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Description

The first compact examples of $G_2$-manifolds were constructed by D. Joyce by desingularising $G_2$-orbifolds with singularities modeled on flat $T^3 \times (\mathbb C^2/{\pm 1})$. T. Walpuski constructed $G_2$-instantons on these compact $G_2$-manifolds. D. Joyce and S. Karigiannis generalised the original construction of $G_2$-manifolds by resolving orbifold singularities modeled on $L \times (\mathbb C^2/{\pm 1})$, where $L$ is an associative submanifold in a $G_2$-manifold. The obvious question is how the original instanton construction can be adapted to the new setting. In the talk I will explain (1) the construction by Joyce and Karigiannis of compact $G_2$-manifolds, (2) the construction by Walpuski of $G_2$-instantons on the original construction, and (3) the construction that I hope will produce $G_2$-instantons on the Joyce-Karigiannis $G_2$-manifolds. I will explain possible sources of examples using desingularisations of $G_2$-orbifolds of the form $T^3 \times (\text{K3-surface/involution})$.