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Chern character for twisted K-theory of orbifolds

Jean-Louis Tu, Ping Xu

TL;DR

The paper develops a comprehensive framework for twisted K-theory on orbifolds via $S^1$-gerbes, the inertia groupoid, and noncommutative geometry. It defines twisted cohomology $H^*_c(\mathfrak{X},\alpha)$ and constructs a Connes-Cern character-based isomorphism $K^*_{\alpha}(\mathfrak{X})\otimes \mathbb{C} \cong H^*_c(\mathfrak{X},\alpha)$ by passing through periodic cyclic homology $HP_*(C_c^{\infty}(\Gamma,L))$ using a trace-based chain map. The work unifies and extends prior results (Baum-Connes, Brylinski-Nistor, Adem-Ruan, Mathai-Stevenson) to the orbifold setting with arbitrary $S^1$-gerbe twists, and provides a local-to-global Mayer-Vietoris strategy backed by Morita invariance. The main technical tool is a chain map between the cyclic complex of the twisted convolution algebra and the twisted cohomology complex, built from a connection, curving, and curvature data of the gerbe.

Abstract

For an orbifold X and $α\in H^3(X, Z)$, we introduce the twisted cohomology $H^*_c(X, α)$ and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups $K_α^* (X) \otimes C$ and twisted cohomology $H^*_c(X, α)$. This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.

Chern character for twisted K-theory of orbifolds

TL;DR

The paper develops a comprehensive framework for twisted K-theory on orbifolds via -gerbes, the inertia groupoid, and noncommutative geometry. It defines twisted cohomology and constructs a Connes-Cern character-based isomorphism by passing through periodic cyclic homology using a trace-based chain map. The work unifies and extends prior results (Baum-Connes, Brylinski-Nistor, Adem-Ruan, Mathai-Stevenson) to the orbifold setting with arbitrary -gerbe twists, and provides a local-to-global Mayer-Vietoris strategy backed by Morita invariance. The main technical tool is a chain map between the cyclic complex of the twisted convolution algebra and the twisted cohomology complex, built from a connection, curving, and curvature data of the gerbe.

Abstract

For an orbifold X and , we introduce the twisted cohomology and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups and twisted cohomology . This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.

Paper Structure

This paper contains 15 sections, 28 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. $S^1$-gerbes over orbifolds
  3. Orbifolds
  4. $S^1$-gerbes over orbifolds
  5. Inertia groupoid
  6. Twisted $K$-theory of orbifolds
  7. Twisted cohomology
  8. De Rham cohomology
  9. Connections and the Dixmier-Douady class
  10. Twisted cohomology
  11. Non-commutative Chern Character
  12. The convolution algebra $\Omega_c^*(\Gamma,L)$
  13. The trace map $Tr:\Omega^*_c(\Gamma,L)\to\Omega_c^*(\Lambda{\mathfrak X},L')$
  14. Chain map
  15. Proof of the main theorem

Key Result

Theorem 1.1

Let ${\mathfrak X}$ be an orbifold and $\alpha\in H^3({\mathfrak X}, {\mathbb Z})$. Assume that ${\widetilde{\Gamma}} \to \Gamma\rightrightarrows M$ is an $S^1$-central extension representing the $S^1$-gerbe determined by the class $\alpha$, which admits a connection $\theta$, a curving $B$, and a c where ${\mathrm{ch}}$ denotes the non-commutative Chern character of Connes-Karoubi.

Theorems & Definitions (41)

  • Theorem 1.1
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Theorem 2.5: Giraud
  • Theorem 2.6
  • Example 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 31 more