F-theory Vacua Research Articles

Small-instanton transitions in F-theory

We study the phase transition between G-instantons and D3-branes quantitatively. A G-instanton is a classical solution to the self-dual equation of the M/F-theory three-form tensor field C in the complex fourfold. This phase transition is dual to that between ‘vertical’ small instantons and 5-branes in the heterotic string. Using G as a background gauge flux, we may dynamically control the gauge symmetry breaking and connect between different vacua of F-theory. We may understand the amount of flux undergoing the phase transition and the resulting number of D3-branes in terms of group-theoretical quantities. We also discuss the resulting chirality change and preservation of anomaly freedom.

Geometric unification of Higgs bundle vacua

Higgs bundles are a central tool used to study a range of intersecting brane systems in string compactifications. Solutions to the internal gauge theory equations of motion for the corresponding worldvolume theories of branes give rise to different low energy effective field theories. This has been heavily used in the study of M-theory on local $G_2$ spaces and F-theory on local elliptically fibered Calabi-Yau fourfolds. In this paper we show that the 3D $\mathcal{N} = 1$ effective field theory defined by M-theory on a local $Spin(7)$ space unifies the Higgs bundle data associated with 4D $\mathcal{N} = 1$ M- and F-theory vacua. This 3D system appears as an interface with finite thickness between different 4D vacua. We develop the general formalism of M-theory on such local $Spin(7)$ spaces, and build explicit interpolating solutions. This provides a complementary local gauge theory analysis of a recently proposed approach to constructing $Spin(7)$ spaces from generalized connected sums.

Constraints on standard model constructions in F-theory

We argue that the following three statements cannot all be true: (i) our vacuum is a type IIB / F-theory vacuum at moderate-to-large $h^{1,1}$, (ii) the $\alpha'$-expansion is controlled via the supergravity approximation, \`a la the KKLT and LVS scenarios, and (iii) there are no additional gauged sectors from seven-branes. Since nearly all known globally consistent F-theory models with the exact chiral spectrum of the Standard Model and gauge coupling unification occur at moderate $h^{1,1}$, this finding calls for new moduli stabilization scenarios or/and a rich seven-brane dark sector.

F-theory vacua and \u03b1\u2032-corrections

In this work we analyze F-theory and Type IIB orientifold compactifications to study α′-corrections to the four-dimensional, mathcal{N} = 1 effective actions. In particular, we obtain corrections to the Kählermoduli space metric and its complex structure for generic dimension originating from eight-derivative corrections to eleven-dimensional supergravity. We propose a completion of the G2R3 and (∇G)2R2-sector in eleven-dimensions relevant in Calabi-Yau fourfold reductions. We suggest that the three-dimensional, mathcal{N} = 2 Kähler coordinates may be expressed as topological integrals depending on the first, second, and third Chern-forms of the divisors of the internal Calabi-Yau fourfold. The divisor integral Ansatz for the Kähler potential and Kähler coordinates may be lifted to four-dimensional, mathcal{N} = 1 F-theory vacua. We identify a novel correction to the Kähler potential and coordinates at order α′2, which is leading compared to other known corrections in the literature. At weak string coupling the correction arises from the intersection of D7-branes and O7-planes with base divisors and the volume of self-intersection curves of divisors in the base. In the presence of the conjectured novel α′-correction resulting from the divisor interpretation the no-scale structure may be broken. Furthermore, we propose a model independent scenario to achieve non-supersymmetric AdS vacua for Calabi-Yau orientifold backgrounds with negative Euler-characteristic.

The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua

We study the structure of gauge and gravitational anomalies in 2d N = (0, 2) theories obtained by compactification of F-theory on elliptically fibered Calabi-Yau 5-folds. Abelian gauge anomalies, induced at 1-loop in perturbation theory, are cancelled by a generalized Green-Schwarz mechanism operating at the level of chiral scalar fields in the 2d supergravity theory. We derive closed expressions for the gravitational and the non-abelian and abelian gauge anomalies including the Green-Schwarz counterterms. These expressions involve topological invariants of the underlying elliptic fibration and the gauge background thereon. Cancellation of anomalies in the effective theory predicts intricate topological identities which must hold on every elliptically fibered Calabi-Yau 5-fold. We verify these relations in a non-trivial example, but their proof from a purely mathematical perspective remains as an interesting open problem. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of anomalies in 6d N = (1, 0) and 4d N = 1 theories obtained from F-theory.

