Spin structures on perfect complexes

TL;DR

This paper extends the concept of spin structures to perfect complexes beyond characteristic two, providing explicit local descriptions and linking them to gerbes, with applications to lifting K-theory classes in DT4 theory.

Contribution

It introduces a new definition of spin structures on perfect complexes, generalizes existing notions, and applies this to K-theory class lifting in algebraic geometry.

Findings

Spin structures on perfect complexes are parametrized by degree 2 gerbes.

Explicit local characterization of spin structures is provided.

Application to lifting K-theory classes in DT4 theory.

Abstract

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.