Kaluza-Klein bundles and manifolds of exceptional holonomy

TL;DR

This work shows that preserving supersymmetry for type IIA backgrounds with RR two-form flux and a nontrivial dilaton reduces to generalized monopole equations on six- and seven-dimensional bases, tightly linked to Kaluza-Klein circle bundles. In six dimensions, imposing $F^{(1,1)}=0$ yields a string-frame Kaehler base with an octonion-derived complex structure and a gauge-covariantly constant spinor, while in seven dimensions a selfduality constraint on the flux makes the internal metric conformal to a $G_2$ metric; the lifted eight-dimensional geometry attains Spin(7) holonomy. These conditions reproduce and illuminate dualities between D6-branes on supersymmetric cycles and M-theory compactifications on manifolds with exceptional holonomy, and they provide a geometric framework for constructing new $G_2$ and Spin(7) metrics via KK monopole bundles. The analysis highlights the pivotal role of three-forms and octonionic structures in governing flux-sourced supersymmetry and suggests avenues for extending to more general fluxes and weak-holonomy settings with potential physical relevance.

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.