# Calabi-Yau Spaces in the String Landscape

# Calabi-Yau Spaces in the String Landscape

- Yang-Hui HeYang-Hui HeOxford University and University of London

### Summary

Calabi-Yau spaces, or Kähler spaces admitting zero Ricci curvature, have played a pivotal role in theoretical physics and pure mathematics for the last half century. In physics, they constituted the first and natural solution to compactification of superstring theory to our 4-dimensional universe, primarily due to one of their equivalent definitions being the admittance of covariantly constant spinors.

Since the mid-1980s, physicists and mathematicians have joined forces in creating explicit examples of Calabi-Yau spaces, compiling databases of formidable size, including the complete intersecion (CICY) data set, the weighted hypersurfaces data set, the elliptic-fibration data set, the Kreuzer-Skarke toric hypersurface data set, generalized CICYs, etc., totaling at least on the order of ${10}^{10}$ manifolds. These all contribute to the vast string landscape, the multitude of possible vacuum solutions to string compactification.

More recently, this collaboration has been enriched by computer science and data science, the former in bench-marking the complexity of the algorithms in computing geometric quantities, and the latter in applying techniques such as machine learning in extracting unexpected information. These endeavours, inspired by the physics of the string landscape, have rendered the investigation of Calabi-Yau spaces one of the most exciting and interdisciplinary fields.

### Keywords

### Subjects

- Particles and Fields

### 1. Introduction and Motivation

The coextensivity between mathematics and theoretical physics is very much a highlight of fundamental science in the 20th and 21st centuries. The two cornerstones of modern physics—general relativity ($\text{GR}$) describing the large structure of space-time, and quantum field theory (QFT) describing the elementary particles that constitute all matter—are well understood as mathematical in nature. The former is the study of how the Ricci curvature of space-time is induced by the energy-momentum tensor dictated by the Einstein-Hilbert action and the latter, of how particles of the Standard Model, furnishing representations of the Lie group $SU(3)\times SU(2)\times U(1)$, interact according to a gauge theory. Both formalisms are tested by experiment to astounding accuracy, with the recent discovery of the Higgs boson and the detection of gravity waves being jewels in the crown of physics.

The brainchild of this tradition of mathematical unification, still dominating mathematical and theoretical physics as the 21st century began, is string theory. It is well known by now that string theory is a “theory of everything” unifying $\text{GR}$ and QFT in 10 space-time dimensions (or, equivalently, by dualities, $\text{M}$-theory in 11 dimensions or $\text{F}$-theory in 12 dimensions). We must therefore account for $10-4=6$ “missing” dimensions. Over the years, a plethora of scenarios have emerged in the interplay between the physics of four dimensions and the geometry of these six dimensions. So great, however, is the number of such scenarios that it poses as one of the greatest theoretical challenges to modern physics. This has come to be known as the “vacuum degeneracy problem,” where one is confronted with how to select our universe amidst an enormous number of solutions, what has come to be known as the string landscape.

As a disclaimer from the outset, this article does not cover the various issues of the string landscape or estimates of its vastness and the probabilities of finding our Standard Model therein. What will constitute the topic of the present article is the most standard and well-studied class of solutions for the six extra dimensions: the so-called Calabi-Yau spaces. The focus here is on the Calabi-Yau landscape, a plethora of geometries in and of themselves. In many ways, the aim is to guide a theoretical physicist through a lightning tour of some contemporary algebraic and differential geometry, with the jargons introduced in a pragmatic manner rather than through formalism.

This article takes a chronological perspective, looks at how Calabi-Yau spaces arise in the physics, and discusses how explicit constructions have been motivated by particle phenomenology and how a plethora of examples has been constructed under a long-term collaboration among mathematicians, physicists, and computer scientists. First, however, the scene is briefly set by introducing a classical mathematical problem dating back to Euler, Gauss, and Riemann. This digression helps to set notation and definition, as well as to gain some intuition.

#### 1.1. A Classical Problem in Mathematics

Consider a surface $\mathrm{\Sigma}$—for example, a sphere ${S}^{2}$ or the surface of a doughnut ${T}^{2}$—and its possible topological types, that is, equivalences up to topology. Restricting to the cases of smooth, compact (no punctures or boundaries), and orientable (nothing like Klein bottles or Möbius strips) surfaces, the classic result is that the shapes that immediately came to mind, ${S}^{2},\phantom{\rule{0.2em}{0ex}}{T}^{2}$, and those with an increasing number of “holes,” are all there is: any smooth, compact, orientable surface can be deformed continuously (topologically homeomorphic) to one of these. The top of Figure 1 illustrates these surfaces. It is clear that a single nonnegative integer classifies the topology of $\mathrm{\Sigma}$, viz., the genus $g(\mathrm{\Sigma})$, referring to the number of “holes.” The quantity $\chi (\mathrm{\Sigma})=2-2g(\mathrm{\Sigma})$ is called the Euler characteristic, or Euler number. Both these quantities are shown in the figure.

As is often the case in mathematics, it is expedient to work over the complex numbers $\u2102$ rather than $\mathbb{R}$ because of the algebraic closure of $\u2102$ (every algebraic equation has solutions over $\text{C}$). One can, in fact, think of $\mathrm{\Sigma}$ as a one-complex-dimensional object (a complex curve) instead of as a 2-dimensional real object (a real surface). For instance, ${S}^{2}$ is the complex plane $\u2102$ with the point at $\infty $ added. Likewise ${T}^{2}$ is $\u2102\simeq {\mathbb{R}}^{2}$ quotiented by a lattice.^{1} What has been done is to endow $\mathrm{\Sigma}$ with so-called complex structure; every orientable surface $\mathrm{\Sigma}$ can be thus equipped and is then called a Riemann surface or a (complex) algebraic curve.^{2} Hence, $\mathrm{\Sigma}$ affords complex local coordinates $(z,\phantom{\rule{0.2em}{0ex}}\overline{z})=(x+iy,\phantom{\rule{0.2em}{0ex}}x-iy)$, where $(x,\phantom{\rule{0.2em}{0ex}}y)$ can be thought of as the usual real coordinates.

In addition to complex structure, there is one more rich property that can be put on $\mathrm{\Sigma}$: the so-called Kähler structure. If the metric ${g}_{\mu \overline{v}}(z,\overline{z})$ comes from^{3} a single scalar function, a (real) potential $\mathcal{K}(z,\phantom{\rule{0.2em}{0ex}}\overline{z})$, via

then, $\mathrm{\Sigma}$ is not only complex but is furthermore Kähler.^{4} For Riemann surfaces, this is always possible.

With the introduction of the metric, it is natural to ask about curvature. This is the classical theorem of Gauss-Bonnet: to relate curvature (differential gemetry) with the Euler number (topology). In fact, a wonderful chain of equalities exists:

The first equality is the definition of the Euler number in terms of the genus. The second relates it to the integration of the Gaussian curvature over $\mathrm{\Sigma}$ and is the content of Gauss-Bonnet, relating the analytic concept of curvature to the topological concept of Euler number. This is the epitome of classical differential geometry.

As one goes further to the right, one enters algebraic geometry. The third equality can be construed as a working definition^{5} for the first Chern class ${c}_{1}(\mathrm{\Sigma})$ of $\mathrm{\Sigma}$: it is essentially the curvature $R$. Meanwhile, higher Chern classes ${c}_{i=\mathrm{2,3},}\dots $ can be written as specific polynomials in $R$. Importantly, these are topological invariants in that small perturbations of the metric do not change the value of the integral ${\int}_{\mathrm{\Sigma}}}R$.

The fourth and last equality relates $\chi $ to the alternating sum^{6} over the so-called Betti numbers ${b}^{i}$. This sum is the generalization of a familiar topological fact: for any convex polyhedron $P$ drawn on the sphere, with $V$ vertices (dimension $0$), $E$ edges (real dimension $1$), and $F$ faces (real dimension $2$), gives that $V-E+F=2$ (taking the cube, e.g., one has $8-12+6=2$). For a convex polyhedron drawn on a genus $g$ Riemann surface, $V-E+F$ gives the Euler number $\chi $. In general, the Betti numbers ${b}^{i}$ of a given space count independent dimension $i$ objects (formally, they are called algebraic cycles) therein.

Equation (1.2) is a special case of the Atiyah-Singer Index Theorem, a deep result relating, as one proceeds from left to right topology, to differential geometry, to algebraic geometry, to combinatorics. From Figure 1, a natural trichotomy of Riemann surfaces can be seen: positive curvature (the sphere), zero curvature (the torus), and negative curvature $(g>1)$. Keep in mind this boundary case of ${T}^{2}$: it is compact, smooth, Ricci-flat, Kähler, and of complex dimension 1.

A central theme of modern geometry is to see how the various connections for Riemann surfaces/complex curves, which have been touched on, generalize to beyond complex dimension 1/real dimension 2. Now, the realm of the geometry of manifolds is entered, that is, shapes that are locally ${\mathbb{R}}^{n}$, just like $\mathrm{\Sigma}$ is locally ${\mathbb{R}}^{2}\simeq \u2102$. As expected, the situation is much more complicated.

Straight away, it is unfortunately *not* the case that every real $2n$-dimensional manifold can admit a complex structure and be turned into a complex $n$-fold^{7} (note that throughout, the notation that an $n$-fold is a complex manifold of complex dimension $n$ and real dimension $2n$ is adhered to). Moreover, it is not true that every -fold admits a Kähler structure. Nevertheless, when a $2n$-manifold admits a complex structure, and further, a Kähler structure, something akin to Figure 1 can be seen, as happens now.

