Spectra of D-branes with Higgs vevs
R. Donagi, S. Katz, E. Sharpe
TL;DR
This work develops a precise dictionary between D-branes with Higgs vevs and sheaves on ambient spaces, showing that BRST spectra computed from worldsheet deformations reproduce Ext groups between the corresponding Higgs-modified sheaves. In particular, nilpotent Higgs vevs are naturally described by nonreduced schemes, providing a physical realization of classical Higgs moduli spaces in orbifolds and a concrete realization of the McKay correspondence. The paper also clarifies how nilpotent vevs relate to exceptional divisors in orbifolds, resolves tensions between viewing orbifolds as resolutions versus quotient stacks, and extends the utility of derived categories in physics by identifying a broad class of physically meaningful sheaves. Together these results establish a consistent, computable framework connecting open string spectra, Higgs field data, and sophisticated algebraic geometry constructions.
Abstract
In this paper we continue previous work on counting open string states between D-branes by considering open strings between D-branes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying D-branes in orbifolds. Ordinarily Higgs vevs can be interpreted as moving the D-brane, but nilpotent Higgs vevs have zero eigenvalues, and so their interpretation is more interesting -- for example, they often correspond to nonreduced schemes, which furnishes an important link in understanding old results relating classical D-brane moduli spaces in orbifolds to Hilbert schemes, resolutions of quotient spaces, and the McKay correspondence. We give a sheaf-theoretic description of D-branes with Higgs vevs, including nilpotent Higgs vevs, and check that description by noting that Ext groups between the sheaves modelling the D-branes, do in fact correctly count open string states. In particular, our analysis expands the types of sheaves which admit on-shell physical interpretations, which is an important step for making derived categories useful for physics.