String Orbifolds and Quotient Stacks
Eric Sharpe
TL;DR
The paper reframes string orbifolds as sigma models on quotient stacks $[X/\Gamma]$, not on the quotient space $X/\Gamma$, by explicitly unraveling the definitions and showing that twisted sectors arise from maps into the stack. It develops the machinery of quotient stacks, their points and maps, and demonstrates that $[X/\Gamma]$ is smooth whenever $X$ is smooth and $\Gamma$ acts by diffeomorphisms, enabling well-behaved CFTs even when $X/\Gamma$ is singular. Twist fields are explained via the inertia stack $I_{[X/\Gamma]}$, yielding the orbifold Euler characteristic via $\chi_{orb}(X,\Gamma)=\sum_{[g]}\chi(X^g/C(g))$, and B-fields are naturally interpreted as residual gerbes on singularities. The work further connects these ideas to M-theory, deformations, and noncommutative geometry, arguing that stacks provide a robust framework for a new class of string compactifications and a deeper understanding of orbifolds.
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.