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.

我们介绍了如何通过Weierstrass模型构造椭圆纤维化的K3表面，该模型可以在对偶杂散弦论中根据Wilson线进行参数化。我们使用反射性多面体的子集来工作，它们允许两条光纤的模空间包含\（E_ {8} \ times E_ {8} \）或\（\ frac {Spin（32）} {{\ mathbb { Z}} _ {2}} \）异质理论在没有威尔逊线的两个圆环上进行了压缩。然后可以将附加模数解释为杂散弦中特定的Wilson线内容。查找此类多面体的简便方法是使用多面体的图，其中链接与不同纤维的模量空间的包含关系有关。然后，我们能够将特定K3曲面的定义方程式中的单项式映射到对偶理论中的Wilson线模。通过开发三个不同的程序来构造图，这些程序给出了模空间中通用点的量规组，Weierstrass模型以及通过将定义K3曲面的超曲面方程的系数发送为零而获得的量规组的基本增强。