Stability conditions and branes at singularities

Abstract

I use Bridgeland's definition of a stability condition on a triangulated category to investigate the stability of D-branes on Calabi-Yau cones given by the canonical line bundle over a del Pezzo surface. In this context, I prove the existence of the decay of a D3-brane into a set of fractional branes. This is an important aspect of the derivation of quiver gauge theories from branes at singularities via the technique of equivalences of categories. Some important technical aspects of this equivalence are discussed. I also prove that the representations corresponding to skyscraper sheaves supported off the zero section are simple.

Moduli stacks and invariants of semistable objects on K3 surfaces

“…(See Definition 2.4.) The precise descriptions of the space Stab(X) have been studied in the articles [6], [5], [9], [27], [26], [24], [15], [2], [29], [28]. In particular when X is a K3 surface or an abelian surface, Bridgeland [6] described Stab * (X), one of the connected components of Stab(X), as a covering space over a certain open subset P + 0 (X) ⊂ N (X) * C , and related its Galois group to the group of autoequivalences of D(X).…”

Probing Geometry with Stability Conditions

“…In this case one can rigorously prove these stability assertions by appealing to Bridgeland's definition of stability [31] (see also [32]). Here one reduces the stability statements to an abelian category given as the heart of a t-structure of D(X).…”

A Note on Support in Triangulated Categories

“…To study D-branes located at a singularity, we need to study the stability of skyscraper sheaves located on the zero section of K X . This is done in [8] following results of Bridgeland [9]. In particular, we will let T be a locally free generator 5 of D(Coh(K X )) such that Ext i (T, T ) = 0 for i = 0.…”

“…This is also Calabi-Yau in the sense that it has a Serre functor equivalent to the shift by three functor. It was conjectured in [8] that this is equivalent to the category D b fd (A − Mod) consisting of objects whose cohomology modules have finite dimension vector. Using the above propositions, we can characterize these subcategories intrinsically as the full subcategories with compact support.…”