More on a dessin on the base: Kodaira exceptional fibers and mutually (non-)local branes

TL;DR

This paper introduces a novel dessin-based method to analyze non-localness and monodromies of 7-branes in F-theory, demonstrated through three detailed examples involving Kodaira fibers and orientifold limits.

Contribution

It develops a new graphical approach using dessins to track non-local branes and compute monodromies in F-theory, enhancing understanding of brane configurations.

Findings

Dessin method identifies non-local brane pairs and their monodromies.

Numerical analysis shows no Hanany-Witten effect in the studied example.

Characteristic O-plane configurations are observed in the orientifold limit.

Abstract

A "dessin d'enfant" is a graph embedded on a two-dimensional oriented surface named by Grothendieck. Recently we have developed a new way to keep track of non-localness among 7-branes in F-theory on an elliptic fibration over $P^1$ by drawing a triangulated "dessin" on the base. To further demonstrate the usefulness of this method, we provide three examples of its use. We first consider a deformation of the $I_0^*$ Kodaira fiber. With a dessin, we can immediately find out which pairs of 7-branes are (non-)local and compute their monodromies. We next identify the paths of string(-junction)s on the dessin by solving the mass geodesic equation. By numerically computing their total masses, we find that the Hanany-Witten effect has not occurred in this example. Finally, we consider the orientifold limit in the spectral cover/Higgs bundle approach. We observe the characteristic configuration presenting the cluster sub-structure of an O-plane found previously.