Mysterious duality and helical line bundles on del Pezzo surfaces

Abstract

Mysterious duality is a relationship, described by Vafa in 2000, between $\frac{1}{2}$-BPS branes in Type II supergravity in dimension $D=d+2$ and rational curves on del Pezzo surfaces of degree $d$. We show that both sides of this correspondence can be linked to a ${\mathbb{Z}}_d$ grading of the Lie algebra $E_8$. In addition, we show that the relevant rational curves correspond to `helical' line bundles, that is, line bundles that can appear in a helix on the del Pezzo surface.