Weierstrass Model Research Articles

Local and global densities for Weierstrass models of elliptic curves

Local and global densities for Weierstrass models of elliptic curves

Elliptic K3 surfaces at infinite complex structure and their refined Kulikov models

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra.

6D anomaly-free matter spectrum in F-theory on singular spaces

In this paper we study the 6d localized charged matter spectrum of F-theory directly on a singular elliptic Calabi-Yau 3-fold, i.e. without smoothing via resolution or deformation of the entire fibration. Given only the base surface, discriminant locus, and the SL(2, ℤ) local system, we propose a general prescription for determining the charged matter spectrum localized at intersections of seven-branes, using the technology of string junctions. More precisely, at each codimension-2 collision of seven-branes, we determine the local seven-brane content and compute the number of massless string junctions modulo the action of the SL(2, ℤ) monodromy. We find agreement with the predicted results from 6d anomaly cancellation in all cases considered. Examples include a generic Weierstrass model with arbitrary Kodaira fiber intersecting an I1, as well as cases with jointly charged matter localized at intersections of non-abelian seven-branes.

On the symplectic type of isomorphisms of the 𝑝-torsion of elliptic curves

Let p ≥ 3 p \geq 3 be a prime. Let E / Q E/\mathbb {Q} and E ′ / Q E’/\mathbb {Q} be elliptic curves with isomorphic p p -torsion modules E [ p ] E[p] and E ′ [ p ] E’[p] . Assume further that either (i) every G Q G_\mathbb {Q} -modules isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] admits a multiple λ ⋅ ϕ \lambda \cdot \phi with λ ∈ F p × \lambda \in \mathbb {F}_p^\times preserving the Weil pairing; or (ii) no G Q G_\mathbb {Q} -isomorphism ϕ : E [ p ] → E ′ [ p ] \phi : E[p] \to E’[p] preserves the Weil pairing. This paper considers the problem of deciding if we are in case (i) or (ii). Our approach is to consider the problem locally at a prime ℓ ≠ p \ell \neq p . Firstly, we determine the primes ℓ \ell for which the local curves E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ E’/\mathbb {Q}_\ell contain enough information to decide between (i) or (ii). Secondly, we establish a collection of criteria, in terms of the standard invariants associated to minimal Weierstrass models of E / Q ℓ E/\mathbb {Q}_\ell and E ′ / Q ℓ E’/\mathbb {Q}_\ell , to decide between (i) and (ii). We show that our results give a complete solution to the problem by local methods away from p p . We apply our methods to show the non-existence of rational points on certain hyperelliptic curves of the form y 2 = x p − ℓ y^2 = x^p - \ell and y 2 = x p − 2 ℓ y^2 = x^p - 2\ell where ℓ \ell is a prime; we also give incremental results on the Fermat equation x 2 + y 3 = z p x^2 + y^3 = z^p . As a different application, we discuss variants of a question raised by Mazur concerning the existence of symplectic isomorphisms between the p p -torsion of two non-isogenous elliptic curves defined over Q \mathbb {Q} .

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’.

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.

F-theory and heterotic duality, Weierstrass models from Wilson lines

We present how to construct elliptically fibered K3 surfaces via Weierstrass models which can be parametrized in terms of Wilson lines in the dual heterotic string theory. We work with a subset of reflexive polyhedras that admit two fibers whose moduli spaces contain the ones of the E_{8}times E_{8} or frac{Spin(32)}{{mathbb {Z}}_{2}} heterotic theory compactified on a two torus without Wilson lines. One can then interpret the additional moduli as a particular Wilson line content in the heterotic strings. A convenient way to find such polytopes is to use graphs of polytopes where links are related to inclusion relations of moduli spaces of different fibers. We are then able to map monomials in the defining equations of particular K3 surfaces to Wilson line moduli in the dual theories. Graphs were constructed developing three different programs which give the gauge group for a generic point in the moduli space, the Weierstrass model as well as basic enhancements of the gauge group obtained by sending coefficients of the hypersurface equation defining the K3 surface to zero.

The duality between F-theory and the heterotic string in $$D=8$$ with two Wilson lines

We construct non-geometric string compactifications by using the F-theory dual of the heterotic string compactified on a two-torus with two Wilson line parameters, together with a close connection between modular forms and the equations for certain K3 surfaces of Picard rank $16$. We construct explicit Weierstrass models for all inequivalent Jacobian elliptic fibrations supported on this family of K3 surfaces and express their parameters in terms of modular forms generalizing Siegel modular forms. In this way, we find a complete list of all dual non-geometric compactifications obtained by the partial higgsing of the heterotic string gauge algebra using two Wilson line parameters.

