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Suppose $(X,\Omega,g)$ is a compact Spin(7)-manifold, e.g. a Riemannian8-manifold with holonomy Spin(7), or a Calabi-Yau 4-fold. Let $G$ be U$(m)$ orSU$(m)$, and $P\to X$ be a principal $G$-bundle. We show that theinfinite-dimensional moduli space ${\mathcal B}_P$ of all connections on $P$modulo gauge is orientable, in a certain sense. We deduce that the moduli space${\mathcal M}_P^{Spin(7)}\subset{\mathcal B}_P$ of irreducibleSpin(7)-instanton connections on $P$ modulo gauge, as a manifold or derivedmanifold, is orientable. This improves theorems of Cao and LeungarXiv:1502.01141 and Mu\~noz and Shahbazi arXiv:1707.02998.  If $X$ is a Calabi-Yau 4-fold, the derived moduli stack $\boldsymbol{\mathscrM}$ of (complexes of) coherent sheaves on $X$ is a $-2$-shifted symplecticderived stack $(\boldsymbol{\mathcal M},\omega)$ byPantev-To\"en-Vaqui\'e-Vezzosi arXiv:1111.3209, and so has a notion oforientation by Borisov-Joyce arXiv:1504.00690. We prove that$(\boldsymbol{\mathscr M},\omega)$ is orientable, by relating algebro-geometricorientations on $(\boldsymbol{\mathscr M},\omega)$ to differential-geometricorientations on ${\mathcal B}_P$ for U$(m)$-bundles $P\to X$, and usingorientability of ${\mathcal B}_P$.  This has applications to the programme of defining Donaldson-Thomas typeinvariants counting moduli spaces of (semi)stable coherent sheaves on aCalabi-Yau 4-fold, as in Donaldson and Thomas 1998, Cao and LeungarXiv:1407.7659, and Borisov and Joyce arXiv:1504.00690.  This is the third in a series arXiv:1811.01096, arXiv:1811.02405 onorientations of gauge-theoretic moduli spaces.

Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds