Mysterious duality and helical line bundles on del Pezzo surfaces

TL;DR

This paper explores the mysterious duality connecting BPS branes in supergravity with rational curves on del Pezzo surfaces, revealing a link to the Lie algebra E8 and introducing the concept of helical line bundles.

Contribution

It uncovers the connection between duality, Lie algebra gradings, and helical line bundles on del Pezzo surfaces, providing new insights into the geometric and algebraic structures involved.

Findings

Linked duality to a Z_d grading of E8

Identified rational curves as helical line bundles

Established geometric interpretation of duality

Abstract

Mysterious duality is a relationship, described by Vafa in 2000, between $\frac12$-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.