Toric Geometry and Calabi–Yau Compactifications

These notes contain a brief introduction to the construction of toric Calabi–Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including tors...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Datum:	2010
1. Verfasser:	Kreuzer, M.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Відділення фізики і астрономії НАН України 2010
Schriftenreihe:	Український фізичний журнал
Schlagworte:	Астрофізика і космологія
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/56206
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:	Toric Geometry and Calabi–Yau Compactifications / M. Kreuzer // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 613-625. — Бібліогр.: 55 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-56206
record_format	dspace
spelling	irk-123456789-562062014-02-14T03:11:24Z Toric Geometry and Calabi–Yau Compactifications Kreuzer, M. Астрофізика і космологія These notes contain a brief introduction to the construction of toric Calabi–Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues, and topological transitions. Цi нотатки мiстять короткий вступ до побудови торiчних Калабi–Яу гiперповерхней та повних перетинiв з акцентом на розрахунках, що стосуються дуальностi струн. Останнi два роздiли можуть бути прочитанi незалежно вiд iнших i присвяченi недавнiм результатам та роботам, якi ще не закiнчено, включаючи кручення в когомологiї, питання класифiкацiї та топологiчних переходiв. 2010 Article Toric Geometry and Calabi–Yau Compactifications / M. Kreuzer // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 613-625. — Бібліогр.: 55 назв. — англ. 2071-0194 PACS 02.40.Tt; 11.25.-w; 11.25.Mj http://dspace.nbuv.gov.ua/handle/123456789/56206 en Український фізичний журнал Відділення фізики і астрономії НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Астрофізика і космологія Астрофізика і космологія
spellingShingle	Астрофізика і космологія Астрофізика і космологія Kreuzer, M. Toric Geometry and Calabi–Yau Compactifications Український фізичний журнал
description	These notes contain a brief introduction to the construction of toric Calabi–Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues, and topological transitions.
format	Article
author	Kreuzer, M.
author_facet	Kreuzer, M.
author_sort	Kreuzer, M.
title	Toric Geometry and Calabi–Yau Compactifications
title_short	Toric Geometry and Calabi–Yau Compactifications
title_full	Toric Geometry and Calabi–Yau Compactifications
title_fullStr	Toric Geometry and Calabi–Yau Compactifications
title_full_unstemmed	Toric Geometry and Calabi–Yau Compactifications
title_sort	toric geometry and calabi–yau compactifications
publisher	Відділення фізики і астрономії НАН України
publishDate	2010
topic_facet	Астрофізика і космологія
url	http://dspace.nbuv.gov.ua/handle/123456789/56206
citation_txt	Toric Geometry and Calabi–Yau Compactifications / M. Kreuzer // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 613-625. — Бібліогр.: 55 назв. — англ.
series	Український фізичний журнал
work_keys_str_mv	AT kreuzerm toricgeometryandcalabiyaucompactifications
first_indexed	2025-07-05T07:26:13Z
last_indexed	2025-07-05T07:26:13Z
_version_	1836790971286683648
fulltext	TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS M. KREUZER Institute for Theoretical Physics, Vienna University of Technology (Wiedner Hauptstr. 8-10, 1040 Vienna, Austria; e-mail: Maximilian. Kreuzer@ tuwien. ac. at ) PACS 02.40.Tt; 11.25.-w; 11.25.Mj c©2010 These notes contain a brief introduction to the construction of toric Calabi–Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sec- tions can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues, and topological transitions Introduction Toric geometry is a beautiful part of mathematics that relates discrete and algebraic geometries and provides an elegant and intuitive construction of many non- trivial examples of complex manifolds [1–6]. Sparked by Batyrev’s construction of toric Calabi–Yau hyper- surfaces, which relates the mirror symmetry to a com- binatorial duality of convex polytopes [7], toric geom- etry also became a pivotal tool in string theory. It provides efficient tools for the construction and analy- sis of large classes of models, and for computing quan- tum cohomology and symplectic invariants [8–11], fi- bration structures [12–15] for non-perturbative dualities [16–19], and Lagrangian submanifolds for open string and D-brane physics [20–22]. F -theory compactifica- tions [23, 24], which are based on elliptic Calabi–Yau 4-folds, are maybe the most promising approach to real- istic unified string models for particle physics [25], but also 3-folds with torsion in cohomology have been used for the successful model building [26]. In the present notes, we describe some tools that are provided by toric geometry and report on some recent results. Section 1 contains the basic definitions and con- structions of toric varieties, working mainly with a ho- mogeneous coordinate ring. In section 2, we recall the string theory context in which Calabi–Yau geometry be- comes important for particle physics and describe the toric construction of hypersurfaces and complete inter- sections. Section 3 explains fibrations and torsion in cohomology in terms of the combinatorics of polytopes. In section 4, we summarize recent results and work in progress and conclude with a list of open problems. 1. Basics of Toric Geometry The (algebraic) n-torus is a product T = (C∗)n of n co- pies of the punctured complex plane C∗ = C \ {0} we regard as a multiplicative Abelian group. It is, hence, a complexification of the real n-torus U(1)n. A toric variety is defined as a (partial) compactification of T in the following sense: It is a (normal) variety X that contains an n-torus T as a dense open subset such that the natural action of the torus T on itself extends to an action of T on the variety X. The beauty of toric geometry comes from the fact that the data is encoded in combinatorial terms and that this structure can be used to derive simple formulas for so- phisticated topological and geometrical objects. More precisely, the data is given by a fan Σ, which is a finite collection of strongly convex rational polyhedral cones (i.e. cones generated by a finite number of lattice points and not containing a complete line) such that all faces of cones and all intersections of any two cones also belong to the fan. The space, in which Σ lives, can be obtained as fol- lows [6]: If we parametrize the torus by coordinates (t1, . . . , tn), the character group M = {χ : T → C∗} of T can be identified with a lattice M ∼= Zn, where m ∈ M corresponds to the character χm((t1, . . . , tn)) = tm1 1 . . . tmnn ≡ tm. Another natural lattice that comes with the torus T can be identified with the group of al- gebraic one-parameter subgroups, N ∼= {λ : C∗ → T}, where u ∈ N corresponds to the group homomorphism λu(τ) = (τu1 , . . . , τun) ∈ T for τ ∈ C∗. The composition (χ ◦λ)(τ) = χ(λ(τ)) = τ 〈χ,λ〉 defines a canonical pairing 〈χm, λu〉 = m ·u which makes N and M ∼= Hom(N,Z) a dual pair of lattices (or free Abelian groups). The char- acters χm for m ∈ M can be regarded as holomorphic functions on the torus T and hence as rational functions on the toric variety X. We will see that the lattice N , ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 613 M. KREUZER or rather its real extension NR = N ⊗Z R ∼= Rn, is where the cones σ ∈ Σ of the fan live. First, we construct the rays ρj with j = 1, . . . , r, i.e. the one-dimensional cones ρj ∈ Σ(1) of the fan, where the n-skeleton Σ(n) contains the n-dimensional cones of Σ. Recall that divisors are formal linear combinations of subvarieties of X of (complex) codimension 1, i.e. of dimension n − 1. It can be shown that normality of X implies that the divisors div(χm), i.e. the hypersurfaces defined by the equations χm = 0, are equal to sums∑r 1 ajDj for some finite set of irreducible divisors Dj . Like div(χm), the Dj are T -invariant and hence unions of complete orbits of the torus action. The coefficients aj(m) are unique so that the decomposition divχm =∑ ajDj defines linear maps m→ aj(m) = 〈m, vj〉. The irreducible divisors Dj define, hence, vectors vj ∈ N for j ≤ r with aj(m) = 〈m, vj〉. These vectors are the primitive generators of the rays ρj that constitute the 1-skeleton Σ(1) of the fan Σ. If we locally write the equa- tion of the divisor as Dj = {zj = 0} with zj a section of some local line bundle, then we can write the torus coordinates, with appropriate choice of normalizations, as ti = ∏ z 