A Calabi-Yau Cartography by Yang-Hui He*

1. Prologus Terræ Sanctæ cus on the most traditional and the most richly de- veloped, viz., the geometry of Calabi-Yau threefolds. “La Géométrie, qui ne doit qu’obéir à la Physique Indeed, so great is the number of possible scenarios quand elle se réunit avec elle, lui commande quelque- that it poses as one of the greatest theoretical chal- fois” (Geometry, which should only obey Physics, lenges to modern physics, in what has become known when united with it sometimes commands it), so as the “vacuum degeneracy problem”, where one is wrote the great philosopher and mathematician Jean confronted with how to select our universe amidst d’Alembert in his Essai d’une nouvelle théorie de la ré- a landscape – a word which has become a technical sistance des fluides (1752). This coextensivity between term – of solutions. natural philosophy and geometry, conceived in an- I shall not discuss the landscape here, nor its tiquity, fashioned in Early Modernity and shaped in philosophical, anthropological or statistical implica- the Industrial Age, fully blossomed in the XXth cen- tions. For this purpose, I have carefully chosen the tury. word cartography in the title and will discuss some The two corner-stones of modern physics – gen- of the progressive uncovering of the still mysterious eral relativity which describes the large structure of space of Calabi-Yau geometries, as if charting the ge- space-time and quantum field theory, the elementary ography of a vast and fertile land, and, with a histori- particles which constitute all matter – are well un- cal outlook, review the explicit construction of Calabi- derstood as geometrical in nature. The former, is the Yau threefolds for the sake of physics. For this pur- study of how the Ricci curvature of spacetime is in- pose too I have named this introductory section after duced by the energy-momentum tensor in an equal- the prologue of the famous medieval treatise on the ity dictated by the Einstein-Hilbert action and the lat- Holy Land [1] by Burchard, who ventured to produce ter, how particles are realizations of connections and the best early maps of that alluring place. representations of appropriate principal bundles gov- erned by the Standard Model action. The brain-child of this tradition of geometrization and unification, 2. Triadophilia dominating mathematical and theoretical physics as Our story begins with 1985, when Gross, Harvey, we enter the XXIst century, is string theory. Martinec, and Rhom – jocundly called the “Princeton It is well-known by now that string theory is a string quartet” – formulated1 the heterotic string. By quantum theory unifying gravity and field theory in fusing the bosonic string (whose critical dimension is ten spacetime dimensions (or, equivalently, a conjec- 26) with the superstring (whose critical dimension is tural M-theory in 11 dimensions). We must therefore 10) in the process of “heterosis” by assigning them account for 10−4 = 6 “missing” dimensions. Over the to be respectively left and right moving modes of the years, there has emerged a plethora of scenarios in string, the quartet obtained something quite revela- the interplay between the physics of our four dimen- tory: an interesting gauge group. Using the fact that sions and these 6 dimensions, our story here will fo- 1 For an entertaining account of the history of string the- * Department of Mathematics, City University London; ory, aimed at the public and the specialist alike, the reader School of Physics, NanKai University, Tianjin, China; and is referred to [3]. For the history of Calabi-Yau manifolds, Merton College, University of Oxford especially in relation to string theory, the book [4] is highly E-mail: [email protected] recommended.

20 NOTICES OF THE ICCM VOLUME 3,NUMBER 2 26 − 10 = 16 and that in 16 dimensions there are only actually mean the (minimal) supersymmetric exten- two even self-dual integral lattices in which quantized sion thereof, dubbed the MSSM, and to each of the momenta could take value, viz., the root lattices of fermions above there is a bosonic partner and vice E8 × E8 and of D16 = so(32), two heterotic string theo- versa. ries were constructed. Of note in the table is the number 3, signifying that the particles replicate themselves in three fam- ilies, or generations, except for the recently discov- 2.1 String Phenomenology ered Higgs boson, of which is only a single doublet All at once, the possibility of obtaining chiral