Toric Divisors Research Articles

Geometric engineering in toric F-theory and GUTs with U(1) gauge factors

An algorithm to systematically construct all Calabi-Yau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. The allowed choices of compactification data are integral points in an auxiliary polytope. In order to ensure the existence of a low-energy gauge theory, the elliptic fibration must be flat, which is reformulated into conditions on the top and its embedding. In particular, flatness of SU(5) fourfolds imposes additional linear constraints on the auxiliary polytope of compactifications, and is therefore non-generic. Abelian gauge symmetries arising in toric F-theory compactifications are studied systematically. Associated to each top, the toric Mordell-Weil group determining the minimal number of U(1) factors is computed. Furthermore, all SU(5)-tops and their splitting types are determined and used to infer the pattern of U(1) matter charges.

Wall-crossing, Toric divisor and Seiberg duality

We study the wall-crossing phenomena of BPS D4-D2-D0 states on the conifold and orbifold $ {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_2}}} \right.} $ , from the viewpoint of the quiver quantum mechanics on the D-branes. The Kahler moduli dependence of the BPS index is translated into the FI parameter dependence of the Witten index. The wall-crossing phenomena are related to the Seiberg dualities of the quiver quantum mechanics. All the differences from the D6-D2-D0 case arise from the additional superpotential and “anti-quark” induced by the D4-brane. When the D-branes are on the conifold, the flop transition changes the duality cascade. When the D-branes are on the orbifold $ {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_2}}} \right.} $ , the generating function of the Witten index is always given by a character of the affine SU(2) algebra. Both are consistent with the wall-crossing formula for BPS indices.

Toric K3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications

AbstractWe analyse several explicit toric examples of compact K3-fibred Calabi-Yau three-folds. These manifolds can be used for the study of string dualities and are crucial ingredients for the construction of LARGE Volume type IIB vacua with promising applications to cosmology and particle phenomenology. In order to build a phenomenologically viable model, on top of the two moduli corresponding to the base and the K3 fibre, we demand also the existence of two additional rigid divisors: the first supporting the non-perturbative effects needed to achieve moduli stabilisation, and the second allowing the presence of chiral matter on wrapped D-branes. We clarify the topology of these rigid divisors by discussing the interplay between a diagonal structure of the Calabi-Yau volume and D-terms. Del Pezzo divisors appearing in the volume form in a completely diagonal way are natural candidates for supporting non-perturbative effects and for quiver constructions, while ‘non-diagonal’ del Pezzo and rigid but not del Pezzo divisors are particularly interesting for model building in the geometric regime. Searching through the existing list of four dimensional reflexive lattice polytopes, we find 158 examples admitting a Calabi-Yau hypersurface with a K3 fibration and four Kähler moduli where at least one of the toric divisors is a ‘diagonal’ del Pezzo. We work out explicitly the topological details of a few examples showing how, in the case of simplicial polytopes, all the del Pezzo divisors are ‘diagonal’, while ‘non-diagonal’ ones appear only in the case of non-simplicial polytopes. A companion paper will use these results in the study of moduli stabilisation for globally consistent explicit Calabi-Yau compactifications with the local presence of chirality.

Fermionic screenings and line bundle twisted chiral de Rham complex on CY manifolds

We present a generalization of Borisov’s construction of the chiral de Rham complex in the case of the line-bundle-twisted chiral de Rham complex on a Calabi-Yau hypersurface in a projective space. We generalize the differential associated with a polytope Δ of the projective space ℙ d − 1 by allowing nonzero modes for the screening currents forming this differential. It is shown that the numbers of screening current modes define the support function of the toric divisor of a line bundle on ℙ d − 1 that twists the chiral de Rham complex on the Calabi-Yau hypersurface.

An invitation to toric degenerations

In [GrSi2] we gave a canonical construction of degenerating families of varieties with effective anticanonical bundle. The central fibre X of such a degeneration is a union of toric varieties, glued pairwise torically along toric prime divisors. In particular, the notion of toric strata makes sense on the central fiber. A somewhat complementary feature of our degeneration is their toroidal nature near the 0-dimensional toric strata of X; near these points the degeneration is locally analytically or in the etale topology given by a monomial on an affine toric variety. Thus in this local model the central fiber is a reduced toric divisor. A degeneration with these two properties is called a toric degeneration. The name is probably not well-chosen as it suggests a global toric nature, which is not the case as we will emphasize below. A good example to think of is a degeneration of a quartic surface in P3 to the union of the coordinate hyperplanes. More generally, any Calabi-Yau complete intersection in a toric variety has toric degenerations [Gr2]. Thus the notion of toric degeneration is a very versatile one, conjecturally giving all deformation classes of Calabi-Yau varieties with maximally unipotent boundary points. Our construction has a number of remarkable features. It generalizes the construction of a polarized toric variety from an integral polyhedron (momentum polyhedron)

The tropical vertex

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.