〈ei,vj〉 j on some dense subset of T ⊆ X. 1.1. Homogeneous coordinates We now regard {zj} as global homogeneous coordinates (z1 : . . . :zr) in a generalization of the construction of the projective space Pn as a C∗-quotient of Cn+1 \ {0} via the identification (z0 : . . . :zn) ≡ (λz0 : . . . :λzn). If all zj are non-zero, then the coordinates (λq1z1 : . . . : λqrzr) ∼ (z1 : . . . : zr), λ ∈ C∗ (1) describe the same point of the torus T with coordinates ti = ∏ z 〈ei,vj〉 j ∈ T = X \ ⋃ Dj (2) if ∑ qjvj = 0, where vj are the generators of the rays ρj ∈ Σ(1) of the fan Σ. Since the vectors vj ∈ N be- long to a lattice of dimension n, the scaling exponents {qj} ∈ ZΣ(1) ∼= Zr in identification (1) are restricted by n independent linear equations. Naively, we might ex- pect therefore that the toric variety X can be written as a quotient (Cr \ Z) / (C∗)r−n, where {(zj)} = Cr \ Z is the set of allowed values for the homogeneous coor- dinates and the (C∗)r−n-action on {(zj)} implements identification (1) with ∑ qjvj = 0. The identification group corresponds, however, to the kernel of the map (zj) → (ti = ∏ z 〈ei,vj〉 j ) for (zj) ∈ (C∗)r. If vj do not span the N -lattice, i.e. if the quotient G ∼= N/(spanZ{v1, . . . , vr}) (3) is a finite Abelian group of order \|G\| > 1, then this kernel is (C∗)r/(C∗)n ∼= (C∗)r−n × G and contains a discrete factor G. The toric variety X can be constructed, hence, in terms of homogeneous coordinates zj , an exceptional set Z ⊂ Cr, and the identification group (C∗)r−n × G as X = (Cr − Z) / ((C∗)r−n ×G), (4) where the quotient can be shown to be “geometrical” if the fan Σ is simplicial [27] (cf., however, Example 2 below), For a given set of generators vj ∈ Rn of the rays ρj ∈ Σ(1), we can construct different toric varieties by choosing different lattices N ⊂ Rn that contain vj as primitive lattice vectors. These are all Abelian quotients of the variety, for which N is the integral span of {vj}. The last piece of information we need is the excep- tional set Z. The limit points (zj) ∈ X \ T = ⋃ Dj that are added to T are determined by the conditions un- der which homogeneous coordinates are allowed to van- ish. This is where the information of the fan Σ enters: A subset of the coordinates zj is allowed to vanish si- multaneously iff there is a cone σ ∈ Σ containing all of the corresponding rays ρj . In geometrical terms, this means that the corresponding divisors Dj intersect in X. The exceptional set Z is, hence, the union of sets ZI = {(z1 : . . . : zr) \| zj = 0 ∀ j ∈ I}, (5) for which there is no cone σ ∈ Σ such that ρj ⊆ σ for all j ∈ I. Minimal index sets I with this property are called primitive collections. They correspond to the maximal irreducible components of Z = ⋃ ZI . 1.2. Torus orbits In terms of homogeneous coordinates, the torus action amounts to the effective part of the (C∗)r action induced on the quotient (4) by independent scalings of zj and thus extends from T to X. The torus orbits, into which X decomposes, are, hence, characterized by the index sets of the vanishing coordinates. It can be shown [3] that the torus orbits Oσ are in one-to-one correspon- dence with the cones σ ∈ Σ, where Oσ ∼= (C∗)n−dimσ is the intersection of all divisors Dj for ρj ⊂ σ with all complements of the remaining divisors. The orbit Oσ is an open subvariety of the orbit closure Vσ, which is the intersection of all divisors Dj for ρj ⊂ σ and which also has dimension n− dimσ. Further important sets are the affine open sets Uσ, which are the intersections of all complements X \ Dj 614 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS with ρj 6∈ σ and which provide a covering of X. The relations between these sets can be summarized as Vσ = ⋃ τ⊇σ Oτ , Uσ = ⋃ τ⊆σ Oτ , X = ⋃ σ∈Σ Uσ, (6) where Vσ and Uσ are disjoint unions and the last union can be restricted to a covering of X by Uσ’s for maximal cones σ. In the traditional approach [1–4], a toric variety X is constructed by gluing its open affine patches Uσ along their intersections Uσ ∩ Uτ = Uσ∩τ ⊇ T = U{0}. (7) The patches Uσ are constructed in terms of their rings of regular functions which are generated by the characters χm that are nonsingular on the relevant patch. Since ti = ∏ z 〈ei,vj〉 j ⇒ χm = tm = ∏ z 〈m,vj〉 j , (8) the relevant exponent vectors m ∈ M are the lattice points in the dual cone σ∨ = {x ∈MR : 〈x, v〉 ≥ 0 ∀ v ∈ σ}. (9) More abstractly, the algebra Aσ = C[σ∨ ∩ M ] of the semigroup σ∨ ∩M is, by definition, the ring of regular functions on Uσ, so that the points of Uσ can be ob- tained as the spectrum Specm(Aσ) of maximal ideals. The Zariski topology of Uσ can be constructed in terms of the prime ideals and the gluing can be worked out by relating the characters in different patches (cf. Example 2 below). We now state two important theorems [1–4]: Theorem 1. A toric variety is compact if and only if the fan is complete, i.e. if the support of the fan covers the N lattice \|Σ\| = ⋃ Σ σ = NR. We prove the only if: For an incomplete fan, we con- sider some u ∈ N \ \|Σ\| and a one-parameter family of points pλ = (λu1t1, . . . , λ untn) ∈ T . The evaluation of χm yields χm(pλ) = λ〈m,u〉χm(p1). But the limit point pλ→0 cannot be contained in any patch Uσ, be- cause χm(pλ) diverges as λ → 0 for m ∈ σ∨ and u 6∈ σ, so that 〈m,u〉 < 0. Theorem 2. A toric variety is non-singular if and only if all cones are simplicial and basic, i.e. if all cones σ ∈ Σ are generated by a subset of a lattice basis of N . To illustrate these theorems, we work out two examples: Example 1. The Hirzebruch surface Hirzebruch surfaces Fn are P1 bundles over P1 that can be defined by v0 = (0,−1), v1 = (1, 0), v2 = (−1, n) and v3 = (0, 1), with linear relations v0 + v3 = 0, nv0 + B B A B s s s s ss sz2 F2 (λ2µz0 :λz1 :λz2 :µw) ∈ (C4−Z)/(C∗)2 F1 @ @ @ @ � �s wAs s BAAs s B z1 Z = {z0 = w=0} ∪ {z1 = z2 =0}F0 z0 Fig. 1. Hirzebruch surface F2 as a blow-up of WP211C C J J J J � � � ��aaaa aaaa�� z1 z3z2 z0(2b)J J J J � � � ��aaaa aaaa�� β α(2a) J J J J � � � ��aaaa aaaa�� xv (2c)uy A A AF as blowup of P . Fig. 2: Toric desingularizations of the conifold.by gluing its open a�netersections (7)in terms of their rings ofgenerated by the charactersrelevant patch. Since We consider the case . If we drop thethe fan would consist of 3 cones and we obtainweighted projective space P with scaling. This space is singular because the coneby has volume 2. Indeed, if we drop thenate and set then is a �xedof the C identi�cation for , i.e. we haquotient singularity. The Hirzebruch surface F Fig. 2. Toric desingularizations of the conifold v1 + v2 = 0 and scaling parameters µ and λ, as shown in Fig. 1. We consider the case n = 2. If we drop the vertex v3, the fan would consist of 3 cones, and we obtain the weighted projective space WP211 with scaling weights (2, 1, 1). This space is singular because the cone spanned by (v1, v2) has volume 2. Indeed, if we drop the coor- dinate w and set µ = 1, then (1 : 0 : 0) is a fixed point of the C∗ identification for λ = −1, i.e. we have an Z2 quotient singularity. The Hirzebruch surface F2 is a desingularization of this surface corresponding to a sub- division of the cone (v1, v2) into two basic cones (v1, v3) and (v3, v2). The exceptional set is accordingly modified to Z = {z0 = w = 0} ∪ {z1 = z2 = 0}. We can now consider two cases: If w 6= 0, then µ = 1/w scales w to w = 1. This yields all points (z0 :z1 :z2 : 1) ≡ (z0 :z1 :z2) of WP211 except for its singular point (1 : 0 : 0) which is excluded due to the subdivision of the cone (v1, v2) by v3. If w = 0, we can scale z0 6= 0 to z0 = 1 and find (1 : z1 : z2 : 0) ∈ F2. We thus observe that the singu- lar point has been replaced by a P1 with homogeneous coordinates (z1 : z2). This process of replacing a point by a projective space is called blow-up. In the present example, it desingularizes a weighted projective space. It can be shown [3] that all singularities of toric varieties can resolved by a sequence of blow-ups that correspond to subdivisions of the fan. Example 2. The conifold singularity According to Theorem 2, the second source of singulari- ties is the nonsimplicity of a cone, which is only possible in at least 3 dimensions. We consider, hence, a quadratic cone σ as displayed in Fig. 2,b with generators v0 = (1, 0, 0), v1 = (0, 1, 0), v2 = (1, 0, 1), v3 = (0, 1,−1) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 615 M. KREUZER and relation ∑ qivi = 0 with q = (1, 1,−1,−1) so that (z0 : z1 : z2 : z3) = (λz0 : λz1, 1 λz2 : 1 λz3), (10) or X = P(1, 1,−1,−1). The dual cone σ∨ has generators m0 =(1, 0, 0),m1 =(0, 1, 