Kaluza-Klein bundles and manifolds of exceptional holonomy

TL;DR

This paper explores how RR two-form fields influence supersymmetry conditions on six- and seven-dimensional manifolds, leading to generalized monopole equations and special holonomy geometries such as G_2 and Spin(7).

Contribution

It introduces new supersymmetry-preserving conditions involving RR fields that generalize monopole equations and relate to special holonomy manifolds.

Findings

Six-dimensional manifolds are Kaehler with complex structures from octonions.

Seven-dimensional solutions are conformal to G_2 metrics with selfdual field strengths.

Solutions lift to geometries with G_2 and Spin(7) holonomy.

Abstract

We show how in the presence of RR two-form field strength the conditions for preserving supersymmetry on six- and seven-dimensional manifolds lead to certain generalizations of monopole equations. For six dimensions the string frame metric is Kaehler with the complex structure that descends from the octonions if in addition we assume F^{(1,1)}=0. The susy generator is a gauge covariantly constant spinor. For seven dimensions the string frame metric is conformal to a G_2 metric if in addition we assume the field strength to obey a selfduality constraint. Solutions to these equations lift to geometries of G_2 and Spin(7) holonomy respectively.