Deformed $\mathrm {G}_2$-instantons on $\mathbb {R}^4 \times S^3$

Abstract:

In this note we construct explicit examples of deformed $\mathrm {G}_2$-instantons, also called Donaldson-Thomas connections, on $\mathbb {R}^4 \times S^3$ endowed with the torsion free $\mathrm {G}_2$-structure found by Brandhuber et al. [Nuclear Phys. B 611 (2001), pp. 179–204] and on $\mathbb {R}^+\times S^3 \times S^3$ endowed with the Bryant-Salamon conical $\mathrm {G}_2$-structure [Duke Math. J. 58 (1989), pp. 829–850]. These are the first such non-trivial examples on a $\mathrm {G}_2$ manifold. As a by-product of our investigation we also construct an associative foliation of $\mathbb {R}^4\times S^3$ by $\mathbb {R}^2 \times S^1$.