Moduli spaces of $G_{2}$ 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.

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