Spin structures on perfect complexes

We define spin structures on perfect complexes outside of characteristic two, generalizing the usual notion for vector bundles. We give an explicit local characterization of spin structures, and show that for an oriented quadratic complex $E$ on an algebraic stack, spin structures on $E$ are parametrized by a degree $2$ gerbe. As an application, we show how to lift the K-theory class of Oh-Thomas in DT4 theory to a genuine (twisted) sheaf.