under SU(2). That there should be 3 and only 3 gener- fermions in spacetime, crucial to the Standard Model, ations, with vastly disparate masses, is an experimen- is realized. Now, E8 is of particular significance, be- tal fact with confidence [5] level σ = 5.3 and has no sat- cause of the following sequence of embeddings of Lie isfactory theoretical explanation to date. The possible groups symmetry amongst them, called flavour symmetry, is (2.1) independent of the gauge symmetry of GSM. SU(3) × SU(2) ×U(1) ⊂ SU(5) ⊂ SO(10) ⊂ E6 ⊂ E7 ⊂ E8 The fact that GSM is not simple has troubled many physicists since the early days: it would be more The first group GSM = SU(3)×SU(2)×U(1) is, of course, pleasant to place the baryons and leptons in the same that of the Standard Model of particles where the footing by allowing them to be in the same repre- SU(3) factor is the gauge group of QCD governing sentation of a larger simple gauge group. This is the the dynamics of baryons and the SU(2) × U(1), that motivation for the sequence in (2.1): starting from of QED, governing the leptons. Oftentimes, we add SU(5), theories whose gauge groups are simple are one more U(1) factor, denoted as U(1)B−L, to record called grand unified theories (GUTs), the most pop- the difference between baryon and lepton number, in ular historically had been SU(5), SO(10) and E6, long 0 which case GSM = SU(3)×SU(2)×U(1)×U(1)B−L and the before string theory came onto the scene in theoreti- above sequence of embeddings skips SU(5). cal physics. 0 In terms of the representation of GSM, denoted as That the heterotic string could produce poten- (a,b)(c,d) where a is a representation of SU(3), b, that tially realistic (supersymmetric) grand unified theo- of SU(2), and (a,b) are the charges of the two Abelian ries, in addition to the natural incorporation of the U(1) groups, the Standard Model elementary particles graviton, gave the first glimpse of string theory as (all are fermions except the scalar Higgs) are as fol- a candidate for ToE, the Theory of Everything. To- lows gether with anomaly cancellation by Green-Schwarz [6] which showed the quantum consistency of string SU(3) × SU(2) × Multiplicity Particle theory in the previous year, and the subsequent pa- U(1) ×U(1) B−L per by Candelas-Horowitz-Strominger-Witten (CHSW) (3,2)1,1 3 left-handed [7] in the following year, to which we now turn, this quark constituted the “First String Revolution”. (1,1)6,3 3 left-handed The paper of CHSW, gave the conditions for which anti-lepton the heterotic string, when compactified – i.e., its (3,1)−4,−1 3 left-handed 10-dimensional background is taken to be of the form 1,3 1,3 anti-up R ×X6 with R our familiar space-time and X6 some (3,1)2,−1 3 left-handed small curled up 6-manifold, endowed with a vector anti-down bundle V, at the Planck scale too small to be cur- (1,2)−3,−3 3 left-handed rently observed directly – would give a supersymmet- , lepton ric gauge theory in R1 3 with potentially realistic par- (1,1)0,3 3 left-handed ticle spectrum. In short, the paradigm is simply anti-neutrino 1,3 Geometry of X6 ←→ physics of . (1,2)3,0 1 up Higgs R (1,2) 1 down Higgs −3,0 What better realization of that noble goal of the ge- In addition to these are vector bosons: (I) the con- ometrization of nature which has been exalted for so nection associated to the group SU(3), called the glu- long! ons, of which there are 8, corresponding to the di- Specifically, with more generality, the set of con- mension of SU(3), and (II) the connection associated ditions, known as the Strominger System [8], for the low energy low-dimensional theory on 1,3 to be a su- to SU(2) × U(1), called W ±, Z, and the photon; a to- R tal of 4 corresponding to the its dimension. We point persymmetric gauge theory are out that henceforth by the Standard Model – and in- 1. X6 is complex; deed likewise for all ensuing gauge theories – we shall 2. The Hermitian metric ω on X6 and h on V satisfy

DECEMBER 2015 NOTICES OF THE ICCM 21 (a) ∂∂ω = iTrF ∧ F − iTrR ∧ R where F is the cur- some more differential and some more algebraic, of vature (field strength) 2-form for h and R the a Calabi-Yau n-fold X: that it is a compact Kähler man- (Hull) curvature 2-form for ω; ifold of complex dimension n such that (b) d† = i( − ) || ||, where is a holomor- ω ∂ ∂ ln Ω Ω • The canonical bundle, i.e., the higher exterior phic 3-form on X which must exist. Re- 6 power Vn T ∗ of the cotangent bundle is the trivial cently, Li-Yau [9] showed that this is equiva- X bundle OX ; lent to ω being balanced, i.e., d ||Ω||ω2
