F-theory Compactifications Research Articles

Gauge groups and matter spectra in $F$-theory compactifications on genus-one fibered Calabi–Yau $4$-folds without section: Hypersurface and double cover constructions

We investigate gauge theories and matter contents in F-theory compactifications on families of genus-one fibered Calabi-Yau 4-folds lacking a global section. To construct families of genus-one fibered Calabi-Yau 4-folds that lack a global section, we consider two constructions: hypersurfaces in a product of projective spaces, and double covers of a product of projective spaces. We consider specific forms of defining equations for these genus-one fibrations, so that genus-one fibers possess complex multiplications of specific orders. These symmetries enable a detailed analysis of gauge theories. $E_6$, $E_7$, and $SU(5)$ gauge groups arise in some models. Discriminant components intersect with one another in the constructed models, and therefore, discriminant components contain matter curves. We deduce potential matter spectra and Yukawa couplings.

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.

Scanning the skeleton of the 4D F-theory landscape

Using a one-way Monte Carlo algorithm from several different starting points, we get an approximation to the distribution of toric threefold bases that can be used in four-dimensional F-theory compactification. We separate the threefold bases into “resolvable” ones where the Weierstrass polynomials (f, g) can vanish to order (4, 6) or higher on codimension-two loci and the “good” bases where these (4, 6) loci are not allowed. A simple estimate suggests that the number of distinct resolvable base geometries exceeds 103000, with over 10250 “good” bases, though the actual numbers are likely much larger. We find that the good bases are concentrated at specific “end points” with special isolated values of h1,1 that are bigger than 1,000. These end point bases give Calabi-Yau fourfolds with specific Hodge numbers mirror to elliptic fibrations over simple threefolds. The non-Higgsable gauge groups on the end point bases are almost entirely made of products of E8, F4, G2 and SU(2). Nonetheless, we find a large class of good bases with a single non-Higgsable SU(3). Moreover, by randomly contracting the end point bases, we find many resolvable bases with h1,1(B) ∼ 50-200 that cannot be contracted to another smooth threefold base.

Algorithmic universality in F-theory compactifications

We study universality of geometric gauge sectors in the string landscape in the context of F-theory compactifications. A finite time construction algorithm is presented for $\frac43 \times 2.96 \times 10^{755}$ F-theory geometries that are connected by a network of topological transitions in a connected moduli space. High probability geometric assumptions uncover universal structures in the ensemble without explicitly constructing it. For example, non-Higgsable clusters of seven-branes with intricate gauge sectors occur with probability above $1-1.01\times 10^{-755}$, and the geometric gauge group rank is above $160$ with probability $.999995$. In the latter case there are at least $10$ $E_8$ factors, the structure of which fixes the gauge groups on certain nearby seven-branes. Visible sectors may arise from $E_6$ or $SU(3)$ seven-branes, which occur in certain random samples with probability $\simeq 1/200$.

Algebraic cycles and local anomalies in F-theory

We introduce a set of identities in the cohomology ring of elliptic fibrations which are equivalent to the cancellation of gauge and mixed gauge-gravitational anomalies in F-theory compactifications to four and six dimensions. The identities consist in (co)homological relations between complex codimension-two cycles. The same set of relations, once evaluated on elliptic Calabi-Yau three-folds and four-folds, is shown to universally govern the structure of anomalies and their Green-Schwarz cancellation in six- and four-dimensional F-theory vacua, respectively. We furthermore conjecture that these relations hold not only within the cohomology ring, but even at the level of the Chow ring, i.e. as relations among codimension-two cycles modulo rational equivalence. We verify this conjecture in non-trivial examples with Abelian and non-Abelian gauge groups factors. Apart from governing the structure of local anomalies, the identities in the Chow ring relate different types of gauge backgrounds on elliptically fibred Calabi-Yau four-folds.

T-branes at the limits of geometry

Singular limits of 6D F-theory compactifications are often captured by T-branes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the F-theory model.

