Hyperk\"ahler Manifolds and Multiply Intersecting Branes

TL;DR

This paper constructs hyper-K"ahler 8-metrics using arrangements of hyperplanes in six-dimensional space, revealing geometric insights into string theory branes and dualities with supersymmetry.

Contribution

It introduces a novel geometric construction of hyper-K"ahler metrics related to string theory branes and dualities, connecting hyperplane arrangements to supersymmetric configurations.

Findings

Constructed hyper-K"ahler 8-metrics from hyperplane arrangements.

Linked geometric configurations to $SL(2,Z)$ actions on branes.

Presented supersymmetric brane configurations with 3/16 SUSY.

Abstract

This is a summary of the work in the author's recent paper with this title written with Jerome Gauntlett, George Papadopoulos and Paul Townsend, hep-th/9702202. We showed how to construct hyper-K\"ahler 8-metrics in terms of arrangements of three-dimensional hyperplanes in six-dimensional euclidean space. The slopes of the planes define two relatively prime integers $(p,q)$. Under reduction to ten dimensions and T-duality we get a geometric picture of the action of $SL(2,Z)$ on the $(NS \otimes NS,q R \otimes R)$ 5-branes of type IIB string theory. Configurations are exhibited with $3 \over 16$'th SUSY.