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

TL;DR

This paper explores the duality between F-theory and heterotic string theory in eight dimensions, constructing explicit models of non-geometric compactifications with two Wilson lines and analyzing their modular properties.

Contribution

It provides explicit Weierstrass models for all relevant K3 fibrations and links their parameters to generalized modular forms, offering a comprehensive classification of dual non-geometric compactifications.

Findings

Explicit Weierstrass models for K3 fibrations

Connection between modular forms and K3 surface equations

Complete classification of dual non-geometric compactifications

Abstract

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.