Gauge backgrounds and zero-mode counting in F-theory

Computing the exact spectrum of charged massless matter is a crucial step towards understanding the effective field theory describing F-theory vacua in four dimensions. In this work we further develop a coherent framework to determine the charged massless matter in F-theory compactified on elliptic fourfolds, and demonstrate its application in a concrete example. The gauge background is represented, via duality with M-theory, by algebraic cycles modulo rational equivalence. Intersection theory within the Chow ring allows us to extract coherent sheaves on the base of the elliptic fibration whose cohomology groups encode the charged zero-mode spectrum. The dimensions of these cohomology groups are computed with the help of modern techniques from algebraic geometry, which we implement in the software gap. We exemplify this approach in models with an Abelian and non-Abelian gauge group and observe jumps in the exact massless spectrum as the complex structure moduli are varied. An extended mathematical appendix gives a self-contained introduction to the algebro-geometric concepts underlying our framework.

The gravitational sector of 2d (0, 2) F-theory vacua

F-theory compactifications on Calabi-Yau fivefolds give rise to two-dimensional N = (0, 2) supersymmetric field theories coupled to gravity. We explore the dilaton supergravity defined by the moduli sector of such compactifications. The massless moduli spectrum is found by uplifting Type IIB compactifications on Calabi-Yau fourfolds. This spectrum matches expectations from duality with M-theory on the same elliptic fibration. The latter defines an N = 2 Supersymmetric Quantum Mechanics related to the 2d (0, 2) F-theory supergravity via circle reduction. Using our recent results on the gravitational anomalies of duality twisted D3-branes wrapping curves in Calabi-Yau fivefolds we show that the F-theory spectrum is anomaly free. We match the classical Chern-Simons terms of the M-theory Super Quantum Mechanics to one-loop contributions to the effective action by S1 reduction of the dual F-theory.

Gauging unbroken symmetries in F-theory

F-theory attempts to include all U-dualities manifestly. Unlike its T-dual manifest partner, which is based on string current algebra, F-theory is based on higher dimensional brane current algebra. Like the T-dual manifest theory, which has $O(D-1,1)^2$ unbroken symmetry, the F-theory vacuum also enjoys certain symmetries ("$H$"). One of its important and exotic properties is that worldvolume indices are also spacetime indices. This makes the global brane current algebra incompatible with $H$ symmetry currents. The solution is to introduce worldvolume covariant derivatives, which depend on the $H$ coordinates even in a "flat" background. We will also give as an explicit example the 5-brane case.

F-theory vacua with [formula omitted] gauge symmetry

Discrete gauge groups naturally arise in F-theory compactifications on genus-one fibered Calabi–Yau manifolds. Such geometries appear in families that are parameterized by the Tate–Shafarevich group of the genus-one fibration. While the F-theory compactification on any element of this family gives rise to the same physics, the corresponding M-theory compactifications on these geometries differ and are obtained by a fluxed circle reduction of the former. In this note, we focus on an element of order three in the Tate–Shafarevich group of the general cubic. We discuss how the different M-theory vacua and the associated discrete gauge groups can be obtained by Higgsing of a pair of five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles in I2-fibers that appear at certain codimension two loci in the base. We explicitly identify all three curves that give rise to the corresponding Higgs fields. In this analysis the investigation of different resolved phases of the underlying geometry plays a crucial rôle.

The Cremmer-Scherk mechanism in F-theory compactifications on K3 manifolds

It is well understood --- through string dualities --- that there are 20 massless vector fields in the spectrum of eight-dimensional F-theory compactifications on smooth elliptically fibered K3 surfaces at a generic point in the K3 moduli space. Such F-theory vacua, which do not have any enhanced gauge symmetries, can be thought of as supersymmetric type IIB compactifications on P1 with 24 (p,q) seven-branes. Naively, one might expect there to be 24 massless vector fields in the eight-dimensional effective theory coming from world-volume gauge fields of the 24 branes. In this paper, we show how the vector field spectrum of the eight-dimensional effective theory can be obtained from the point of view of type IIB supergravity coupled to the world-volume theory of the seven-branes. In particular, we first show that the two-forms of the type IIB theory absorb the seven-brane world-volume gauge fields via the Cremmer-Scherk mechanism. We then proceed to show that the massless vector fields of the eight-dimensional theory come from KK-reducing the SL(2,Z) doublet two-forms of type IIB theory along SL(2,Z) doublet one-forms on the P1. We also discuss the relation between these vector fields and the "eaten" world-volume vector fields of the seven-branes.

