F-theory and heterotic duality, Weierstrass models from Wilson lines

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 $$E_{8}\times E_{8}$$ E 8 × E 8 or $$\frac{Spin(32)}{{\mathbb {Z}}_{2}}$$ S p i n ( 32 ) 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.

TYPE I ON (GENERALIZED) VOISIN–BORCEA ORBIFOLDS AND NONPERTURBATIVE ORIENTIFOLDS

We consider nonperturbative four-dimensional [Formula: see text] space–time supersymmetric orientifolds corresponding to Type I compactifications on (generalized) Voisin–Borcea orbifolds. Some states in such compactifications arise in "twisted" open string sectors which lack world sheet description in terms of D-branes. Using Type I-heterotic duality as well as the map between Type IIB orientifolds and F theory we are able to obtain the massless spectra of such orientifolds. The four-dimensional compactifications we discuss in this context are examples of chiral [Formula: see text] supersymmetric string vacua which are nonperturbative from both orientifold and heterotic points of view. In particular, they contain both D9- and D5-branes as well as nonperturbative "twisted" open string sector states. We also explain the origins of various inconsistencies arising in such compactifications for certain choices of the gauge bundle.