Mirror Symmetry and Polar Duality of Polytopes

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.


Introduction
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.

Reflexive Polytopes
A polytope P ⊆ R d is the convex hull of finitely many points in R d .If 0 ∈ Int(P ), then its polar or dual is It is well-known that P • is a polytope with 0 ∈ Int(P • ) and (P • ) • = P (see, for instance, ([3] 2.3)).
Figure 1 shows a classic example of a polytope and its dual in three dimensions.The polytopes relevant to mirror symmetry have dimension four.In Example 2.1, we give the polytopes that underlie the quintic threefold and its mirror.As polytopes, P and P • are simplices.But in terms of lattice points (points with integer coordinates, i.e., Z 4 ⊆ R 4 ), there is a substantial difference: • P has 125 lattice points, i.e., |P ∩ Z 4 | = 125.
We will see later that P and P • give rise to mirror manifolds where a hard problem on P transforms into a simpler problem on P • because of the small number of lattice points.
This example suggests that lattice points have an important role to play.In general, a lattice polytope or integer polytope is the convex hull of a finite subset of Z d ⊆ R d .The book [5] gives a nice introduction to lattice polytopes and their associated counting problems.
The cube P in Figure 1 becomes a lattice polytope when we choose coordinates so that its vertices are ±e 1 ± e 2 ± e 3 for the standard basis e 1 , e 2 , e 3 of R 3 .Then P • is the lattice polytope with vertices ±e 1 , ±e 2 , ±e 3 .
In general, polar duality interacts poorly with lattice polytopes-what happened in Figure 1 is rather special.For example, if we double the size of the cube to get 2P , then its dual is which fails to be a lattice polytope.This leads to the following definition, due to Batyrev [2].
Definition 2.1.A lattice polytope P is reflexive if 0 ∈ Int(P ) and P • is also a lattice polytope.
Figure 2 shows a less obvious example of a reflexive polytope.If P is reflexive, then it is straightforward to show that 0 is the unique interior lattice point of P .For another way to see what reflexive means, we first recall another description of the dual polytope.Given an arbitrary polytope with 0 ∈ Int(P ), every facet F of P has a unique inward-pointing facet normal u F with the property that Then one can show that Hence, a lattice polytope P with 0 ∈ Int(P ) is reflexive if and only if u F ∈ Z d for all facets F of P .Classifying reflexive polytopes in R d is an important problem because of their relevance to mirror symmetry.By "classify", we mean up to a lattice equivalence of Z d , i.e., up to coordinate change by an element of GL(d, Z).In dimension two, the 16 classes of reflexive polygons are shown in Figure 3.It is a fun exercise to match polygons with their duals.In some cases, one needs to change coordinates by an element of GL(2, Z) to identify the dual.Some of the polygons are self-dual up to GL(2, Z).
In dimension three, there are 4319 classes of reflexive polytopes, and such number balloons to 473,800,776 in dimension four, an impressive calculation done by Kreuzer and Skarke [7] in 2002.As we will soon see, 4-dimensional reflexive polytopes are important in mirror symmetry.

