F-theory and Heterotic Duality, Weierstrass Models from Wilson lines

TL;DR

This paper constructs elliptically fibered K3 surfaces using Weierstrass models parametrized by Wilson lines in heterotic string theory, revealing a geometric approach to understanding dualities and moduli spaces.

Contribution

It introduces a method to relate Weierstrass models of K3 surfaces to heterotic Wilson line moduli using reflexive polyhedra and graph-based algorithms.

Findings

Mapped monomials in K3 equations to Wilson line moduli

Developed algorithms for gauge group determination

Connected polyhedral geometry to heterotic moduli spaces

Abstract

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.