On the flux vacua in F-theory compactifications

We study moduli stabilization of the F-theory compactified on an elliptically fibered Calabi–Yau fourfold. Our setup is based on the mirror symmetry framework including brane deformations. The complex structure moduli dependence of the resulting 4D N=1 effective theory is determined by the associated fourfold period integrals. By turning on appropriate G-fluxes, we explicitly demonstrate that all the complex structure moduli fields can be stabilized around the large complex structure point of the F-theory fourfold.

Terminal singularities, Milnor numbers, and matter in F-theory

We initiate a systematic investigation of F-theory on elliptic fibrations with singularities which cannot be resolved without breaking the Calabi–Yau condition, corresponding to Q-factorial terminal singularities. It is the purpose of this paper to elucidate the physical origin of such non-crepant singularities in codimension two and to systematically analyze F-theory compactifications containing such singularities. The singularities reflect the presence of localized matter states from wrapped M2-branes which are not charged under any massless gauge potential. We identify a class of Q-factorial terminal singularities on elliptically fibered Calabi–Yau threefolds for which we can compute the number of uncharged localized hypermultiplets in terms of their associated Milnor numbers. These count the local complex deformations of the singularities. The resulting six-dimensional spectra are shown to be anomaly-free. We exemplify this in a variety of cases, including models with non-perturbative gauge groups with both charged and uncharged localized matter. The underlying mathematics will be discussed further in a forthcoming publication.

Machine learning in the string landscape

We utilize machine learning to study the string landscape. Deep data dives and conjecture generation are proposed as useful frameworks for utilizing machine learning in the landscape, and examples of each are presented. A decision tree accurately predicts the number of weak Fano toric threefolds arising from reflexive polytopes, each of which determines a smooth F-theory compactification, and linear regression generates a previously proven conjecture for the gauge group rank in an ensemble of frac{4}{3}times 2.96times {10}^{755} F-theory compactifications. Logistic regression generates a new conjecture for when E6 arises in the large ensemble of F-theory compactifications, which is then rigorously proven. This result may be relevant for the appearance of visible sectors in the ensemble. Through conjecture generation, machine learning is useful not only for numerics, but also for rigorous results.

Dual little strings from F-theory and flop transitions

A particular two-parameter class of little string theories can be described by M parallel M5-branes probing a transverse affine AN − 1 singularity. We previously discussed the duality between the theories labelled by (N, M) and (M, N). In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by (N, M) and left(frac{NM}{k},kright) are dual to each other, where k = gcd(N,M). To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications on toric, non-compact Calabi-Yau threefolds XN,M which have a double elliptic fibration structure. We show explicitly for a number of examples that XNM/k,k is part of the extended moduli space of XN,M, i.e. the two are related through symmetry transformations and flop transitions. By working out the full duality map, we provide a simple check at the level of the free energy of little string theories.

Discrete symmetries in Heterotic/F-theory duality and mirror symmetry

We study aspects of Heterotic/F-theory duality for compactifications with Abelian discrete gauge symmetries. We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with n-sections, associated with the Tate-Shafarevich group {mathbb{Z}}_n . Such models are obtained by studying first a specific toric set-up whose associated Heterotic vector bundle has structure group {mathbb{Z}}_n . By employing a conjectured Heterotic/F-theory mirror symmetry we construct dual geometries of these original toric models, where in the stable degeneration limit we obtain a discrete gauge symmetry of order two and three, for compactifications to six dimensions. We provide explicit constructions of mirror-pairs for symmetric examples with {mathbb{Z}}_2 and {mathbb{Z}}_3 , in six dimensions. The Heterotic models with symmetric discrete symmetries are related in field theory to a Higgsing of Heterotic models with two symmetric abelian U(1) gauge factors, where due to the Stückelberg mechanism only a diagonal U(1) factor remains massless, and thus after Higgsing only a diagonal discrete symmetry of order n is present in the Heterotic models and detected via Heterotic/F-theory duality. These constructions also provide further evidence for the conjectured mirror symmetry in Heterotic/F-theory at the level of fibrations with torsional sections and those with multi-sections.

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.

