BPS Open Strings and A-D-E-singularities in F-theory on K3

TL;DR

This paper refines the graphical representation of 7-brane solutions in F-theory on K3 to analyze open string descriptions of A-D-E singularities, revealing the relationship between BPS strings and gauge group generators.

Contribution

It introduces an improved graphical method to study 7-brane solutions, clarifies the correspondence between BPS strings and gauge group generators, and highlights differences across A-D-E singularities.

Findings

Complete global branch cut structure identified

Sufficient BPS strings found for gauge generators

One-to-one correspondence established only for D-series and A-series

Abstract

We improve on a recently constructed graphical representation of the supergravity 7-brane solution and apply this refined representation to re-study the open string description of the A-D-E-singularities in F-theory on K3. A noteworthy feature of the graphical representation is that it provides the complete global branch cut structure of the 7-brane solution which plays an important role in our analysis. We first identify those groups of branes which when made to coincide lead to the A-D-E-gauge groups. We next show that there is always a sufficient number of open BPS strings to account for all the generators of the gauge group. However, as we will show, there is in general no one-to-one relation between BPS strings and gauge group generators. For the D_{n+4}- and E-singularities, in order to relate BPS strings with gauge group generators, we make an SU(n+4), respectively SU(5) subgroup of the D_{n+4}- and E-gauge groups manifest. We find that only for the D-series (and for the standard A-series) this is sufficient to identify, in a one-to-one manner, which BPS strings correspond to which gauge group generators.