0),m2 =(0, 1, 1),m3 =(1, 0,−1). According to (8), the coordinate ring Aσ is generated by x = χm0 = z0z2, y = χm1 = z1z3, (11) u = χm2 = z1z2, v = χm3 = z0z3, (12) which are regular, invariant under the scaling (10), and obey the relation xy = uv. Hence, Aσ = C[σ∨ ∩M ] ∼= C[x, y, u, v]/〈xy−uv〉 and X=Uσ can be identified with the hypersurface xy = uv in C4 which has a “conifold singularity” at the origin. As shown in Fig. 2, the cone σ can be triangulated in two different ways. In the first case, σα = 〈v0, v2, v3〉, σβ = 〈v1, v2, v3〉, and the dual cones are σ∨α = (mα,m0,m3), σ∨β = (mβ ,m1,m2) with mα = −mβ = (−1, 1, 1), so that we obtain the algebras Aα = C(mα,m0,m3) 3 (z1/z0, x = z0z2, v = z0z3), (13) Aβ = C(mβ ,m2,m1) 3 (z0/z1, u = z1z2, y = z1z3). (14) With the exceptional set Z = {z0 = z1 = 0}, we ob- serve that the singular point x = y = u = v = 0 has been replaced by a P1 3 (z0 : z1), and the transition functions show that X{σα,σβ} can be identified with the total space of the rank two bundle O(−1)⊕O(−1)→ P1. The reader may verify that the homogeneous coordi- nates work straightforwardly (geometrical) in the sim- plicial cases (2,a) and (2,c). In the non-simplicial case (2,b), quotient (4) has to be taken in the “GIT–sense” [27], because all C∗ orbits (λz0 : λz1 : 0 : 0) and (0 : 0 : 1 λz2 : 1 λz3) map to the tip 0 ∈ Uσ of the coni- fold (the categorial quotient of geometric invariant theory (GIT) involves the dropping of “bad” orbits). The two toric resolutions correspond to blowups re- placing the singular point by two different P1’s (which topologically are 2-spheres S2). We can go, hence, from one “small resolution” to the other via the conifold by blowing down the P1 to a point and blowing up that point in a different way. This is called a flop transition. There is also a non-toric possibility to resolve the singularity by deforming the hypersurface equation to xy − uv = ε. Topologically, this amounts to replacing the singularity by a 3-sphere S3. This can be seen as follows: by a linear change of variables {x∓y2 , u±v2 } ↔ {ilwl}, we can write the deformed conifold equation as∑4 l=1 w 2 l = ε. With wl = al + ibl, its real and imaginary part become∑ l≤4 a2 l = ε+ ∑ l≤4 b2l , ∑ l≤4 albl = 0. (15) For ε > 0, the four real variables a′k = ak/ √ ε+ ∑ l b 2 l parametrize a 3-sphere, and bk with ∑ l a ′ lbl = 0 parametrize the fibers of the cotangent bundle T ∗S3. The topology change between the small resolution and the deformation is called conifold transition. 1.3. Line bundles The conifold is the standard example of a non-compact (local) Calabi–Yau geometry. A compact toric varieties, on the other hand, never have c1 = 0. Hence, we will not only be interested in toric varieties themselves but also in hypersurfaces or complete intersections thereof, which are smooth Calabi–Yau spaces under appropri- ate conditions. Their defining equations will be sections of non-trivial line bundles. The relevant data of these bundles are the transition functions between different patches. These data are closely related to the topolog- ical data of Cartier divisors which are locally given, by definition, in terms of the rational equations fα = 0 with fα/fβ regular and nonzero on the overlap of two patches. Since the multiplication by a rational function does not change the line bundle, we are interested in the classes of divisors with respect to linear equivalence, i.e. mod- ulo addition of principal divisors div(f), which are the divisors of rational functions f . Hence, Cartier divisor classes determine the Picard group Pic(X) of holomor- phic line bundles. Finite formal sums of irreducible varieties of codimen- sion one are called Weil divisors (which may not be Cartier, i.e. locally principal, on singular varieties). On a toric variety, it can be shown that the Chow group An−1(X) of Weil divisors modulo linear equiva- lence is generated by the T -invariant irreducible divisors Dj modulo the principal divisors div(χm) with m ∈M , i.e. there is an exact sequence 0→M → ZΣ(1) → An−1(X)→ 0, (16) where M 3 m → (〈m, vj〉) ∈ ZΣ(1) and ZΣ(1) 3 (aj) →∑ ajDj . Hence, the Chow group An−1(X) has rank r−n. It contains the Picard group as a subgroup, which is torsion-free if XΣ is compact [3]. A Weil divisor of the form D = ∑ ajDj is Cartier and, hence, defines a line bundle O(D) ∈ Pic(X), if there exists an mσ ∈ M for each maximal cone σ ∈ Σ such that 〈mσ, vj〉 = −aj for all ρj ∈ σ. The transition functions of O(D) between the patches Uσ and Uτ are then given by χmσ−mτ . If X is smooth, then all Weil divisors are Cartier. For a simplicial fan, kD is Cartier for some positive integer k. For Cartier divisors, the 616 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS Σ-piecewise linear real function ψD on NR defined by ψD(v) = 〈mσ, v〉 for v ∈ σ (17) is called support function. If X is compact and D =∑ ajDj is Cartier, then O(D) is generated by global sections iff the support function ψD is convex, and D is ample iff ψD is strictly convex, i.e. if 〈mσ, vj〉 > −aj for dimσ = n and ρj 6⊂ σ. For convex support functions, ΔD = {m ∈MR : 〈m, vj〉 ≥ −aj ∀ j ≤ r} (18) = {m ∈MR : 〈m,u〉 ≥ ψD(u) ∀ u ∈ N} (19) defines a convex lattice polytope ΔD ⊂ MR, whose lat- tice points provide the global sections of the line bun- dle O(D) corresponding to a divisor D (the first equal- ity defines ΔD also if D is not Cartier). In particular, ΔkD = kΔD and ΔD+div(χm) = ΔD−m so that the poly- tope can be translated in the M lattice without changing the divisor class and the transition functions. In terms of polytope (18), D is generated by global sections iff ΔD is the convex hull of {mσ}, and D is ample iff ΔD is n-dimensional with vertices mσ for σ ∈ Σ(n) and with mσ 6= mτ for σ 6= τ ∈ Σ(n). In the latter case, there is a bijection between faces of ΔD and cones in Σ or, more pricisely, Σ is the normal fan of ΔD: By definition, the cones στ of the normal fan ΣΔ of a polytope Δ are the dual cones of the cones over Δ − x, where x ∈ MR is any point in the relative interior of a face τ ⊂ Δ. If 0 ∈M is in the interior of Δ, as can always be achieved by a rational translation of Δ by δx ∈ MQ, then the normal fan ΣΔ coincides with the fan of cones over the faces of the polar polytope Δ◦ ⊆ NR defined by Δ◦ = {y ∈ NR : 〈x, y〉 ≥ −1 ∀x ∈ Δ}. (20) On smooth compact toric varieties, it can be shown that every ample T -invariant divisor is very ample. The sec- tions χm of such O(D) provide an embedding of XΣ into PK−1 via (χm1 : . . . : χmK ), where K = \|ΔD ∩M \| is the dimension of the space of global sections of O(D). Therefore, a toric variety XΣ is projective iff Σ is the normal fan of a lattice polytope Δ ⊂MR [2, 3]. Summarizing, the equations defining Calabi–Yau hy- persurfaces or complete intersections will be sections of the line bundles O(D) given by Laurent polynomials f = ∑ m∈ΔD∩M cmχ m = ∑ m∈ΔD∩M cm ∏ j z 〈m,vj〉 j , (21) whose exponent vectors m span the convex lattice poly- topes ΔD ⊆ MR defined in eq. (18). In an affine patch Uσ, the local section fσ = f/χmσ is a regular function fσ ∈ Aσ because ΔD −mσ ⊂ σ∨. 1.4. Intersection ring and Chern classes If a collection ρj1 , . . . , ρjk of rays is not contained in a single cone, then the corresponding homogeneous coor- dinates zjl are not allowed to vanish simultaneously, and the corresponding divisorsDjl have no common intersec- tion. For the intersection ring, we expect the non-linear relations RI = Dj1 · . . . · Djk = 0, where it is sufficient to consider the primitive collections I = {j1 . . . jk} as defined by Batyrev, i.e. the minimal index sets such that the corresponding rays do not all belong to the same cone (cf. the definition of the exceptional set Z = ⋃ ZI in Section 1 1.1). The ideal in Z[D1, . . . , Dr] generated by these RI is called Stanley–Reisner ideal J , and Z[D1, . . . , Dr]/J is a Stanley–Reisner ring. The Chow groups Ak(X) of a variety X are gener- ated by k-dimensional irreducible closed subvarieties of X modulo rational equivalence by divisors of rational functions on subvarieties of dimension k + 1. For an ar- bitrary toric variety XΣ, it can be shown that Ak(X) is generated by the equivalence classes of orbit closures Vσ for cones σ ∈ Σ(n−k). The intersection ring of a non- singular compact toric variety XΣ is [1] A∗(XΣ) = Z[D1, . . . , Dr] / 〈 RI , ∑ j〈m, vj〉Dj 〉 (22) (for a definition of the intersection product see [3]). The intersection ring can be obtained from the Stanley–Reisner ring by adding the linear relations∑ j〈m, vj〉Dj ' 0, where it is sufficient to take a set of basis vectors of the M -lattice for