Narrow Results

After a brief review of string and M-theory we point out some deficiencies. Partly to rectify them, we present several arguments for "F-theory", enlarging spacetime to (2, 10) signature, following the original suggestion of C. Vafa. We introduce a suggestive supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space E₆/Spin^c(10). Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that F-theory is yet just a very provisional attempt, lacking clear dynamical principles.

According to the open-closed mirror symmetry and generalized Gel'fand–Kapranov–Zelerinski (GKZ) system, we calculate off-shell D-brane/F-theory effective superpotentials of four compact Calabi–Yau (CY) manifolds by integrating special periods of subsystem, and extracting the open Ooguri–Vafa invariants from the expansion of off-shell D-brane superpotentials.

We calculate the D-brane superpotentials for two Calabi–Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials 𝒲_GVW of the dual F-theory compactification on the relevant Calabi–Yau fourfolds in the weak decoupling limit. We also compute the Ooguri–Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov–Witten invariants.

In this paper, by generalizing the GKZ system for Picard–Fuchs equations under the toric description of compact complete intersection Calabi–Yau (CICY) construction, we extend the calculation of effective superpotentials to the case of compact CICY. As seen in several one and two bulk deformation moduli models, this generalized method is effective to calculate both on-shell and off-shell superpotentials. Furthermore, through the mirror symmetry technique, the Ooguri–Vafa invariants are extracted from the A-model instanton expansion. It is also noteable that the chosen discrete symmetrical groups, ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_3$, between the holomorphic curves wrapped by D-branes, play important roles in superpotential calculations.

Diaconescu, Moore and Witten proved that the partition function of type IIA string theory coincides (to the extent checked) with the partition function of M-theory. One of us (Kriz) and Sati proposed in a previous paper a refinement of the IIA partition function using elliptic cohomology and conjectured that it coincides with a partition function coming from F-theory. In this paper, we define the geometric term of the F-theoretical effective action on type IIA compactifications. In the special case when the first Pontrjagin class of space–time vanishes, we also prove a version of the Kriz–Sati conjecture by extending the arguments of Diaconescu–Moore–Witten. We also briefly discuss why even this special case allows interesting examples.

We evolve our Multiverse Blueprints to characterize our local neighborhood of the String Landscape and the Multiverse of plausible string, M- and F-theory vacua. Building upon the tripodal foundations of (i) the Flipped SU(5) Grand Unified Theory (GUT), (ii) extra TeV-Scale vector-like multiplets derived out of F-theory, and (iii) the dynamics of No-Scale supergravity, together dubbed No-Scale , we demonstrate the existence of a continuous family of solutions which might adeptly describe the dynamics of distinctive universes. This Multiverse landscape of solutions, which we shall refer to as the -Landscape, accommodates a subset of universes compatible with the presently known experimental uncertainties of our own universe. We show that by secondarily minimizing the minimum of the scalar Higgs potential of each solution within the -Landscape, a continuous hypervolume of distinct minimum minimorum can be engineered which comprise a regional dominion of universes, with our own universe cast as the bellwether. We conjecture that an experimental signal at the LHC of the No-Scale framework's applicability to our own universe might sensibly be extrapolated as corroborating evidence for the role of string, M- and F-theory as a master theory of the Multiverse, with No-Scale supergravity as a crucial and pervasive reinforcing structure.

We review particle physics model building in type IIB string theory and F-theory. This is a region in the landscape where in principle many of the key ingredients required for a realistic model of particle physics can be combined successfully. We begin by reviewing moduli stabilization within this framework and its implications for supersymmetry breaking. We then review model building tools and developments in the weakly coupled type IIB limit, for both local D3-branes at singularities and global models of intersecting D7-branes. Much of recent model building work has been in the strongly coupled regime of F-theory due to the presence of exceptional symmetries which allow for the construction of phenomenologically appealing Grand Unified Theories. We review both local and global F-theory model building starting from the fundamental concepts and tools regarding how the gauge group, matter sector and operators arise, and ranging to detailed phenomenological properties explored in the literature.

In the context of F-theory, we study the related eight-dimensional super-Yang–Mills theory and reveal the underlying supersymmetric quantum mechanics algebra that the fermionic fields localized on the corresponding defect theory are related to. Particularly, the localized fermionic fields constitute a graded vector space, and in turn this graded space enriches the geometric structures that can be built on the initial eight-dimensional space. We construct the implied composite fiber bundles, which include the graded affine vector space and demonstrate that the composite sections of this fiber bundle are in one-to-one correspondence to the sections of the square root of the canonical bundle corresponding to the submanifold on which the zero modes are localized.