#### 1.2. A Modern Breakthrough

In the 1950s, E. Calabi (1954, 1957) conjectured that for $\text{a}$ (compact) Kähler manifold, the first Chern class uniquely governs^{8} the Ricci curvature of the Kähler metric (1.1), much in the spirit of (1.2). In particular, ${c}_{1}(M)=0$ is the “boundary” situation where the metric is Ricci flat, much like the ${T}^{2}$ example for Riemann surfaces.

It took several decades for the existence part of this conjecture to be settled (the uniqueness is not too hard and follows from standard contradiction arguments). S.-T. Yau (1977, 1978) proved the conjecture in 1978, using analysis of a partial differential equation (PDE) of Monge-Ampère type. The fact that Yau was immediately awarded the Fields Medal (1982) shows the importance of the result. In honor of the two, the critical case of zero Ricci curvature Kähler manifolds (i.e., with ${c}_{1}=0$) are called Calabi-Yau manifolds. Henceforth, a Calabi-Yau $n$-fold is denoted as ${\text{CY}}_{n}$. The familiar torus ${T}^{2}$ from the discussions in Section 1.1, “A Classical Problem in Mathematics,” is thus $\text{a}$ (in fact, *the*) ${\text{CY}}_{1}$. For readers’ convenience, the chain of specializations is summarized as follows:

#### 1.3. Topological Quantities

Section 1.2, a “Modern Breakthrough” introduced the notion of the Betti number ${b}^{i}$ with $i$ ranging from $0$ to the (real) dimension of the manifold. As a complex number is paired with its conjugate, for a complex $n$-fold $X,\phantom{\rule{0.2em}{0ex}}{b}^{i}$ is refined into a doubled-indexed object ${h}^{j,k}$, counting the holomorphic (corresponding to the $z$-variables) and anti-holomorphic (corresponding to the $\overline{z}$-variables) pieces; these are so-called Hodge numbers, satisfying

For smooth, compact $n$-folds, complex conjugation renders ${h}^{j,k}={h}^{k,j}$ and something called Poincaré duality renders ${h}^{j,k}={h}^{n-j,n-k}$.

Moreover, assuming that $X$ is connected (every point is linked to another by a path in $X$) renders ${b}^{0}={h}^{\mathrm{0,0}}=1$, and assuming that it is simply connected (all closed paths can be retracted without breaking) forces ${b}^{1}={h}^{\mathrm{1,0}}={h}^{\mathrm{0,1}}=0$. Finally, if $X$ is $\text{CY}$, then^{9} ${h}^{n,0}={h}^{0,n}=1$.

Putting all these constraints together, the point is that for (compact, smooth, simply connected) ${\text{CY}}_{3}$, there are only a few independent Hodge numbers, as illustrated by the following:

Here, it is customary to write the Hodge numbers into a rhombic shape called the Hodge diamond. It can be seen that ${\text{CY}}_{3}$ is characterized by only two integers, ${h}^{\mathrm{1,1}}$ and ${h}^{\mathrm{2,1}}$, which count precisely the number of Kähler and complex structure deformations, respectively. In section 2.4, “Finiteness?,” other quantities that completely characterize a Calabi-Yau manifold topologically are addressed. Importantly, a consequence of (1.4) and (1.2) is that

which, interestingly, has physical significance, as is discussed in 1.4 “A Timely Relation to Physics”.

#### 1.4. A Timely Relation to Physics

What does all this beautiful mathematics have to do with physics? First, Ricci-flat manifolds have long been known to physicists because they satisfy vacuum Einstein equations in $\text{GR}$. Perhaps less familiar is the complex (and, moreover, Kähler) condition.

This side of the story begins with Candelas, Horowitz, Strominger, and Witten (CHSW) [3] in 1985, a culmination of the first string revolution, when Green and Schwarz (1984) canceled anomalies and when the heterotic string of Gross, Harvey, Martinec, and Rohm (1985) naturally gave an ${E}_{8}$ gauge group to string theory. Recall that ${E}_{8}$ is of particular significance because of the following sequence of embeddings of Lie groups:

The first group ${G}_{SM}=SU(3)\times SU(2)\times U(1)$ is, of course, that of the Standard Model ($\text{SM}$). One could also include the baryon-lepton symmetry $U{(1)}_{B-L}$ and write ${{G}^{\prime}}_{SM}=SU(3)\times SU(2)\times U(1)\times U{(1)}_{B-L}$, in which case the above sequence of embeddings skips $SU(5)$.

The fact that ${G}_{SM}$ is not simple has troubled many physicists since the early days: it would be more pleasant to place the baryons and leptons in the same footing by allowing them to be in the same representation of a larger simple gauge group. This is the motivation for the sequence in (1.6): starting from $SU(5)$, the gauge groups are simple and give archetypal grand unified theories (GUTs); the most popular historically had been $SU(5)$, $\text{S}O(10)$, and ${E}_{6}$, long before string theory came onto the scene in theoretical physics. Finally, throughout supersymmetric extensions of $\text{SM}$ are always considered, such as the minimally supersymmetric standard model (MSSM).^{10}

In the compactification scenario, the 10-dimensional background is taken to be of the form ${\mathbb{R}}^{\mathrm{1,3}}\times {X}_{6}$, with ${\mathbb{R}}^{\mathrm{1,3}}$ the familiar space-time and ${X}_{6}$ some small, curled-up 6-manifold too small to be currently observed directly, and possibly endowed with extra structure such as a vector bundle $\mathcal{V}$ (this subject of bundles is returned to later). CHSW gave the conditions for which the heterotic string, when compactified, would give a supersymmetric gauge theory in ${\mathbb{R}}^{\mathrm{1,3}}$, with a potentially realistic particle spectrum. Heuristically, one needs ${X}_{6}$ to be complex in order to have chiral fermions, and Kähler and Ricci flat, to preserve supersymmetry and to solve vacuum Einstein’s equations. In other words, ${X}_{6}$ is a Calabi-Yau threefold.^{11}

It should be emphasized that there are many possible solutions for ${X}_{6}$ as well as its associated further geometric structures. Indeed, for the duality-equivalent, 11-dimensional-theory as well as 12-dimensional-theory formulations of string theory, the compactification manifolds ${X}_{7}$ and ${X}_{8}$ allow even further possible solutions. All these contribute to the vastness of the string landscape, which is not addressed here. It can now be seen that even for ${\text{CY}}_{3}$, a substantial multitude of possibilities already exist.

### 2. Calabi-Yau Constructions and Databases

As with any manifold, an important structure is its tangent bundle (for a sphere, think of this as the space of tangent planes). For $X$, a ${\text{CY}}_{3}$, the tangent bundle $\mathcal{V}={T}_{X}$ has an extra property that it has $SU(3)$ structure, meaning that the group for parallel transport^{12} of vectors (the holonomy group) is $SU(3)$. This $SU(n)$ holonomy is, in fact, another equivalent definition of a $\text{CYn}$.

Now, from Lie theory, it is understood that $SU(3)$ has commutant ${E}_{6}$ in ${E}_{8}$; that is, a maximal commuting pair within the group ${E}_{8}$ is $SU(3)\times {E}_{6}$. In relation to the compactification scenario in Section 1.4, “A Timely Relation to Physics,” this means that from a ${\text{CY}}_{3}$ compactification one naturally has an (supersymmetric) ${E}_{6}$ GUT theory in ${\mathbb{R}}^{\mathrm{1,3}}$. The particle content is readily determined from group theory. More importantly, this in turn determines the vector bundle cohomology group that is associated with the particles. The general paradigm is that

Thus is born the subject of string phenomenology.

Specifically, one has that the decomposition of the adjoint 248 of ${E}_{8}$ breaks into $SU(3)\times {E}_{6}$ as $248\to (\mathrm{1,78})\oplus (\mathrm{3,27})\oplus (\overline{3},\phantom{\rule{0.2em}{0ex}}\overline{27})\oplus (\mathrm{8,1})$. Thus the SM particles, which in an ${E}_{6}$ GUT all reside in its 27 representation, are associated with the fundamental 3 of $SU(3)$. The 10-dimensional fermions are eigenfunctions of the Dirac operator, which splits into the 4-dimensional one and the 6-dimensional one on the ${\text{CY}}_{3}$. The former gives the fundamental particles that are seen, and the latter is governed by the geometry of $X$. Later, in Section 2.2.1, “Recent Developments,” it is demonstrated how this setup can be generalized to other GUT groups and to the $\text{SM}$.

Here, for example, it turns out that the 27 representation is precisely associated to ${h}^{\mathrm{2,1}}(X)$, and the conjugate $\overline{27}$, to ${h}^{\mathrm{1,1}}(X)$ (q.v. Griffiths & Harris, 1978). As promised at the end of Section 1.4, “A Timely Relation to Physics,” the topological invariants of $X$ determine the particle content of the 4-dimensional physics. It can be summarized that for the fundamental fermions (the choice of which is antigeneration and which is generation by convention):

An immediate constraint is that there should be three net generations (Tanabashi et al., 2019), meaning that

where (1.5) has been used. Thus the endeavor of finding Calabi-Yau threefolds with the property (2.8) began in 1986. This geometrical “love for threeness”^{13} has been more recently dubbed by Candelas et al. as triadophilia (Candelas, de la Ossa, He, & Szendroi, 2008).

Having extracted a mathematical problem from physical constraints, a practical quest in geometry resulted, which has prompted some 40 years of research. This was one of the first times theoretical physics gave a precise homework problem to modern geometry: Does there exist $a$ (compact, smooth, simply connected) Calabi-Yau threefold with $\chi =\pm 6?$ Indeed, how does one construct a CYn at all?