Structure in 6D and 4D $ \mathcal{N}=1 $ supergravity theories from F-theory

We explore some aspects of 4D supergravity theories and F-theory vacua that are parallel to structures in the space of 6D theories. The spectrum and topological terms in 4D supergravity theories correspond to topological data of F-theory geometry, just as in six dimensions. In particular, topological axion-curvature squared couplings appear in 4D theories; these couplings are characterized by vectors in the dual to the lattice of axion shift symmetries associated with string charges. These terms are analogous to the Green-Schwarz terms of 6D supergravity theories, though in 4D the terms are not generally linked with anomalies. We outline the correspondence between F-theory topology and data of the corresponding 4D supergravity theories. The correspondence of geometry with structure in the low-energy action illuminates topological aspects of heterotic-F-theory duality in 4D as well as in 6D. The existence of an F-theory realization also places geometrical constraints on the 4D supergravity theory in the large-volume limit.

THE ${\mathcal F}$-LANDSCAPE: DYNAMICALLY DETERMINING THE MULTIVERSE

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 [Formula: see text], we demonstrate the existence of a continuous family of solutions which might adeptly describe the dynamics of distinctive universes. This Multiverse landscape of [Formula: see text] solutions, which we shall refer to as the [Formula: see text]-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 [Formula: see text]-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 [Formula: see text] 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.

Massive Abelian gauge symmetries and fluxes in F-theory

F-theory compactified on a Calabi-Yau fourfold naturally describes non-Abelian gauge symmetries through the singularity structure of the elliptic fibration. In contrast Abelian symmetries are more difficult to study because of their inherently global nature. We argue that in general F-theory compactifications there are massive Abelian symmetries, such as the uplift of the Abelian part of the U(N) gauge group on D7-branes, that arise from non-Kahler resolutions of the dual M-theory setup. The four-dimensional F-theory vacuum with vanishing expectation values for the gauge fields corresponds to the Calabi-Yau limit. We propose that fluxes that are turned on along these U(1)s are uplifted to non-harmonic four-form fluxes. We derive the effective four-dimensional gauged supergravity resulting from F-theory compactifications in the presence of the Abelian gauge factors including the effects of possible fluxes on the gauging, tadpoles and matter spectrum.

The No-Scale Multiverse at the LHC

We present a contemporary perspective on the String Landscape and the Multiverse of plausible string, M- and F-theory vacua, seeking to demonstrate a non-zero probability for the existence of a universe matching our own observed physics within the solution ensemble. We argue for the importance of No-Scale Supergravity as an essential common underpinning for the spontaneous emergence of a cosmologically flat universe from the quantum "nothingness". Our context is a highly detailed phenomenological probe of No-Scale ℱ-SU(5), a model representing the intersection of the ℱ-lipped SU(5) × U(1)x Grand Unified Theory (GUT) with extra TeV-Scale vector-like multiplets derived out of ℱ-theory, and the dynamics of No-Scale Supergravity. The latter in turn imply a very restricted set of high energy boundary conditions.We present a highly constrained "golden point" located near M1/2 = 455 GeV and tan β = 15 in the tan β – M1/2 plane, and a highly non-trivial "golden strip" with tan β ≃ 15, mt = 173.0-174.4 GeV, M1/2 = 455-481 GeV, and Mv = 691-1020 GeV, which simultaneously satisfies all the known experimental constraints, featuring also an imminently observable proton decay rate. We supplement this bottom-up phenomenological perspective with a top-down theoretical analysis of the one-loop effective Higgs potential. A striking consonance is achieved via the dynamic determination of tan β and M1/2 for fixed Z-boson mass at the local minimum minimorum of the potential, that being the secondary minimization of the spontaneously broken electroweak Higgs vacuum Vmin. By also indirectly determining the electroweak scale, we suggest that this constitutes a complete resolution of the Standard Model gauge hierarchy problem.Finally, we present the distinctive collider level signatures of No-Scale ℱ-SU(5) for the √s = 7 TeV LHC, with 1 fb−1 of integrated luminosity. The characteristic feature is a light stop and gluino, both sparticles lighter than all other squarks, generating a surplus of ultra-high multiplicity (≥ 9) hadronic jet events. We propose modest alterations to the canonical background selection cut strategy which are expected to yield significantly enhanced resolution of the characteristic ultra-high jet multiplicity ℱ-SU(5) events, while readily suppressing the contribution of all Standard Model processes, and allowing moreover a clear differentiation from competing models of new physics, most notably minimal supergravity. Detection by the LHC of the ultra-high jet signal would constitute a suggestive evocation of the intimately linked stringy origins of ℱ-SU(5), and could possibly provide a glimpse into the underlying structure of the fundamental string moduli, possibly even opening a darkened glass upon the hidden workings of the No-Scale Multiverse.