Mirror Symmetry
String theory from mathematical physics is based on a 10-dimensional universe, where four dimensions are the familiar space-time of general relativity and the remaining six dimensions are where the quantum effects take place.The Elegant Universe by Greene [8] describes this model of the universe for a general audience.
The 6-dimensional quantum piece is a (very small) compact manifold, about the size of Planck's constant.To make this manifold support the kind of quantum field theory required by string theory, the manifold needs to have a complex structure with a trivial canonical bundle and vanishing first Betti number.Six real dimensions mean three complex dimensions, and the complex manifolds that arise are called Calabi-Yau threefolds.We recommend The Shape of Inner Space by Yau and Nadis [9] for a non-technical account of these spaces.
Example 3.1.The simplest Calabi-Yau threefold is the quintic threefold.We start with P 4 , the 4-dimensional projective space over the complex numbers.Points in 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 defines the quintic threefold V ⊆ P 4 .For most choices of F , V is smooth, i.e., is a manifold.Then having degree five guarantees that V is a Calabi-Yau threefold.
Mirror symmetry involves some sophisticated physics and algebraic geometry.The string theories used in mirror symmetry are superconformal field theories (SCFTs) with twisted versions called the A-model and the B-model.The SCFT on a Calabi-Yau threefold V depends on two types of parameters: • 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.
The number of parameters of each type is determined by the Hodge numbers of V , defined by In particular, h 11 (V ) is the number of Kähler parameters and h 21 (V ) is the number of complex parameters.The central idea of mirror symmetry is that given a family of Calabi-Yau threefolds V , there should be a mirror family of Calabi-Yau threefolds V • such that the corresponding SCFTs are isomorphic in a way that interchanges the A-and B-models and hence interchanges Kähler and complex moduli.In particular, V and V • satisfy The first explicit example of mirror symmetry involves the quintic threefold V ⊆ P 4 and its mirror, which is defined as follows.
Example 3.2.To construct the quintic mirror, we begin with the threefold in W ⊆ P 4 defined by Let ζ = e 2πi/5 and note that the group acts on W .The quotient W/G is not smooth, so one needs to do a resolution of singularities to produce a smooth variety.If done carefully, the result is Calabi-Yau.This is the quintic mirror V • .The single parameter ψ in the above equation gives the complex moduli parameter λ = ψ 5 , so that h 21 (V • ) = 1.This is consistent with (2) since the quintic threefold V has h 11 (V ) = 1.
As explained in the 1991 paper [1], we have the following miracle: • 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 • , where the complex moduli and B-model can be studied by the differential equations that arise in the variation of Hodge structure on V • .This is straightforward to study since • 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).
This formula astounded the algebraic geometers: a far-out theory from physics turns a hard problem in enumerative algebraic geometry into a solvable problem in Hodge theory.Mind blowing!Since SCFTs are not mathematically rigorous, the formulas in [1] were regarded as conjectures by the mathematics community.These formulas are now theorems (see ( [10] Chapter 11) for proofs and references to the original papers), though the full story of the relation between rational curves on the quintic threefold and Gromov-Witten invariants is still not fully understood.

Mirror Symmetry and Reflexive Polytopes
After the quintic threefold, many other examples of mirror manifolds were discovered using weighted projective spaces.However, there were some examples where the mirrors seemed to be missing.This was rectified in 1994 when Batyrev [2] pointed out that a 4-dimensional reflexive polytope gives a pair of Calabi-Yau threefolds that are natural candidates for a mirror pair.
An elementary approach to the transition from 4-dimensional reflexive polytopes to Calabi-Yau threefolds can be found in the paper [11], which is accessible to undergraduates.A more sophisticated approach uses the theory of toric varieties, which is an important part of modern algebraic geometry.A brief introduction to toric varieties can be found in [12], while [4] gives a comprehensive treatment.
For us, the starting point is that a lattice polytope P ⊆ R d gives a d-dimensional toric variety X P as follows.Let t 1 , . . ., t d be coordinates on the torus (C * ) d , where C * = C \ {0}.Then a lattice point m = (a 1 , . . ., a d ) ∈ Z d gives the Laurent monomial Negative exponents are allowed, but this is fine since t i ∈ C * .If P has "enough lattice points" (this can be made precise) and P ∩ Z d = {m 0 , . . ., m s }, then the map is injective, and the toric variety X P ⊆ P s is defined to be the closure of the image.Note that X P contains the torus (C * ) d , hence the name "toric variety".(When P does not have enough lattice points, one uses nP for n 0 in Z-see [4] §2.2 and §2.3.)Besides helping to define X P , the lattice points of P also give some interesting hypersurfaces of X P .Namely, if P ∩ Z d = {m 0 , . . ., m s } as above, then the equation defines a hypersurface in (C * ) d .Its closure in X P is the desired hypersurface V ⊆ X P .
If instead we use P • = Conv(e 1 , e 2 , e 3 , e 4 , (−1, −1, −1, −1)), then we get the toric variety X P • = P 4 /G, where G is the group defined in Example 3.2.The six lattice points of P • consist of the origin and the five vertices, so that Equation (4) reduces to When we apply the homogenization process described in ( [12] Section 13) and ([4] §5.4), this equation becomes Rescaling the coordinates x 0 , . . ., x 5 appropriately, one can assume that Hence we recover the Equation ( 3), which here defines the hypersurface of X P • = P 4 /G that was denoted W/G in Example 3.2.As explained in that example, this gives the quintic mirror V • after a suitable resolution of singularities.
It is satisfying to see how P and P • lead naturally to the quintic threefold and its mirror.In 1994, Batyrev [2] (see also ([10] Section 4.1)) proved the following general results: • 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 • gives a Calabi-Yau threefold V • , and the Hodge numbers of V and V • are related by as required by mirror symmetry.
In 1995, the paper [13] used reflexive polytopes to supply the "missing mirrors" mentioned at the beginning of this section.This was first clear indication of the deep relation between mirror symmetry and 4-dimensional reflexive polytopes.However, there are two things to keep in mind: • 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 • 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.