#### 2.1. The Quintic

Constructing shapes from Cartesian geometry is a familiar concept. For example, a quadratic equation in two real variables $(x,\phantom{\rule{0.2em}{0ex}}y)$ is a conic section, such as a circle, in ${\mathbb{R}}^{2}$. Thus, two variables with one polynomial constraint gives a $2-1=1$ dimensional real manifold in an ambient ${\mathbb{R}}^{2}$. This simple idea is the beginnings of algebraic geometry.

Because one is looking for complex manifolds, they can simply be constructed as the zero locus of multivariate polynomials in complex variables. From the introduction, recall that a Calabi-Yau onefold is nothing but a Riemann surface of zero curvature, namely, the torus ${T}^{2}={S}^{1}\times {S}^{1}$. This can be realized as a cubic in two complex variables given by the so-called Weierstraß equation: ${T}^{2}\simeq \{x,\phantom{\rule{0.2em}{0ex}}y\in \u2102|{y}^{2}={x}^{3}-{g}_{2}x-{g}_{4}\}\subset {\u2102}^{2}$, where ${g}_{\mathrm{2,4}}$ are complex constants.^{14}

Now, compactness is needed, which means the point at infinity must be included, where $(x,\phantom{\rule{0.2em}{0ex}}y)=(\infty ,\phantom{\rule{0.2em}{0ex}}\infty )$. One can do this by so-called projectivization, where, instead of ${\u2102}^{2}$, one more complex coordinate, $z$, is introduced, such that the point $(x,\phantom{\rule{0.2em}{0ex}}y,\phantom{\rule{0.2em}{0ex}}z)\in {\u2102}^{3}$ is to be identified with $\lambda (x,\phantom{\rule{0.2em}{0ex}}y,\phantom{\rule{0.2em}{0ex}}z)$ for any nonzero $\lambda \in \u2102$. This scale invariance brings the point at infinity to a finite point, rendering the resulting ambient ${\u2102}^{2}$ and the subsequent torus compact. What has been done is to construct, from ${\u2102}^{3}$ with coordinates $(x,\phantom{\rule{0.2em}{0ex}}y,\phantom{\rule{0.2em}{0ex}}z)$, the complex projective space $\u2102{\mathbb{P}}^{2}$ with scale-invariant (or so-called homogeneous) coordinates $[x:y:z]$. Formally, one can define $\u2102{\mathbb{P}}^{n}$ from ${\u2102}^{n+1}$ with coordinates $({z}_{0},\phantom{\rule{0.2em}{0ex}}{z}_{1},\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.2em}{0ex}}{z}_{n})$ via the quotient by the equivalence relation$\sim $ as

Note that the origin $\left\{\overrightarrow{0}\right\}$ is taken out because it is a fixed point under $\sim $ and would give rise to a singularity; $\u2102{\mathbb{P}}^{n}$ is a smooth complex manifold of complex dimension $n$ with $n+1$ homogeneous coordinates.

In summary, one can construct a ${\text{CY}}_{1}$ as the zero locus of a homogeneous cubic polynomial (the Weierstraß equation) inside $\u2102{\mathbb{P}}^{2}$

where the homogeneous coordinates $[x:y:z]$ means that the point $(x,\phantom{\rule{0.2em}{0ex}}y,\phantom{\rule{0.2em}{0ex}}z)$ is to be identified with $\lambda (x,\phantom{\rule{0.2em}{0ex}}y,\phantom{\rule{0.2em}{0ex}}z)$ for any nonzero $\lambda $, and ${g}_{2}$ and ${g}_{4}$ are complex coefficients.

In a nutshell, algebraic geometry is the study of how complex manifolds arise as zero locus of multivariate complex polynomials in projective spaces. When there is only a single polynomial, it is called a hypersurface. In general, there could be many polynomials defining the manifold as an algebraic variety. For hypersurfaces, the dimension is simply that of the ambient space minus one. For instance, in the above example, the ${T}^{2}$ is of complex dimension $2-1=1$, as required. Luckily, loci of homogeneous polynomials in complex projective space are guaranteed to be Kähler manifolds, so one is already halfway there.

This construction, of having a degree $n+1$ homogeneous polynomial in $\u2102{\mathbb{P}}^{n}$ is indeed valid in general: one can show that if the number of projective coordinates, here $n+1$, equals the degree of the hypersurface, this implies the vanishing of the first Chern class. In other words, the hypersurface defined by a homogeneous polynomial of degree $n+1$ in $\u2102{\mathbb{P}}^{n}$ is a Calabi-Yau $(n-1)$-fold. Thus one arrives at the first, and perhaps most famous, example of a Calabi-Yau threefold: the quintic hypersurface in $\u2102{\mathbb{P}}^{4}$. There are many degree 5 monomials (the number of these different monomials is roughly the complex structure) one could compose of five coordinates; the most well studied is the so-called Fermat quintic:

What are the topological numbers of $Q$? It turns out^{15} that ${h}^{\mathrm{2,1}}(Q)=101$ and ${h}^{\mathrm{1,1}}(Q)=1$ so that $\chi (Q)=2(1-101)=-200$, and this is quite far from $\pm 6$.

#### 2.2. The CICY Database

To continue to address the question raised in equation (2.8), an algorithmic generalization of the construction for the quintic was undertaken: Instead of a single $\u2102{\mathbb{P}}^{n}$, what about embedding a collection of (homogeneous) polynomials into a product $A$ of projective spaces? For further simplification, consider only complete intersections, which means the optimal case where the number of equations is 3 less than the dimension of the ambient space $A$, so that each polynomial reduces exactly one degree of freedom.

In other words, let $A=\u2102{\mathbb{P}}^{{n}_{1}}\times \dots \times \u2102{\mathbb{P}}^{{n}_{m}}$, of dimension $n={n}_{1}+{n}_{2}+\dots +{n}_{m}$ and each having homogeneous coordinates $[{x}_{1}^{(r)}:{x}_{2}^{(r)}:\dots :{x}_{{n}_{r}}^{(r)}]$ with the superscript $(r)={n}_{1},\phantom{\rule{0.2em}{0ex}}{n}_{2},\dots ,{n}_{m}$ indexing the projective space factors. Our ${\text{CY}}_{3}$ is then defined as the intersection of $K=n-3$ homogeneous polynomials in the coordinates ${x}_{j}^{(r)}$. Clearly this is a generalization of the quintic, for which $r=m,\phantom{\rule{0.2em}{0ex}}{n}_{r}=4$, and $K=1$, and more generally a hypersurface is thus trivially a complete intersection with one defining polynomial.

The Calabi-Yau condition of the vanishing of ${c}_{1}({T}_{X})$ generalizes analogously to the condition that for each $r=1,\dots ,m$, one has $\sum _{j=1}^{K}{q}_{j}^{r}={n}_{r}+1$. Succinctly, this information can be written into an $m\times K$ configuration matrix (to which the first column could be adjoined, designating the ambient product of projective spaces, for clarity; this is redundant because one can extract ${n}_{r}$ from one less than the row sum):

For example, the matrix configuration [5], or $[4|5]$, denotes the quintic. Two more immediate examples are

The first is called the Schoen Manifold, and the second, the Yau-Tian. Specifically, the configuration $S$ means that the ambient space is $\u2102{\mathbb{P}}^{1}\times \u2102{\mathbb{P}}^{2}\times \u2102{\mathbb{P}}^{2}$ of dimension 5. A check shows that because there are two polynomials (columns), $5-2=3$ gives a ${\text{CY}}_{3}$. The first column $(1,\phantom{\rule{0.2em}{0ex}}3,\phantom{\rule{0.2em}{0ex}}0)$ means that the first polynomial is linear in $\u2102{\mathbb{P}}^{1}$ and cubic in the first $\u2102{\mathbb{P}}^{2}$ while having no dependence on the second $\u2102{\mathbb{P}}^{2}$ The second column is likewise defined. Another lesson, as can be induced from $S$ and $\tilde{S}$, is the rather cute fact that the transpose gives a new valid configuration, though the pair can be of wildly different geometry. The topological numbers are often attached as ${X}_{\chi}^{{h}^{\mathrm{1,1}},{h}^{\mathrm{2,1}}}$ for completeness, so one can write, for instance, ${Q}_{-200}^{\mathrm{1,101}},\phantom{\rule{0.2em}{0ex}}{S}_{0}^{\mathrm{19,19}}$ and and ${\tilde{S}}_{-18}^{\mathrm{14,23}}{\tilde{S}}_{-18}^{\mathrm{14,23}}$. Importantly, the Chern classes and the Euler number can be read off the matrix configuration explicitly. The individual terms $({h}^{\mathrm{1,1}},\phantom{\rule{0.2em}{0ex}}{h}^{\mathrm{2,1}})$ in (1.5), however, cannot be deduced from the configuration matrix directly.^{16}

Manifolds defined by (2.11) were considered and explicitly constructed by Candelas, Dale, Lutken, and Schimmrigk (1988) (q.v. Hübsch’s classic book [1994]) in the early 1990s and were affectionately called CICYs (complete intersection Calabi-Yau manifolds). Classifying these above matrices, up to geometrical equivalence, was thus a relevant affair for physics: one could, for instance, read off the Euler number to see whether any of them had magnitude of 6. Obviously it was also relevant for mathematics: until then, creating data sets of algebraic varieties was not the style of questions for pure mathematicians. The combinatorial problem for these integer matrices turned out to be rather nontrivial and one of the most powerful supercomputers then available was recruited. This was perhaps the first time when heavy machine computation was done for the sake of algebraic geometry.

In all, CICYs were shown to be finite in number, a total of 7,890 inequivalent configurations, with a maximum of 12 rows, a maximum of 15 columns, and all having entries ${q}_{j}^{r}\in [\mathrm{0,5}]$. There are 266 distinct Hodge pairs $({h}^{\mathrm{1,1}},\phantom{\rule{0.2em}{0ex}}{h}^{\mathrm{2,1}})=(\mathrm{1,65}),\dots ,\left(\text{19},19\right)$, giving 70 distinct Euler numbers $\chi \in [-\mathrm{200,0}]$.

