Christoph Mayrhofer

G(4)-flux, chiral matter and singularity resolution in F-theory compactifications

We construct a set of chirality inducing G(4)-fluxes in global F-theory compactifications on Calabi-Yau four-folds. Special emphasis is put on models with gauge group SU(5) x U(1)(X) relevant in the context of F-theory GUT model building, which are described in terms of a U(1)-restricted Tate model. A G(4)-flux arises in a manner completely analogous to the U(1)x gauge potential. We describe in detail the resolution by blow-up of the various singularities responsible for the U(1)(X) factor and the standard SU(5) gauge group and match the result with techniques applied in the context of tonic geometry. This provides an explicit identification of the structure of the resolved fibre over the matter curves and over the enhancement points relevant for Yukawa couplings. We compute the flux-induced chiral index both of SU(5) charged matter and of SU(5) singlets charged only under U(1)(X) localised on curves which are not contained in the SU(5) locus. We furthermore discuss global consistency conditions such as D3-tadpole cancellation, D-term supersymmetry and Freed-Witten quantisation. The U(1)(X) gauge flux is a global extension of a class of split spectral cover bundles. It constitutes an essential ingredient in the construction of globally defined F-theory compactifications with chiral matter. We exemplify this in a three-generation SU(5) x U(1)(X) model whose flux satisfies all of the above global consistency conditions. We also extend our results to chiral fluxes in models without U(I) restriction

U(1) symmetries in F-theory GUTs with multiple sections

A bstractWe present a systematic construction of F-theory compactifications with Abelian gauge symmetries in addition to a non-Abelian gauge group G. The formalism is generally applicable to models in global Tate form but we focus on the phenomenologically interesting case of G = SU(5). The Abelian gauge factors arise due to extra global sections resulting from a specific factorisation of the Tate polynomial which describes the elliptic fibration. These constructions, which accommodate up to four different U(1) factors, are worked out in detail for the two possible embeddings of a single U(1) factor into E8, usually denoted SU(5) × U(1)X and SU(5) × U(1)PQ. The resolved models can be understood either patchwise via a small resolution or in terms of a

D-branes at del Pezzo singularities: global embedding and moduli stabilisation

{{\mathbb{P}}_{1,1,2 }}

SU(5) tops with multiple U(1)s in f-theory

[4] description of the elliptic fibration. We derive the U(1) charges of the fields from the geometry, construct the U(1) gauge fluxes and exemplify the structure of the Yukawa interaction points. A particularly interesting result is that the global SU(5) × U(1)PQ model exhibits extra SU(5)-singlet states which are incompatible with a single global decomposition of the 248 of E8. The states in turn lead to new Yukawa type couplings which have not been considered in local model building.

Moduli stabilisation for chiral global models

A bstractIn the context of type IIB string theory we combine moduli stabilisation and model building on branes at del Pezzo singularities in a fully consistent global compactification. By means of toric geometry, we classify all the Calabi-Yau manifolds with 3 < h1,1 < 6 which admit two identical del Pezzo singularities mapped into each other under the orientifold involution. This effective singularity hosts the visible sector containing the Standard Model while the Kähler moduli are stabilised via a combination of D-terms, perturbative and non-perturbative effects supported on hidden sectors. We present concrete models where the visible sector, containing the Standard Model, gauge and matter content, is built via fractional D3-branes at del Pezzo singularities and all the Kähler moduli are fixed providing an explicit realisation of both KKLT and LARGE volume scenarios, the latter with D-term uplifting to de Sitter minima. We perform the consistency checks for global embedding such as tadpole, K-theory charges and Freed-Witten anomaly cancellation. We briefly discuss phenomenological and cosmological implications of our models.

Gauge Fluxes in F-theory and Type IIB Orientifolds

Abstract We study F-theory compactifications with up to two Abelian gauge group factors that are based on elliptically fibred Calabi–Yau 4-folds describable as generic hypersurfaces. Special emphasis is put on elliptic fibrations based on generic Bl 2 P 2 [ 3 ] -fibrations. These exhibit a Mordell–Weil group of rank two corresponding to two extra rational sections which give rise to two Abelian gauge group factors. We show that an alternative description of the same geometry as a complete intersection makes the existence of a holomorphic zero-section manifest, on the basis of which we compute the U ( 1 ) generators and a class of gauge fluxes. We analyse the fibre degenerations responsible for the appearance of localised charged matter states, whose charges, interactions and chiral index we compute geometrically. We implement an additional SU ( 5 ) gauge group by constructing the four inequivalent toric tops giving rise to SU ( 5 ) × U ( 1 ) × U ( 1 ) gauge symmetry and analyse the matter content. We demonstrate that notorious non-flat points can be avoided in well-defined Calabi–Yau 4-folds. These methods are applied to the remaining possible hypersurface fibrations with one generic Abelian gauge factor. We analyse the local limit of our SU ( 5 ) × U ( 1 ) × U ( 1 ) models and show that one of our models is not embeddable into E 8 due to recombination of matter curves that cannot be described as a Higgsing of E 8 . We argue that such recombination forms a general mechanism that opens up new model building possibilities in F-theory.

