Dessin d’enfant and a description of mutually nonlocal 7-branes without using branch cuts

Abstract

We consider the special roles of the zero loci of the Weierstrass invariants g 2 ðτðzÞÞ and g 3 ðτðzÞÞ 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 fðzÞ and gðzÞ of a Weierstrass equation. They are thought of as complex codimension-1 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 nonlocalness among 7-branes without employing unphysical branch cuts or their base point. With the dessin, we can see that weak-and strongcoupling 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.

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

“…We observe the characteristic configuration presenting the cluster sub-structure of an Oplane, which was found previously in [3].…”

“…Let us summarize how a "dessin on the base" can be drawn on P 1 [3]. We consider F-theory on an elliptic fibration over P 1 with a section, given by a Weierstrass equation…”

“…Let us now spell out how our new method of reading off the monodromies works. Following the notation used in [3], we denote a T -wall as G (for Green), an S-wall as B (for Blue) and a T -wall as dG (for dashed Green). To compute a monodromy along a given path, one first lists the walls crossed by the path in the order that they appear along the path.…”

“…A "dessin d'enfant" is a graph embedded on a two-dimensional oriented surface named by Grothendieck [1,2]. Recently, we have shown that such a drawing on a base P 1 of an elliptic fibration, along with a triangulation, is a convenient tool for computing the monodromies of the elliptic fiber [3]. In this set-up, we introduce two kinds of new codimension-one objects defined by the zero loci of the coefficient functions f (z) and g(z) of the Weierstrass equation.…”

“…The plan of this paper is as follows. In the next section, we give a review of the dessin on the base developed in [3]. In section 3, we consider an example of a deformation of a I * 0 Kodaira fiber.…”