F-theory, SO(32) and Toric Geometry

TL;DR

We investigate F-theory duals of the heterotic string with unbroken $Spin(32)/Z_2$ symmetry using toric geometry. The same three-dimensional reflexive polyhedron that encodes $E_8\times E_8$ can also realize $SO(32)$ gauge groups when different elliptic fibrations are chosen, with the dual polyhedron providing two complementary fibrations. Gauge enhancements are read from toric divisor intersections via an intersection pattern that reproduces ADE Dynkin diagrams, and the authors introduce 'tops' to generalize the reading of these patterns. By applying these ideas to compactifications to six and four dimensions, the paper reproduces the known six-dimensional 'record gauge group' and identifies a new four-dimensional one, illustrating the power of toric methods to capture large non-perturbative gauge sectors in F-theory/heterotic dual pairs.

Abstract

We show that the F-theory dual of the heterotic string with unbroken Spin(32)/Z_2 symmetry in eight dimensions can be described in terms of the same polyhedron that can also encode unbroken E_8\times E_8 symmetry. By considering particular compactifications with this K3 surface as a fiber, we can reproduce the recently found `record gauge group' in six dimensions and obtain a new `record gauge group' in four dimensions. Our observations relate to the toric diagram for the intersection of components of degenerate fibers and our definition of these objects, which we call `tops', is more general than an earlier definition by Candelas and Font.

Figures (3)

• Figure 1: The reflexive polyhedron that contains both the Weierstrass and the new $SO(32)$ triangle.
• Figure 2: The fan of the base of the $SO(32)$-fibration for the threefold. The rays give rise to the indicated groups.
• Figure 3: The 'top' for the group $Sp(24)$. There are 25 points in a straight line which form the extended Dynkin diagram of the group.