The Satake sextic in F-theory

Malmendier, Andreas; Shaska, Tony

doi:10.1016/j.geomphys.2017.06.010

Cited by 33 publications

(47 citation statements)

References 17 publications

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“…In Section 5 we discuss the specialization of six-line configurations tangent to a common conic and the associated K3 surfaces. We find perfect agreement in this case with the results in [7,24,27]. In Section 6 we discuss the construction of F-theory models, dual to the heterotic string with two non-vanishing Wilson line parameters that naturally follows from our geometric construction.…”

Section: Introductionsupporting

confidence: 75%

Six Line Configurations and String Dualities

Clingher

Malmendier

Shaska

2019

Commun. Math. Phys.

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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 ⊕ E 7 (−1) ⊕ 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.

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Section: Introductionsupporting

confidence: 75%

Six Line Configurations and String Dualities

Clingher

Malmendier

Shaska

2019

Commun. Math. Phys.

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“…Malmendier and Shaska in [33] proved that as a genus two curve C varies the Shioda-Inose K3 surface SI(Jac(C)) fits into the following forur parameter family in P 3 given in terms of the variables [W : X : Y : Z] ∈ P 3 by the equation…”

Section: Kummer and Shioda-inose Surfaces Of Reducible Jacobiansmentioning

confidence: 99%

“…, 6; see [26], when E 1 and E 2 were isogenous and show that in this case both F 5 and F 6 have rank 18. Such elliptic fibrations are studied extensively from other authors; see [4,9,10,18,33]. Perhaps the most interest is due to the promising post-quantum cryptography applications of such varieties.…”

Section: Introductionmentioning

confidence: 99%

Isogenous components of Jacobian surfaces

Beshaj

Elezi

Shaska

2019

European Journal of Mathematics

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Let C be a genus 2 curve defined over a field K, char K = p ≥ 0, and Jac(C, ι) its Jacobian, where ι is the principal polarization of Jac(C) attached to C. Assume that Jac(C) is (n, n)-geometrically reducible with E 1 and E 2 its elliptic components. We prove that there are only finitely many curves C (up to isomorphism) defined over K such that E 1 and E 2 are N -isogenous for n = 2 and N = 2, 3, 5, 7 with Aut (Jac C) ∼ = V 4 or n = 2, N = 3, 5, 7 with Aut (Jac C) ∼ = D 4 . The same holds if n = 3 and N = 5. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above Jac C and show how such results in positive characteristic p > 2 suggest nice applications in cryptography.

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“…E 8 E 7 singularity and non-geometric heterotic strings with O + (L 2,3 ) symmetry 5 . We also refer the reader to [33,34,35,36,37,38,39] for recent studies on non-geometric strings.…”

Section: Introductionmentioning

confidence: 99%

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups

Kimura

2019

J. High Energ. Phys.

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Eight-dimensional nongeometric heterotic strings were constructed as duals of Ftheory on Λ 1,1 ⊕ E 8 ⊕ E 7 lattice polarized K3 surfaces by Malmendier and Morrison. We study the structure of the moduli space of this construction. There are special points in this space at which the ranks of the non-Abelian gauge groups on the 7-branes in F-theory are enhanced to 18. We demonstrate that the enhanced rank-18 non-Abelian gauge groups arise as a consequence of the coincident 7-branes, which deform stable degenerations on the F-theory side. This observation suggests that the non-geometric heterotic strings include nonperturbative effects of the coincident 7-branes on the Ftheory side. The gauge groups that arise at these special points in the moduli space do not allow for perturbative descriptions on the heterotic side.We also construct a family of elliptically fibered Calabi-Yau 3-folds by fibering K3 surfaces with enhanced singularities over P 1 . Highly enhanced gauge groups arise in F-theory compactifications on the resulting Calabi-Yau 3-folds. 5 Connections of lattice polarized K3 surfaces, O + (Λ 2,2 )-modular forms, and non-geometric heterotic strings were discussed in [31]. K3 surfaces with Λ 1,1 ⊕ E 7 ⊕ E 7 lattice polarization and the construction of nongeometric heterotic strings were studied in [32].6 See [42] for the construction of the Jacobians of elliptic curves.

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The Satake sextic in F-theory

Cited by 33 publications

References 17 publications

Six Line Configurations and String Dualities

Six Line Configurations and String Dualities

Isogenous components of Jacobian surfaces

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups

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