$\mathrm{G}_2$–instantons over twisted connected sums

Abstract

We introduce a method to construct $G_2$–instantons over compact $G_2$–manifolds arising as the twisted connected sum of a matching pair of building blocks. Our construction is based on gluing $G_2$–instantons obtained from holomorphic vector bundles over the building blocks via the first author’s work. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a $K3$ surface.