Duality and Symmetry in Mirror Symmetry
Given the themes of duality and symmetry, it makes sense to say more about what they mean in the context of mirror symmetry.The duality aspect is clear, given how P and P • give a mirror pair V and V • .But what about symmetry?
When considering a smooth projective threefold V , its Hodge numbers h pq = h pq (V ) from Equation ( 1) are often represented in the Hodge diamond shown in Figure 4. h 01 h 00 (a) In this figure, the Hodge diamond on the right follows from standard facts about smooth projective threefolds and the definition of Calabi-Yau.In particular, the Hodge diamond of a Calabi-Yau threefold is completely determined by h 11 = h 11 (V ) and h 21 = h 21 (V ).Since mirror symmetry gives we see that the Hodge diamond of V is the mirror image of the Hodge diamond of V • about the 45 • line through the center of the diamond.This is the origin of the name "mirror symmetry".
The Hodge diamond symmetry applies to a single mirror pair V and V • .If we take the Hodge numbers of all mirror pairs, then another remarkable picture emerges.We noted in Section 2 that there were 473,800,776 4-dimensional reflexive polytopes.This gives a lot of mirror pairs, all of which have symmetric of Hodge numbers.When plotted in two dimensions, we get Figure 5.This iconic image is taken from [14].Since the Betti numbers are the row sums of the of Hodge diamond, the topological Euler characteristic is χ = 2(h 11 −h 21 ).This invariant from algebraic topology is important in mathematical physics, where it shows up in various guises (central charge of a Virasoro algebra, ±2× number of fermion generations, etc.).Mirror symmetry interchanges h 11 and h 21 .This leaves the vertical coordinate h 11 + h 21 of Figure 5 unchanged but replaces the horizontal coordinate χ = 2(h 11 − h 21 ) with its negative.Hence mirror symmetry underlies the bilateral symmetry of Figure 5.
The 2015 paper [15] presents a state-of-the-art discussion of Figure 5, which includes data for other mirror pairs beyond those arising from 4-dimensional reflexive polytopes.That paper also describes the website [16] where the reader can find the most current version of Figure 5.

