Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
Yalong Cao, Jacob Gross, Dominic Joyce
TL;DR
The paper investigates orientability of moduli spaces of Spin(7)-instantons on compact Spin(7) manifolds and of moduli spaces of coherent sheaves on Calabi–Yau 4-folds, a key step for defining DT-like invariants in higher dimensions. It develops the orientation theory via determinant line bundles on gauge-theoretic moduli spaces and through shifted-symplectic derived geometry on moduli stacks of coherent sheaves, relating these orientations to topological and algebraic data. A central result asserts orientability of the gauge-theory moduli spaces ${\\mathcal B}_P$ for G = U(m) or SU(m) under suitable hypotheses, and a secondary result connects orientation on the derived moduli stack ${\boldsymbol{\mathcal M}}$ to a mapping space ${\mathcal C}$, yielding trivial orientation for Calabi–Yau 4-folds. The erratum highlights a flaw in a prior claim and points to Joyce–Upmeier’s corrected theory, underscoring the need to adopt their framework for rigorous orientation results, with implications for DT4-type invariants.
Abstract
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.