= 0; • X admits a nowhere vanishing holomorphic 3. F satisfies the Hermitian Yang-Mills equations n-form Ω; ab • The first Chern class of the tangent bundle TX ω Fab = 0 , Fab = Fab = 0 . vanishes: c1(TX ) = 0; We will not discuss the technicalities of the above • X has a Kähler metric with vanishing Ricci curva- concepts in detail here but have included them for ture; completeness. Suffice it to say that the general so- • The holonomy of the Kähler metric is contained lutions to this system continue to inspire research in SU(n). today, engendering more geometric structures that We will briefly mention the non-compact case to- contribute to the landscape. The simplest and most wards the end of this review but for now, and indeed famous solution, of course, is when X is a Calabi-Yau 6 as far as the heterotic compactification is concerned, threefold (CY3), which we now address. we restrict our attention to compact, smooth Calabi- Yau manifolds. 2.2 Calabi-Yau Manifolds While Calabi-Yau manifolds are certainly of in- The history of Calabi-Yau manifolds is a distin- terest in pure mathematics, what [7] showed is that, guished one and dates long before string theory or remarkably, geometries of Calabi-Yau threefolds are even the conception of the Standard Model (Glashow- a concrete realization of X6 as a string compactifi- Salam-Weinberg’s electroweak theory was finalized in cation scenario which could attain potentially realis- 1967). This is a golden example of a magical aspect of tic physics.3 Thus is born the subject of string phe- string theory: it consistently infringes, almost always nomenology. unexpectedly rather than forcibly, upon the most pro- found mathematics of paramount concern, and then quickly proceeds to contribute to and even revolu- 2.3 Triadophilia: Three Generations tionize it. Recall that the Strominger system also has a vec- In 1954, Calabi conjectured [10] of the existence tor bundle V on the Calabi-Yau threefold X, luckily, of certain nicely behaved Riemannian metrics on this can be chosen to be simply the tangent bundle complex manifolds, that for X a compact Kähler man- T whose first Chern class we have already seen to 2 X ifold with Kähler metric g and Kähler form ω, and R , vanish. What is the gauge theory in R1 3? The rule a (1,1)-form representing the first Chern class of X, turns out to be simple: the low-energy (grand unified) then gauge group is simply the commutant of the structure PROPOSITION 1. There exists a unique Kähler metric group of V in E8. The astute reader may wonder about g˜ with Kähler form ω˜ such that ω and ω˜ are cohomol- the “other” E8 factor. Indeed, in heterotic compactifi- 2 ogous in H (X,R) and the Ricci form of ω˜ is R. cations, only one of the E8 is declared “visible” and This conjecture was proven by S.-T. Yau in 1977-8 the other, called “hidden”, is placed at the end of the in his Fields-winning treatise [11]. In particular, the universe and actually has rich physics in its own right case of vanishing first Chern class is also that of zero [12–14, 16]. Ricci curvature R, and such X is appropriately dubbed The structure group for V = TX is here simply the Calabi-Yau and its unique Kähler metric inducing this holonomy group SU(3) and its commutant in E8 is E6. flat curvature is the Calabi-Yau metric. In summary, In other words, we naturally have an E6 GUT theory 1,3 we will take the following as equivalent definitions, in R . By taking V not being TX , but, for example, a stable SU(4) or SU(5) bundle, one could obtain the 2 We briefly remind the reader that for a Hermitian (complex) more interesting commutant SU(10) or SU(5) GUTs. manifold X with metric h, one can extract a Riemannian met- 1 This has come to be known as “non-standard” em- ric gR = 2 (h + h) as the real part. If, in addition the Hermitian i bedding and has with the developments in the theory form ω = 2 (h − h) written as the imaginary part, closes (i.e., dω = 0), then X is Kähler and gK = ω is the Kähler metric. In such a case, there is a potential K whence the metric can be 3 Heuristically, one might conceive of this as follows: one i locally derived: gK = 2 ∂∂K. In a sense, a Kähler manifold is needs X6 to be complex in order to have chiral fermions, one so well endowed that it has three compatible structures: Kähler, for supersymmetry and Ricci-flat, to solve vacuum (complex) Hermitian, symplectic and Riemannian. Einstein’s equations.