General F-theory models with tuned (SU(3) \xd7 SU(2) \xd7 U(1))/\u21246 symmetry

We construct a general form for an F-theory Weierstrass model over a general base giving a 6D or 4D supergravity theory with gauge group (SU(3) × SU(2) × U(1))/ℤ6 and generic associated matter, which includes the matter content of the standard model. The Weierstrass model is identified by unHiggsing a model with U(1) gauge symmetry and charges q ≤ 4 previously found by the first author. This model includes two distinct branches that were identified in earlier work, and includes as a special case the class of models recently studied by Cvetič, Halverson, Lin, Liu, and Tian, for which we demonstrate explicitly the possibility of unification through an SU(5) unHiggsing. We develop a systematic methodology for checking that a parameterized class of F-theory Weierstrass models with a given gauge group G and fixed matter content is generic (contains all allowed moduli) and confirm that this holds for the models constructed here.

Modular curves and Mordell-Weil torsion in F-theory

In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set of torsion groups which can occur on a smooth elliptic Calabi-Yau n-fold is contained in the set of subgroups which appear on a rational elliptic surface if n ≥ 3 and is slightly larger for n = 2. The key idea in our proof is showing that any elliptic fibration with sufficiently large torsion1 has singularities in codimension 2 which do not admit a crepant resolution. We prove this by explicitly constructing and studying maps to a modular curve whose existence is predicted by a universal property. We use the geometry of modular curves to explain the minimal singularities that appear on an elliptic fibration with prescribed torsion, and to determine the degree of the fundamental line bundle (hence the Kodaira dimension) of the universal elliptic surface which we show to be consistent with explicit Weierstrass models. The constraints from the modular curves are used to bound the fundamental group of any gauge group G in a supergravity theory obtained from F-theory. We comment on the isolated 8-dimensional theories, obtained from extremal K3’s, that are able to circumvent lower dimensional bounds. These theories neither have a heterotic dual, nor can they be compactified to lower dimensional minimal SUGRA theories. We also comment on the maximal, discrete gauged symmetries obtained from certain Calabi-Yau threefold quotients.

Euler Characteristics of Crepant Resolutions of Weierstrass Models

Based on an identity of Jacobi, we prove a simple formula that computes the pushforward of analytic functions of the exceptional divisor of a blowup of a projective variety along a smooth complete intersection with normal crossing. We apply this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate's algorithm. Since these Euler characteristics depend only on the sequence of blowups and not on the Kodaira fiber itself, nor the associated group, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi-Yau threefolds, we also compute the Hodge numbers. For elliptically fibered Calabi-Yau fourfolds, our results also prove a conjecture of Blumenhagen-Grimm-Jurke-Weigand based on F-theory/heterotic string duality.

Six Line Configurations and String Dualities

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with $H \oplus E_7(-1) \oplus E_7(-1)$ lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type $IV$. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions. As a result, we obtain novel F-theory models, dual to non-geometric heterotic string compactifications in eight dimensions with two non-vanishing Wilson line parameters.

Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers

We compare the sets of Calabi-Yau threefolds with large Hodge numbers that are constructed using toric hypersurface methods with those can be constructed as elliptic fibrations using Weierstrass model techniques motivated by F-theory. There is a close correspondence between the structure of “tops” in the toric polytope construction and Tate form tunings of Weierstrass models for elliptic fibrations. We find that all of the Hodge number pairs (h1,1, h2,1) with h1,1 or h2,1 ≥ 240 that are associated with threefolds in the Kreuzer-Skarke database can be realized explicitly by generic or tuned Weierstrass/Tate models for elliptic fibrations over complex base surfaces. This includes a relatively small number of somewhat exotic constructions, including elliptic fibrations over non-toric bases, models with new Tate tunings that can give rise to exotic matter in the 6D F-theory picture, tunings of gauge groups over non-toric curves, tunings with very large Hodge number shifts and associated nonabelian gauge groups, and tuned Mordell-Weil sections associated with U(1) factors in the corresponding 6D theory.

The Geometry of the SU(2) \xd7 G2-model

We study elliptic fibrations that geometrically engineer an SU(2) ×G2 gauge theory realized by a Weierstrass model for the collision III + I0* ns. We find all the distinct crepant resolutions of such a model and the flops connecting them. We compute the generating function for the Euler characteristic of the SU(2) ×G2-model. In the case of a Calabi-Yau threefold, we consider the compactification of M-theory and F-theory on an SU(2) × G2-model to a five and six-dimensional supergravity theory with eight super-charges. By matching each crepant resolution with each Coulomb chamber of the five-dimensional theory, we determine the number of multiplets and compute the prepotential in each Coulomb chamber. In particular, we discuss the counting numbers of hypermultiplets in the presence of singularities. We discuss in detail the cancellation of anomalies of the six-dimensional theory.

Flopping and slicing: $\operatorname{SO}(4)$ and $\operatorname{Spin}(4)$-models

We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graph the affine $A_1$ Dynkin diagram. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell-Weil group of the elliptic fibration is required to be trivial for Spin(4) and Z/2Z for SO(4). Each of these models have two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi-Yau threefolds, we also compute their Hodge numbers, and check the cancellations of anomalies in a six-dimensional supergravity theory.