m. The Chow ring also determines the homology groups H2k(XΣ,Z) = Ak(X,Z). These results actually generalize to the sim- plicial projective case with the exception that one needs to admit rational coefficients [2, 3]. In particular, for a maximal-dimensional simplicial cone σ spanned by vj1 , . . . , vjn , the intersection number of the correspond- ing divisors is Dj1 · . . . ·Djn = 1/Vol(σ), (23) where Vol(σ) is the lattice-volume (i.e. the geometrical volume divided by the volume 1/n! of a basic simplex). Having discussed the cycles, we did not turn to dif- ferential forms. The canonical bundle of a non-singular toric variety can be obtained by considering the rational form ω = dx1 x1 ∧ . . . ∧ dxn xn (24) which, by an appropriate choice of the orientation (i.e. the order of the local coordinates xi in the affine ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 617 M. KREUZER patches), is a rational section of ΩnX . This implies ΩnX = OX(− r∑ j=1 Dj) (25) and for the canonical divisor −D = −∑Dj . The com- putation of the total Chern class requires an expression for the (co)tangent bundle, for which there is an exact sequence 0 → Ω1 X → Ω1 X(logD) res→ ⊕ j O(Dj), (26) where Ω1 X(logD) turns out to be trivial, and the residue map takes ω = ∑ fj dzj/zj res→ ⊕fj \|Dj [3]. A calculation yields the total Chern class of the tangent bundle c(TX) = ∏r 1(1 +Dj) = ∑ σ∈Δ[Vσ] (27) and the Todd class td(TX) = ∏r 1 Dj 1−exp(−Dj) = 1 + 1 2 c1 + 1 12 c 2 1c2 + . . . (28) The first Chern class c1 = ∑ Dj is positive for compact toric varieties, but it vanishes for the conifold, because of the linear relations D0 +D2 ∼ 0, D1 +D3 ∼ 0 and D1 ∼ D3. (Implications for volumes and numbers of lattice points can now be derived by applying the Hirzebruch– Riemann–Roch formula χ(X,E) = ∫ ch(E) Td(X) to the case of line bundles of Cartier divisors as described, for example, in the last chapter of [3].) 1.5. Symplectic reduction There is another approach to toric geometry in terms of symplectic instead of complex geometry, which is impor- tant because, in addition to the complex structure, we will also need the Kähler metric. Moreover, the symplec- tic approach can be given a direct physical interpretation in terms of supersymmetric gauged linear sigma models [28]. The idea is that the C∗-quotient can be performed in two steps: we first divide out the phase parts which amount to compact U(1) quotients and then – instead of a radial identification – fix the values of appropriate “radial” variables to certain sizes ta that will parametrize the Kähler metric. In order that the quotient inherits a Kähler form (and, hence, a symplectic structure) from the natural Kähler form on Cr ω = i ∑ dzj ∧ dz̄j = 2 ∑ dxj ∧ dyj = ∑ dr2j ∧ dϕj , (29) with z = x+ iy = reiϕ, we use the symplectic reduction formalism. This requires that the G-action is Hamil- tonian, i.e. given by a moment map µ : Cr → g∗ to the dual g∗ of the Lie algebra g ofG = U(1)r−n such that the Hamiltonian flows defined by µ generate the infinitesi- mal G-transformations. Then the symplectic reduction theorem guarantees that the restriction of the image of a moment map to fixed values ta for a = 1, . . . , n − r induces a symplectic structure on the quotient of the preimage µ−1(ta)/G for regular values of ta. In toric geometry, we consider the moment maps µa = ∑ j q (a) j \|zj \|2 with ∑ j q (a) j vj = 0. (30) With ω−1 ∼ ∑j ∂ ∂ϕj ∧ ∂ ∂r2j , the corresponding Hamilto- nian flows ω−1(µa) ∼ ∑ j q (a) j ∂ ∂ϕj generate the compact subgroups of the C∗ actions of the homomorphic quo- tients (4). In the gauged linear sigma model [28], the quantities q(a)j are the charges of r chiral superfields zj under a U(1)r−n gauge group, and the moment maps µa are D-terms in the superpotential. The imaginary parts of the complexified radii ta thus correspond to θ-angles. Under the symplectic reduction, the holomorphic r- form Ω = ∏ dzj on Cr descends to a holomorphic n- form iff it is invariant under the group action, i.e. if∑ j q (a) j = 0 for all a ≤ r−n. These equations can be in- terpreted as U(1) gauge anomaly cancellation conditions in the linear sigma model [28]. Since the existence of a holomorphic n-from on XΣ is equivalent to c1 = 0, we thus obtain a simple form of the Calabi–Yau condition (cf. the vanishing of ∑ j qj for a conifold). Instead of the linear combinations (30), we can con- sider all moment maps µj = \|zj \|2, whose flows are phase rotations of the homogeneous coordinates zj . After the symplectic reduction, the effective part G ∼= U(1)n of this U(1)r-action yields the compact part of the torus action G ⊂ T . The image of the corresponding momentum map is a convex polytope, the Delzant polytope Δ(ta), whose corners correspond to fixed points of G. Since the Lie algebra g of G can be identified with the the real exten- sion of the N -lattice, Δ(ta) is a polytope in g∗ ≡MR. Example 3. For the projective space Pn, all qj = 1, and, with ta = r2, we obtain the symplectic quotient as {\|z0\|2 + . . . + \|zn\|2 = r2}/U(1). Hence, the value ta = r2 of the moment map of the symplectic reduction parametrizes the size of the projective space. The image of the moment map for G ⊂ T on the resulting toric va- riety is the simplex {ti ≥ 0, r2 −∑n 1 ti = t0 ≥ 0} ⊂MR, whose faces of codimension k correspond to the vanish- ing of k moment maps ti and, hence, to fixed points of a U(1)k subgroup of G. The fan of the toric variety Pn is the normal fan of Δ. We thus can construct Pn as a (compact) torus fibration over a polytope Δ ∈ Rn, 618 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS whose fibers degenerate to lower-dimensional tori over the faces of Δ. This fibration structure has been used by Strominger, Yau, and Zaslow [20] for an interpretation of the mirror symmetry as the T -duality on a torus-fibered Calabi–Yau manifold. Example 4. As a non-compact example, we consider the conifold, whose Kähler metric is parametrized by t = \|x0\|2 + \|x1\|2 − \|x2\|2 − \|x3\|2. Obviously, t = 0 is a singular value, while, for t = ±ε2 → 0, the size ε of one of the blown-up P1’s shrinks to 0. Regular val- ues of the moment maps ta lead to a smooth symplectic quotient and, in particular, to a (projective) triangu- lation of the fan. The corresponding smooth Kähler metric is parametrized by the r − n values ta which can be interpreted as sizes of certain two-cycles, in ac- cord with the dimension r − n of H2(XΣ). The reg- ular values correspond to open cones of the secondary fan which parametrizes the Kähler moduli spaces and whose chambers are separated by walls that correspond to flop transitions [29] between different smooth phases (in the physicist’s language [28]). At the transition, a cycle shrinks to a point that is blown up according to a different triangulation ΣΔ(t) on the other side of the wall [29]. 2. Strings, Geometry, and Reflexive Polytopes At observable energy scales, string theory leads to an ef- fective theory that corresponds to a 10-dimensional su- pergravity compactified on a 6-dimensional manifold K. At small distances, space-time looks, hence, like M4×K, where M4 is our 4-dimensional Minkowski space, as long as quantum fluctuations of the metric are sufficiently small to allow for a semiclassical geometrical interpreta- tion. For phenomenological reasons, we require usually that supersymmetry survives the compactification, which im- plies the existence of a covariantly constant spinor ∇η = 0 on the internal manifold K. In the simplest situation, RR background fields and the B field vanish. Candelas, Horowitz, Strominger, and E. Witten [30] showed that this implies that K is a complex Kähler manifold with the vanishing first Chern class. (The inclusion of B fields was already discussed in a beautiful paper by Strominger [31], but RR fluxes were largely omitted for a long time, until their importance for moduli stabilization in type II theories was recognized [32]. The investigation of their geometry leads to the important new concept of general- ized complex structures [33–35].) Explicitly, the Kähler form ω and the holomorphic 3-form Ω of the Calabi–Yau hypersurface can be constructed in terms of η as ωij = iη†γ[iγj]η, Ω ∼ η†γ[iγjγk]η, (31) and the integrability condition Nij k = 0 for the com- plex structure Ji l = ωijg jl with the Nijenhuis tensor Nij k = Ji l∂lJj k − Jj l∂lJi k − ∂iJj lJl k + ∂jJi lJl k is a trivially satisfied for the torsion-free metric-compatible connection that stabilizes η = 0. The condition c1 = 0, which is equivalent to the exis- tence of a holomorphic 3-form Ω, has been conjectured by Calabi and proven by Yau to be also equivalent to the existence of a Ricchi-flat Kähler metric, so that the vacuum Einstein equations are satisfied. In the standard construction of anomaly-free heterotic strings with the gauge group E6, it turns out that charged particles and anti-particles show up in conjunc- tion with elements of the Dolbeault cohomology groups H11 and H21, respectively. While H11 parametrizes the Kähler metric, H21 can be related to complex struc- ture deformations via contraction with the holomorphic 3-form, ΩµνλδJλρ̄ ∈ H21. Since the exchange of par- ticles and anti-particles, as well as the corresponding sign of a U(1) charge in the sigma model description of the Calabi–Yau compactifications, are mere conventions, physicists came up with the idea of mirror symmetry [36] which was used by Candelas et al. [37] to construct a mirror map between the Kähler and complex structure moduli spaces of a Calabi–Yau manifoldX and its mirror dual X∗, whose topologies are related by h11(X) = h21(X∗), h21(X) = h11(X∗). (32) The power series expansions of this map could be in- terpreted as instanton corrections in the quantum the- ory and thus lead to a prediction of numbers of rational curves [8, 9]. 