Unfortunately, none of the 7,890 had $\chi =\pm 6$. Although this was initially disappointing, it was soon realized that circumventing this problem gave rise to the resolution of another important physical question. A freely acting order-3 symmetry was found on $\tilde{S}$; the freely acting is important, because it means that the quotient ${S}^{\prime}=\tilde{S}/{\mathbb{Z}}_{3}$ is also a smooth ${\text{CY}}_{3}$, albeit not a CICY. For such smooth quotients, the Euler divides (the individual Hodge numbers do not and it turns out that one has $({h}^{\mathrm{1,1}},\phantom{\rule{0.2em}{0ex}}{h}^{\mathrm{2,1}})=(\mathrm{6,9})$ for ${S}^{\prime}$), and ${S}^{\prime}$ became the first three-generation manifold.

Now, the quotienting is crucial for another reason. In the sequence (1.6), the focus has been on GUTs. What about the Standard Model itself? It so happens that one standard way of obtaining ${G}_{SM}=SU(3)\times SU(2)\times U(1)$ from any of the GUT groups is precisely by quotienting. Grouped theoretically, this amounts to finding a discrete group whose generators can be embedded into the GUT group, so that the commutant is the the desired ${G}_{SM}$.

Geometrically, this is the action of the Wilson Line, where a ${\text{CY}}_{3}$ with nontrivial fundamental group admits a nontrivial loop which, coupled with the discrete group action, decomposes the ${E}_{8}$ to ${G}_{SM}$. In our example above, $\tilde{S}$ has trivial fundamental group ${\pi}_{1}$, but the quotient ${S}^{\prime}$ has, by construction, ${\pi}_{1}({S}^{\prime})\simeq {\mathbb{Z}}_{3}$, thereby admitting a ${\mathbb{Z}}_{3}$ Wilson line. This, for the early models, can be used to break the ${E}_{6}$ GUT down to the Standard Model. Not surprisingly, the manifold ${S}^{\prime}$ became central to string phenomenology in the early days (Greene, Kirklin, Miron, & Ross, 1986, 1987).

##### 2.2.1. Recent Developments

As mentioned, the compactification scenario can, in general, involve Calabi-Yau manifolds as well as extra structures thereon, such as fluxes and bundles. Indeed, CHSW gave a more general set of solutions in terms of the ${\text{CY}}_{3}X$ as well as a field strength $F$. Specifically, the set of conditions, known as the Hull-Strominger system (Hull, 1986; Strominger, 1986), for the low-energy, low-dimensional theory on ${\mathbb{R}}^{\mathrm{1,3}}$ to be a supersymmetric gauge theory are (rather schematically as this article does not delve into the details of equations):

${X}_{6}$ is complex

2.The (Hermitian) metric $g$ (as well as its Ricci curvature $R$) on ${X}_{6}$ and field strength $F$ satisfy

$\partial \overline{\partial}g=i\phantom{\rule{0.2em}{0ex}}\text{Tr}\phantom{\rule{0.2em}{0ex}}F\wedge F-i\phantom{\rule{0.2em}{0ex}}\text{Tr}\phantom{\rule{0.2em}{0ex}}R\wedge R$

$(\partial +\overline{\partial})(g\mathrm{\Omega}\wedge \overline{\mathrm{\Omega}})=0$, where $\mathrm{\Omega}$ is a holomorphic 3-form on ${X}_{6}$

$F$ satisfies the Hermitian Yang-Mills equations: ${\omega}^{\mu \overline{\nu}}{F}_{a\overline{\nu}}=0,\phantom{\rule{0.2em}{0ex}}{F}_{\mu \overline{\nu}}={F}_{\overline{\mu \nu}}=0$

In the above, $\wedge $ is the wedge product of differential forms, and $\partial $ is the derivative with respect to the holomorphic coordinate (appropriately generalized for forms). The general solutions to this system continue to inspire research today, engendering more geometric structures that contribute to the landscape.

Technically, the field strength lives on a vector bundle $\mathcal{V}$ on $X$, and the case discussed at the beginning of this section, of $X$ being ${\text{CY}}_{3}$ and $\mathcal{V}={T}_{X}$ came to be known as the “standard embedding” and gave rise to ${E}_{6}$ GUT theories. Nonstandard embedding, by taking the vector bundle $\mathcal{V}$ to have structure group $SU(4)$ or $SU(5)$, gives $SO(10)$ and $SU(5)$ GUTs, respectively. Equations (2.7) and (2.8) generalize to so-called computations of bundle cohomology groups of $\mathcal{V},\phantom{\rule{0.2em}{0ex}}{\mathcal{V}}^{*}$ and their tensor powers, which give the low-energy particle spectrum. The cubic Yukawa couplings in the Lagrangian constituted by these particles (fermion-fermion-Higgs) are appropriate trilinear maps^{17} taking these cohomology groups to $\u2102$. The group theory^{18} can be summarized as follows:

${E}_{8}\to G\times H$ |
Breaking Pattern |
---|---|

$\begin{array}{l}\phantom{\rule{0.1em}{0ex}}SU(3)\times \phantom{\rule{0.2em}{0ex}}{E}_{6}\phantom{\rule{0.1em}{0ex}}\hfill \\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.2em}{0ex}}SU(4)\times \phantom{\rule{0.2em}{0ex}}SO(10)\phantom{\rule{0.1em}{0ex}}\hfill \\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.2em}{0ex}}SU(5)\times \phantom{\rule{0.2em}{0ex}}SU(5)\phantom{\rule{0.1em}{0ex}}\hfill \end{array}$ |
$\begin{array}{l}\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.2em}{0ex}}248\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\to \phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}(1,\phantom{\rule{0.2em}{0ex}}78)\oplus (\mathrm{3,27})\oplus (\overline{3},\phantom{\rule{0.2em}{0ex}}\overline{27})\oplus (\mathrm{8,1})\phantom{\rule{0.1em}{0ex}}\hfill \\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.2em}{0ex}}248\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\to \phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}(1,\phantom{\rule{0.2em}{0ex}}45)\oplus (\mathrm{4,16})\oplus (\overline{4},\phantom{\rule{0.2em}{0ex}}\overline{16})\oplus (\mathrm{6,10})\oplus (\mathrm{15,1})\phantom{\rule{0.1em}{0ex}}\hfill \\ \phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.2em}{0ex}}248\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\to \phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}(1,\phantom{\rule{0.2em}{0ex}}24)\oplus (5,\phantom{\rule{0.2em}{0ex}}\overline{10})\oplus (\overline{5},10)\oplus (\mathrm{10,5})\oplus (\overline{10},\overline{5})\oplus (\mathrm{24,1})\phantom{\rule{0.1em}{0ex}}\hfill \end{array}$ |

Constructing bundles over ${\text{CY}}_{3}$ satisfying the Hull-Strominger system has become an industry of realistic model building (q.v. a long program initiated by Ovrut et al. [Donagi, Ovrut, Pantev, & Waldram, 2002; Donagi, He, Ovrut, & Reinbacher, 2004, 2005]). As mentioned at the end of Section 2.2, “The CICY Database,” one could proceed one more step by Wilson line projection from the GUT to the $\text{SM}$.^{19} The first heterotic compactifications with exact MSSM particle content were constructed by finding appropriate bundles on the quotient of the Schoen manifold $S$, thereby answering the 20-year-old challenge (Bouchard & Donagi, 2006; Braun, He, Ovrut, & Pantev, 2005, 2006). Another was found over the bi-cubic from the CICY database in Anderson, Gray, He, and Lukas (2010). Exploring the “zoo” of small Hodge Calabi-Yau threefolds that seems to be a fertile ground for exact $\text{SM}$ solutions itself became a program of Braun, Candelas, and Davies (2010); Candelas, Constantin, and Mishra (2018); Candelas and Davies (2010); and Candelas, de la Ossa, He, and Szendroi (2008).

All discrete, freely acting symmetries for all CICYs were classified with an impressive computer search using the computer algebra packages (SageMath, 2020; The Gap Group, 2018) by Braun (2011). Subsequently, a systematic scan over bundles (of the form of direct sum of line bundles and with chosen Kähler parameters in the stability region) over all CICYs (up to ${h}^{\mathrm{1,1}}=6$) was performed by Anderson et al., with the right cohomology groups computed, so that the result is the exact particle content of the $\text{SM}$ (Anderson, Gray, Lukas, & Palti, 2011, 2012). This is a tremendous task, and some 200 models were found in about ${10}^{10}$ candidates. This is curiously in line with the l-in-a-billion statistic from the type II scan for finding the $\text{SM}$ (Gmeiner, Blumenhagen, Honecker, Lust, & Weigand, 2006).

Meanwhile, from a mathematical point of view, several generalizations of the CICY data were undertaken. CICY4, the fourfold version of (2.11) was completed by Gray, Haupt, and Lukas (2013, 2014), and 921,497 were found. Furthermore, it was found that one can in fact relax the condition that the configuration matrix entries be nonnegative (Anderson, Apruzzi, Gao, Gray, & Lee, 2016), giving so-called gCICYs (generalized CICYs), thereby generating vast numbers of new ${\text{CY}}_{3}\text{s}$.

