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

Abstract

We 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 nonperturbative 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.

Poly-instanton inflation

“…The new ingredient is the inclusion of poly-instanton corrections to the superpotential [24] for Calabi-Yau manifolds with a K3 or T 4 fibre over a P 1 base [25,26]. The main difference with the model of [14] is the topological nature of the inflaton field.…”

“…These manifolds are characterised by the fact that their volume is linear in the two-cycle giving the volume of the P 1 base. Explicit examples of this kind of Calabi-Yau three-folds with additional del Pezzo divisors have been analysed in [26] using the powerful tools of toric geometry. Here we shall just focus on the simplest of such manifolds whose volume takes the form: 12) where t 1 is the volume of the P 1 base, τ 1 = λ 1 t 2 2 is the size of the K3 or T 4 fibre, and τ 3 = 3λ 2 t 2 3 controls the volume of a blow-up mode (the other four-cycle volume is given by τ 2 = 2λ 1 t 1 t 2 ).…”

“…However, as argued in [25], the poly-instantons might get generated either for a fluxed instanton on a K3 four-cycle or for an instanton on a T 4 divisor with deformations fixed by the background fluxes. We shall therefore focus on Calabi-Yau three-folds which admit a K3 or T 4 fibration over a P 1 base similarly to the toric constructions of [26].…”

Explicit de Sitter flux vacua for global string models with chiral matter

“…• Using toric geometry, we constructed explicit compact Calabi-Yau (CY) orientifold compactifications [19,20].…”

Anisotropic modulus stabilisation: strings at LHC scales with micron-sized extra dimensions

“…For explicit Calabi-Yau constructions via toric geometry which exhibit this form of the overall volume see [32]. The volumes, τ i , of the 4-cycles dual to these 2-cycles are defined by τ i = ∂V/∂t i , and so…”