On the differential $K$-theory of moduli stacks

TL;DR

This paper calculates the differential K-theory and cohomology of the moduli stack of principal G-bundles with connection, using homotopy theory and invariant polynomials.

Contribution

It introduces a homotopy-theoretic approach to compute differential K-theory of moduli stacks, connecting it with invariant polynomials and representation rings.

Findings

Computed connective differential K-theory of moduli stacks

Expressed results in terms of invariant polynomials and representation rings

Established a homotopy-theoretic framework for these calculations

Abstract

We compute the connective differential $K$-theory and the differential cohomology of the moduli stack of principal $G$-bundles with connection. The results are formulated in terms of invariant polynomials and the representation ring of $G$. We use the homotopy theory of presheaves of spaces and presheaves of spectra to establish the results.