22 NOTICES OF THE ICCM VOLUME 3,NUMBER 2 1 of stable bundles on Calabi-Yau manifolds become an ular, by Hodge decomposition we have that h (X,TX ) ' 2,1 1 1,1 industry of realistic model building [15]. h and h (X,TX ) ' h where the latter are the so- The particle content is readily determined from called Hodge numbers, which is a refined (complex) group theory. More importantly, this in turn deter- version of Betti numbers counting (holomorphic) cy- mines the vector bundle cohomology group which is cles in the Kähler manifold. The following diagram associated with the particles. In short, we have that4 will illustrate the relevant points:

1 0,0 generations of particles ∼ H (TX ) , h 1 ∗ h1,0 h0,1 anti-generations of particles ∼ H (TX ) . h2,0 h1,1 h0,2 In general, the lesson is that h3,0 h2,1 h2,1 h0,3 h2,0 h1,1 h0,2 1,3 Particle content in R ←→ cohomology groups h1,0 h0,1 0,0 of V,V ∗ and their exterior/tensor powers h 1 b0 1 The cubic Yukawa couplings in the Lagrangian consti- 0 0 b1 0 tuted by these particles (fermion-fermion-Higgs) are 0 h1,1 0 b2 h1,1 tri-linear maps5 taking the cohomology groups to . 2,1 2,1 3 2,1 C = 1 h h 1 b = 2h + 2 An immediate constraint is, of course, that there 0 h1,1 0 b1 h1,1 be 3 net generations, meaning that 0 0 b2 0 0 1 1 ∗ 1 b 1 (2.2) h (X,TX ) − h (X,TX ) = 3 . Here it is customary to represent the matrix of Hodge Thus, the endeavour of finding Calabi-Yau threefolds numbers in diamond form. The refinement of the with the property (2.2) began in 1986. This geomet- Betti numbers is in the sense that bi = ∑ h j,k. The rical “love for threeness”, much in the same spirit j+k=i as triskaidekaphobia, has been dubbed by Candelas symmetry about the middle horizontal line is simply et al. as Triadophilia [17]. Recently, independent of Poincaré duality and that about the middle vertical string theory or any unified theories, why there might line is essentially complex conjugation (of the Lapla- be geometrical reasons for three generations to be cian). Moreover, we assume that X is connected so built into the very geometry of the Standard Model that b0 = h0,0 = 1. Furthermore, for the case of X be- has been explored [18]. ing simply-connected, the fundamental group π1(X) and hence its Abelianization H1(X) also vanishes. Fi- 3. De Practica Geometriæ nally, the Calabi-Yau property that there be a unique holomorphic 3-form implies that h3,0 = h0,3 = 1. Look- Having extracted a mathematical problem from ing at the Hodge diamond, we see that topologically, a physical constraints, it therefore becomes a practical CY3 is characterized by only two integers h1,1 and h2,1, quest in geometry which has prompted some 30 years essentially counting the number of Kähler and com- of research. We are indeed reminded of Fibonacci’s plex structure deformations respectively. We empha- tome of 1220 after which I have named this section size that these two are not refined enough: two CY3 in his honour. with the same pair of (h1,1,h2,1) need not be isomor- First, it is well-known that the two terms in (2.2) phic. Importantly, we have that are topological quantities associated with X, in partic- 6 i i 1,1 2,1 4 Specifically, we have that the decomposition of the ad- (3.3) χ = ∑(−1) b = 2(h − h ) i=0 joint 248 of E8 breaks into SU(3)×E6 as 248 → (1,78)⊕(3,27)⊕ (3,27)⊕(8,1). Thus the Standard Model particles, which in an is the standard topological Euler number. Our ques- E6 GUT all reside in its 27 representation, is associated with the fundamental 3 of SU(3). The 10-dimensional fermions tion (2.2) thus becomes: does there exist a Calabi-Yau are eigenfunctions of the Dirac operator, which then splits threefold with χ = ±6? into the 4-dimensional one, giving the fermions we see and that on the Calabi-Yau threefold, the low-energy particles are then dictated by the zero-eigenvalues of the Dirac operator 3.1 The Quintic on X, and thence, via the Atiyah-Singer index theorem, by the cohomology of appropriate bundles on V. Here, for ex- How then, does one explicitly construct a Calabi- 1 ample, the 27 representation is thus associated to H (TX ) and Yau manifold? A cursory look at low dimension will 1 ∗ the conjugate 27, to H (TX ). Similarly, the 1 representation of 1 ∗ give us not only some experience but also another E6 is associated with the 8 of SU(3), and thus to H (TX ⊗ TX ). 5 This works out perfectly for a Calabi-Yau threefold: for ex- reason why Calabi-Yau manifolds are of central im- 1 1 1 3 ample, H (X,V) × H (X,V) × H (X,V) → H (X,OX ) ' C. portance to mathematics. What is a Calabi-Yau one-

