F-theory Compactifications Research Articles

Non-minimal elliptic threefolds at infinite distance. Part I. Log Calabi-Yau resolutions

We study infinite-distance limits in the complex structure moduli space of elliptic Calabi-Yau threefolds. In F-theory compactifications to six dimensions, such limits include infinite-distance trajectories in the non-perturbative open string moduli space. The limits are described as degenerations of elliptic threefolds whose central elements exhibit non-minimal elliptic fibers, in the Kodaira sense, over curves on the base. We show how these non-crepant singularities can be removed by a systematic sequence of blow-ups of the base, leading to a union of log Calabi-Yau spaces glued together along their boundaries. We identify criteria for the blow-ups to give rise to open chains or more complicated trees of components and analyse the blow-up geometry. While our results are general and applicable to all non-minimal degenerations of Calabi-Yau threefolds in codimension one, we exemplify them in particular for elliptic threefolds over Hirzebruch surface base spaces. We also explain how to extract the gauge algebra for F-theory probing such reducible asymptotic geometries. This analysis is the basis for a detailed F-theory interpretation of the associated infinite-distance limits that will be provided in a companion paper [1].

The frozen phase of heterotic F-theory duality

We study the duality between the Spin(32)/ℤ2 heterotic string without vector structure and F-theory with frozen singularities. We give a complete description in theories with 6d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (1, 0) supersymmetry and identify the duals of Spin(32)/ℤ2-instantons on ADE singularities without vector structure in the frozen phase of F-theory using an ansatz introduced by Bhardwaj, Morrison, Tachikawa, and Tomasiello. As a consequence, we obtain a strongly coupled description of orbifold phases of type I string theory without vector structure, substantially expanding the list of known examples of 6d F-theory compactifications with frozen singularities. Supergravity theories can be fused from these instanton theories, in a way that commutes with switching off vector structure, which we use to propose new consistency checks via neutral hypermultiplet counting. Finally, we describe various Higgsings of this duality, and comment on constraints on higher form symmetries.

On the frozen F-theory landscape

We study 6d \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = (1, 0) supergravity theories arising in the frozen phase of F-theory. For each of the known global models, we construct an F-theory compactification in the unfrozen phase with an identical non-abelian gauge algebra and massless matter content. Two such low energy effective theories are then distinguished through gauge enhancements in moduli space. We study potentially new global models obtained via compact embeddings of a plethora of 6d \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = (1, 0) superconformal field theories and little string theories constructed using frozen 7-branes. In some cases, these provably do not exist, and in other cases, we explicitly construct a compact embedding, yielding 6d supergravity theories with new massless spectra. Finally, by using gravitational anomaly cancellation, we conjecture the existence of localized neutral hypermultiplets along frozen 7-brane loci.

On the string landscape without hypermultiplets

In this work we study interesting corners of the quantum gravity landscape with 8 supercharges pushing the boundaries of our current understanding. Calabi-Yau threefolds compactifications of F/M/type II theories to 6, 5 and 4 dimensions are the most prominent examples of this class, and these always lead to a universal hypermultiplet coming from the volume/string coupling constant. We find new examples of asymmetric orbifold constructions which have no hypermultiplets in 4 or 5 dimensions and no neutral hypers in 6d. We argue that these theories can also be obtained by going to strong coupling/small volume regions of geometric constructions where a new Coulomb branch opens up and moving in this direction freezes the volume/string coupling constant. Interestingly we find that the Kodaira condition encountered in geometric limits of F-theory compactifications to 6 dimensions is violated in these corners of the landscape due to strong quantum corrections. We also construct a theory in 3 dimensions which if it were to arise by toroidal compactifications from 5d, it would have to come from pure N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with no massless scalar fields.

Affine characters at negative level and elliptic genera of non-critical strings

We study the elliptic genera of the non-critical strings of six dimensional superconformal field theories from the point of view of the strings’ worldsheet theory. We formulate a general ansatz for these in terms of characters of the affine Lie algebra associated to the 6d gauge group at negative level, and provide ample evidence for the validity of this ansatz for 6d theories obtained via F-theory compactification on elliptically fibered Calabi-Yau manifolds over a Hirzebruch base. We obtain novel closed form results for many elliptic genera in terms of our ansatz, and show that our results specialize consistently when moving along Higgsing trees.