Towards exotic matter and discrete non-abelian symmetries in F-theory

We present a prescription in F-theory for realizing matter in “exotic” representations of product gauge groups. For 6D vacua, bifundamental hypermultiplets are engineered by starting at a singular point in moduli space which includes 6D superconformal field theories coupled to gravity. A deformation in Higgs branch moduli space takes us to a weakly coupled gauge theory description. In the corresponding elliptically fibered Calabi-Yau threefold, the minimal Weierstrass model parameters (f, g, Δ) vanish at collisions of the discriminant at least to order (4, 6, 12), but with sufficiently high order of tangency to ensure the existence of T-brane deformations to a weakly coupled gauge theory with exotic bifundamentals. We present explicit examples including bifundamental hypermultiplets of {mathfrak{e}}_7times mathfrak{s}{mathfrak{u}}_2 and {mathfrak{e}}_6times mathfrak{s}{mathfrak{u}}_3 , each of which have dual heterotic orbifold descriptions. Geometrically, these matter fields are delocalized across multiple points of an F-theory geometry. Symmetry breaking with such representations can be used to produce high dimension representations of simple gauge groups such as the four-index symmetric representation of mathfrak{s}{mathfrak{u}}_2 and the three-index symmetric representation of mathfrak{s}{mathfrak{u}}_3 , and after further higgsing can yield discrete non-abelian symmetries.

Integrable Systems of Algebraic Origin and Separation of Variables

A plane algebraic curve whose Newton polygone contains d lattice points can be given by d points it passes through. Then the coefficients of its equation Poisson commute having been regarded as functions of coordinates of those points. It is observed in the work by O.Babelon and M.Talon, 2002. We formulate a generalization of this fact in terms of separation of variables and prove relations implying the Poisson commutativity. The examples of the integrable systems obtained this way include coefficients of the Lagrange and Hermit interpolation polynomials, coefficients of the Weierstrass models of curves.

Large U(1) charges in F-theory

We show that massless fields with large abelian charges (up to at least q = 21) can be constructed in 6D F-theory models with a U(1) gauge group. To show this, we explicitly construct F-theory Weierstrass models with nonabelian gauge groups that can be broken to U(1) theories with a variety of large charges. Determining the maximum abelian charge allowed in such a theory is key to eliminating what seems currently to be an infinite swampland of apparently consistent U(1) supergravity theories with large charges.

Abelian F-theory models with charge-3 and charge-4 matter

This paper analyzes U(1) F-theory models admitting matter with charges q = 3 and 4. First, we systematically derive a q = 3 construction that generalizes the previous q = 3 examples. We argue that U(1) symmetries can be tuned through a procedure reminiscent of the SU(N ) and Sp(N ) tuning process. For models with q = 3 matter, the components of the generating section vanish to orders higher than 1 at the charge-3 matter loci. As a result, the Weierstrass models can contain non-UFD structure and thereby deviate from the standard Morrison-Park form. Techniques used to tune SU(N ) models on singular divisors allow us to determine the non-UFD structures and derive the q = 3 tuning from scratch. We also obtain a class of a q=4 models by deforming a prior U(1) × U(1) construction. To the author’s knowledge, this is the first published F-theory example with charge-4 matter. Finally, we discuss some conjectures regarding models with charges larger than 4.

Classifying global symmetries of 6D SCFTs

We characterize the global symmetries for the conjecturally complete collection of all six dimensional superconformal field theories (6D SCFTs) which are realizable in F-theory and have no frozen singularities. We provide comprehensive checks of earlier 6D SCFT classification results via an alternative geometric approach yielding new restrictions which eliminate certain theories. We achieve this by directly constraining elliptically fibered Calabi-Yau (CY) threefold Weierstrass models and find this allows bypassing all anomaly cancellation machinery. This approach reduces the problem of classifying which 6D SCFT gauge and global symmetries are realizable in F-theory models before RG-flow to characterizing features of elliptic fibrations associated to these theories obtained by analysis of polynomials determining their local models. We supply an algorithm with implementation producing from a given SCFT base an explicit listing of all compatible gauge enhancements and their associated global symmetry maxima consistent with the geometric constraints we derive while making manifest the corresponding geometric ingredients for these symmetries including any possible Kodaira type realizations of each algebra summand. In mathematical terms, this amounts to determining all potentially viable non-compact CY threefold elliptic fibrations at finite distance in the moduli space with Weil-Petersson metric which meet certain requirements including the transverse pairwise intersection of singular locus components. We provide local analysis exhausting nearly all CY consistent transverse singular fiber collisions, global analysis concerning all viable gluings of these local models into larger configurations, and many novel constraints on singular locus component pair intersections and global fiber arrangements. We also investigate which transitions between 6D SCFTs can result from gauging of global symmetries and find that continuous degrees of freedom can be lost during such transitions.