Formula omitted] vacua in F-theory

The Standard Model group and matter spectrum is obtained in vacua of F-theory, without resorting to an intermediate unification group. The group SU ( 3 ) × SU ( 2 ) × U ( 1 ) Y is the commutant to SU ( 5 ) ⊥ × U ( 1 ) Y structure group of a Higgs bundle in E 8 and is geometrically realized as a deformation of I 5 singularity. Lying along the unification groups of E n , our vacua naturally inherit their unification structure. By modding SU ( 5 ) ⊥ out by Z 4 monodromy group, we can distinguish Higgses from lepton doublets by matter parity. Turning on universal G-flux on this part, the spectrum contains three generations of quarks and leptons, as well as vectorlike pairs of electroweak and colored Higgses. Minimal Yukawa couplings is obtained at the renormalizable level.

GUT relations from string theory compactifications

Wilson line on a non-simply connected manifold is a nice way to break SU(5) unified symmetry, and to solve the doublet–triplet splitting problem. This mechanism also requires, however, that the two Higgs doublets are strictly vector-like under all underlying gauge symmetries, and consequently there is a limit in a class of modes and their phenomenology for which the Wilson line can be used. An alternative is to turn on a non-flat line bundle in the U ( 1 ) Y direction on an internal manifold, which does not have to be non-simply connected. The U ( 1 ) Y gauge field has to remain in the massless spectrum, and its coupling has to satisfy the GUT relation. In string theory compactifications, however, it is not that easy to satisfy these conditions in a natural way; we call it U ( 1 ) Y problem. In this article, we explain how the problem is solved in some parts of moduli space of string theory compactifications. Two major ingredients are an extra strongly coupled U(1) gauge field and parametrically large volume for compactification, which is also essential in accounting for the hierarchy between the Planck scale and the GUT scale. Heterotic M-theory vacua and F-theory vacua are discussed. This article also shows that the toroidal orbifold GUT approach using discrete Wilson lines corresponds to the non-flat line-bundle breaking above when orbifold singularities are blown up. Thus, the orbifold GUT approach also suffers from the U ( 1 ) Y problem, and this article shows how to fix it.

A note on toroidal compactifications of the type I superstring and other superstring vacuum configurations with 16 superchanges

We show that various disconnected components of the moduli space of superstring vacua with 16 superchanges admit a rationale in terms of BPS unorientifolds, i.e. type I toroidal compactifications with constant non-vanishing but quantized vacuum expectation values of the NS × NS antisymmetric tensor. These include various heterotic vacua with reduced rank, known as CHL strings, and their dual type II (2, 2) superstrings in D ⩽ 6. Type I vacua without open strings allow for an interpretation of several disconnected components with N v = 10 − D. An adiabatic argument relates these unconventional type I superstrings to type II (4, 0) superstrings without D-branes. The latter are connected by U-duality in D ⩽ 6 to type II (2, 2) superstrings. We also comment on the relation between some of these vacua and compactifications of the putative M-theory on unorientable manifolds as well as F-theory vacua.

On the complementarity of F-theory, orientifolds, and heterotic strings

We study F-theory duals of six-dimensional heterotic vacua in extreme regions of moduli space where the heterotic string is very strongly coupled. We demonstrate how to use orientifold limits of these F-theory duals to regain a perturbative string description. As an example, we reproduce the spectrum of a T 4/ Z 4 orientifold as an F-theory vacuum with a singular K3 fibration. We relate this vacuum to previously studied heterotic E 8 × E 8 compactifications on K3.

Orientifold and F-theory duals of CHL strings

We present orientifold and F-theory duals of the heterotic string compactification constructed by Chaudhuri, Hockney and Lykken (CHL) which has the maximal supersymmetry but gauge group of reduced rank. The 8-dimensional dual is given by the Type IIA orientifold on the M\"{o}bius band. We show the non-trivial monodromy on the base induces non-simply laced gauge groups on the F-theory orbifolds dual to the CHL strings. The F-theory models dual to the CHL strings in six dimensions are examples of N=2 F-theory vacua. We discuss the other N=2 F-theory vacua in six dimensions.

U-manifolds

We use non-perturbative U-duality symmetries of type II strings to construct new vaccum solutions. In some ways this generalizes the F-theory vacuum constructions. We find the possibilities of new vacuum constructions are very limited. Among them we construct new theories with N = 2 supersymmetry in 3 dimensions and (1, 1) supersymmetry in 2 dimensions.