Generalised Chern–Simons Theory and $\mathrm{G}_2$-Instantons over Associative Fibrations

Adjusting conventional Chern–Simons theory to $\mathrm{G}_2$-manifolds, one describes $\mathrm{G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover, if $c_2\neq0$, any (generic) deformation of the $\mathrm{G}_2$-structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations $f:Y^7\to X^4$ relates $\mathrm{G}_2$-instantons on pullback bundles $f^*E\to Y$ and self-dual connections on the bundle $E\to X$ over the base, a fact which may be of independent interest.