A Note on Computations of D-brane Superpotential

We develop some computational methods for the integrals over the 3-chains on the compact Calabi-Yau 3-folds that plays a prominent role in the analysis of the topological B-model in the context of the open mirror symmetry. We discuss such 3-chain integrals in two approaches. In the first approach, we provide a systematic algorithm to obtain the inhomogeneous Picard-Fuchs equations. In the second approach, we discuss the analytic continuation of the period integral to compute the 3-chain integral directly. The latter direct integration method is applicable for both on-shell and off-shell formalisms.Comment: 61 pages, 5 figures; v2: typos corrected, minor changes, references adde

Five-Brane Superpotentials, Blow-Up Geometries and SU(3) Structure Manifolds

We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading N=1 scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane by background fluxes. This translates the original brane-bulk system into a unifying geometrical formulation. Using this blow-up geometry we compute the complete set of open-closed Picard-Fuchs differential equations and identify the brane superpotential at special points in the field space for five-branes in toric Calabi-Yau hypersurfaces. This has an interpretation in open mirror symmetry and enables us to list compact disk instanton invariants. As a first step towards promoting the blow-up geometry to a supersymmetric heterotic background we propose a non-Kaehler SU(3) structure and an identification of the three-form flux.Comment: 95 pages, 4 figures; v2: Minor corrections, references update

Yukawa hierarchies at the point of E8E_8 in F-theory

We analyse the structure of Yukawa couplings in local SU(5) F-theory models with $E_8$ enhancement. In this setting the $E_8$ symmetry is broken down to SU(5) by a 7-brane configuration described by T-branes, all the Yukawa couplings are generated in the vicinity of a point and only one family of quarks and leptons is massive at tree-level. The other two families obtain their masses when non-perturbative effects are taken into account, being hierarchically lighter than the third family. However, and contrary to previous results, we find that this hierarchy of fermion masses is not always appropriate to reproduce measured data. We find instead that different T-brane configurations breaking $E_8$ to SU(5) give rise to distinct hierarchical patterns for the holomorphic Yukawa couplings. Only some of these patterns allow to fit the observed fermion masses with reasonable local model parameter values, adding further constraints to the construction of F-theory GUTs. We consider an $E_8$ model where such appropriate hierarchy is realised and compute its physical Yukawas, showing that realistic charged fermions masses can indeed be obtained in this case.Comment: 46 pages + appendices, 5 figures. v2, added references and typos corrected, version accepted on JHEP. v3, typos correcte

Explicit de Sitter Flux Vacua for Global String Models with Chiral Matter

We address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kaehler 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. 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 Kaehler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative alpha'-corrections as in the LARGE Volume Scenario. In the considered example the visible sector lives at a dP_6 singularity which can be higgsed to the phenomenologically interesting class of models at the dP_3 singularity.Comment: 49 pages, 5 figures; v2: references adde

Fluxes and Warping for Gauge Couplings in F-theory

We compute flux-dependent corrections in the four-dimensional F-theory effective action using the M-theory dual description. In M-theory the 7-brane fluxes are encoded by four-form flux and modify the background geometry and Kaluza-Klein reduction ansatz. In particular, the flux sources a warp factor which also depends on the torus directions of the compactification fourfold. This dependence is crucial in the derivation of the four-dimensional action, although the torus fiber is auxiliary in F-theory. In M-theory the 7-branes are described by an infinite array of Taub-NUT spaces. We use the explicit metric on this geometry to derive the locally corrected warp factor and M-theory three-from as closed expressions. We focus on contributions to the 7-brane gauge coupling function from this M-theory back-reaction and show that terms quadratic in the internal seven-brane flux are induced. The real part of the gauge coupling function is modified by the M-theory warp factor while the imaginary part is corrected due to a modified M-theory three-form potential. The obtained contributions match the known weak string coupling result, but also yield additional terms suppressed at weak coupling. This shows that the completion of the M-theory reduction opens the way to compute various corrections in a genuine F-theory setting away from the weak string coupling limit.Comment: 46 page

Anomaly Cancelation in Field Theory and F-theory on a Circle

We study the manifestation of local gauge anomalies of four- and six-dimensional field theories in the lower-dimensional Kaluza-Klein theory obtained after circle compactification. We identify a convenient set of transformations acting on the whole tower of massless and massive states and investigate their action on the low-energy effective theories in the Coulomb branch. The maps employ higher-dimensional large gauge transformations and precisely yield the anomaly cancelation conditions when acting on the one-loop induced Chern-Simons terms in the three- and five-dimensional effective theory. The arising symmetries are argued to play a key role in the study of the M-theory to F-theory limit on Calabi-Yau manifolds. For example, using the fact that all fully resolved F-theory geometries inducing multiple Abelian gauge groups or non-Abelian groups admit a certain set of symmetries, we are able to generally show the cancelation of pure Abelian or pure non-Abelian anomalies in these models.Comment: 48 pages, 2 figures; v2: typos corrected, comments on circle fluxes adde

Computing Brane and Flux Superpotentials in F-theory Compactifications

In four-dimensional F-theory compactifications with N=1 supersymmetry the fields describing the dynamics of space-time filling 7-branes are part of the complex structure moduli space of the internal Calabi-Yau fourfold. We explicitly compute the flux superpotential in F-theory depending on all complex structure moduli, including the 7-brane deformations and the field corresponding to the dilaton-axion. Since fluxes on the 7-branes induce 5-brane charge, a local limit allows to effectively match the F-theory results to a D5-brane in a non-compact Calabi-Yau threefold with threeform fluxes. We analyze the classical and instanton contributions to the F-theory superpotential using mirror symmetry for Calabi-Yau fourfolds. The F-theory compactifications under consideration also admit heterotic dual descriptions and we discuss the identification of the moduli in this non-perturbative duality.Comment: 75 pages, 1 figure; typos corrected, references adde