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.
Highlights
F-theory compactified on elliptically fibered K3 surfaces is believed to be dual at the quantum level to the heterotic string compactified on a two torus with Wilson lines [1,2,3,4,5,6]
We show that additional monomials in the hypersurface equation which defines the elliptically fibered K3s on which we compactify on correspond to additional Wilson line moduli in both the E8 × E8 and heterotic strings
We constructed a graph of polytopes with two fibers where the links can be considered in this specific case as inclusion relations between the moduli spaces of the elliptically fibered K3s associated with each fiber
Summary
F-theory compactified on elliptically fibered K3 surfaces is believed to be dual at the quantum level to the heterotic string compactified on a two torus with Wilson lines [1,2,3,4,5,6]. Wilson line/monomial duality we can construct Weierstrass models of elliptically fibered K3s which are not directly obtained from reflexive polyhedra They can be interpreted as a certain Wilson line content in the dual heterotic theories. We show that in some cases, this notion of Wilson line description of K3 surfaces can be extended to polytopes with more than two fibers This should be helpful to explicitly understand the duality between F-theory compactified on K3s and heterotic string on a two torus, and eventually in compactifications to lower dimensions involving K3 surfaces. Weierstrass model and finds the enhancements one can obtain by sending the coefficients which parametrize the hypersurface equation of the K3 in some toric varieties to zero This can be useful to construct graphs of polytopes and we show how one can link polytopes up to three moduli. We show how to construct Weierstrass models of elliptically fibered K3s which one can directly interpret in the dual theory as a particular Wilson line content
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