A dessin on the base: a description of mutually non-local 7-branes without using branch cuts
arXiv:1808.04135 · doi:10.1103/PhysRevD.100.126025
Abstract
We consider the special roles of the zero loci of the Weierstrass invariants , in F-theory on an elliptic fibration over or a further fibration thereof. They are defined as the zero loci of the coefficient functions and 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 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 -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 -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.
arXiv comment27 pages, 9 figures. v3: Title changed, section 3 added to explain the notion of a "dessin d'enfant", sections 4,6 and 7 in v2 removed
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