Deformed $G_2$-instantons on ${\mathbb{R}}^4\times S^3$

We construct explicit examples of deformed $G_2$-instantons, also called Donaldson-Thomas connections, on ${\mathbb{R}}^4\times S^3$ endowed with the torsion free $G_2$-structure found by Brandhuber et al. and on ${\mathbb{R}}^{+}\times S^3\times S^3$ endowed with the Bryant-Salamon conical $G_2$-structure. These are the first such non-trivial examples on a $G_2$ manifold. As a by-product of our investigation we also find an associative foliation of ${\mathbb{R}}^4\times S^3$ by ${\mathbb{R}}^2\times S^1$.