Quotient Stacks and String Orbifolds

TL;DR

This paper clarifies that string orbifolds are better described as sigma models on quotient stacks rather than quotient spaces, providing a geometric understanding of their properties and features.

Contribution

It demonstrates that string orbifolds correspond to sigma models on quotient stacks, offering a new geometric perspective that explains their well-behaved CFT and topological features.

Findings

String orbifolds are sigma models on quotient stacks, not quotient spaces.

This perspective explains features like orbifold Euler characteristics.

Provides a geometric interpretation of string orbifold properties.

Abstract

In this short review we outline some recent developments in understanding string orbifolds. In particular, we outline the recent observation that string orbifolds do not precisely describe string propagation on quotient spaces, but rather are literally sigma models on objects called quotient stacks, which are closely related to (but not quite the same as) quotient spaces. We show how this is an immediate consequence of definitions, and also how this explains a number of features of string orbifolds, from the fact that the CFT is well-behaved to orbifold Euler characteristics. Put another way, many features of string orbifolds previously considered ``stringy'' are now understood as coming from the target-space geometry; one merely needs to identify the correct target-space geometry.