When studying Calabi-Yau manifolds, there is a particular interest at looking at the complex and Kähler moduli of the Calabi–Yau manifold which is denoted X6. Reason being, there exists Calabi-Yau manifolds
which arise in paired or mirror families with dual geometric properties to some other set of Calabi-Yau mani- folds [25]. How complex structure is varied in one family should compare to how Kähler structure is varied in the other family. The concept of mirror symmetry stems from the fact that the geometrical properties of two families X6 and X6oyield two equivalent physical theories[26]. The study of the mathematical consequences
of such a duality, is what we refer to as mirror symmetry. Consider a complex one dimensional torus (see Section1.1.5.1), the geometry of the torus is parametrized by two parameters, the complex structure τ and the Kähler class r. This is not a unique feature of a one dimensional complex torus. As was shown, in general, Calabi-Yau manifolds are parametrized by a complex structure and Kähler form. The moduli space of Calabi-Yau manifolds is thus referring to the various Kähler class and complex structure of Calabi-Yau manifolds.
An insight by Baytrev was that mirror families of Calabi-Yau manifolds can be described using combi- natorial objects called reflexive polytopes [20]. Consider a lattice polygon:
Figure 1.4: We call the collection of integer coordinate points in the plane a lattice N . A
lattice polygon is a polygon with vertices on these integer coordinate points.
A polygon is reflexive if it contains if the single lattice point in its interior is the origin. We can describe a reflexive polygon by listing the vertices and equations of each edge line. Let M be another copy of the integer lattice. The dot product allows the pairing of points in N with points in M :
(n1· n2) · (m1· m2) = n1m2+ n2m2. (1.1.171)
If we call the lattice polygon in N which contains the point (0, 0) ∆, then the polar polygon ∆o, is the polygon in M that is given by:
{(m1, m2) : (n1· n2) · (m1· m2) ≥ −1 ∀ (n1, n2) ∈ ∆}. (1.1.172)
Figure 1.5: Lattice polygon ∆ with 3 lattice points on the boundary.
by first finding the vertex coordinates, the equations of the edges become:
−x − y = −1, (1.1.173)
2x − y = −1, (1.1.174)
−x + 2y = −1. (1.1.175)
To find the polar lattice polygon, we use (1.1.172). This then gives us:
(x · y) · (−1, −1) = −1, (1.1.176)
(x · y) · (2, −1) = −1, (1.1.177)
(x · y) · (−1, 2) = −1, (1.1.178)
who’s edge equations describe the new polar lattice polygon ∆o in Figure1.6below
Figure 1.6: Polar lattice polygon ∆0 with 9 lattice points on the boundary.
This is to show that if ∆ is a reflexive polygon, then ∆o is also reflexive,
(∆o)o= ∆, (1.1.179)
Extending this concept to higher dimensions, if we have a vector {~v1, ~v2, . . . , ~vq} be a set of points inRk,
the polytope with vertices {~v1, ~v2, . . . , ~vq} is the convex hull of these points. If we let N be the lattice of
points with integer coordinates inRk, a lattice polytope has vertices in N . Similarly, if we have the dual lattice M and a dot product
(n1, . . . , nk) · (m1, . . . , mk) = n1m2+ . . . + nkmk, (1.1.180)
then we can define the polar polytope ∆o as:
{(m1, . . . , mk) : (n1, . . . , nk) · (m1, . . . , mk) ≥ −1 ∀ (n1, . . . , nk) ∈ ∆}. (1.1.181)
This lattice polytope is said to be reflexive if ∆ois also reflexive, in which case, ∆ and ∆oare mirror pairs.
It turns out that polytopes can be translated to polynomials [20,27]. This can be done in the following manner:
• Associate the zth
i complex variable with the ith standard basis vector in the lattice N ;
(1, 0, . . . , 0) ↔ z1 (1.1.182)
(0, 1, . . . , 0) ↔ z2 (1.1.183)
.. .
(0, 0, . . . , 1) ↔ zn. (1.1.184)
• For each lattice point within the polar polytope ∆o, one can define a monomial using
(m1, . . . , mk) ↔Z1(1, 0, . . . , 0) · (m1, . . . , mk)Z2(0, 1, . . . , 0) · (m1, . . . , mk) . . .Zk(0, 0, . . . , 1) · (m1, . . . , mk).
(1.1.185)
• Multiply each monomial by a complex parameter αj, and add up the monomials. This gives us a
family of polynomials parametrized by the αj.
For continuation of the examples, the Laurent polynomials for the polygon shapes in Figure1.6 are of the form given by:
α1z1−1z 2 2+ α2z−11 z2+ α3z2+ α4z−11 + α5+ α6z1+ α7z−11 z −1 2 + α8z2−1+ α9z1z2−2α1+ z12z −1 2 ,
and for Figure1.5
β1w1−1w −1
2 + β2w2+ β3+ β4w1.
Solutions to the Laurent polynomials pα constitute a geometric space. These geometric spaces, when
compactified by using techniques from algebraic geometry, form spaces Xαwhich are exactly the Calabi-Yau
varieties of of dimension d = k − 1. For k ≥ 4, the possible deformations of complex structure of Xα for a
complex vector space of dimension hd−1,1(Xα) are given by
h1,1(Xα) = l(∆) − k − 1 − X Γ l∗(Γo) +X Θ l∗(Θo)l∗( bΘo), (1.1.186) hd−1,1(Xα) = l(∆o) − k − 1 − X Γo l∗(Γo) +X Θo l∗(Θo)l∗( bΘo), (1.1.187) where
• l() = the number of lattice points.
• l∗() = the number of lattice points in the relative interior of a polytope or face.
• Γo are the codimension 1 of faces of ∆o.
• Θo are the codimensions 2 of faces of Θo.
• bΘo is the face of ∆ the dual to Θo.
When comparing this to the expressions for Xo
αone finds the following relations
h1,1(Xα) = hd−1,1(Xαo), (1.1.188)
hd−1,1(Xα) = h1,1(Xαo). (1.1.189)
This represents the mirror families of Calabi-Yau varieties Xα and Xαoof dimension d = k − 1. The classifi-
cation of these Calabi–Yau manifolds thus amounts to that of reflexive polytopes in various dimensions, and the intense computer work of Kreuzer and Skarke was to combinatorially find such polytopes. For n = 1, there are 16 such polytopes in R2, and we have Calabi–Yau onefolds, or elliptic curves. For n = 2, there
are 4319 such polytopes in R3, and we have Calabi–Yau twofolds, or K3 surfaces. For n = 3, there are
473, 800, 776 such polytopes, and we have the Calabi–Yau threefolds. In principle, the same Calabi–Yau geometry can arise from different reflexive polytopes or triangulations of a given reflexive polytope. For the Calabi–Yau threefolds the Hodge number pairs h1,1 and h1,2 have become a topic of large interest, mainly as by looking at their distribution, "experimental evidence" of mirror symmetry was seen for the first time.