Orientation of the fixed locus in the Calabi–Yau threefold case

Determine whether the fixed locus $\bar{\mathcal{X}}^{\mathrm{sd}}$ of the $(-1)$-shifted symplectic moduli stack $\bar{\mathcal{X}}$ of perfect complexes on a Calabi–Yau threefold admits an orientation (i.e., whether its canonical bundle admits a square root compatible with the symplectic structure), enabling the construction of motivic self-dual DT invariants.

Background

For Calabi–Yau threefolds, the moduli stack of perfect complexes $\bar{\mathcal{X}}$ carries a natural $(-1)$-shifted symplectic structure and an orientation data is known in the linear case. The self-dual fixed locus $\bar{\mathcal{X}}^{\mathrm{sd}}$ also inherits a $(-1)$-shifted symplectic structure.

However, to define motivic self-dual DT invariants on $\bar{\mathcal{X}}^{\mathrm{sd}}$, an orientation of this fixed locus is required. The existence of such an orientation is highlighted by the authors as presently unknown.