Geometry and Topology of String Junctions
Antonella Grassi, James Halverson, Julius L. Shaneson
TL;DR
This work develops a geometric framework for string junctions in elliptic fibrations by linking vanishing cycles to Lie algebra structures via deformations. It introduces junctions with asymptotic charge and shows that junctions with a(J)=0 realize root lattices corresponding to ADE types, with explicit constructions recovering A_r from I_{r+1}, D4 and G2 via monodromy, and IV giving su(3); weights arise from nonzero asymptotic charges in higher dimensions. The method extends to higher-dimensional and higher-codimension settings, including two-pronged examples that decompose I0* into I1 fibers and reveal monodromy that reduces D4 to G2, aligning geometric junction data with gauge and matter structures in F-theory. Overall, the approach provides a robust geometric tool for analyzing generalized seven-branes and string junction states in F-theory and related string compactifications, connecting deformations, vanishing cycles, and Lie algebra representations in a concrete, computable framework.
Abstract
We study elliptic fibrations by analyzing suitable deformations of the fibrations and vanishing cycles. We introduce geometric string junctions and describe some of their properties. We show how the structure of the geometric string junctions is naturally related to the Lie algebra structures of the associated singularities. One application in physics is in F-theory, where our novel approach connecting deformations and Lie algebras describes the structure of generalized type IIB seven-branes and string junction states which end on them.