2.1. Toric hypersurfaces The beauty of the toric construction of Calabi–Yau spaces is based on the fact that it relates the mirror symmetry to a combinatorial duality of lattice polytopes, as was discovered by Batyrev [7]. He showed that the Calabi–Yau condition for a hypersurface, i.e. the van- ishing of the first Chern class, requires as a necessary and sufficient condition that the polytope ΔD ⊆ MR of the line bundle O(D), whose section defines the hyper- surface, is polar to the lattice polytope Δ∗ = Δ◦D ⊆ NR, where Δ∗ is the convex hull of the generators vj of rays ρj ∈ Σ(1) of the fan of the ambient toric variety XΣ. A lattice polytope, whose polar polytope (20) is again a ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 619 M. KREUZER @ @ @sss s sss s sss s sss s � � ��sss s sss s sss s sss s � � �@ @ @s s s s ss s s s ss s s s s � � � A A Ass s ss s ss s ss s ss s ss s @@ @@�� ss s ss s ss s @@ss s ss s ss s @@�� ss s ss s ss s @ @ @s s s ss s s ss s s s @@ � � � ��ss s ss s ss s @ @ @��s s s ss s s ss s s s � � � @@ss s ss s ss s � � �@ @ @s s s ss s s ss s s s A A Ass s ss s ss s ��@@ss s ss s ss s @@ @@ss s ss s ss sFig. 3: All 16 re�exive polygons in 2D: The �rst 3 dual pairs are maximal/minimal andcontain all others as subpolygons, while the last 4 polygons are selfdual.terms in the second sum can be understood as multipli-cities of toric divisors and their presence indicates thatonly a subspace of the K�ahler moduli is accessible totoric methods. for the reconstruction of the data as wellother purposes like the analysis of �brationscohomology (see section 3.), as well ason of higher-dimensional examples. Allaccessed on the web page [43] and we justnumbers of re�exive polytopes in 3d and 4d Fig. 3. All 16 reflexive polygons in 2D: the first 3 dual pairs are maximal/minimal and contain all others as subpolygons, while the last 4 polygons are self-dual lattice polytope, is called reflexive. Batyrev also derived a combinatorial formula for the Hodge numbers h11(XΔ) = h2,1(XΔ◦) = l(Δ◦)− 1− dim Δ− (33) −sumcodim(θ◦)=1l ∗(θ◦) + ∑ codim(θ◦)=2 l∗(θ◦)l∗(θ) where θ and θ◦ is a dual pair of faces of Δ and Δ◦, respectively. The quantity l(θ) is the number of lattice points of a face θ, and l∗(θ) is the number of its interior lattice points. Mirror symmetry now amounts to the exchange of Δ and Δ◦, and the formula for the Hodge data makes the topological duality (32) manifest. Formula (33) has a simple interpretation: the princi- pal contributions to h11 come from the toric divisors Dj that correspond to lattice points in Δ◦ different from the origin. There are dim(Δ) linear relations among these divisors. The first sum corresponds to the subtraction of interior points of facets. The corresponding divisors of the ambient space do not intersect a generic Calabi–Yau hypersurface. Lastly, the bilinear terms in the second sum can be understood as multiplicities of toric divi- sors, and their presence indicates that only a subspace of the Kähler moduli is accessible to toric methods. The enumeration of all reflexive polygons has been achieved by Batyrev many years ago (see Fig. 3). In dimensions 3 and 4, which are relevant for K3 surfaces and Calabi–Yau 3-folds, respectively, the enumeration required the extensive use of computers and was achieved in [38, 39] and [40, 41]. The code was later included into the software package PALP [42] which can be used for the reconstruction of the data, as well as for many other purposes like the analysis of fibrations and integral co- homology (see Section 3) and the construction of higher- dimensional examples. All results can be accessed on the web page [43] and we just note that the numbers of re- flexive polytopes in 3d and 4d are 4319 and 473 800 776, respectively. The resulting 30108 different Hodge data of 3-folds amount to 15122 mirror pairs as shown in Fig. 4. 2.2. Complete intersections Soon after the hypersurface case, Batyrev and Borisov discovered another beautiful combinatorial duality that corresponds to the mirror symmetry of toric complete in- tersections [44]. In this generalization, two polar pairs of reflexive lattice polytopes are involved with the defining conditions summarized in the following equations, Δ = Δ1 + . . .+ Δr Δ◦ = 〈∇1, . . . ,∇r〉conv (Δl,∇m) ≥ −δlm (34) ∇◦ = 〈Δ1, . . . ,Δr〉conv ∇ = ∇1 + . . .+∇r, where r is the codimension of the Calabi–Yau variety, and the defining equations fi = 0 are sections of O(Δi). The decomposition of the M -lattice polytope Δ ⊂ MR into a Minkowski sum Δ = Δ1 + . . .+Δr is now dual to a nef (numerically effective) partition of the vertices of Δ◦ ⊂ NR such that the convex hulls ∇i of the respective vertices and 0 ∈ N only intersect at the origin [5, 44]. ∇ = ∇1 + . . .+∇r is another reflexive polytope, whose dual ∇◦ has a nef partition in terms of the vertices of Δi. The Hodge numbers hpq of the corresponding com- plete intersections have been computed and shown to obey (32) in [45]. They are summarized for arbitrary di- mension n− r of the Calabi–Yau variety in a generating polynomial E(t, t̄) as E(t, t̄) = ∑ (−1)p+q hpq tp t̄q = (35) = ∑ I=[x,y] (−)ρxtρy (tt̄)r S(Cx, t̄ t )S(C∗y , tt̄)BI(t −1, t̄) 620 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS in terms of the combinatorial data of the n + r dimen- sional Gorenstein cone Γ({Δi}) spanned by vectors of the form (ei, v), where ei is a unit vector in Rr and v ∈ Δi. In this formula, x, y label faces Cx of dimen- sion ρx of Γ({∇i}), and C∨x denotes the dual face of the dual cone of Γ({Δi}). The interval I = [x, y] labels all cones that are faces of Cy containing Cx. The Batyrev– Borisov polynomials BI(t, t̄) encode certain combina- torial data of the face lattice [45]. The polynomials S(Cx, t) = (1− t)ρx∑n≥0 t nln(Cx) of degree ρx − 1 are related to the numbers ln(Cx) of lattice points at de- gree n in Cx and, hence, to the Ehrhart polynomial of the Gorenstein polytope generating Cx. (The Goren- stein polytope ΔC consists of the degree 1 points of a Gorenstein cone C. In the hypersurface case, r = 1 and ΔC = Δ = Δ1.) Without going into details, let us emphasize that for- mula (35) contains positive and negative contributions, whose interpretation in terms of individual contributions from toric divisors Dj is, in contrast to the hypersurface formula (33), unfortunately unknown. In additional to efficiency problems in specific calculations (the formula is implemented in the nef-part of PALP [42], but be- comes quite slow for codimension r > 2), this entails important theoretical problems which will be comment below. 3. Fibrations and Torsion in Cohomology Fibration structures play an important role in string the- ory, like, e.g., in heterotic-type II duality [10, 16–18], F-theory [23–25], but also for the construction of vec- tor bundles in heterotic compactifications, where they are often combined with non-trivial fundamental groups [26]. We now discuss how these topological properties manifest themselves in combinatorial properties of the polytopes that define toric Calabi–Yau varieties. 