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

Abstract Suppose ( X , Ω , g ) is a compact Spin ( 7 ) -manifold, e.g. a Riemannian 8-manifold with holonomy Spin ( 7 ) , or a Calabi–Yau 4-fold. Let G be U ( m ) or SU ( m ) , and P → X be a principal G-bundle. We show that the infinite-dimensional moduli space B P of all connections on P modulo gauge is orientable, in a certain sense. We deduce that the moduli space M P Spin ( 7 ) ⊂ B P of irreducible Spin ( 7 ) -instanton connections on P modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung [9] and Munoz and Shahbazi [42] . If X is a Calabi–Yau 4-fold, the derived moduli stack M of (complexes of) coherent sheaves on X is a −2-shifted symplectic derived stack ( M , ω ) by Pantev–Toen–Vaquie-Vezzosi [46] , and so has a notion of orientation by Borisov–Joyce [7] . We prove that ( M , ω ) is orientable, by relating algebro-geometric orientations on ( M , ω ) to differential-geometric orientations on B P for U ( m ) -bundles P → X , and using orientability of B P . This has applications to defining Donaldson–Thomas type invariants counting semistable coherent sheaves on a Calabi–Yau 4-fold, as in Donaldson and Thomas [15] , Cao and Leung [8] , and Borisov and Joyce [7] .