Refined BPS invariants of 6d SCFTs from anomalies and modularity

F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function Ztop of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing Ztop in terms of contributions Zβ at base degree β of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kähler class of the elliptic fiber and elliptic parameters the couplings and mass parameters. The indices with regard to the multiple elliptic parameters are fixed by the refined holomorphic anomaly equations, which we show to be completely determined from knowledge of the chiral anomaly of the corresponding SCFT. We express Zβ as a quotient of weak Jacobi forms, with a universal denominator inspired by its pole structure as suggested by the form of Ztop in terms of 5d BPS numbers. The numerator is determined by modularity up to a finite number of coefficients, which we prove to be fixed uniquely by imposing vanishing conditions on 5d BPS numbers as boundary conditions. We demonstrate the feasibility of our approach with many examples, in particular solving the E-string and M-string theories including mass deformations, as well as theories constructed as chains of these. We make contact with previous work by showing that spurious singularities are cancelled when the partition function is written in the form advocated here. Finally, we use the BPS invariants of the E-string thus obtained to test a generalization of the Göttsche-Nakajima-Yoshioka K-theoretic blowup equation, as inspired by the Grassi-Hatsuda-Mariño conjecture, to generic local Calabi-Yau threefolds.

Shift-symmetries and gauge coupling functions in orientifolds and F-theory

We investigate the field dependence of the gauge coupling functions of four-dimensional Type IIB orientifold and F-theory compactifications with space-time filling seven-branes. In particular, we analyze the constraints imposed by holomorphicity and covariance under shift-symmetries of the bulk and brane axions. This requires introducing quantum corrections that necessarily contain Riemann theta functions on the complex torus spanned by the D7-brane Wilson line moduli. Our findings hint towards a new underlying geometric structure for gauge coupling functions in string compactifications. We generalize this discussion to a genuine F-theory compactification on an elliptically fibered Calabi-Yau fourfold. We perform the first general dimensional reduction of eleven-dimensional super-gravity and dualization to the F-theory frame. The resulting effective action is compared with the circle reduction of a four-dimensional mathcal{N} = 1 supergravity theory. The F-theory geometry elegantly unifies bulk and brane degrees of freedom and allows us to infer non-trivial results about holomorphicity and shift-symmetries. For instance, we gain new insight into kinetic mixing of bulk and brane gauge fields.

Three-form periods on Calabi-Yau fourfolds: toric hypersurfaces and F-theory applications

The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. This work introduces the mathematical machinery to derive the complete moduli dependence of the periods of non-trivial three-forms for fourfolds realized as hypersurfaces in toric ambient spaces. It sets the stage to determine Picard-Fuchs-type differential equations and integral expressions for these forms. The key tool is the observation that non-trivial three-forms on fourfold hypersurfaces in toric ambient spaces always stem from divisors that are build out of trees of toric surfaces fibered over Riemann surfaces. The three-form periods are then non-trivially related to the one-form periods of these Riemann surfaces. In general, the three-form periods are known to vary holomorphically over the complex structure moduli space and play an important role in the effective actions arising in fourfold compactifications. We discuss two explicit example fourfolds for F-theory compactifications in which the three-form periods determine axion decay constants.

Discrete gauge groups in F-theory models on genus-one fibered Calabi-Yau 4-folds without section

We determine the discrete gauge symmetries that arise in F-theory compactifications on examples of genus-one fibered Calabi-Yau 4-folds without a section. We construct genus-one fibered Calabi-Yau 4-folds using Fano manifolds, cyclic 3-fold covers of Fano 4-folds, and Segre embeddings of products of projective spaces. Discrete ℤ5, ℤ4, ℤ3 and ℤ2 symmetries arise in these constructions. We introduce a general method to obtain multisections for several constructions of genus-one fibered Calabi-Yau manifolds. The pullbacks of hyperplane classes under certain projections represent multisections to these genus-one fibrations. We determine the degrees of these multisections by computing the intersection numbers with fiber classes. As a result, we deduce the discrete gauge symmetries that arise in F-theory compactifications. This method applies to various Calabi-Yau genus-one fibrations.