3.1. Torsion in cohomology We begin with a discussion of the fundamental group which is trivial for every compact toric variety [1] but may become non-trivial for hypersurfaces and complete intersections. First, we need to discuss smoothness con- ditions and to focus on the hypersurface case. If we consider the normal fan of a reflexive polytope Δ ⊆MR, then XΣ will generically have singularities which have to be resolved if they have positive dimension, while point- like singularities can be avoided by a generic choice of the hypersurface equation. The resolution can be performed by the choice of a convex (or coherent) triangulation of - χ100 500 900 6h11 + h12 ≤ 502 pp ppppp pppppp ppppppppp pppppppppp pppppppppppp ppppppppppppp ppppppppppppppp pppppppppppppppp ppppppppppppppppp ppppppppppppppppppp pppppppppppppppppppp pppppppppppppppppppppp pppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppp pppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp p ppppppppppppppppppppppppppppppppppppppppppppp ppp pppppppppppppppppppppppppppppppppppppppppppppp ppp pppppppppppppppppppppppppppppppppppppppppppppp ppp pppppppppppppppppppppppppppppppppppppppppppppp ppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp p pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp p pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp pppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp pppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppp pppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp pppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp pppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp pppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppp pppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp ppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppp ppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppp ppppppppppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppp pppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pppppppppppppppp p pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp pppppppppppppppppp p ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppp ppppppppppppppppppp p ppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp pppppppppppppppppppp p ppppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppppp ppppppppppppp pppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppppppppp ppppppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp pppppppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp ppppppppppp p ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppp ppppppppppp p pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppp pppppppppppp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppp pp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppp pppppppppp pp ppppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp pppppppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp ppppppppppp pp ppppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppppp pppppppppp ppppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppp ppppppppp ppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppp pppppppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppppppp pppppppp pppppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp ppppppppppp ppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp ppppppppp ppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp pppppppp pp ppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp ppppppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppp ppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp ppppppppppp pp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp pppppppppppp pp pppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppp pppppppppp p pppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp ppppppppppppp pppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppp ppppppppppp ppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppppppp pppppppp ppppppppppppppppppppppppppppppppppppppp pppppppppppppppppppppppp ppppppppp pppppppppppppppppppppppppppppppppppp ppppppppppppppppppppppp pppppppp ppppppppppppppppppppppppppppppppppp ppppppppppppppppppppp ppppppp ppppppppppppppppppppppppppppppppppp pppppppppppppppppppp ppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppp ppppppppppppppppppp ppppppp pppppppppppppppppppppppppppppppppppp ppppppppppppppppp ppppppp ppppppppppppppppppppppppppppppppppppp ppppppppppppppppp pppppp pppppppppppppppppppppppppppppppppppp pppppppppppppppppppp pppppppppp ppppppppppppppppppppppppppppppppppp pppppppppppppppppppppp pppppp ppppppppppppppppppppppppppppppppppp pppppppppppppppppp ppp ppppppppppppppppppppppppppppppppp pppppppppppppppp ppppp ppppppppppppppppppppppppppppp ppppppppppppppp ppppp pppppppppppppppppppppppppppppppp pppppppppppppppppp pppppp ppppppppppppppppppppppppppppppppp ppppppppppppppppppppp ppppppppppppppppppppppppppppppppppp pppppppppppp ppppp ppppppppppppppppppppppppppppppp ppppppppppppppppp pppp ppppppppppppppppppppppppppppppppp ppppppppppppppp ppp pppppppppppppppppppppppppppp pppppppppppppppp p ppppppppppppppppppppppppppppp pppppppppppppppp pp ppppppppppppppppppppppppp pppppppppppppppppppppp ppppppppppppppppppppppppppp pppppppppppppp ppppp ppppppppppppppppppppppppp ppppppppppppp ppppp pppppppppppppppppppppppppppp pppppppppppppppp pppp pppppppppppppppppppppppppp pppppppppppppppp pppp ppppppppppppppppppppppppppp ppppppppppppppp ppppp pppppppppppppppppppppppppppppp pppppppppppppp pppp pppppppppppppppppppppppppppp ppppppppppppppppp pppp ppppppppppppppppppppppppppppp pppppppppppppppppp ppp pppppppppppppppppppppppppppppp pppppppppppppp ppp ppppppppppppppppppppppppp ppppppppppppp ppp ppppppppppppppppppp ppppppppppppppp pp ppppppppppppppppppppppppppp ppppppppppppppp ppp ppppppppppppppppppppppppp pppppppppppppppp p pppppppppppppppppppppppppppp ppppppppppppp pp pppppppppppppppppppppppppppp pppppppppp ppppppppppppppppppppppppp pppppppppp pppppppppppppppppppppp ppppppppppppp pppppppppppppppppppppppppp ppppppppppppp pp ppppppppppppppppppppppppp ppppppppppp p pppppppppppppppppppppp ppppppppppp pppppppppppppppppppppp pppppp pppppppppppppppppppppp pppppppp pppppppppppppppppppppp pppppppppp pppppppppppppppppppppp ppppppp pp ppppppppppppppppppppp pppppppppppp ppppppppppppppppppppp ppppppp p ppppppppppppppppppppp pppppp p pppppppppppppppppppp pppppppp pp pppppppppppppppppppp pppppppppppp pppppppppppppppp pppppppppppp ppppppppppppp ppppppppppppp pppppppppppppppp ppppppppppp ppppppppppppppp ppppppppp p ppppppppppppppppppp ppppppppppppp ppppppppppppppppppp pppppppppp pppppppppppppppppppp ppppppppppp p ppppppppppppppppp pppppppppp ppppppppppppppppppp ppppppppp pp ppppppppppppppppp pppppppppp p pppppppppppppppp ppppppppppp pppppppppppppp ppppppppp pppppppppppppppp ppppppppp ppppppppppppp ppppppp ppppppppppppppppppp ppppppp ppppppppppppppppp ppppppp ppppppppppppppppp ppppp ppppppppppppppppp pppppppp ppppppppppppppp ppppppp pppppppppppppppp pppppp pppppppppppppppp ppppp pppppppppppppppp ppppppp ppppppppppppppp pppppp ppppppppppppp pppp pp pppppppppppp pppppppp p ppppppppppppppppp p pppppppppppp ppp p pppppppppppp pp pp ppppppppppppp pp ppp ppppppppppp pppp pppppppppp pppppp ppppppppppp pppppppp ppppppppp pppppp ppppppppppppp pppppppp pppppppppppp pppppppp ppppppppppp ppppppp pppppppppppppp pppppppp pppppppppppppp pppppp pppppppppppp ppppp ppppppppppp ppppppp ppppppppppppp pp pppppppp pppp ppppppppppp pppppppppp ppp ppppppppppppp pppppppppppppppp pppppppppp pp ppppppppp pp pppppppppppp pp ppppppppppp ppppppppp ppppppppppp ppppppppp ppppppppp pppppp pppppppp pppppp ppppp ppppppp ppp ppppppp ppppp ppppp pppppp pppppp pppppp pppppp pppppp pppppp pppppp ppp pp pppp pppp ppppp ppppp pp ppp ppp pp pppp pppp pppp ppp ppp pp ppp ppp pp pp pp pp ppppp ppp pppp pppppp p ppp pp pp pp p pp ppp pppp p pp p pp pp p p p p p p Fig. : Hypersurface spectra for . The maximalcomes from (251,251) and (491,11).The enumeration of all re�exive polygons has beenachieved by Batyrev many years ago (see �gure 3).dimensions 3 and 4, which are relevant for K3 surfacesand Calabi�Yau 3-folds, respectively, the enumerationrequired extensive use of computers and was achievin [38, 39] and [40, 41]. The code was later included Fig. 4. Hypersurface spectra for h11≤h12. The maximal h11+h12 comes from (251,251) and (491,11) the fan ΣΔ [10, 29, 46], whose rays should consist of all rays over lattice points of Δ◦ (this amounts to a max- imal star triangulation of Δ◦; rays of lattice points in N \ Δ◦ would contribute to c1 and, hence, destroy the Calabi–Yau condition if the corresponding divisors in- tersect the hypersurface). For K3-surfaces, such a trian- gulation already leads to a smooth toric ambient space, because reflexivity implies that the facets are at distance one from the origin, and every maximal triangulation of a polygon consists of basic simplices. For Calabi– Yau 3-folds, only