A dessin on the base: a description of mutually non-local 7-branes without using branch cuts

TL;DR

This paper introduces a novel dessin-based method to analyze mutually non-local 7-branes in F-theory, avoiding unphysical branch cuts and revealing the coexistence of weak and strong coupling regions.

Contribution

It presents a new dessin d'enfant approach to track 7-brane non-localness and compute monodromies without using branch cuts in F-theory.

Findings

Dessin provides a visual tool for 7-brane analysis.

Weak and strong coupling regions coexist across S-walls.

A simple method for monodromy computation is introduced.

Abstract

We consider the special roles of the zero loci of the Weierstrass invariants $g_2(\tau(z))$, $g_3(\tau(z))$ in F-theory on an elliptic fibration over $P^1$ or a further fibration thereof. They are defined as the zero loci of the coefficient functions $f(z)$ and $g(z)$ of a Weierstrass equation. They are thought of as complex co-dimension one objects and correspond to the two kinds of critical points of a dessin d'enfant of Grothendieck. The $P^1$ base is divided into several cell regions bounded by some domain walls extending from these planes and D-branes, on which the imaginary part of the $J$-function vanishes. This amounts to drawing a dessin with a canonical triangulation. We show that the dessin provides a new way of keeping track of mutual non-localness among 7-branes without employing unphysical branch cuts or their base point. With the dessin we can see that weak- and strong-coupling regions coexist and are located across an $S$-wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through.