D-branes, orbifolds, and Ext groups

TL;DR

This paper extends the analysis of massless open string spectra from D-branes on complex manifolds to orbifolds, showing they are counted by Ext groups on quotient stacks and relating them to resolutions via the McKay correspondence.

Contribution

It introduces the use of Ext groups on quotient stacks to count spectra of D-branes on orbifolds and discusses their relation to resolutions, including pedagogical explanations of stacks.

Findings

Massless spectra are counted by Ext groups on quotient stacks.

Ext groups on stacks relate to those on resolutions via McKay correspondence.

Subtleties in previous work also appear in the orbifold context.

Abstract

In this note we extend previous work on massless Ramond spectra of open strings connecting D-branes wrapped on complex manifolds, to consider D-branes wrapped on smooth complex orbifolds. Using standard methods, we calculate the massless boundary Ramond sector spectra directly in BCFT, and find that the states in the spectrum are counted by Ext groups on quotient stacks (which provide a notion of homological algebra relevant for orbifolds). Subtleties that cropped up in our previous work also appear here. We also use the McKay correspondence to relate Ext groups on quotient stacks to Ext groups on (large radius) resolutions of the quotients. As stacks are not commonly used in the physics community, we include pedagogical discussions of some basic relevant properties of stacks.