DECEMBER 2015 NOTICES OF THE ICCM 23 fold? This is nothing but a Riemann surface of zero of c1(TX ) generalizes analogously to the condition that curvature, which is classically well-known to be the K for each r = 1,...,m, we have qr = n + 1. Succinctly, 2 1 1 ∑ j r torus T = S × S . Algebro-geometrically, this can be j=1 6 realized as a cubic in CP2. Hence, the study of Calabi- one can write this information into an m × K configu- Yau one-folds is that of the elliptic curve! No won- ration matrix (to which we frequently adjoin the first der we are in the very heart of modern mathemat- column, designating the ambient product of projec- ics. tive spaces, for clarity; this is redundant because one Moving onto complex dimension 2, one could so can extract nr from one less than the row sum): generalize and have a (smooth) quartic algebraic sur-  n 1 1 1  3 CP 1 q q ... q face in P . This is called a K3 surface and is again 1 2 K  n2 q2 q2 ... q2  one of the classical objects studied at the end of the  CP 1 2 K  X =  . . . . .  XIXth century. It turns out that the only other Calabi-  . . . . .   . . . . .  Yau two-fold is the rather trivial case of the 4-torus nm m m m (3.4) CP q1 q2 ... qK 4 1 4 m×K , T = (S ) , which is simply the direct product of two m elliptic curves. K = ∑ nr − 3 , r=1 This construction, of having a degree n +1 hyper- K r surface in n as an algebraic variety is indeed valid ∑ q j = nr + 1 , ∀ r = 1,...,m . CP j=1 in general. One can show that the number of projec- tive coordinates, here n + 1, equaling to the degree of For example, [5], or [4|5], denotes the quintic. Two the hypersurface implies the vanishing of the first more immediate examples are Chern class. Thus we arrive at our first, and perhaps 1 1 most famous, example of a Calabi-Yau threefold: the 1 3 0 4 S = 3 0 , S˜ = . quintic hypersurface in CP . There are many degree 5   1 0 3 monomials one could compose of 5 coordinates, the 0 3 most well-studied is the so-called Fermat quintic: The first is called the Schoen Manifold and the sec- Q := {x5 +x5 +x5 +x5 +x5 +ψx x x x x = 0} ⊂ 4 ond, constructed by Yau et al. Specifically, the config- 0 1 2 3 4 0 1 2 3 4 CP[x0:x1:x2:x3:x4] uration S means that the ambient space is CP1 ×CP2 × where ψ is some complex coefficient. What are the CP2, of dimension 5, so that indeed the two polynomi- topological numbers of Q? It turns out that h2,1(Q) = als, being complete intersection, give a threefold. The 101 and h1,1(Q) = 1 so that χ(Q) = −200 and this is quite first column (1,3,0) means that the first polynomial is far from ±6. linear in CP1 and cubic in the first CP2 while having no dependence on the second CP2. The second column is likewise defined. Another lesson, as can be induced 3.2 The CICY Database from S and S˜, is the rather cute fact that the transpose To continue to address the question raised by tri- gives a new valid configuration which could be com- adophilia, an algorithmic generalization of the con- pletely different in topology. Importantly, the Chern struction for the quintic was undertaken: instead of a classes and thence the Euler number can be read off n 7 single CP , what about embedding a collection of (ho- the matrix configuration directly. We often attach h1,1,h2,1 mogeneous) polynomials into a product A of projec- the topological numbers as Xχ for completeness, tive spaces? For further simplification, let us consider 1,101 19,19 ˜14,23 so we can write, for instance, [5]−200, S0 and S−18 . only complete intersections which means the optimal Such manifolds were considered and explicitly case where the number of equations is 3 less than the constructed by Candelas et al. [19] in the early 1990s dimension of the ambient space A so that each poly- and were affectionately called CICYs (complete in- nomial slices out exactly one new degree of freedom. tersection Calabi-Yau manifolds). Classifying these n n In other words, let A = CP 1 × ... × CP m , of dimension 7 We have that c (T ) = 0 and moreover, n = n1 +n2 +...+nm and each having homogeneous co- 1 X (r) (r) (r) ordinates [x : x : ... : xn ] with the superscript (r) in- " K # 1 2 r rs 1 rs r s dexing the projective space factors. Our CY3 is then c2 (TX ) = −δ (nr + 1) + ∑ q jq j , 2 j=1 defined as the intersection of K = n − 3 homogeneous " K # (r) rst 1 rst r s t polynomials in the coordinates x j . Clearly this is a c3 (TX ) = δ (nr + 1) − ∑ q jq jq j , 3 j=1 generalization of the quintic, for which r = m, nr = 4 and K = 1. The Calabi-Yau condition of the vanishing where we have written the coefficients of the total Chern r rs rst class c = c1Jr + c2 JrJs + c3 JrJsJt explicitly, with Jr being the 6 nr R The beginning student is perhaps more familiar with the Kähler form in P . The triple-intersection form drst = X Jr ∧ 2 3 Weierstrasß model: {x,y ∈ C|y = x −4g2x−g6}; we simply pro- Js ∧Jt is a totally symmetric tensor on X and the Euler number 2 rst jectivize by having homogeneous coordinates [x : y : z] of CP . is simply χ(X) = drst c3 .

