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Spin structures on perfect complexes

Nikolas Kuhn

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.

Spin structures on perfect complexes

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 on an algebraic stack, spin structures on are parametrized by a degree gerbe. As an application, we show how to lift the K-theory class of Oh-Thomas in DT4 theory to a genuine (twisted) sheaf.

Paper Structure

This paper contains 67 sections, 68 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Construction of a DT4 twisted virtual structure sheaf.
  4. Further Directions
  5. Characteristic $2$.
  6. $-2$-shifted symplectic stacks.
  7. Derived Clifford Algebras.
  8. Existence of Spin structures in DT4 theory.
  9. Generalized principal bundles.
  10. Sketch of the construction
  11. Self-dual Representatives.
  12. Isotropic Reductions.
  13. Orientations.
  14. Associated principal bundles.
  15. Spin structures with a twist.
  16. ...and 52 more sections

Key Result

Theorem 1.1

Let $M$ be a quasi-projective scheme with an oriented DT4 obstruction theory $\mathbb{E}$. Then a choice of spin structure on $\mathbb{E}$ gives rise to a lift of $[\widehat{\mathcal{O}}_M^{\mathrm{vir}}]$ to a $\mathbb{Z}_2$-graded sheaf.

Theorems & Definitions (183)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Conjecture 1.2
  • Definition 1.5
  • Theorem 1.6: cf. Definition \ref{['def:spin-functor']}
  • Definition 1.7
  • Theorem 1.8: cf. Corollary \ref{['cor:restriction-iso']}
  • Definition 1.9
  • Theorem 1.10: Proposition \ref{['prop:spin-gerbe']} & Corollary \ref{['cor:spin-gerbe']}
  • ...and 173 more