Light strings and strong coupling in F-theory

We consider 4d mathcal{N} = 1 theories arising from F-theory compactifications on elliptically-fibered Calabi-Yau four-folds and investigate the non-perturbative structure of their scalar field space beyond the large volume/large complex structure regime. We focus on regimes where the F-theory field space effectively reduces to the deformation space of the worldsheet theory of a critical string obtained from a wrapped D3-brane. In case that this critical string is a heterotic string with a simple GLSM description, we identify new strong coupling singularities in the interior of the F-theory field space. Whereas from the perturbative perspective these singularities manifest themselves through a breakdown of the perturbative α′-expansion, the dual GLSM perspective reveals that at the non-perturbative level these singularities correspond to loci in field space along which the worldsheet theory of the critical D3-brane string breaks down and a 7-brane gauge theory becomes strongly coupled due to quantum effects. Therefore these singularities signal a transition to a strong coupling phase in the F-theory field space which can be shown to arise due to the failure of the F-theory field space to factorize between complex structure and Kähler sector at the quantum level. Such singularities are hence a feature of a genuine mathcal{N} = 1 theory without a direct counterpart in mathcal{N} = 2 theories in 4d. By relating our setup to recent studies of global string solutions associated to axionic strings we argue that the D3-brane string dual to the perturbative heterotic string leaves the spectrum of BPS strings when traversing into the strong coupling phase. The absence of the perturbative, critical heterotic string then provides a physical explanation for the breakdown of the perturbative expansion and the obstruction of certain classical infinite distance limits in accordance with the Emergent String Conjecture.

Brill-Noether-general limit root bundles: absence of vector-like exotics in F-theory Standard Models

Root bundles appear prominently in studies of vector-like spectra of 4d F-theory compactifications. Of particular importance to phenomenology are the Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze a superset of the physical root bundles whose cohomologies encode the vector-like spectra for the matter representations (3,2)1/6, ( overline{textbf{3}} ,1)−2/3 and (1,1)1. For the family B3( {Delta }_4^{{}^{circ}} ) consisting of mathcal{O} (1011) F-theory QSM geometries, we argue that more than 99.995% of the roots in this superset have no vector-like exotics. This indicates that absence of vector-like exotics in those representations is a very likely scenario in the mathcal{O} (1011) QSM geometries B3( {Delta }_4^{{}^{circ}} ).The QSM geometries come in families of toric 3-folds B3(∆°) obtained from triangulations of certain 3-dimensional polytopes ∆°. The matter curves in XΣ ∈ B3(∆°) can be deformed to nodal curves which are the same for all spaces in B3(∆°). Therefore, one can probe the vector-like spectra on the entire family B3(∆°) from studies of a few nodal curves. We compute the cohomologies of all limit roots on these nodal curves.In our applications, for the majority of limit roots the cohomologies are determined by line bundle cohomology on rational tree-like curves. For this, we present a computer algorithm. The remaining limit roots, corresponding to circuit-like graphs, are handled by hand. The cohomologies are independent of the relative position of the nodes, except for a few circuits. On these jumping circuits, line bundle cohomologies can jump if nodes are specially aligned. This mirrors classical Brill-Noether jumps. B3( {Delta }_4^{{}^{circ}} ) admits a jumping circuit, but the root bundle constraints pick the canonical bundle and no jump happens.

Generalized Symmetries in F-theory and the Topology of Elliptic Fibrations

We realize higher-form symmetries in F-theory compactifications on non-compact elliptically fibered Calabi-Yau manifolds. Central to this endeavour is the topology of the boundary of the non-compact elliptic fibration, as well as the explicit construction of relative 2-cycles in terms of Lefschetz thimbles. We apply the analysis to a variety of elliptic fibrations, including geometries where the discriminant of the elliptic fibration intersects the boundary. We provide a concrete realization of the 1-form symmetry group by constructing the associated charged line operator from the elliptic fibration. As an application we compute the symmetry topological field theories in the case of elliptic three-folds, which correspond to mixed anomalies in 5d and 6d theories.

Faking gauge coupling unification in string theory

Gauge coupling unification misleads infrared observers if new gauge bosons do not simultaneously come into the spectrum. Though easy to engineer in gauge theory, the situation in string theory is nuanced due to moduli dependence. We study the possibility of faking gauge coupling unification in the context of 4D F-theory compactifications. Specifically, we formulate a sufficient condition that we call a strong calibration, under which seven-brane gauge couplings on homologically distinct divisors become equal at codimension one in K\"ahler moduli space. We prove that a strong calibration is preserved under appropriate topological transitions and that a pair of nonintersecting divisors each admitting a contraction can always be strongly calibrated. Within the tree ensemble [J. Halverson et al., Phys. Rev. D 96, 126006 (2017)], we find that $\ensuremath{\approx}77.12%$ of pairs of intersecting toric divisors can be strongly calibrated and $\ensuremath{\approx}3.22%$ can never be calibrated. Physically, this means that gauge coupling unification can be faked in most cases that we study, but in others it can not, which is surprising from a gauge theoretic perspective.