We study the cosmological inflation within a recently proposed framework of perturbative moduli stabilization in type IIB/F-theory compactifications on Calabi–Yau threefolds. The stabilization mechanism utilizes three stacks of magnetized 7-branes and relies on perturbative corrections to the Kähler potential that grow logarithmically in the transverse sizes of co-dimension two due to local tadpoles of closed string states in the bulk. The inflaton is the Kähler modulus associated with the internal compactification volume that starts rolling down the scalar potential from an initial condition around its maximum. Although the parameter space allows moduli stabilization in de Sitter space, the resulting number of e-foldings is too low. An extra uplifting source of the vacuum energy is then required to achieve phenomenologically viable inflation and a positive (although tiny) vacuum energy at the minimum. We discuss a class of uplifting potentials arising from strongly coupled matter fields. In a particular case, they reproduce the effect of the new Fayet–Iliopoulos term recently discussed in a supergravity context, that can be written for a non-R-symmetry U(1) and is gauge invariant at the Lagrangian level.

We observe a new puzzling physical phenomenon in F-theory on the multisection geometry, wherein a model without a gauge group transitions to another model with a discrete ${\mathbb{Z}}_n$ gauge group via Higgsing. This phenomenon may suggest an unknown aspect of F-theory compactification on multisection geometry lacking a global section. A possible interpretation of this puzzling physical phenomenon is proposed in this note. We also propose a possible interpretation of another unnatural physical phenomenon observed for F-theory on four-section geometry, wherein a discrete ${\mathbb{Z}}_2$ gauge group transitions to a discrete ${\mathbb{Z}}_4$ gauge group via Higgsing as described in the previous literature.

In this paper, we analyze gauge groups in six-dimensional $N=1$ F-theory models. We construct elliptic Calabi–Yau 3-folds possessing various singularity types as double covers of “1/2 Calabi–Yau 3-folds,” a class of rational elliptic 3-folds, by applying the method discussed in a previous study to classify the singularity types of the 1/2 Calabi–Yau 3-folds. One to three U(1) factors are formed in six-dimensional F-theory on the constructed Calabi–Yau 3-folds. The singularity types of the constructed Calabi–Yau 3-folds corresponding to the non-Abelian gauge group factors in six-dimensional F-theory are deduced. The singularity types of the Calabi–Yau 3-folds constructed in this work consist of $A$- and $D$-type singularities.

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.

We discuss local F-theory geometries and their gauge theory dualities in terms of intersecting D7-branes wrapped four-cycles in Type IIB superstring. The manifolds are built as elliptic K3 surface fibrations over intersecting F₀ = CP¹ × CP¹ base geometry according to ADE Dynkin Diagrams. The base is obtained by blowing up the extended ADE hyper-Kähler singularities of eight-dimensional manifolds considered as sigma model target spaces with eight supercharges. The resulting gauge theory of such local F-theory models are given in terms of Type IIB D7-branes wrapped intersecting F₀. The four-dimensional N = 1 anomaly cancelation requirement translates into a condition on the associated affine Lie algebras.

We discuss quiver gauge models with bi-fundamental and fundamental matter obtained from F-theory compactified on ALE spaces over a four-dimensional base space. We focus on the base geometry which consists of intersecting F₀ = CP¹ × CP¹ Hirzebruch complex surfaces arranged as Dynkin graphs classified by three kinds of Kac–Moody (KM) algebras: ordinary, i.e. finite-dimensional, affine and indefinite, in particular hyperbolic. We interpret the equations defining these three classes of generalized Lie algebras as the anomaly cancelation condition of the corresponding N = 1 F-theory quivers in four dimensions. We analyze in some detail hyperbolic geometries obtained from the affine base geometry by adding a node, and we find that it can be used to incorporate fundamental fields to a product of SU-type gauge groups and fields.

We study the Kähler potential of charged matter fields, whose profiles have a peak on their matter curve — on an "intersection" of 7-branes, in an F-theory compactification. It is shown that the Kähler potential is exactly given by the integral over the matter curve, but not by the integral over the whole GUT surface of 7-branes. This talk is based on our paper "A Note on Kahler Potential of Charged Matter in F-theory," in collaboration with Yoichi Tsuchiya and Taizan Watari.

We have established a relation between the four-dimensional chiral index in F-theory compactifications and the three-dimensional Chern-Simons coupling in M-theory compactifications by using F-theory - M-theory duality. The content of this article is based on our recent paper,¹ which is in collaboration with Thomas W. Grimm.

We show that the split/non-split transition is, with some exceptions, a conifold transition at a D_2k (k ≥ 2) or an E₇ codimension-two singularity, and these non-split models have local deformed conifolds. These clarify, in these cases, that the origin of non-local matter generation and that the deformations of conifold singularities correspond to diagram automorphisms of the expected simply-laced Dynkin diagrams in the split sides.