Six Line Configurations and String Dualities

Abstract

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.

Finite Groups for the Kummer Surface: the Genetic Code and a Quantum Gravity Analogy

“…It is worthwhile to mention that a K 3 surface appears in string theory with the concept of 'string duality'-how distinct string theories are related-, see Ref. [23,28]. Another work relating quantum gravity and K 3 surfaces is in Ref.…”

“…the Grassmann quadric Gr(2,4) in the five-dimensional Plücker space with another quadratic hypersurface W. The set of lines in X is parametrized by the Jacobian Jac(C) of a Riemann surface of genus 2 ramified along 6 points corresponding to 6 singular quadrics. See[23] for the relation to string dualities.Nowadays, in the classification by algebraic geometry, the Kummer surface is an exemple of a K 3 surface built from the quotient of an abelian variety A by the action from a point a to its opposite −a, resulting in 16 singularities[24,25]. The minimal resolution is the Kummer surface.…”

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

“…The Jacobian fibration (45) and the elliptic fibration (32) are two elliptic fibrations of the same attractive K3 surface S [2 1 2] . Therefore, there exists a birational map that transforms the Jacobian fibration (45) into the elliptic fibration (32). Because the Jacobian fibration (45) and the elliptic fibration (32) are birational, the F-theory compactification on the Jacobian (45) relates to a nongeometric E 8 × E 8 heterotic string.…”

“…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.…”

Isogenous components of Jacobian surfaces

“…, 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.…”