[Paper Review] Gerbes over Orbifolds and K-Theory

Ernesto Lupercio, Bernardo Uribe|arXiv (Cornell University)|May 5, 2001

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper constructs a geometric model for gerbes over orbifolds using Chern-Weil theory to derive their characteristic class in H³(X). It introduces a twisting L Korb(X) of orbifold K-theory that generalizes Witten's smooth case construction, providing a unified framework for twisted K-theory in the orbifold setting.

ABSTRACT

In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H³(X) via Chern-Weil theory. For an arbitrary gerbe L, a twisting L Korb(X) of the orbifold K-theory of X is constructed, and shown to generalize previous twistings by Witten [21] in the smooth case

Motivation & Objective

• To develop a geometric model for gerbes over orbifolds that is amenable to curvature-based analysis.
• To establish a connection between the curvature of a gerbe and its characteristic class in H³(X) using Chern-Weil theory.
• To construct a twisting of the orbifold K-theory Korb(X) for any gerbe L, generalizing prior constructions.
• To extend Witten's twisting of K-theory in the smooth case to the orbifold setting.

Proposed method

• The authors use Chern-Weil theory to associate a characteristic class in H³(X) to the curvature of a gerbe over an orbifold.
• They define a geometric model for gerbes on orbifolds using connections and curvatures in the orbifold context.
• The twisting of Korb(X) is constructed via the gerbe L, generalizing the smooth case via a canonical construction.
• The orbifold structure is preserved throughout the construction, ensuring compatibility with the stacky nature of orbifolds.
• The method relies on differential geometric techniques adapted to the singular structure of orbifolds.

Experimental results

Research questions

• RQ1 How can a geometric model for gerbes over orbifolds be explicitly constructed?
• RQ2 How does the curvature of a gerbe on an orbifold yield its characteristic class in H³(X)?
• RQ3 Can a twisting of orbifold K-theory be defined for any gerbe, generalizing Witten's construction?
• RQ4 What is the relationship between the Chern-Weil construction and the characteristic class of a gerbe in the orbifold setting?

Key findings

• A geometric model for gerbes over orbifolds is successfully constructed using differential geometric methods.
• The curvature of a gerbe gives rise to a characteristic class in H³(X) via Chern-Weil theory.
• A twisting L Korb(X) of the orbifold K-theory is defined for any gerbe L.
• This twisting construction generalizes Witten's twisting in the smooth case to the orbifold setting.