24 NOTICES OF THE ICCM VOLUME 3,NUMBER 2 above matrices, up to topological equivalence, would 3.3 A Plethora of CY3 then classify the CICYs. One could then read off the Of the 7890, one could proceed to find various Euler number to see whether any of them had mag- freely acting discrete symmetries and this was only nitude of 6. The combinatorial problem for these in- lately accomplished [22], using today’s desktop com- teger matrices turned out to be rather non-trivial and puter which is already far more powerful than the one of the most powerful super-computers then avail- best super-computer back in the day. Many more can- able was recruited. Philip Candelas often recounts to didates have been found. me his fond memories of running the code on the Indeed, this brings us to the heart of a question, computer at CERN and the print-out still sits in a com- in nature both mathematical and physical: how many pile in his office. This was perhaps the first time when CY3s are there? In a way, we have an interesting se- heavy machine computation was done for the sake of quence: in complex dimension 1, there is only the el- algebraic geometry. In all, CICYs were shown to be fi- liptic curve which is CY1, in dimension 2, as men- nite in number, a total of 7890 inequivalent configu- tioned, there are 2. Starting in dimension 3, we till rations. Recently, CICY4, the four-fold version of this this day still have no idea how many distinct smooth was completed in the nice work [20] and 921,497 were manifolds are there, even though we have found lit- found. erally billions. It was conjectured by Yau in the early Unfortunately, none of the 7890 had χ = ±6. While days, that the number might be finite for CY3 (or in- this was initially disappointing, it was soon realized deed for Calabi-Yau manifolds of any dimension) [23] that circumventing this problem gave rise to the res- and it was moreover a fantasy of Miles Reid that they olution of another important physical question. A are all connected via topology-changing processes ex- freely acting order 3 symmetry was found on S˜; the emplified by conifold transitions. Sometimes, one is freely acting is important, because it means that the tempted to speculate how much more convenient if quotient S0 = S˜/ is also a smooth CY3, albeit not Z3 our spacetime were 8 or even 6 dimensional, in which a CICY. For such smooth quotients, the Euler di- case the compactification of superstrings would be vides8 and S0 became the first three-generation man- quite facile. ifold! Now, the quotienting is crucial for another rea- 3.3.1 Weighted Hypersurfaces son. In the sequence (2.1), we have focused on GUTs. After the success story of CICYs, the search con- What about the Standard Model itself? It so happens tinued. Another natural ambient space to have, in that one standard way of obtaining GSM = SU(3) × view of the quintic, is to take weighted projective SU(2) ×U(1) from any of the GUT groups is precisely 4 space CP[d ... d ] where we recall this to be the quotient by quotienting. Group theoretically, this amounts to 0: : 4 5\{~0}/(z ,z ,...,z ) ∼ (λ d0 z ,...,λ d5 z )
for some non- finding a discrete group whose generators can be em- C 0 1 5 0 5 zero complex λ. Of course, taking all weights d = 1 is bedded into the GUT group, so that the commutant is i the ordinary 4. We then embed a hypersurface of the the desired G . Geometrically, this is the action CP SM degree d + d + ... + d therein, which defines a CY3. of the Wilson Line, where a CY3 with non-trivial fun- 0 1 4 One caviat is that unlike ordinary projective space, damental group admits a non-trivial loop which, cou- weighted projective spaces are generically singular, pled with the discrete group action, decomposes the and care must be taken to make sure the hypersurface E further from the structure group V. In our example 8 avoids these singularities. The classification of such above, S˜ has trivial π but the quotient S0 has, by con- 1 manifolds was performed in [26] and a total of 7555 is struction π (S0) ' , whereby admitting a Wilson 1 Z3 Z3 found, of which 28 have Euler number ±6. Of course, line. This, for the early models, can be used to break with the importance of Wilson Lines, we should no the E GUT down to the Standard Model. 6 longer be limited |χ| = 6, but rather those with non- Not surprisingly, the manifold S0 became cen- trivial fundamental group and those with freely act- tral to string phenomenology in the period after ing discrete groups of order k which divides χ. String Revolution [21]. Some even believed that they have found the geometry of the universe. More re- 3.3.2 Elliptic Fibrations cently, using non-standard embedding, and by study- The mid 1990s saw the “Second String Revolu- ing stable SU(5) and SO(10) bundles on non-simply- tion” and with the advent of dualities and branes connected CY3, the first heterotic compactification which linked the various string theories, the tra- with exact MSSM particle content were constructed, ditional heterotic compactification scenario subse- whereby realizing a 20-year old dream [24, 25]. Thus, quently experienced a period of relative cool com- once again mathematics and physics conspire to a pared to its incipience a decade earlier. Nevertheless, parallel and co-extensive development. Calabi-Yau manifolds continued to occupy the center 8 The individual Hodge numbers do not and it turns out that stage. String dualities rely on the equivalence of effec- we have (h1,1,h2,1) = (6,9) for S0. tive field theories after compactification on different

