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

This paper concerns orientability of moduli spaces of Spin(7)-instantons on compact 8-manifolds $X$ with Spin(7)-structure for the Lie groups SU($m$) and U($m$), and of moduli spaces of coherent sheaves on Calabi-Yau 4-folds. Such orientations are needed to define enumerative invariants 'counting' Spin(7) instantons, or coherent sheaves on Calabi-Yau 4-folds $X$. The previous version of the paper, version 2, published in Advances in Mathematics 368 (2020), claimed to prove all these moduli spaces are orientable. VERSION 3 BEGINS WITH AN ERRATUM. THERE IS A MISTAKE IN THE PROOF OF THEOREM 1.11 OF VERSION 2, AND THE THEOREM ITSELF, ONE OF OUR MAIN RESULTS, IS FALSE. THE 8-MANIFOLD SU(3) IS A COUNTEREXAMPLE. COROLLARIES 1.12 AND 1.17 OF VERSION 2 DEPEND ON THEOREM 1.11, AND SO MAY ALSO BE FALSE, THOUGH WE DO NOT HAVE COUNTEREXAMPLES. OUR OTHER MAIN RESULT, THEOREM 1.15, IS UNAFFECTED BY THE MISTAKE. THE AUTHORS APOLOGIZE FOR THIS. Joyce-Upmeier arXiv:2503.20456 (197 pages) gives a new theory for studying orientability of moduli spaces using 'bordism categories'. Amongst other results they prove corrected versions of Theorem 1.11 and Corollaries 1.12 and 1.17, which hold with an extra assumption on $H^3(X,\mathbb Z)$. In version 3, we highlight and explain the mistakes, but we do not correct them, as this would take many pages. Except for Theorem 1.15, readers are advised to read, and cite, Joyce-Upmeier arXiv:2503.20456 instead of this paper.