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On the differential $K$-theory of moduli stacks

Daniel Grady

TL;DR

The paper computes differential cohomology and connective differential K-theory for the moduli stack of smooth principal $G$-bundles with connection, $\mathbf{B}_{\nabla}G$, using a homotopy-theoretic framework of presheaves of spaces and spectra. It establishes curvature and forgetful maps, identifies integral invariant-polynomial classes, and gives explicit short exact sequences connecting torsion, differential refinements, and Chern–Weil data, with sharp results in the compact, connected case. A main achievement is showing that the connective differential K-theory $\widehat{k}^0(\mathbf{B}_{\nabla}G)$ is isomorphic to the completed representation ring $R(G)^{\wedge}$, aligning differential refinements with classical representation-theoretic invariants via the Atiyah–Segal completion. The work integrates Čech–de Rham theory for stacks, basic forms, and a Hopkins–Singer model to provide a comprehensive description of differential refinements on moduli stacks.

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.

On the differential $K$-theory of moduli stacks

TL;DR

The paper computes differential cohomology and connective differential K-theory for the moduli stack of smooth principal -bundles with connection, , using a homotopy-theoretic framework of presheaves of spaces and spectra. It establishes curvature and forgetful maps, identifies integral invariant-polynomial classes, and gives explicit short exact sequences connecting torsion, differential refinements, and Chern–Weil data, with sharp results in the compact, connected case. A main achievement is showing that the connective differential K-theory is isomorphic to the completed representation ring , aligning differential refinements with classical representation-theoretic invariants via the Atiyah–Segal completion. The work integrates Čech–de Rham theory for stacks, basic forms, and a Hopkins–Singer model to provide a comprehensive description of differential refinements on moduli stacks.

Abstract

We compute the connective differential -theory and the differential cohomology of the moduli stack of principal -bundles with connection. The results are formulated in terms of invariant polynomials and the representation ring of . We use the homotopy theory of presheaves of spaces and presheaves of spectra to establish the results.
Paper Structure (7 sections, 20 theorems, 85 equations)

This paper contains 7 sections, 20 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Differential forms on smooth stacks
  4. The differential cohomology of $\mathbf{B}_{\nabla}G$
  5. The differential $K$-theory of $\mathbf{B}_{\nabla}G$
  6. Sheaves of spectra
  7. A lemma about derived mapping spaces

Key Result

Theorem 1.1

Let $A$ be a graded ring of finite type that is torsion free in each degree (as a $\mathbb Z$-module). Let $G$ be any Lie group and let $I_{A}^*(\mathfrak{g}^{\vee})_H\subset I_{A}^*(\mathfrak{g}^{\vee})$ denote the subring of $H$-integral elements. Then the curvature map defines a surjective homomo Moreover, if $G$ is compact and connected, then the forgetful map $\mathcal{I}:\widehat{H}^*(\mathb

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • Definition 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 33 more