Asymptotic cohomological functions of toric divisors

We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartier divisor as a linear combination of the volumes of the bounded regions in the corresponding hyperplane arrangement and prove an asymptotic converse to Serre vanishing.

Toric varieties whose blow-up at a point is Fano

We classify smooth toric Fano varieties of dimension $n\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\geq 3$ with a fixed point $x\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved using toric Mori theory due to Reid.

The signature of a toric variety

We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold. We prove positive (or nonnegative) lower bounds for this quantity under geometric hypotheses on the polytope and, in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch signature formula. Moreover, we show that under these hypotheses on the polytope, the ith L-class of the corresponding toric variety is (−1)i times an effective class for any i.

The signature of a toric variety

We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold. We prove positive (or nonnegative) lower bounds for this quantity under geometric hypotheses on the polytope and, in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch signature formula. Moreover, we show that under these hypotheses on the polytope, the ith L-class of the corresponding toric variety is (−1)i times an effective class for any i.

Gromov-Witten invariants of a class of toric varieties.

1.1. Background. Toric varieties admit a combinatorial description, which allows many invariants to be expressed in terms of combinatorial data. Batyrev [Ba2] and Morrison and Plesser [MP] describe the quantum cohomology rings of certain toric varieties, in terms of generators (divisors and formal q variables) and relations (linear relations and q-deformed monomial relations). The relations are easily obtained from the combinatorial data. Unfortunately, the relations alone do not tell us how to multiply cohomology classes in the quantum cohomology ring QH∗(X), or even how to express ordinary cohomology classes in H∗(X,Q) in terms of the given generators. In this paper, we give a formula that expresses any class inH∗(X,Q)—as a polynomial in divisor classes and formal q variables—for anyX belonging to a certain class of toric varieties. These expressions, along with the presentation of QH∗(X) via generators and relations, permit computation of any product of cohomology classes in QH∗(X). Let X be a complete toric variety of dimension n over the complex numbers (all varieties in this paper are over the complex numbers). This means X is a normal variety with an action by the algebraic torus (C∗)n and a dense equivariant embedding (C∗)n → X. By the theory of toric varieties (cf. [F]), such X are characterized by a fan ∆ of strongly convex polyhedral cones in N ⊗Z R, where N is the lattice Z. The cones are rational, that is, generated by lattice points. In particular, to every ray (1-dimensional cone) σ there is a unique generator ρ ∈ N such that σ ∩N = Z>0 · ρ. There is a one-to-one correspondence between such ray generators ρ and toric (i.e., torus-invariant) divisors of X. Given toric divisors D1, . . . , Dk, with corresponding ray generators ρ1, . . . , ρk, we have D1 ∩ · · · ∩Dk 6= ∅ if and only if ρ1, . . ., ρk span a cone in ∆. Hypotheses on X translate as follows into conditions on ∆: (i) X is nonsingular if and only if every cone is generated by a part of a Z-basis of N ; (ii) given that X is nonsingular: X is Fano (i.e., X has ample anticanonical class) if and only if the set of ray generators is strictly convex.

Mirror symmetry and the web of Landau-Ginzburg string vacua

We present some mathematical aspects of Landau-Ginzburg string vacua in terms of toric geometry. The one-to-one correspondence between toric divisors and some of the (−1, 1) states in the Landau-Ginzburg model is presented for superpotentials of typical types. The Landau-Ginzburg interpretation of non-toric divisors is also presented. Using this interpretation, we propose a method to solve the so-called “twisted sector problem” by orbifold construction. Moreover, this construction shows that the moduli spaces of the original Landau-Ginzburg string vacua and their orbifolds are connected. By considering the mirror map of the Landau-Ginzburg models, we obtain the relation between Mori vectors and the twist operators of our orbifoldization. This consideration enables us to argue the embedding of the Seiberg-Witten curve in the defining equation of the Calabi-Yau manifolds on which the type II string gets compactified. Related topics concerning the Calabi-Yau fourfolds and the extremal transition are discussed.

Enhanced gauge symmetry in type II and F-theory compactifications: Dynkin diagrams from polyhedra

We explain the observation by Candelas and Font that the Dynkin diagrams of non-abelian gauge groups occurring in type IIA and F-theory can be read off from the polyhedron Δ ∗ that provides the toric description of the Calabi-Yau manifold used for compactification. We show how the intersection pattern of toric divisors corresponding to the degeneration of elliptic fibers follows the ADE classification of singularities and the Kodaira classification of degenerations. We treat in detail the cases of elliptic K3 surfaces and K3 fibered threefolds where the fiber is again elliptic. We also explain how even the occurrence of monodromy and non-simply laced groups in the latter case is visible in the toric picture. These methods also work in the fourfold case.