Instantons in $7$-dimensions: contact, Hermitian Yang-Mills and G_2

We study a natural contact instanton equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (HYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. We derive cohomological conditions for smoothness, and we express its dimension in terms of the index of a transverse elliptic operator. As an instance of concrete interest, we specialise to transversely holomorphic Sasakian bundles over 7-dimensional Calabi-Yau links and we show that in this context the notions of contact instanton, integrable G_2-instanton and HYM connection coincide.