D-branes, orbifolds, and Ext groups

TL;DR

This paper provides a first-principles BCFT derivation showing that massless boundary Ramond states of open strings between D-branes on orbifolds are counted by Ext groups on quotient stacks [X/G], and uses the McKay correspondence to relate these to Ext groups on large-radius resolutions. It extends the established manifold results to orbifolds, introduces stack-theoretic machinery to describe fractional branes, and furnishes numerous explicit computations for both abelian and nonabelian orbifolds, as well as nonsupersymmetric cases. The analysis demonstrates that spectral sequences linking sheaf cohomology to Ext groups are physically realized in BRST cohomology, and highlights the necessity of stacks to capture fractional branes and other orbifold features. The work also clarifies how the McKay correspondence preserves Ext groups, enabling a bridge between orbifold physics and large-radius geometry, and it discusses the broader implications for the role of derived categories and stacks in string theory.

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.