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A Calabi-Yau Cartography by Yang-Hui He*

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A Calabi-Yau Cartography by Yang-Hui He* 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 s = 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 w on X6 and h on V satisfy DECEMBER 2015 NOTICES OF THE ICCM 21 (a) ¶¶w = 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 w; ifold of complex dimension n such that (b) d† = i( − ) jj jj, where is a holomor- w ¶ ¶ ln W W • 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 w being balanced, i.e., d jjWjjw2 = 0; • X admits a nowhere vanishing holomorphic 3.
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