On the Duality of F-Theory and the CHL String in Seven Dimensions

We show that the duality between F-theory and the CHL string in seven dimensions defines algebraic correspondences between K3 surfaces polarized by the rank-ten lattices $H \oplus N$ and $H\oplus E_8(-2)$. In the special case when the F-theory admits an additional anti-symplectic involution or, equivalently, the CHL string admits a symplectic one, both moduli spaces coincide. In this case, we derive an explicit parametrization for the F-theory compactifications dual to the CHL string, using an auxiliary genus-one curve, based on a construction given by Andr\'e Weil.

On a Lefschetz-type phenomenon for elliptic Calabi-Yaus

We consider 18 families of elliptic Calabi-Yaus which arise in constructing F-theory compactifications of string vacua, and show in each case that the upper Hodge diamond of a crepant resolution of the associated Weierstrass model coincides with the upper Hodge diamond of the (blown up) projective bundle in which the crepant resolution is naturally embedded. Such results are unexpected, as each crepant resolution we consider does not satisfy the hypotheses of the Lefschetz hyperplane theorem. In light of such findings, we suspect that all elliptic Calabi-Yaus satisfy such a ‘Lefschetz-type phenomenon’.

D-instanton superpotential in string theory

We study the non-perturbative superpotential generated by D(-1)-branes in type IIB compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. To compute the D-instanton superpotential, we study F-theory compactification on toric complete intersection elliptic Calabi-Yau fourfolds. We take the Sen-limit, but with finite gs, in F-theory compactifications with the restriction that all D7-branes are carrying SO(8) gauge groups, which we call the global Sen-limit. In the global Sen-limit, the axio-dilaton is not varying in the compactification manifold. We compute the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, and show that the Picard-Fuchs equations of the elliptic fourfolds split into that of the underlying Calabi-Yau threefolds and of the elliptic fiber. We then demonstrate that this splitting property of the Picard-Fuchs equation implies that the fourform period of the elliptic Calabi-Yau fourfolds in the global Sen-limit does not contain exponentially suppressed terms mathcal{O}left({e}^{-pi /{g}_s}right) . With this result, we finally show that in the global Sen-limit, the superpotential of the underlying type IIB compactification does not receive D(-1)-instanton contributions.

Emergent strings, duality and weak coupling limits for two-form fields

We systematically analyse weak coupling limits for 2-form tensor fields in the presence of gravity. Such limits are significant for testing various versions of the Weak Gravity and Swampland Distance Conjectures, and more broadly, the phenomenon of emergence. The weak coupling limits for 2-forms correspond to certain infinite-distance limits in the moduli space of string compactifications, where asymptotically tensionless, solitonic strings arise. These strings are identified as weakly coupled fundamental strings in a dual frame, which makes the idea of emergence manifest. Concretely we first consider weakly coupled tensor fields in six-dimensional compactifications of F-theory, where the arising tensionless strings play the role of dual weakly coupled heterotic strings. As the main part of this work, we consider certain infinite distance limits of Type IIB strings on K3 surfaces, for which we show that the asymptotically tensionless strings describe dual fundamental Type IIB strings, again on K3 surfaces. By contrast the analogous weak coupling limits of M-theory compactifications are found to correspond to an F-theory limit where an extra dimension emerges rather than tensionless strings. We comment on extensions of our findings to four-dimensional compactifications.

Elliptic threefolds with high Mordell–Weil rank

We present the first examples of smooth elliptic Calabi-Yau threefolds with Mordell-Weil rank 10, the highest currently known value. They are given by the Schoen threefolds introduced by Namikawa. We explicitly compute the Shioda homomorphism for the generators of the Mordell-Weil group and their induced height pairing. There are six isolated fibers of Kodaira Type IV. Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the charges and multiplicities of the massless hypermultiplets as Gopakumar-Vafa invariants and show that they satisfy the gravitational and abelian anomaly cancellation conditions. We explicitly describe a Weierstrass model over $\mathbb P^2$ of the Calabi-Yau threefolds as a log canonical model and relate it to a construction by Elkies.

$\mathbb{P}^1$-fibrations in F-theory and string dualities

In this work we study F-theory compactifications on elliptically fibered Calabi-Yau n-folds which have $\mathbb{P}^1$-fibered base manifolds. Such geometries, which we study in both 4- and 6-dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of $\mathbb{P}^1$-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of this study, we find a number of new phenomena. For example, in both 4- and 6-dimensions we find transitions whereby the base of a $\mathbb{P}^1$-bundle can change nature, or jump, at certain loci in complex structure moduli space. We discuss the implications of this jumping for the associated heterotic duals. We argue that $\mathbb{P}^1$-bundles with only rational sections lead to heterotic duals where the Calabi-Yau manifold is elliptically fibered over the section of the $\mathbb{P}^1$- bundle, and not its base. As expected, we see that degenerations of the $\mathbb{P}^1$-fibration of the F-theory base correspond to 5-branes in the dual heterotic physics, with the exception of cases in which the fiber degenerations exhibit monodromy. Along the way, we discuss a set of useful formulae and tools for describing F-theory compactifications on this class of Calabi-Yau manifolds.