#### 2.3. A Plethora of CY_{3}

Indeed, this brings up the heart of a question, in essence both mathematical and physical: How many $C{Y}_{3}s$ are there? In a way, this presents an interesting sequence: in complex dimension 1, there is only the torus that is ${\text{CY}}_{1}$; in dimension 2, there turn out to be two (the 4-dimensional torus ${T}^{4}=({S}^{1}{)}^{4}$ and a so-called K3 surface). Starting in dimension 3, and until now, no idea exists as to how many distinct smooth manifolds there are, even though literally billions have been found, as can now be seen. In this section, the various databases constructed since the CICYs are presented, which progressed in approximate five-year periods since the late 1980s.

##### 2.3.1. Weighted Hypersurfaces

After the success story of CICYs, the search continued, partially because all $\chi =2({h}^{\mathrm{1,1}}-{h}^{\mathrm{2,1}})$ for CICYs were negative, but it was already suspected that there should be mirror symmetry so that each ${\text{CY}}_{3}$ with Hodge pair $({h}^{\mathrm{1,1}},\phantom{\rule{0.2em}{0ex}}{h}^{\mathrm{2,1}})$ has a mirror with these numbers reversed. Thus there should be, for each negative $\chi $, another ${\text{CY}}_{3}$ with positive $\chi $. Mirror symmetry is one of the most exciting subjects of modern mathematics and theoretical physics; unfortunately this article cannot delve into this subject because of limitations of space, and readers are instead referred to the authoritative volumes of Hori et al. (2003).

Recall that the first example of the quintic $Q$ had $\u2102{\mathbb{P}}^{4}$ as its ambient space $A$. Another natural generalization is to take weighted projective space $\u2102{\mathbb{P}}_{[{d}_{0}:\dots :{d}_{4}]}^{4}$ as $A$; this generalizes (2.9) by having integer “weights” $({d}_{0},\phantom{\rule{0.2em}{0ex}}{d}_{1},\phantom{\rule{0.2em}{0ex}}{d}_{2},\phantom{\rule{0.2em}{0ex}}{d}_{3},\phantom{\rule{0.2em}{0ex}}{d}_{4})\in {\mathbb{Z}}_{+}$ as

Of course, taking all weights ${d}_{i}=1$ is the ordinary $\u2102{\mathbb{P}}^{4}$. As with $Q$, if a hypersurface of degree ${d}_{0}+{d}_{1}+\dots +{d}_{4}$ is embedded into $\u2102{\mathbb{P}}_{[{d}_{0}:\dots :{d}_{4}]}^{4}$, it defines a ${\text{CY}}_{3}$.

One caveat is that unlike ordinary projective space, weighted projective spaces are generically singular, and care must be taken to make sure the hypersurface avoids these singularities. The classification of such manifolds was performed by Candelas, Lynker, and Schimmrigk (1990), and a total of 7,555 was found, of which 28 have^{20} Euler number $\pm 6$. In all, there are 2,780 distinct Hodge pairs, and with a more balanced $\chi \in [-\mathrm{960,960}]$.

##### 2.3.2. Elliptic Fibrations

The mid-1990s saw the “Second String Revolution,” and with the advent of dualities and branes that linked the various string theories, the traditional heterotic compactification scenario subsequently experienced a period of relative cool compared to its birth a decade earlier. Because of the web of dualities, in particular that between the heterotic string and $\text{F}$-theory, there emerged another family of ${\text{CY}}_{3}$ studied in the 1990s; this is the class of elliptically fibered ${\text{CY}}_{3}$ (Grassi & Morrison, 2012; Morrison & Vafa, 1996).

This is a generalization of our familiar ${\text{CY}}_{1}$, the torus, by allowing the elliptic curve to vary over a complex base surface. Algebro-geometrically, this amounts to allowing the coefficients ${g}_{2},\phantom{\rule{0.2em}{0ex}}{g}_{4}$ in the Weierstraß equation in (2.10) to depend on some appropriate coordinates of the base surface (or more strictly, to take values in dual of the canonical bundle of the base), arriving at an overall ${\text{CY}}_{3}$ as the total space. What are the possible bases? Once again, this turned out to be a finite set,^{21} which was much explored in the 1990s (Candelas, Perevalov, & Rajesh, 1997), uncovering a wealth of elegant structure.

In light of $\text{F}$-theory model building, elliptic ${\text{CY}}_{3}$ and ${\text{CY}}_{4}$ have recently been investigated with renewed zest (Beasley, Heckman, & Vafa, 2009; Marsano, Saulina, & Schafer-Nameki, 2009), and a new program in identifying elliptically fibered CYs amongst the established databases has been ongoing with the help of modern computing (Braun, 2011, 2013; Huang & Taylor, 2019a, 2019b, 2019c; Taylor, 2012). It is widely believed that, in fact, the large majority of $\text{CYn}$ are elliptic fibrations over some appropriate $(n-1)$-fold. For example it was found in Anderson, Gao, Gray, and Lee (2017) and Gray, Haupt, and Lukas (2014) that, among the CICYs, more than 99% of the 7,890 are elliptic fibrations; likewise, among the some ${10}^{6}$ CICY fourfolds, more than 99.9% are elliptic.

##### 2.3.3. Toric Hypersurfaces: The KS Database

So far, the ambient space $A$ of ${\text{CY}}_{3}$ being $\u2102{\mathbb{P}}^{4}$ (the quintic) has been seen, as well as such generalizations to products of $\u2102{\mathbb{P}}^{n}$ (CICY) and weighted $\u2102{\mathbb{P}}^{4}$ One systematic generalization of weighted projective space is a toric variety, which, instead of having a single list of weights as in (2.13), has a list of $m$ weights (giving a so-called charge matrix) acting on ${\u2102}^{n+m}$ to give an *n*-fold. Founded on the theoretic development of Batyrev-Borisov (1994), Kreuzer and Skarke spent almost a decade^{22} explicitly constructing such Calabi-Yau manifolds, culminating in the early 2000s with the construction of the most extensive database of ${\text{CY}}_{3}$ so far, the toric hypersurfaces (Avram, Kreuzer, Mandelberg, & Skarke, 1997; Kreuzer & Starke, 1997, 1998, 2002a, 2002b).

In brief, the ambient space is a toric fourfold $A$, and the charge matrix of weights can be repackaged into the vertices of an integer polytope (i.e., a convex body whose vertices are lattice points) living in ${\mathbb{R}}^{4}$. This is the key to toric geometry: to translate algebraic geometry to the combinatorics of integer lattices and polytopes.

In particular, the focus is on an integer polytope $\mathrm{\Delta}\subset {\mathbb{R}}^{4}$ that is reflexive, meaning that $\mathrm{\Delta}$ has a single interior point (which can be taken to be the origin), and all bounding hyperplanes are distance 1 from this point. Equivalently, from $\mathrm{\Delta}$ one can define^{23} the dual polytope ${\mathrm{\Delta}}^{\circ}:=\{\overrightarrow{v}\in {\mathbb{R}}^{4}|\overrightarrow{m}\cdot \overrightarrow{v}\ge -1\phantom{\rule{0.2em}{0ex}}\forall \overrightarrow{m}\in \mathrm{\Delta}\}$. Then, $\mathrm{\Delta}$ is reflexive if ${\mathrm{\Delta}}^{\circ}$ also has integer vertices. In such a case, a hypersurface in the toric variety $A$, constructed from the polytope data, is a $\text{CY}$ hypersurface.

Specifically, the defining equation of the ${\text{CY}}_{3}$ is given by

with ${x}_{j}$ coordinates of the ambient toric fourfold, ${c}_{\overrightarrow{m}}$ complex coefficients, and ${\overrightarrow{v}}_{j}$ the (integer) vertices of ${\mathrm{\Delta}}^{\circ}$. Weighted $\u2102{\mathbb{P}}^{4}$ (and of course ordinary $\u2102{\mathbb{P}}^{4}$) are archetypal examples of reflexive toric fourfolds. For our familiar example of $Q\subset \u2102{\mathbb{P}}^{4},\phantom{\rule{0.2em}{0ex}}\mathrm{\Delta}=\left[\begin{array}{lllll}-1\hfill & \phantom{\rule{0.2em}{0ex}}4\hfill & -1\hfill & -1\hfill & -1\hfill \\ -1\hfill & -1\hfill & \phantom{\rule{0.2em}{0ex}}4\hfill & -1\hfill & -1\hfill \\ -1\hfill & -1\hfill & -1\hfill & \phantom{\rule{0.2em}{0ex}}4\hfill & -1\hfill \\ -1\hfill & -1\hfill & -1\hfill & -1\hfill & \phantom{\rule{0.2em}{0ex}}4\hfill \end{array}\right]$ and ${\mathrm{\Delta}}^{\circ}=\left[\begin{array}{lllll}1\hfill & 0\hfill & 0\hfill & 0\hfill & -1\hfill \\ 0\hfill & 1\hfill & 0\hfill & 0\hfill & -1\hfill \\ 0\hfill & 0\hfill & 1\hfill & 0\hfill & -1\hfill \\ 0\hfill & 0\hfill & 0\hfill & 1\hfill & -1\hfill \end{array}\right]$, where the columns are the vertices of the polytopes, and one can check that the dot product of each column of one with that of the other is $\ge -1$, in accord with the definition. Equation (2.14) is precisely the quintic hypersurface, and the ambient toric variety here is exactly $\u2102{\mathbb{P}}^{4}$.

Thus the question of finding toric hypersurface ${\text{CY}}_{3}$ is the classification of reflexive integer 4-polytopes (up to redefinition by $SL(4;\mathbb{Z})$, under which the toric fourfolds are equivalent). In ${\mathbb{R}}^{1}$, there is trivially one reflexive polytope (the pair of points $\pm 1$). In ${\mathbb{R}}^{2}$, it is classically known that there are 16 reflexive polygons up to integer linear transformations with unit determinant (which give equivalent toric varieties). For reference, these are drawn in Figure 2; hypersurfaces in the toric twofolds created from these give 16 distinguished ${\text{CY}}_{1}$ as elliptic curves or tori.

