Existence of self-dual orientation data for coherent sheaf moduli on Calabi–Yau threefolds

Establish the existence of a self-dual orientation data for the $(-1)$-shifted symplectic moduli stack of perfect complexes on a Calabi–Yau threefold. Concretely, construct an orientation $o_{\bar{\mathcal{X}}}$ on the derived moduli stack of perfect complexes $\bar{\mathcal{X}}$ and a compatible orientation $o_{\bar{\mathcal{X}}^{\mathrm{sd}}}$ on its fixed locus $\bar{\mathcal{X}}^{\mathrm{sd}}$ under the self-dual structure, such that the induced orientations satisfy the compatibility conditions with the graded-points isomorphisms (namely, agree with product orientations under the identifications of $\mathrm{Grad}(\bar{\mathcal{X}})$ and $\mathrm{Grad}(\bar{\mathcal{X}}^{\mathrm{sd}})$ described by the authors).

Background

The paper defines a self-dual orientation data for a self-dual $(-1)$-shifted symplectic linear stack as a pair of orientations on the stack and its fixed locus that satisfy explicit compatibility requirements with the induced orientations on graded-points stacks. Such orientations are essential to define motivic DT invariants in the self-dual setting.

While orientation data exists in the linear case for moduli of coherent sheaves on Calabi–Yau threefolds (Joyce–Upmeier), the authors emphasize that extending this to the self-dual fixed locus requires additional structure and compatibility that are currently not known to exist.