D7 moduli stabilization: the tadpole menace

D7-brane moduli are stabilized by worldvolume fluxes, which contribute to the D3-brane tadpole. We calculate this contribution in the Type IIB limit of F-theory compactifications on Calabi-Yau four-folds with a weak Fano base, and are able to prove a no-go theorem for vast swathes of the landscape of compactifications. When the genus of the curve dual to the D7 worldvolume fluxes is fixed and the number of moduli grows, we find that the D3 charge sourced by the fluxes grows faster than 7/16 of the number of moduli, which supports the Tadpole Conjecture of ref. [1]. Our lower bound for the induced D3 charge decreases when the genus of the curves dual to the stabilizing fluxes increase, and does not allow to rule out a sliver of flux configurations dual to high-genus high-degree curves. However, we argue that most of these fluxes have very high curvature, which is likely to be above the string scale except on extremely large (and experimentally ruled out) compactification manifolds.

Extremal $1/2$ Calabi–Yau $3$-folds and six-dimensional F-theory applications

We discuss a method for classifying the singularity types of 1/2 Calabi-Yau 3-folds, a family of rational elliptic 3-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi-Yau 3-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi-Yau 3-folds yield elliptic Calabi-Yau 3-folds and applications to six-dimensional $N = 1$ F-theory on the Calabi-Yau 3-folds are also discussed. The deduced singular fibers have applications in studying the gauge groups formed in 6D F-theory compactifications. The blow-up methods used to analyze the singular fibers and sections utilized in this research might have applications in studying the U(1) factors and hypermultiplets charged under U(1) in 6D F-theory.

Root bundles and towards exact matter spectra of F-theory MSSMs

Motivated by the appearance of fractional powers of line bundles in studies of vector-like spectra in 4d F-theory compactifications, we analyze the structure and origin of these bundles. Fractional powers of line bundles are also known as root bundles and can be thought of as generalizations of spin bundles. We explain how these root bundles are linked to inequivalent F-theory gauge potentials of a G4-flux.While this observation is interesting in its own right, it is particularly valuable for F-theory Standard Model constructions. In aiming for MSSMs, it is desired to argue for the absence of vector-like exotics. We work out the root bundle constraints on all matter curves in the largest class of currently-known F-theory Standard Model constructions without chiral exotics and gauge coupling unification. On each matter curve, we conduct a systematic “bottom”-analysis of all solutions to the root bundle constraints and all spin bundles. Thereby, we derive a lower bound for the number of combinations of root bundles and spin bundles whose cohomologies satisfy the physical demand of absence of vector-like pairs.On a technical level, this systematic study is achieved by a well-known diagrammatic description of root bundles on nodal curves. We extend this description by a counting procedure, which determines the cohomologies of so-called limit root bundles on full blow-ups of nodal curves. By use of deformation theory, these results constrain the vector-like spectra on the smooth matter curves in the actual F-theory geometry.

F-theory flux vacua at large complex structure

We compute the flux-induced F-term potential in 4d F-theory compactifications at large complex structure. In this regime, each complex structure field splits as an axionic field plus its saxionic partner, and the classical F-term potential takes the form V = ZABρAρB up to exponentially-suppressed terms, with ρ depending on the fluxes and axions and Z on the saxions. We provide explicit, general expressions for Z and ρ, and from there analyse the set of flux vacua for an arbitrary number of fields. We identify two families of vacua with all complex structure fields fixed and a flux contribution to the tad- pole Nflux which is bounded. In the first and most generic one, the saxion vevs are bounded from above by a power of Nflux. In the second their vevs may be unbounded and Nflux is a product of two arbitrary integers, unlike what is claimed by the Tadpole Conjecture. We specialise to type IIB orientifolds, where both families of vacua are present, and link our analysis with previous results in the literature. We illustrate our findings with several examples.

Determining F-theory Matter Via Gromov-Witten Invariants

We show how to use Gromov-Witten invariants to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau manifolds X over Hirzebruch surfaces. To determine the representations of these matter multiplets under the gauge algebra \({\mathfrak {g}}\), we use toric methods to embed the weight lattice of \({\mathfrak {g}}\) into the integer homology lattice of X. We then apply mirror symmetry to determine whether classes in this lattice which correspond to weights of given representations are represented by irreducible curves. Applying mirror symmetry efficiently to such geometries requires obtaining good approximations to their Mori cones. We propose an algorithm for obtaining such approximations. When the algorithm yields a smooth cone, we find that the latter in fact coincides with the Mori cone of X and already contains information on the matter content of compactifications on X. Our algorithm relies on studying toric ambient spaces for the Calabi-Yau hypersurface X which are merely birationally equivalent to fibrations over Hirzebruch surfaces. We study the flops relating such varieties in detail.