Reflexive polytopes in ${\mathbb{R}}^{n\ge 3}$ were unknown to mathematicians until the work of Kreuzer and Skarke. They found 4,319 in ${\mathbb{R}}^{3}$, and the computational challenge was to find all reflexive integer polytopes in ${\mathbb{R}}^{4}$. The actual calculation was performed on an SGI origin 2000 machine with about 30 processors (quite the state of the art in the 1990s), which took approximately six months; 473,800,776 were found. Each of these gives a hypersurface ${\text{CY}}_{3}$, and thus from the database of tens of thousands established by the early 1990s, the list of ${\text{CY}}_{3}$ suddenly grew, with this tour de force, to half a billion.

It should be emphasized that most of the ambient spaces $A$ (as with weighted $\u2102{\mathbb{P}}^{4}$) from the 500 million polytopes are not smooth and require smoothing or resolution of singularities: different resolutions give rise to potentially different ${\text{CY}}_{3}\text{s}$. Interestingly, in this large family, only 125 have smooth ambient space $A$, and more remarkably, only 16 have a nontrivial fundamental group. Though, of course, a classification of discrete freely acting symmetries has yet to be systematized, from which one could potentially extract many more nonsimply connected ${\text{CY}}_{3}$ by quotienting, these special 16 are quite interesting (He, Lee, Lukas, & Sun, 2014).

Now, although for a given $\mathrm{\Delta}$, the Hodge pair will be the same, different resolutions will give different intersection numbers and Chern classes. Up to ${h}^{\mathrm{1,1}}=7$, this was done exhaustively by Altman et al. (2015), and for the highest ${h}^{\mathrm{1,1}}\sim 490$, it was done by Braun, Long, McAllister, Stillman, and Sung (2017). The full list of ${\text{CY}}_{3}\text{s}$, after all the resolutions, has been recently estimated to be as large as ${10}^{{10}^{5}}$ (Altman, Carifio, Halverson, & Nelson, 2019). In any case, the $\text{KS}$ data set produced 30,108 distinct Hodge pairs and $\chi \in [-\mathrm{960,960}]$. The extremal values of $\pm 960$ are actually the weighted $\u2102{\mathbb{P}}^{4}$ cases. No $\text{CY}$ construction so far has ever produced an Euler number whose magnitude exceeds 960.

**A Statistical Plot:** There is an iconic plot: the data set is drawn with ${h}^{\mathrm{1,1}}(X)+{h}^{\mathrm{2,1}}(X)$ in the vertical versus $\chi =2({h}^{\mathrm{1,1}}(X)-{h}^{\mathrm{2,1}}(X))$ in the horizontal. This is shown in part (b) of Figure 3. In part (a), the data sets discussed so far are indicated in a Venn diagram. Several properties are of note. Because there are only a total of 30,108 distinct points, the some half-billion ${\text{CY}}_{3}$ are severely degenerate in $({h}^{\mathrm{1,1}},\phantom{\rule{0.2em}{0ex}}{h}^{\mathrm{1,2}})$. The funnel shape delineating the lower extremes is just due to theivet plotting difference versus sum of the (nonnegative) Hodge numbers. The fact that the figure is left–right symmetric is perhaps the best “experimental” evidence for mirror symmetry: that to each point with $\chi $ there should be one with $-\chi $, coming from the interchange of the two Hodge numbers. There is a paucity of ${\text{CY}}_{3}$ near the corners, near the very bottom tip and the top of the funnel shape, and a huge concentration resides near the bottom center (note this is a log-density plot). In fact, the most “typical” ${\text{CY}}_{3}$ thus far known is one with Hodge numbers (27, 27), numbering about 1 million. The distribution of the Hodge numbers is studied in detail in He, Jejjala, and Pontiggia (2017).

#### 2.4. Finiteness?

A multitude of ${\text{CY}}_{3}\text{s}$ have been seen, with astronomical numbers. Now, as seen from Figure 1, the topological type of Riemann surfaces (Kähler manifold of complex dimension 1 is determined by a single integer, the Euler number; the analogue in complex dimension 3 is given by a theorem of Wall (1966) that the topology of a Kähler threefold is determined by (a) the Hodge numbers; (b) the triple intersection numbers ${d}_{rst}$; and (c) the first Pontrjagin class ${p}_{1}({T}_{M})={c}_{1}{({T}_{M})}^{2}-2{c}_{2}({T}_{M})$, where ${c}_{2}$ is the second Chern class, which was alluded to in Section 1.1, “A Classical Problem in Mathematics,” For Calabi-Yau threefolds, this amounts to the list of integers

This list should completely characterize the topology.

In the early days, it was conjectured by Yau that the number of possible values for (2.15) is finite for ${\text{CY}}_{3}$ (or indeed for Calabi-Yau manifolds of any dimension). It is furthermore a conjecture of Miles Reid—the Reid Fantasy—that all Calabi-Yau manifolds are connected via topology-changing processes.

#### 2.5. Flux Compactifications and Beyond

It is worth reiterating the point that ${\text{CY}}_{3}\text{s}$ and even ${\text{CY}}_{3}\text{s}$ with vector bundle data, as abundant as they are, comprise a small corner of the string landscape. Indeed, even as far back as the mid-1980s, the number of vacua was already expected to be astronomical from the context of lattices (Lerche, Lust, & Schellekens, 1986) (indeed, one of the authors, Schellekens, speculated on the “landscape” some time before its popularity today [Schellekens, 1998]).

The often-quoted ${10}^{500}$ vacua first came from a rough estimate by Douglas (2004) from the flux compactification scenario of Bousso & Polchinski (2000) and Kachru, Kallosh, Linde, & Trivedi (2003). In brief, one turns on various form fields (generalizing the 1-form of electromagnetism) and integrates them on cycles (counted by our familiar Betti numbers) within Calabi-Yau spaces, giving us effective Lagrangians in terms of these “fluxes” (see Balasubramanian, Berglund, Conlon, & Quevedo, 2005). Thus, one can arrive at an approximate number of vacua, for example, from the typical size of the Betti numbers in known ${\text{CY}}_{3}\text{s}$.

It should be mentioned in passing that everything thus far discussed has been compact CYs. Much like Liouville’s theorem that bounded entire functions are constant, once the condition of compactness is relaxed, finiteness in classification is no longer expected. Indeed, the landscape of noncompact ${\text{CY}}_{3}\text{s}$ is infinite and in the context of the $\text{AdS}/\text{CFT}$ correspondence (Maldacena, 1999), such spaces provide the transverse direction for -brane world-volume gauge theories that are of potential phenomenological interest.^{24}

The simplest example is the trivially Ricci-flat ${\u2102}^{3}$ (as an affine variety). Any (singular) quotient of ${\u2102}^{3}$, by discrete finite subgroups of $SU(3)$, is also locally ${\text{CY}}_{3}$. More importantly, any lattice cone (in that the generators are lattice vectors) in ${\mathbb{R}}^{3}$ with coplanar generating vectors gives a toric variety that is a noncompact ${\text{CY}}_{3}$ space. In other words, any lattice polygon gives an affine toric ${\text{CY}}_{3}$. Thus, an infinite family is readily obtained. For instance, the 16 reflexive polygons in Figure 2 are perfect examples of noncompact Calabi-Yau threefolds (e.g., number [Calabi, 1954] corresponds to a ${\mathbb{Z}}_{3}$ quotient of ${\u2102}^{3}$). The mapping between the world-volume gauge theory and the polytope data is a beautiful one, involving the combinatorics of dimers, the physics of brane tilings and the algebraic geometry of singularity resolutions (Feng, He, Kennaway, & Vafa, 2008; Franco, Hanany, Kennaway, Vegh, & Wecht, 2006a; Franco et al., 2006b).

In addition to geometrical compactifications, be they the ${\text{CY}}_{3}$ scenarios discussed in this article or (in theory duality-equivalent) scenarios such as $\text{F}$-theory compactifications on fourfolds, $\text{M}$ theory compactification on ${G}_{2}$ manifolds, and so on, there are still nongeometrical situations where one only considers the world-sheet conformal field theory such as Gepner models. All these have brought the string landscape to such a size^{25} that a re-examination of whether all effective field theories coupled to gravity can have such $\text{UV}$ completions as string theory is investigated in the so-called Swampland conjectures by Vafa (2005); (see also Brennan, Carta, & Vafa, 2017; Ooguri & Vafa, 2007).

### 3. Conclusions and Prospectus

With the ever-growing number of Calabi-Yau structures, the number of ways to reach phenomenologically viable four-dimensional effective theories is staggering. In some sense, string theory has traded one difficult problem, quantization of gravity, with another, vacuum selection. The approach over the last few decades has been, as discussed above, to (a) establish concrete databases; (b) take advantage of the latest development in computer algebra (Decker, Greuel, Pfister, & Schönemann, 2018; Grayson, Stillman, & Eisenbud, 1993; Harrington, He, & Ruehle, 2019; Hauenstein, Sommese, & Wampler, 2013; He & Mehta, 2013; He, Seong, & Stillman, 2015; SageMath, 2020; The GAP Group, 2018; Wolfram Research, 2018) and exhaustively calculate the relevant physical quantities (such as bundle cohomology for particle content); and (c) sift for vacua akin to our universe.

In the end, when all the constraints such as the right Yukawa coupling and masses, and so on, have been enforced, it could still be the case that our universe is very special and rare (q.v. Candelas et al., 2008). However, confronted with the enormity of the landscape, this exhaustive search is computationally unfeasible, especially given that the key algorithms, be they Groöbner bases, which are at the heart of computational algebraic geometry, or lattice polytope triangulations, which are at the core of combinatorial geometry, are known to be exponentially expensive. Thus, statistical and data-science methods (Cicoli, Krippendorf, Mayrhofer, Quevedo, & Valandro, 2013; Cole & Shiu, 2019; Conlon & Quevedo, 2004; Denef & Douglas, 2007; Dienes, 2006; Douglas, 2004) appear to be the way forward.