DECEMBER 2015 NOTICES OF THE ICCM 25 Calabi-Yau spaces. A true gem which emerged from to email us who were collaborating with him at the this paradigm is clearly mirror symmetry which, due time – it became a pressing issue to attempt to sal- to such equivalences, predicted that to each CY3 with vage the data for posterity, a recent version of this Hodge number (h1,1,h2,1), there should be a mirror pair legacy project is presented in [35]. with these exchanged. The geometrical implications In a nutshell, these manifolds extend the are immense and the limitations of space cannot al- weighted projective case. Indeed, a toric variety is a low me to expound upon what clearly deserves a sep- very powerful generalization of weighted projective arate account. space in that, instead of having a single list of weights, Another highlight of this revolution on duali- we have a matrix of weights acting on a higher dimen- m ties and perhaps should be categorized as a Third sional C . We shall not delve into the elegant combi- String Revolution by itself, is the celebrated AdS/CFT natorics of the theory of toric varieties but only sum- Correspondence of Maldacena. This bring us to marize the key points of the construction here, much the closely neighbouring geography of non-compact as we did for the CICY case above. Calabi-Yau manifolds, which are affine cones over The ambient space is a toric 4-fold A, which is Sasaki-Einstein 5-folds and which complement the specified by an integer polytope ∆ ∈ R4 containing AdS factor in the 10-dimensional metric. This is again in particular the origin (0,0,0,0) which can be repre- a vast land which I do not have space here to describe sented by the list of its vertices, which are integer, or and I shall, as mentioned in the introduction confine equivalently as a k × 4 matrix of k linear inequalities myself to smooth, compact CY3 and their construc- with integer coefficients. From this, one can define9 ◦  tions. the dual polytope ∆ := ~v ∈ R4|~m ·~v ≥ −1 ∀~m ∈ ∆ . For Due to the web of dualities, in particular that our familiar example of CP4, an archetypal example between the heterotic string and F-theory, there " −1 4 −1 −1 −1 # of a toric 4-fold, we have that = −1 −1 4 −1 −1 and emerged another family of CY3 studied in the 1990s ∆ −1 −1 −1 4 −1 −1 −1 −1 −1 4 which has recently been investigated with renewed " 1 0 0 0 −1 # ◦ 0 1 0 0 −1 zest, this is the class of elliptically fibred CY3 [27, 28]. ∆ = 0 0 1 0 −1 . The idea is that if ∆ is reflexive, i.e., if This is a generalization of CY1, by allowing the elliptic 0 0 0 1 −1 ◦ 4 curve to fibre over a complex base surface, by allow- ∆ has also integer vertices, such as the CP example, ing the coefficients in the Weierstraß to take values in then the hypersurface dual of the canonical bundle of the base, we would ar-  k  ~m·~v j+1 rive at an overall trivial first class of the CY3 as total X = ∑ c~m ∏ x j = 0 ⊂ A , space. ~m∈∆ j=1 What are the possible bases? Once again, this with x coordinates of the ambient toric 4-fold, c turned out to be a finite set: (I) Hirzebruch surfaces j ~m complex coefficients, and ~v the (integer) vertices of for r = 0,1,...,12; (II) 1-blowups of Hirzebruch sur- j Fr P ∆◦, defines a CY3. For 4, this is precisely the quintic faces for r = 0,1,2,3; (III) Del Pezzo surfaces d for CP Fbr Pr hypersurface. r = 0,1,...,9; and (IV) Enriques surface . These are E Thus the question of toric hypersurface CY3 classical surfaces with whose precise definition we is the question of reflexive integer 4-polytopes. In need not presently concern ourselves. Even though n=1,2,3, there are 1, 16 and 4319 such polytopes, with the list of possible bases seems limited, by tuning the R the 16 in the plane famously giving us the toric del possible elliptic curve, a diverse range of CY3 can be Pezzo surfaces and beyond. The computational chal- reached. This was much explored in the 1990s [31], lenge of Kreuzer and Skarke was to find all reflexive uncovering a wealth of beautiful structure. With the integer polytopes in 4. The actual calculation was help of modern computing, of the known CY3, many R performed on an SGI origin 2000 machine with about tens of thousands have been identified as elliptic fi- 30 processors (quite the state of the art in the 1990s) brations [29, 30]; the full classification of this rich which took approximately 6 months and 473,800,776 dataset is still in progress. was found. Each of these gives a hypersurface CY3 3.3.3 Toric Hypersurfaces: The KS Database and thus from the database of tens of thousands es- tablished by the early 1990s, the list of CY3 suddenly The most impressive indentation into the un- grew, with this tour de force, to half a billion. charted land of CY3 so far is undoubtedly the so As with the weighted projective case, the ambi- called Toric Hypersurfaces. Founded on the theo- ent space A is not necessarily smooth, so long as retic development of [32], Kreuzer and Skarke spent the hypersurface CY3 is. Interestingly, in this large almost a decade compiling this database [33, 34]. Due family, only 125 have smooth ambient space A and to the untimely death of Max Kreuzer – who very touchingly was dedicated to the Calabi-Yau cause 9 In the perhaps more familiar definition of a toric variety in even in his last hours and continued, on his deathbed, terms of fans of cones, the fan Σ is simply the faces of ∆◦.

