Primary operations in differential cohomology

TL;DR

This paper characterizes primary operations in differential cohomology using stacks, refining classical Steenrod operations and developing computational techniques with applications in higher geometry and physics.

Contribution

It introduces a novel framework for primary operations in differential cohomology and refines Steenrod squares and powers explicitly within this context.

Findings

Explicit differential refinements of Steenrod operations.

Development of computational tools like K"unneth decomposition.

Applications to higher geometry and mathematical physics.

Abstract

We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. Along the way we develop computational techniques in differential cohomology, including a K\"unneth decomposition, that should also be useful in their own right, and point to applications to higher geometry and mathematical physics.