String Orbifolds and Quotient Stacks

TL;DR

This paper clarifies that string orbifolds are better understood as describing strings on quotient stacks rather than quotient spaces, providing new insights into their structure, twisted sectors, and implications for string compactifications.

Contribution

It demonstrates that string orbifolds correspond to quotient stacks, introduces a stack-based framework for understanding twisted sectors and B-fields, and extends the concept to M-theory and new string compactifications.

Findings

String orbifolds describe strings on quotient stacks, not just quotient spaces.

Sigma models on quotient stacks have natural twisted sectors and nonsingular CFTs.

The approach enables understanding orbifolds in M-theory and new string compactifications.

Abstract

In this note we observe that, contrary to the usual lore, string orbifolds do not describe strings on quotient spaces, but rather seem to describe strings on objects called quotient stacks, a result that follows from simply unraveling definitions, and is further justified by a number of results. Quotient stacks are very closely related to quotient spaces; for example, when the orbifold group acts freely, the quotient space and the quotient stack are homeomorphic. We explain how sigma models on quotient stacks naturally have twisted sectors, and why a sigma model on a quotient stack would be a nonsingular CFT even when the associated quotient space is singular. We also show how to understand twist fields in this language, and outline the derivation of the orbifold Euler characteristic purely in terms of stacks. We also outline why there is a sense in which one naturally finds B nonzero on exceptional divisors of resolutions. These insights are not limited to merely understanding existing string orbifolds: we also point out how this technology enables us to understand orbifolds in M-theory, as well as how this means that string orbifolds provide the first example of an entirely new class of string compactifications. As quotient stacks are not a staple of the physics literature, we include a lengthy tutorial on quotient stacks.