Derived Spin structures and moduli of sheaves on Calabi-Yau fourfolds

I will present a notion of spin structure on a perfect complex in characteristic zero, generalizing the classical notion for an (algebraic) vector bundle. For a complex $E$ on $X$ with an oriented quadratic structure one obtains an associated ${\mathbb Z}/2{\mathbb Z}$-gerbe over X which obstructs the existence of a spin structure on $E$. This situation arises naturally on moduli spaces of coherent sheaves on Calabi-Yau fourfolds. Using spin structures as orientation data, we construct a categorical refinement of a K-theory class constructed by Oh-Thomas on such moduli spaces.