Moduli Spaces of $$G_{2}$$-Instantons and Spin(7)-Instantons on Product Manifolds

Let X be a closed 6-dimensional manifold with a half-closed SU(3)-structure. On the product manifold $$X\times S^{1}$$ , with respect to the product $$G_{2}$$ -structure and on a pullback vector bundle from X, we show that any $$G_{2}$$ -instanton is equivalent to a Hermitian Yang–Mills connection on X via a “broken gauge”. This result reveals the topological type of the moduli of $$G_{2}$$ -instantons on $$X\times S^{1}$$ . In dimension 8, similar result holds for moduli of Spin(7)-instantons. A generalization and an example are given.