Dirac geometry I: Commutative algebra

TL;DR

This paper introduces Dirac geometry, a new geometric framework based on commutative algebra in spectra, emphasizing the role of grading and spin-like structures derived from anima rather than sets.

Contribution

It develops the foundational aspects of Dirac geometry from commutative algebra in spectra, highlighting the significance of grading and spin-like features.

Findings

Homotopy groups form graded commutative algebras

Dirac geometry encodes spin-like structures via grading

Development of coherent cohomology with half-integer Serre twists

Abstract

The homotopy groups of a commutative algebra in spectra form a commutative algebra in the symmetric monoidal category of graded abelian groups. The grading and the Koszul sign rule are remnants of the structure encoded by anima as opposed to sets. The purpose of this paper and its sequel is to develop the geometry built from such algebras. We name this geometry Dirac geometry, since the grading exhibits the hallmarks of spin. Indeed, it is a reflection of the internal structure encoded by anima, and it distinguishes symmetric and anti-symmetric behavior, as does spin. Moreover, the coherent cohomology, which we develop in the sequel admits half-integer Serre twists.