the codimension-two cones of the tri- angulation are basic, while maximal-dimensional cones may contain point-like singularities. This is still o.k., be- cause point-like singularities can be avoided by a generic hypersurface. Toric 4-fold hypersurfaces, on the other hand, may have terminal singularities that cannot be avoided, so that many 5-dimensional reflexive polytopes cannot be used for the construction of smooth Calabi– Yau hypersurfaces. For complete intersections, the sit- uation is analogous: 3-folds are generically smooth, be- cause the codimension dim Δ− 3 of the Calabi–Yau hy- persurface is larger than the dimension dim Δ− 4 of the singular locus of XΣ. If we now consider a fixed polytope Δ ⊂ R4 without specification of the lattice, then the reflexivity, i.e. the integrality of the vertices of Δ and of Δ◦, implies that N is a sublattice of the dual of the lattice MV generated by the vertices of Δ and that N contains the lattice NV ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 621 M. KREUZER generated by the vertices of Δ◦ NV ⊆ N ⊆M◦V . (36) A refinement of the N lattice amounts to a geometrical quotient by a group action G ⊂ T we call toric, because it acts diagonally on the homogeneous coordinates. Such a refinement always entails additional quotient singular- ities in the ambient space but no contributions to its fundamental group [1]. If, however, a Calabi–Yau hy- persurface does not intersect the singular locus of that quotient, then the group acts freely on that variety and contributes to π1. This is the case if the refinement of the lattice does not lead to additional lattice points of Δ◦ (more precisely, lattice points in the interior of facets can be ignored, because the corresponding divisors do not intersect the hypersurface, according to Eq. (33). For a given pair of reflexive polytopes, there are only a finite number of lattices N that obey (36) and, hence, only a finite number of possible toric free quotients. In [47], we have shown that the fundamental group of a toric Calabi–Yau hypersurface is isomorphic to the lat- tice quotient of the N -lattice divided by its sublattice N (3) generated by the lattice points on 3-faces of Σ. All fundamental groups of toric hypersurfaces thus come from toric quotients and are Abelian, so that π1 is iso- morphic to the torsion in H2. We also found a combi- natorial formula for the Brauer group B, which is the torsion in the third cohomology H3, in terms of the sub- lattice N (2) generated by lattice points on 2-faces of Σ. Here, however, B ×B must be a subgroup of N/N (2). Various dualities imply that the complete torsion in the cohomology groups is determined in terms of π1 and B, and we conjectured, based on some K-theory argu- ments, that these groups are exchanged with the du- als of each other under the mirror duality [47]. This conjecture could be verified for all toric hypersurfaces by explicit calculation. Two well-known examples are the free Z5 quotient of the quintic and the free Z3 quo- tient of the Calabi–Yau hypersurface in P2 × P2. For the complete list of 473 800 776 reflexive polytopes, one finds 14 more examples of toric free quotients [42]: the elliptically fibered Z3 quotient of the degree 9 surface in P4 11133, whose group action on the homogeneous co- ordinates is given by the phases (1, 2, 1, 2, 0)/3, and 13 elliptic K3 fibrations, for which the lattice quotient has index 2. The 16 non-trivial Brauer groups showed up, as expected, exactly for the 16 polytopes that are polar to the ones that lead to a non-trivial fundamental group. For complete intersections, it is possible to have both a non-trivial fundamental group and a non-trivial Brauer group at the same time, and our conjecture was verified for a codimension-2 Calabi–Yau hypersurface, for which both groups are Z3 × Z3 [48]. 3.2. Fibrations For general K3 surfaces and Calabi–Yau 3-folds, there exists a criterion by Oguiso for the existence of ellip- tic and K3 fibrations in terms of intersection numbers [18, 49]. In the toric context, the data of a given reflex- ive polytope Δ◦ ⊆ NR have to be supplemented by a triangulation of the fan, as discussed above, and fibra- tion properties, as well as intersection numbers, depend on the chosen triangulation. Computation of all intersection numbers for all tri- angulations is computationally quite expensive, but, for toric Calabi–Yau spaces, there is, fortunately, a more direct way to search for fibrations that manifest them- selves in the geometry of the polytope and to single out the appropriate triangulations [12–15, 41]. These fibra- tions descend from toric morphisms of the ambient space [3, 4], which correspond to a map φ : Σ → Σb of fans in N and Nb, respectively, where φ : N → Nb is a lattice homomorphism such that, for each cone σ ∈ Σ, there is a cone σb ∈ Σb that contains the image of σ. The lattice Nf for the fiber is the kernel of φ in N . If we are interested in fibrations, whose fibers are Calabi–Yau varieties of lower dimension, then the re- striction of the defining equations to the fan Σf in Nf needs to satisfy Batyrev’s criterion. We hence require that the intersection Δ◦f = Δ◦ ∩ Nf is reflexive, like the intersection with the horizontal plane in the exam- ple of Fig. 5. The search for toric fibrations amounts to a search for reflexive sections of Δ◦ with appropriate dimension (or, equivalently, for reflexive projections in the M -lattice, which was used in a search for K3 fibra- tions in [12]). In order to guarantee the existence of the projection, we choose a triangulation of Δ◦f and then extend it to a triangulation of Δ◦ (this may not always be possible if the codimension is larger that 1, as was pointed out and analyzed by Rohsiepe [15]). For each such choice, we can interpret the homogeneous coordi- nates that correspond to rays in Δ◦f as coordinates of the fiber and the others as parameters of the equations and, hence, as moduli of the fiber space. For hypersurfaces, the geometry of the resulting fibra- tion has been worked out in detail in [14]. Even in the case of complete intersections, the reflexivity of the fiber polytope Δ◦f ensures that the fiber also is a complete intersection of the Calabi–Yau hypersurface, because a nef partition of Δ◦ automatically induces a nef partition of Δ◦f [10]. The codimension rf of the fiber generically 622 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONSthe Brauercohomology ,lattice poi-ust be atorsion interms of-theorywith the[47]. Thisypersurfacesexamples are thequotientcomplete14 moreelliptically �-, whosegiven by l l l l l , , , , , e e e % % % , , , , , l l l l l � � � � � � L L L L L L �� HHHHHH ��Fig. 5: CY �bration from re�exive sectionIf we are interested in �brations whoseCalabi�Yau varieties of lower dimensionction of the de�ning equations to theneeds to satisfy Batyrev's criterion. Wthat the intersection is re�exivintersection with the horizontal plane in�gure 5. The search for toric �brations Fig. 5. Calabi–Yau fibration from a reflexive section of Δ◦ ⊆ NR coincides with the codimension r of the fibered space. But, for r > 1, it may happen that Σ1 f does not intersect one (or more) of the Δi of the nef partition, in which case the codimension decreases [10]. In [10], we per- formed an extensive search for K3-fibrations in complete intersections which, due to modular properties, could be used for all-genus calculations of topological string am- plitudes. In that search, we also encountered an example where the fibration does not extend to a morphism of the ambient spaces, because some exceptional points do not intersect the Calabi–Yau hypersurface. Even in that ex- ample, however, the K3 fiber is realized by a fan on a sublattice. 