symmetry* 7, no. 3: 1633-1645.
https://doi.org/10.3390/sym7031633