Generalised Chern-Simons Theory and G₂-Instantons over Associative Fibrations

Abstract

Adjusting conventional Chern-Simons theory to G₂-manifolds, one describes G₂-instantons on bundles over a certain class of 7-dimensional flat tori which fiber non-trivially over T⁴, by a pullback argument. Moreover, if c₂≠0, any (generic) deformation of the G₂-structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations f:Y⁷→X⁴ relates G₂-instantons on pullback bundles f∗E→Y and self-dual connections on the bundle E→X over the base, a fact which may be of independent interest.