CICYs and Duality of Cones
We mentioned earlier that not all Calabi-Yau threefolds come from 4-dimensional reflexive polytopes.Currently, there is no classification of Calabi-Yau threefolds, although many constructions are known.Some involve toric varieties, while others, such as those described in ([10] Section 4.4), have nothing to do with toric methods.
Within the toric realm, there is more to the story than just hypersurfaces in 4-dimensional toric varieties.For example: • A generic intersection of two cubic hypersurfaces in P 5 is a Calabi-Yau threefold.
• A generic intersection of two quadric hypersurfaces and a cubic hypersurface in P 6 is a Calabi-Yau threefold.
There are similar examples in higher dimensional projective spaces.These are example of complete intersection Calabi-Yau threefolds, often written CICYs.The term "complete intersection" refers to the fact that the number of defining equations equals the codimension.
These examples can be generalized to the toric setting where higher dimensional reflexive polytopes take center stage.The construction is based on work of Batyrev and Borisov.We will follow the version presented in [17], focusing on the case of CICY threefolds.The reader should consult [17] for references to the original papers.
Before we can begin, we need some tools from polyhedral geometry: • The Minkowski sum of polytopes P 1 , P 2 ⊆ R d is defined by Note that P 1 + P 2 is a lattice polytope whenever P 1 and P 2 are.• Points m 1 , . . ., m s ∈ Z d generate the rational convex polyhedral cone • Given such a cone σ, its dual is Then σ ∨ is again a rational convex polyhedral cone, and we have the duality (σ ∨ ) ∨ = σ.
Proofs of these standard facts can be found in [3].
Now suppose that we have a (r + 3)-dimensional reflexive polytope P that is a Minkowski sum where each P i is a lattice polytope containing the origin.This is called a nef-partition for reasons having to do with numerically effective divisors on toric varieties.Given a nef-partition Equation ( 5), we get the toric variety X P of dimension r+3, and using the lattice points of the Minkowski summands P 1 , . . ., P r as in Equation ( 4), we get hypersurfaces Y 1 , . . ., Y r ⊆ X P .If the equations of the Y i are sufficiently generic, then the intersection is complete intersection threefold whose resolution of singularities (carefully done) is a Calabi-Yau threefold V .This is CICY threefold determined by the nef-partition Equation (5).The examples presented at the beginning of this section are instances of this construction.
To create the mirror family, we use duality, but we have to be careful since in Equation ( 5), the P i contain 0, but not as an interior point.This means that the dual P • i is an unbounded polyhedron.The key idea of the Batyrev-Borisov construction is to define "dual" polytopes Q 1 , . . ., Q r as follows: Here are the key properties of Q 1 , . . ., Q r : • Q 1 , . . ., Q r are lattice polytopes containing the origin.
In other words, a nef-partition, called the dual nef-partition.This also works in reverse, since the toric variety X Q and the hypersurfaces coming from Q 1 , . . ., Q r .The resulting CICY threefold V • is a candidate for the mirror of the CICY V of the original nef-partition To get a better sense of what Equation ( 6) means from the point of view of duality, let us focus on Q 1 .First observe that the cone is rational polyhedral since it is the cone generated by the vertices of P 2 , . . ., P r .Then we can write Q 1 as follows: Individually, P • 1 and Cone(P 2 ∪ • • • ∪ P r ) ∨ are unbounded polyhedra.Their intersection is bounded, i.e., is a polytope, because 0 is an interior point of P 1 + • • • + P r .This works not just for Q 1 but for all of the Q i .The surprise, as noted in the above bullets, is that the Q i are lattice polytopes with reflexive Minkowski sum.
The resulting "duality" between P and Q is remarkable: we take P and decompose it into pieces via P = P 1 + • • • + P r .For each piece P i , we modify the usual dual P • i using the cone dual to the remaining pieces.This gives Q i , and then we assemble the Here is an example of what this looks like in an especially simple case.
Figure 6 shows a nef-partition of P and the resulting dual nef-partition.In the figure, note that Q 1 is a line segment.The "dual" Q is equivalent to the polygon labeled "5a" in Figure 3, while the usual dual P • of P is the polygon labeled "3" in Figure 3.The cones C {P i } and C {Q i } are examples of dual reflexive Gorenstein cones.The duality of these cones leads to additional examples relevant to mirror symmetry.We refer the reader to [17] for details and further reading.

Conclusions
In this article we have explored several types of duality of polytopes that relate to mirror symmetry.Although both physics and algebraic geometry have moved beyond the "classical" picture of mirror symmetry presented here, we hope to have made a convincing demonstration that something wonderful happened in 1994 when Batyrev suggested that the duality of reflexive polytopes was relevant to mirror symmetry.

Figure 1 .
Figure 1.A cube P ⊆ R 3 and its dual octahedron P • .Reprinted from [4] (p. 81) with permission of the American Mathematical Society.

Figure 2 .
Figure 2. A reflexive polytope in R 3 with 14 vertices.Reprinted from [6] (p. 180) with permission of the American Mathematical Society.

Figure 4 .
Figure 4.The Hodge diamond of an arbitrary smooth projective threefold (a); the Hodge diamond of a Calabi-Yau threefold (b).