Toric construction of global F-theory GUTs

A bstractWe combine moduli stabilisation and (chiral) model building in a fully con-sistent global set-up in Type IIB/F-theory. We consider compactifications on Calabi-Yau orientifolds which admit an explicit description in terms of toric geometry. We build globally consistent compactifications with tadpole and Freed-Witten anomaly cancellation by choosing appropriate brane set-ups and world-volume fluxes which also give rise to SU(5) or MSSM-like chiral models. We fix all the Kähler moduli within the Kähler cone and the regime of validity of the 4D effective field theory. This is achieved in a way compatible with the local presence of chirality. The hidden sector generating the non-perturbative effects is placed on a del Pezzo divisor that does not have any chiral intersection with any other brane. In general, the vanishing D-term condition implies the shrinking of the rigid divisor supporting the visible sector. However, we avoid this problem by generating r < n D-term conditions on a set of n intersecting divisors. The remaining (n − r) flat directions are fixed by perturbative corrections to the Kähler potential. We illustrate our general claims in an explicit example. We consider a K3-fibred Calabi-Yau with four Kähler moduli, that is a hypersurface in a toric ambient space and admits a ‘simple’ F-theory up-lift. We present explicit choices of brane set-ups and fluxes which lead to three different phenomenological scenarios: the first with GUT-scale strings and TeV-scale SUSY by fine-tuning the background fluxes; the second with an exponentially large value of the volume and TeV-scale SUSY without fine-tuning the background fluxes; and the third with a very anisotropic configuration that leads to TeV-scale strings and two micron-sized extra dimensions. The K3 fibration structure of the Calabi-Yau three-fold is also particularly suitable for cosmological purposes.

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

A bstractWe provide a detailed correspondence between G4 gauge fluxes in F-theory compactifications with SU(n) and SU(n) × U(1) gauge symmetry and their Type IIB orientifold limit. Based on the resolution of the relevant F-theory Tate models, we classify the factorisable G4-fluxes and match them with the set of universal D5-tadpole free U(1)-fluxes in Type IIB. Where available, the global version of the universal spectral cover flux corresponds to Type IIB gauge flux associated with a massive diagonal U(1). In U(1)-restricted Tate models extra massless abelian fluxes exist which are associated with specific linear combinations of Type IIB fluxes. Key to a quantitative match between F-theory and Type IIB is a proper treatment of the conifold singularity encountered in the Sen limit of generic F-theory models. We also shed further light on the brane recombination process relating generic and U(1)-restricted Tate models.

Toric K3-Fibred Calabi-Yau Manifolds with del Pezzo Divisors for String Compactifications

We systematically construct a large number of compact Calabi-Yau fourfolds which are suitable for F-theory model building. These elliptically fibered Calabi-Yaus are complete intersections of two hypersurfaces in a six dimensional ambient space. We first construct three-dimensional base manifolds that are hypersurfaces in a toric ambient space. We search for divisors which can support an F-theory GUT. The fourfolds are obtained as elliptic fibrations over these base manifolds. We find that elementary conditions which are motivated by F-theory GUTs lead to strong constraints on the geometry, which significantly reduce the number of suitable models. The complete database of models is available at [1]. We work out several examples in more detail.

D3/D7 branes at singularities: Constraints from global embedding and moduli stabilisation

A bstractWe address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kähler moduli) in fluxed type IIB Calabi-Yau compactifications with chiral matter. Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content or some close extensions. In order to control complex structure moduli stabilisation we consider Calabi-Yau manifolds which exhibit a discrete symmetry that reduces the effective number of complex structure moduli. We calculate the corresponding periods in the symplectic basis of invariant three-cycles and find explicit flux vacua for concrete examples. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kähler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative α′ corrections as in the LARGE Volume Scenario. In the considered example the visible sector lives at a dP6 singularity which can be higgsed to the phenomenologically interesting class of models at the dP3 singularity.