Horizontal and Vertical Five-Branes in Heterotic/F-Theory Duality

TL;DR

This paper explores heterotic string theory with five-branes on elliptic Calabi-Yau three-folds, establishing a duality with F-theory that accounts for both horizontal and vertical brane configurations, and confirms the matching of five-branes and three-branes.

Contribution

It extends heterotic/F-theory duality to include five-branes on both horizontal and vertical curves, revealing consistent brane counting in this generalized setup.

Findings

Number of five-branes on elliptic fibers matches F-theory three-branes.

Duality relation holds for combined horizontal and vertical five-brane configurations.

Generalization broadens understanding of heterotic/F-theory duality with branes.

Abstract

We consider the heterotic string on an elliptic Calabi-Yau three-fold with five-branes wrapping curves in the base ('horizontal' curves) of the Calabi-Yau as well as some elliptic fibers ('vertical' curves). We show that in this generalized set-up, where the association of the heterotic side with the $F$-theory side is changed relative to the purely vertical situation, the number of five-branes wrapping the elliptic fibers still matches the corresponding number of $F$-theory three-branes.