#### 3.1. Deep Learning the Landscape

In He (2017a, 2017b), a paradigm was proposed to attempt to use machine learning (the collection of techniques such as neural networks, support vector machines, or decision trees, and so forth, is loosely referred to here as simply $\text{ML}$) to bypass the aforementioned standard algorithms, in particular to study the string landscape and beyond. It was found that central problems such as computing cohomology of vector bundles appear to be machine-learnable to very high precision. Indeed, He (2017a, 2017b); Krefl and Seong (2017); Ruehle (2017); and Carifio, Halverson, Krioukov, and Nelson (2017) brought machine learning to the string landscape in 2017, and there has been a host of activity since.

The basic idea is that whatever the problem at hand, in algebraic or combinatorial geometry, quantity (usually a positive integer) $O$ is typically computed from a given configuration (usually some integer matrix) $I$, using sophisticated and computationally expensive methods from modern mathematics. This has been done by brute force over the decades, with the help of computer algebra to establish data sets of the form $\mathcal{D}=\{{I}_{i}\to {O}_{i}{\}}_{i=\mathrm{1,2},\dots ,N}$ for a large number $N$ of cases. Such a situation is perfectly adapted for supervised machine learning ($\text{ML}$)^{26} (the reader is referred to the canonical introduction [Mitchell, 1997]).

At the fundamental level, supervised learning by $\text{ML}$ is simply an intricate form of (nonlinear) regression: (a) a host of internal parameters and a directed graph of nodes, each of which is some function, (b) are fitted to available data in order to specify the parameters by minimizing an appropriate cost-function (such as sum of errors squared), and (c) prediction and extrapolation are then attempted. To avoid overfitting and to see how well the $\text{ML}$ is performing, as there could be literally thousands of parameters, the canonical method is cross-validation. The data $\mathcal{D}$ is a disjoint union of a training set and a validation set: $D:=T\bigsqcup V$. The parameters in the $\text{ML}$ on $\mathcal{T}$ are optimized and checked as to how it performs on $\mathcal{V}$, which it has not “seen” before. Usually, one steps gradually in the percentage of $\mathcal{T}$ in $\mathcal{D}$ (with a few rounds of random samples to estimate error bars) and plots the accuracy measure, which gives a so-called learning curve. To entice the reader, this is illustrated here with two case studies.

##### 3.1.1. Learning Hodge Numbers

Consider the CICYs. Although the Euler number is easily computed from the configuration matrix analytically (see Hubsch, 1994), the individual terms $({h}^{\mathrm{1,1}},\phantom{\rule{0.2em}{0ex}}{h}^{\mathrm{2,1}})$ involve exact sequence chasing, in particular determining the (co-)kernels of the maps in the sequence that very quickly becomes of high dimension. This was nevertheless performed in Candelas, Dale, Lutken, and Schimmrigk (1988); Candelas, Lutken, and Schimmrigk (1988); and Gagnon and Ho-Kim (1994) over the course of the years. Can $\text{ML}$ “guess” at the right value of ${h}^{\mathrm{1,1}}$ by simply “looking” at the configuration matrix? In principle, there might be some form of an equation of ${h}^{\mathrm{1,1}}$ in terms of ${q}_{r}^{j}$, but it would involve so many conditionals on how ${q}_{j}^{r}$ fits into an induced long exact sequence in cohomology that such equations would be of tremendous length and rather unhelpful.

In He (2017a, 2017b), this exercise of pattern recognition of ${h}^{\mathrm{1,1}}$ from ${q}_{r}^{j}$ was done, and the learning curve is shown in Figure 4 (a). Three different methods are compared: a neural network, a support vector machine, and a neural classifier. The accuracy measure is the percentage in $\mathcal{V}$ of when the ${h}^{\mathrm{1,1}}$ agrees between the actual and the predicted (note that among the 7,890 manifolds, ${h}^{\mathrm{1,1}}$ can range from 1 to 19). It can be seen that the classifier performs the best, and at 50% training data, there is already over 80% correct prediction for ${h}^{\mathrm{1,1}}$ The error bars are due to different random sampling of $\mathcal{T}$. Crucially, instead of taking the many hours to compute exactly, $\text{ML}$ is guessing the result in a matter of seconds.

##### 3.1.2. Learning Elliptic Fibrations

Continuing with the CICYs as a playground, a substantial fraction of which are, in fact, elliptically fibered. By an exhaustive search using the theorems and conjectures of Kollár (2015); Oguiso (1993); and Wilson (1994), the elliptically fibered ones were distinguished (Anderson et al., 2017; Gray et al., 2014). Can $\text{ML}$ do this identification as a 1/0 binary query (yes/no to elliptic fibration) merely by “looking” at the configuration matrix, but without any knowledge of the underlying mathematics? Again, there is no known simple formula of the 1/0 in terms of ${q}_{r}^{j}$, nor is there expected to be one.

Using a support vector machine that finds an optimal hyperplane in the very high-dimensional vector space whose entries are the flattened configuration matrix (considering the maximum number of rows and columns of ${q}_{r}^{j}$, this is ${\mathbb{Z}}^{12\times 15}$) between 0/1, He and Lee (2019) constructed the learning curve. This is shown in Figure 4 (b), which plots both the precision in yellow (percentage of 1/0 agreed between actual and predicted) as well as so-called Matthew’s $\varphi $-coefficient in blue (this is a scaled version of the chi-squared that is well adapted to binary classification problems in order to account for false positives; the closer $\varphi $ is to 1, the better the confidence of the agreement). It can be seen that even at as small as 30% training data, $\text{ML}$ can predict which are elliptically fibered to 95% precision and confidence over the entire CICY data set.^{27} Again, the guessing is done in seconds rather than in many hours of computation.

### 4. Epilogue

This article has taken a promenade in the space of ${\text{CY}}_{3}$ in the context of the string landscape, in a more or less historical order of the appearance of the constructions. From the initial triadophiliac query of whether there exists smooth compact ${\text{CY}}_{3}$ with Euler number $\pm 6$, to the proliferation of ${\text{CY}}_{3}\text{s}$ constructed from algebro-geometric and combinatorial methods, to the first exact $\text{SM}$ particle content from a ${\text{CY}}_{3}$ that endowed a vector bundle that solves the Hull-Strominger system, and to the clear future of applying data-scientific and $\text{ML}$ techniques, this journey has spanned some four decades. Throughout, attention has been directed to the intricate interplay between the mathematics and physics, emboldened by the plenitude of data and results, and inspired by glimpses of the yet inexplicable. The cartography of Calabi-Yau manifolds will certainly continue to provoke further exploration in an ever-growing interdisciplinary research in physics, mathematics, and computer science and artificial intelligence.

#### Further Reading

The interested reader is referred to the following reviews, lecture notes and books for further reading:

- Aspinwall, P. S. (2004). D-branes on Calabi-Yau manifolds. [hep-th/0403166].

Lecture notes on CY and string theory

- Candelas, P. (1987). Lectures on complex manifolds. In proceedings, superstrings.

For a rapid introduction of algebraic geometry for physicists

- Denef, F. (2008). Les houches lectures on constructing string vacua.
*Les Houches*,*87*, 483. [arXiv:0803.1194 [hep-th]].

Counting string vacua

- Grana, M. (2006). Flux compactifications in string theory: A comprehensive review.
*Physics Report*,*423*, 91–158. [hep-th/0509003].

For string phenomenology and flux compactifications on CY

- Greene, B. R. (1997). String theory on Calabi-Yau manifolds. [hep-th/9702155].

Lecture notes on CY and string theory

- Griffiths, P., & Harris, J. (1978).
*Principles of algebraic geometry: Pure and applied mathematics*. New York, NY: Wiley-Interscience.

A classic textbook on algebraic geometry

- Gross, M., Huybrechts, D., & Joyce, D. (2003).
*Calabi-Yau manifolds and related geometries*. Universitext. Berlin, Germany: Springer.

A more recent compendium on CY and string theory

- Hartshorne, R. (1977).
*Algebraic geometry*. New York, NY: Springer-Verlag.

A classic textbook on algebraic geometry

- He, Y. H. (2018). The Calabi-Yau landscape: From geometry, to physics, to machine-learning. arXiv:1812.02893 [hep-th].

For a pedagogical introduction for machine learning CY

- Hübsch, T. (1994).
*Calabi-Yau manifolds: A bestiary for physicists*. Singapore: World Scientific.

The classic book on CY manifolds

- Ibáñez, L., & Uranga, A. L. (2012). String theory and particle physics—an introduction to string phenomenology. Chicago, IL: Chicago University Press.

For string phenomenology and flux compactifications on CY

- Mitchell, T. M. (1997).
*Machine learning*. New York, NY: McGraw-Hill.

The standard introduction to machine learning

- Schenck, H. (2003).
*Computational algebraic geometry*. Chicago, IL: University of Chicago Press.

An invitation to computational algebraic geometry

- Yau, S.-T. (2009). A survey of Calabi–Yau manifolds. Geometry, analysis, and algebraic geometry: Forty years of the
*Journal of Differential Geometry*.*Surveys in Differential Geometry*,*13*, 277–318.

A survey of CY

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### Notes

1. This is seen as follows. The plane $\u2102$ can be tessellated by an infinite number of equal parallelograms $P$, the vertices of which constitute the lattice. Thus, the quotient first reduces the plane to a single $P$, the so-called fundamental domain. Next, each pair of the opposite sides of $P$ is pasted to give a torus.