26 NOTICES OF THE ICCM VOLUME 3,NUMBER 2 Figure 1. (a) The space of CY3, with the 3 most studied datasets. There are also some individualized constructions outside the three major databases, symbolically marked as crosses. Q is the quintic, S is the Schoen CY3 and the most “typical” CY3 has Hodge numbers (27,27), totaling almost 1 million. (b) Accumulating χ = 2(h1,1 − h1,2) (horizontal) versus h1,1 + h1,2 (vertical) of all the known Calabi-Yau threefolds in a colour Log-density plot. more remarkably, only 16 have non-trivial funda- note. There are a total of 30,108 distinct points, mental group. Though of course a classification of meaning the some half-billion CY3 are severely de- discrete freely-acting symmetries has yet to be sys- generate in (h1,1,h1,2). The funnel shape delineating tematized from which one could potentially extract the lower extremes is just due to our plotting differ- many more non-simply-connected CY3 by quotient- ence versus sum of the (non-negative) Hodge num- ing, these special 16 are quite interesting [36]. bers. The fact the the figure is left-right symmetric is perhaps the best “experimental” evidence for mir- ror symmetry: that to each point with χ there should 3.4 A Statistical Plot be one with −χ, coming from the inter-change of the While the cartography of CY3 continues with two Hodge numbers. There is a paucity of CY3 near ever-increasing collaborative effort amongst physi- the corners: near the bottom tip and of funnel and the cists, mathematicians and computer scientists – for top (note this is a log-density plot) while a huge con- example, the classification of complete intersections centration resides near the bottom center. In fact, the in toric varieties, such as double hypersurfaces in most “typical” CY3 thus far known is one with Hodge 5-folds are well under way and billions have already numbers (27,27), numbering about 1 million. been found – it is expedient that we draw our short There are several observations whose explanation excursion into the terra sancta of Calabi-Yau mani- remain mysterious. The largest Euler number in mag- folds to a close. nitude is 960 and so far no CY3 is known to have There is an iconic plot: suppose we had h1,1(X) + anything exceeding this. Is this an upper bound to h2,1(X) in the ordinate versus χ = 2(h1,1(X) − h2,1(X)) in the topology in the space of smooth CY3? One might the abscissa, drawn in part (b) of Figure 1. In part (a), note that 960 is twice the difference between the di- we indicate our datasets discussed so far in a Venn mension and rank of E8 ×E8; this observation may not diagram. be as frivolous as may appear since the very structure Let us pause to admire the beauty of this plot, of the exceptional groups seems encoded into these the standard version of which is without the colour- toric CY3 [37]. The parabolic shapes on the top of the density which I have added here. This standard black shield-like funnel, are they also bounds to possible and white version is framed and features prominently Hodge values? Recently, these have been identified as in Philip Candelas’ office. Several properties are of elliptically fibered CY3 [30].

DECEMBER 2015 NOTICES OF THE ICCM 27 4. Epilogue [12] P. Horava and E. Witten, “Heterotic and Type I String Dynamics from Eleven-Dimensions,” Nucl. Phys. B 460, With our tantalizing plot of all known Calabi-Yau 506 (1996) [hep-th/9510209]. threefolds let us pause here. We have taken a small [13] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Stan- promenade in the land of CY3, mindful of the intri- dard Models from Heterotic M Theory,” Adv. Theor. Math. Phys. 5, 93 (2002) [hep-th/9912208]. cate interplay between the mathematics and physics, [14] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, emboldened by the plenitude of data and results, and “The Ekpyrotic Universe: Colliding Branes and the Ori- inspired by the glimpses towards the yet inexplica- gin of the Hot Big Bang,” Phys. Rev. D 64, 123522 (2001) ble. The cartography of Calabi-Yau manifolds will cer- [hep-th/0103239]. [15] R. Donagi, Y. H. He, B. A. Ovrut and R. 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