Spectra of D-branes with Higgs vevs

TL;DR

This paper explores the spectra of D-branes with Higgs vacuum expectation values, especially nilpotent ones, using sheaf theory to connect mathematical structures with physical open string states.

Contribution

It provides a sheaf-theoretic framework for D-branes with Higgs vevs, including nilpotent cases, enhancing the understanding of their physical and mathematical properties.

Findings

Sheaf description correctly counts open string states via Ext groups.

Expands the class of sheaves with physical interpretations.

Links Higgs vevs to nonreduced schemes and orbifold resolutions.

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.