4. Work in Progress and Open Problems For toric Calabi–Yau hypersurfaces in 3 dimensions, the enumeration and the computation of the integral coho- mology has been completed. But, for the case of com- plete intersections, only the surface has been scratched [10, 50]. While the number of reflexive polytopes in 5 dimensions, which would be relevant for 4-folds as used in F-theory, is simply too large (maybe something like 1018), there is some hope for that a classification of complete intersection 3-folds may be feasible, at least for small codimensions, via an enumeration of reflex- ive Gorenstein cones [51, 52]. On the theoretical side, it would be important to find a better formula for the Hodge data that allows a direct interpretation of the Pi- card number in terms of toric divisors (for codimension r > 1, even divisors that correspond to vertices of Δ◦ may not intersect the Calabi–Yau hypersurface [10]). A related issue is the search for a combinatorial formula for the torsion in cohomology, which would also be very useful for model building. In spite of the fact that the toric construction yields by far the largest class of known Calabi–Yau spaces, it is unclear how generic these spaces are, and it is not even known whether the total number of topological types is finite [53]. A first step beyond the toric realm along the lines of [54] has been taken recently when we studied � � � � � � � � � � � �� @ @ @ @ @ @ @ @ @ @ @ @@ c cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc cc c c cc cc cc cc cc cc cc cc c c cc cc cc c c cc cc c rrrr rr rr rr rrrr rr rr rr rr rrrr rr rr rr rr rr rr rr rrrr rr rr rr rr rr rr rr rr rrrr rr rr rr rr rr rr rr rr rr rr rrrr rr rr rr rr rr rr rr rr rr rr rr rrrr rr rr rr rr rr rr rr rr rr rr rr rr rr rrrr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rrrr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr rr Fig. 6: Deformed conifold Hodge data (circles) and toricCY hypersurfaces (dots) with .Inspite of the fact that the toric construction yieldsby far the largest class of known Calabi�Yau spaces, itis unclear how generic these spaces are and it is noteven known whether the total number of topologicaltypes is �nite [53]. A �rst step beyond the toric realmalong the lines of [54] has been taken recently whenwe studied conifold transition to non-toric Calabi�Yau Fig. 6. Deformed conifold Hodge data (circles) and toric Calabi– Yau hypersurfaces (dots) with h11 + h12 ≤ 46 the conifold transition to non-toric Calabi–Yau spaces [22]. As shown in Fig. 6, this construction, which still uses toric tools, yields a surprisingly rich class of new Calabi–Yau spaces with small Picard number h11. The realm with small h11 + h21, on the other hand, seems to be populated by varieties with non-trivial fundamental group [55]. Systematic studies of free quotients, however, so far have only been performed in special cases. I would like to thank Stepan Moskaliuk for the invita- tion to the Bogolyubov Institute for Theoretical Physics in Kiev, where one part of these notes were presented at the Kummer Symposium “Fundamental Problems in Modern Quantum Theories and Experiments III” and the other part at the Bogolyubov Conference in 2009. It is also a pleasure to acknowledge many vital and enjoy- able discussions with my collaborators Victor Batyrev, Albrecht Klemm, Erwin Riegler, Emanuel Scheidegger, and Harald Skarke. This work was supported in part by the Austrian Research Funds FWF under grant number P18679-N16. 1. V.I. Danilov, Russian Math. Survey 33, 97 (1978). 2. T. Oda, Convex Bodies and Algebraic Geometry (Springer, Berlin, 1988). 3. W. Fulton, Introduction to Toric Varieties (Princeton Univ. Press, Princeton, 1993) 4. G. Ewald, Combinatorial Convexity and Algebraic Geom- etry (Springer, Berlin, 1996). 5. D. Cox, AMS Proc. Symp. Pure Math. 62, 389 (1997) [alg-geom/9606016]; Update on toric geometry, Séminaires et Congrés 6, SMF (2002). 6. D. Cox, Minicourse on Toric Varieties, http://www.amherst.edu/~dacox/. 7. V.V. Batyrev, J. Alg. Geom. 3 493(1994) [arXiv:alg- geom/9310003]. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 623 M. KREUZER 8. D.A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry (American Mathematical Society, Providence, RI, 1999). 9. K. Hori, S. Katz, A. Klemm, R. Pandhirapande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Sym- metry, (AMS, Providence, RI, 2003). 10. A. Klemm, M. Kreuzer, E. Riegler, and E. Scheideg- ger, J. High Energy Phys. 0505 023(2005) [arXiv:hep- th/0410018]. 11. A. Klemm, Topological string theory on Calabi–Yau threefolds, PoS RTN2005, 002 (2005). 12. A. Avram, M. Kreuzer, M. Mandelberg, and H. Skarke, Nucl. Phys. B 494, 567(1997); [arXiv:hep-th/9610154]. 13. M. Kreuzer, H.Skarke, J. Geom. Phys. 466, 1 (1997); [arXiv:hep-th/9701175]. 14. Y. Hu, C.H. Liu, and S.T. Yau, Adv. Theor. Math. Phys. 6, 457 (2003); [arXiv:math.ag/0010082]. 15. F. Rohsiepe, [arXiv:hep-th/0502138]. 16. S. Kachru and C. Vafa, Nucl. Phys. B 450, 69 (1995); [arXiv:hep-th/9505105]. 17. A. Klemm, W. Lerche and P. Mayr, Phys. Lett. B 357, 313 (1995); [arXiv:hep-th/9506112]. 18. P.S. Aspinwall, J. Louis, Phys. Lett. B 369, 233 (1996); [arXiv:hep-th/9510234]. 19. P. Candelas and A. Font, Nucl. Phys. B 511, 295 (1998); [arXiv:hep-th/9603170]. 20. A. Strominger, S.T. Yau, and E. Zaslow, Nucl. Phys. B 479, 243 (1996); [arXiv:hep-th/9606040]. 21. M. Aganagic, C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs [arXiv:hep-th/0012041]. 22. V. Batyrev and M. Kreuzer, work in progress. 23. C.Vafa, Nucl. Phys. B 469, 403 (1996); [arXiv: hep- th/9602022]. 24. D.R. Morrision and C. Vafa, Nucl. Phys. B 473, 74 (1996);[arXiv:hep-th/9602114]; Nucl. Phys. B 476, 437 (1996); [arXiv:hep-th/9603161]. 25. F. Denef, M.R. Douglas, B. Florea, A. Grassi, and S. Kachru, Fixing all moduli in a simple F-theory compact- ification, [arXiv:hep-th/0503124]. 26. V.Braun, Y.H. He, B.A.Ovrut, and T. Pantev, Vector bundle extensions, sheaf cohomology, and the heterotic standard model, [arXiv:hep-th/0505041]; Heterotic stan- dard model moduli, [arXiv:hep-th/0509051]. 27. D.A. Cox, J. Alg. Geom 4, 17 (1995); [arXiv:alg- geom/9210008]. 28. E. Witten, Nucl. Phys. B 403, 159 (1993); [arXiv:hep- th/9301042]. 29. I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, Dis- criminants, Resultants, and Multidimensional Determi- nants (Birkhäuser, Boston, 1994). 30. P. Candelas, G.T. Horowitz, A. Strominger, and E. Wit- ten, Nucl. Phys. B 258, 46 (1985). 31. A. Strominger, Nucl. Phys. B 274, 253 (1986). 32. S. Kachru, M.B. Schulz, and S. Trivedi, JHEP 0310, 007 (2003); [arXiv:hep-th/0201028]. 33. N. Hitchin, Quart. J. Math. Oxford Ser. 54, 281 (2003); [arXiv:math.DG/0209099]. 34. M. Gualtieri, Generalized complex geometry, Oxford Uni- versity PhD thesis 107,[arXiv:math.DG/0401221]. 35. M. Grana, Flux compactifications in string theory: A comprehensive review, [arXiv:hep-th/0509003]. 36. W. Lerche, C. Vafa, and N. Warner, Nucl. Phys. B 324, 427 (1989). 37. P. Candelas, X.C. De la Ossa, P.S. Green, and L. Parkes, Phys. Lett. B 258, 118 (1991). 38. M. Kreuzer and H. Skarke, Commun. Math. Phys. 185, 495 (1997); [arXiv:hep-th/9512204]. 39. M. Kreuzer and H. Skarke, Adv. Theor. Math. Phys. 2, 847 (1998); [arXiv:hep-th/9805190]. 40. M. Kreuzer and H. Skarke, Adv. Theor. Math. Phys. 4, 6 (2000); [arXiv:hep-th/0002240]. 41. M. Kreuzer and H. Skarke, Rev. Math. Phys. 14, 343 (2002); [arXiv:math.AG/0001106]. 42. M. Kreuzer, H. Skarke, Comput. Phys. Comm. 157, 87 (2004); [arXiv:math.SC/0204356]. 43. http://hep.itp.tuwien.ac.at/~kreuzer/CY.html 44. V.V. Batyrev and L.A. Borisov, in Higher-Dimensional Complex Varieties (de Gruyter, Berlin, 1996), p. 39; [arXiv:alg-geom/9412017]. 45. V.V. Batyrev and L.A. Borisov, Invent. Math. 126, 183 (1996); [arXiv:alg-geom/9509009]. 46. L.J. Billera, P. Filliman, and B.Sturmfels, Adv. in Math. 83, 155 (1990). 47. V. Batyrev and M. Kreuzer, in Mirror Symmetry V, Proceedings of BIRS workshop on Calabi–Yau Va- rieties and Mirror Symmetry, December 6-11, 2003 [arXiv:math.AG/0505432]. 48. V. Braun, private communication. 49. K. Oguiso, Int. J. of Math. 4, 439 (1993). 50. M. Kreuzer, E. Riegler, and D. Sahakyan, J. Geom. Phys. 46, 159–173 (2003); [arXiv:math.AG/0103214]. 51. V.V. Batyrev, L.A. Borisov, in Mirror Symmetry II, edited by B. Greene, S.T. Yau, Adv. Math. 1, 71 (1997); [arXiv:alg-geom/9402002]. 624 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 TORIC GEOMETRY AND CALABI–YAU COMPACTIFICATIONS 52. M. Kreuzer, work in progress. 53. M. Gross, A finiteness theorem for elliptic Calabi–Yau threefolds [arXiv:alg-geom/9305002]. 54. V.V. Batyrev, I. Ciocan-Fontanine, B. Kim, and D. van Straten, Nucl. Phys. B 514, 640 (1998); [alg- geom/9710022]. 55. M. Gross and S. Pavanelli, A Calabi–Yau threefold with Brauer group (Z/8Z)2 [arXiv:math.AG/0512182]. Received 28.05.09 ТОРIЧНА ГЕОМЕТРIЯ I КАЛАБI–ЯУ КОМПАКТИФIКАЦIЇ М. Кройцер Р е з ю м е Цi нотатки мiстять короткий вступ до побудови торiчних Калабi–Яу гiперповерхней та повних перетинiв з акцентом на розрахунках, що стосуються дуальностi струн. Останнi два роздiли можуть бути прочитанi незалежно вiд iнших i при- свяченi недавнiм результатам та роботам, якi ще не закiнчено, включаючи кручення в когомологiї, питання класифiкацiї та топологiчних переходiв. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 625

Toric Geometry and Calabi–Yau Compactifications

Institution

Ähnliche Einträge