Strong coupling in F-theory and geometrically non-Higgsable seven-branes

Geometrically non-Higgsable seven-branes carry gauge sectors that cannot be broken by complex structure deformation, and there is growing evidence that such configurations are typical in F-theory. We study strongly coupled physics associated with these branes. Axiodilaton profiles are computed using Ramanujan's theories of elliptic functions to alternative bases, showing explicitly that the string coupling is O(1) in the vicinity of the brane; that it sources nilpotent SL(2,Z) monodromy and therefore the associated brane charges are modular; and that essentially all F-theory compactifications have regions with order one string coupling. It is shown that non-perturbative SU(3) and SU(2) seven-branes are related to weakly coupled counterparts with D7-branes via deformation-induced Hanany–Witten moves on (p,q) string junctions that turn them into fundamental open strings; only the former may exist for generic complex structure. D3-brane near these and the Kodaira type II seven-branes probe Argyres–Douglas theories. The BPS states of slightly deformed theories are shown to be dyonic string junctions.

Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications

We study seven-branes in $O(10^{15})$ four-dimensional F-theory compactifications where seven-brane moduli must be tuned in order to achieve non-abelian gauge symmetry. The associated compact spaces $B$ are the set of all smooth weak Fano toric threefolds. By a study of fine star regular triangulations of three dimensional reflexive polytopes, the number of such spaces is estimated to be $5.8\times 10^{14}\lesssim N_\text{bases}\lesssim 1.8\times 10^{17}$. Typically hundreds or thousands of moduli must be tuned to achieve symmetry for $h^{11}(B)<10$, but the average number drops sharply into the range $O(25)$-$O(200)$ as $h^{11}(B)$ increases. For some low rank groups, such as $SU(2)$ and $SU(3)$, there exist examples where only a few moduli must be tuned in order to achieve seven-brane gauge symmetry.

Gauge symmetries and matter fields in $\mathrm{F}$-theory models without section — compactifications on double cover and Fermat quartic $\mathrm{K}3$ constructions times $\mathrm{K}3$

We investigate gauge theories and matter fields in F-theory compactifications on genus-one fibered Calabi-Yau 4-folds without a global section. In this study, genus-one fibered Calabi-Yau 4-folds are built as direct products of a genus-one fibered K3 surface that lacks a section times a K3 surface. We consider i) double covers of $\mathbb{P}^1\times\mathbb{P}^1$ ramified along a bidegree (4,4) curve, and ii) complete intersections of two bidegree (1,2) hypersurfaces in $\mathbb{P}^1\times\mathbb{P}^3$ to construct genus-one fibered K3 surfaces without a section. $E_7$ gauge group arises in some F-theory compactifications on double covers times K3. We show that the tadpole can be cancelled for an F-theory compactification on complete intersection K3 times K3, when complete intersection K3 is isomorphic to the Fermat quartic, and the complex structure of the other K3 surface in the direct product is appropriately chosen.

Mordell-Weil torsion in the mirror of multi-sections

We give further evidence that genus-one fibers with multi-sections are mirror dual to fibers with Mordell-Weil torsion. In the physics of F-theory compactifications this implies a relation between models with a non-simply connected gauge group and those with discrete symmetries. We provide a combinatorial explanation of this phenomenon for toric hypersurfaces. In particular this leads to a criterion to deduce Mordell-Weil torsion directly from the polytope. For all 3134 complete intersection genus-one curves in three-dimensional toric ambient spaces we confirm the conjecture by explicit calculation. We comment on several new features of these models: The Weierstrass forms of many models can be identified by relabeling the coefficient sections. This reduces the number of models to 1024 inequivalent ones. We give an example of a fiber which contains only non-toric sections one of which becomes toric when the fiber is realized in a different ambient space. Similarly a singularity in codimension one can have a toric resolution in one representation while it is non-toric in another. Finally we give a list of 24 inequivalent genus-one fibers that simultaneously exhibit multi-sections and Mordell-Weil torsion in the Jacobian. We discuss a self-mirror example from this list in detail.