2. It is rather confusing to the outsider that in algebraic geometry, because one usually works over $\u2102$, a “curve” means a complex curve of real dimension 2 and is really one’s familiar surface; likewise, a “surface” usually means a complex surface of real dimension 4.

3. Here, the indices $\mu ,\phantom{\rule{0.2em}{0ex}}\overline{\nu}$ are just $z$ and $\overline{z}$ because one is in complex dimension 1; in complex dimension $n$, one needs $n$ pairs of coordinates $({z}^{\mu},\phantom{\rule{0.2em}{0ex}}{\overline{z}}^{\overline{\nu}})$.

4. The formal definition is the existence of a closed 2-form, the Kähler form $\omega $, associated with the Hermitian metric, which implies condition (1.1).

5. Strictly, the Chern class is defined for the tangent bundle ${\tau}_{\mathrm{\Sigma}}$ of $\mathrm{\Sigma}$; this point is returned to in Section 2.2.1. “Recent Developments.” Furthermore, Equation (1.2) has written the term $[{c}_{1}(\mathrm{\Sigma})][\mathrm{\Sigma}]$ more formally as an inner product, in intersection theory, of the first Chern class ${c}_{1}$ (as an element, hence the square brackets, of the second cohomology ${H}^{2}(\mathrm{\Sigma})$), of the tangent bundle ${T}_{\mathrm{\Sigma}}$ with the second homology class ${H}_{2}(\mathrm{\Sigma})$ of $\mathrm{\Sigma}$.

6. Formally, ${b}^{i}$ is the rank of the

*i*-th homology group ${H}_{i}(\mathrm{\Sigma})$, which is also that of the*i*-th cohomology group ${H}_{i}(\mathrm{\Sigma})$.7. There are some further mild conditions, akin to the Cauchy-Riemann equations that are needed to define complex analytic functions.

8. More technically, the conjecture is as follows. Let $(M,\phantom{\rule{0.2em}{0ex}}g,\phantom{\rule{0.2em}{0ex}}\omega )$ be a compact Kähler manifold and $R$ a $(\mathrm{1,1})$-form such that $[R]=[{c}_{1}(M)]$. Then there exists a unique Kähler metric $\tilde{g}$ and Kähler form $\tilde{\omega}$ such that $[\omega ]=[\tilde{\omega}]\in {H}^{2}(M;\mathbb{R})$ with $R=Ric(\tilde{\omega})$, the Ricci form of $\tilde{\omega}$.

9. An equivalent definition for Calabi-Yau is that there is a unique holomorphic -form, the volume form. This unique form compels ${h}^{n,0}$ to equal 1.

10. Although one might be skeptical of supersymmetry on grounds of there being still no experimental evidence, from a theoretical point of view it is definitely the right way of thinking. An analogy with the real or complex number system is a fitting one. Problems in mathematics are far better behaved over $\u2102$ than over $\mathbb{R}$ because the former is, as mentioned earlier, algebraically closed. Moreover, $\u2102$ is in some sense the unique extension of $\mathbb{R}$ with this closure property (otherwise, it is necessary to sacrifice commutativity and associativity). Likewise, supersymmetry, by the theorem of Haag, Lopuszański, and Sohnius (1975), is in some sense the unique extension for space-time symmetry under which QFTs are much better behaved.

11. It just so happened that Strominger, of CHSW, was sharing an office with Yau at the Institute for Advanced Study, Princeton, as they arrived at their condition, not long after Yau got the Fields medal for . This juxtaposition engendered the perfect discussion between the physicist and the mathematician. This is a golden example of a magical aspect of string theory: it repeatedly infringes, almost always unexpectedly rather than forcibly, on the most profound mathematics of paramount concern, and often proceeds to contribute to it.

12. This is unlike for an arbitrary Kähler threefold, which has $U(3)$, or a Riemannian 6-manifold, which has $SO(6)$

13. Recently, independent of string theory or any unified theories, why there might be geometrical reasons for three generations to be built into the very geometry of the Standard Model has been explored (He, Jejjala, Matti, Nelson, & Stillman, 2015).

14. One can check by writing out $(x,\phantom{\rule{0.2em}{0ex}}y)$ in their real and imaginary parts, and the Weierstraß equation becomes two real constraints. We can then solve this numerical and plot the result to see a torus emerge.

15. In this case, one can heuristically obtain the 1 as being descended from the Kähler class of the ambient $\u2102{\mathbb{P}}^{4}$ and the $101$ as follows: ${h}^{\mathrm{2,1}}$ counts the inequivalent complex deformations, there are $\left(\begin{array}{l}5+5-1\hfill \\ \phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}5\hfill \end{array}\right)=125$ degree $5$ monomials in $5$ variables, subtract this by the action of linear redefinition of variables which amounts to ${(5}^{2}-1)=24$, and then by an overall scaling gives us $125-24-1=101$. In general, however, this naive counting does not work.

16. This is one of the short-comings of the index theorem: the integral of curvature and the intersection of the Chern classes give only the alternating sum (Euler number) in (co-)homology, but not the individual terms.

17. This works out perfectly for a Calabi-Yau 3-fold: for example, ${H}^{1}(X,v)\times {H}^{1}(X,v)\times {H}^{1}(X,v)\to {H}^{3}(X,{\mathcal{O}}_{X})\simeq \u2102$.

18. The corresponding cohomology groups are as follows. For $SU(5)$ we have $\begin{array}{l}\#(\overline{10})={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}\mathcal{V})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(10)={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}{\mathcal{V}}^{*})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(5)={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}{\wedge}^{2}\mathcal{V})\phantom{\rule{0.2em}{0ex}},\\ \#(\overline{5})={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}{\wedge}^{2}{\mathcal{V}}^{*})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(1)={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}\mathcal{V}\otimes {\mathcal{V}}^{*})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(24)=1,\end{array}$ where the MSSM quarks/lepton live in the $10\oplus \overline{5}$, and the Higgs, in the 5. Similarly, for $SO(10)$, we have $\begin{array}{l}\#(\overline{16})={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}{\mathcal{V}}^{*})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(16)={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}\mathcal{V})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(10)={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}{\wedge}^{2}\mathcal{V})={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}{\wedge}^{2}{\mathcal{V}}^{*})\phantom{\rule{0.2em}{0ex}},\\ \phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(1)={h}^{1}(X,\phantom{\rule{0.2em}{0ex}}\mathcal{V}\otimes {\mathcal{V}}^{*})\text{,}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\#(45)=1,\end{array}$ where the $\text{SM}$ quarks/lepton live in the 16, and the Higgs, in the 10.

19. Geometrically, this means a Calabi-Yau 3-fold ${X}^{\prime}$ with freely acting discrete group $\mathrm{\Gamma}$ must be found so that the smooth quotient $X={X}^{\prime}/\mathrm{\Gamma}$ has fundamental group $\mathrm{\Gamma}={\pi}_{1}(X)$ and admits a stable $\mathrm{\Gamma}$-equivariant bundle $\mathcal{V}$.

20. Of course, with the importance of Wilson Lines, one should no longer be limited to $\left|\chi \right|=6$, but rather those with freely acting discrete groups of order $k$ and $\chi =\pm 3k$.

21. Namely, (I) Hirzebruch surfaces ${\mathbb{F}}_{r}$ for $r=\mathrm{0,1},\phantom{\rule{0.2em}{0ex}}\dots ,\text{12}$; (II) So-called $\u2102{\mathbb{P}}^{1}$-blowups of Hirzebruch surfaces ${\widehat{\mathbb{F}}}_{r}$ for $r=\mathrm{0,1,2,3}$; (III) Del Pezzo surfaces $d{\mathbb{P}}_{r}$ for $r=\mathrm{0,1},\phantom{\rule{0.2em}{0ex}}\dots ,\phantom{\rule{0.2em}{0ex}}\text{9}$; and (IV) Enriques surface E. These are classical complex surfaces that are the likes of $\u2102{\mathbb{P}}^{1}$ fibered over $\u2102{\mathbb{P}}^{1}$, as well as blowups of $\u2102{\mathbb{P}}^{2}$.

22. Due to the untimely death of Max Kreuzer, it became a pertinent issue to attempt to salvage the data for posterity, a recent version of this legacy project is presented in Braun, Knapp, Scheidegger, Skarke, and Walliser (2013); and Braun and Walliser (2011). One is also referred to Qureshi and Szendroi (2011) where the ambient space is more general than a toric variety

23. In the perhaps more familiar definition of a toric variety in terms of fans of cones, the fan $\mathrm{\Sigma}$ is simply the faces of ${\mathrm{\Delta}}^{\circ}$.

24. Geometrically, these noncompact ${\text{CY}}_{3}$ spaces can be seen as cones over Sasaki-Einstein manifolds (Gauntlett, Martelli, Sparks, & Yau, 2007; He, Seong, & Yau, 2018; Martelli, Sparks, & Yau, 2008) of real dimension 5.

25. For instance, a recent estimate of consistent F-theory flux compactifications numbers as high as ${10}^{{10}^{5}}$ by Taylor and Wang (2015). Even models with exact SM particle content are estimated to be very high powers of 10 (Constantin, He, & Lukas, 2019; Cvetic, Halverson, Lin, Liu, & Tian, 2019).

26. Likewise, one could have an incomplete classification of configurations $\left\{{I}_{i}\right\}$ and attempt to extrapolate to the remaining by unsupervised learning.

27. One might worry that the elliptic data set is highly imbalanced in that the vast majority are elliptically fibered; balanced data were ensured by enhancing the positives with permutations, which are equivalent representations of the same manifold. In other words, the ML is not predicting 1 all the time.