Deformations and D-branes

Abstract

I discuss the relation of Hochschild cohomology to the physical states in the closed topological string. This allows a notion of deformation intrinsic to the derived category. I use this to identify deformations of a quiver gauge theory associated to a D-branes at a singularity with generalized deformations of the geometry of the resolution of the singularity. An explicit map is given from noncommutative deformations (i.e., B-fields) to terms in the superpotential.

Stability conditions and branes at singularities

“…As discussed in [6], when C ∼ = D b (Coh(X)), HH 0 (C) = Γ(O X ), i.e., global functions on X. On the other hand, when C ∼ = D b (A−fgmod), we have that HH 0 (D b (A−fgmod)) = Z(A), the center of the algebra A.…”

“…In this case, the quiver has loops, reflecting the infinite dimensionality of A. For a discussion of the construction of this quiver, please see [3,4,6,8]. As discussed therein, we have the correspondence π * E i ↔ P i where P i is the projective representation associated to the node i of the quiver.…”

“…In fact, it is not difficult to see that the central charge map defined above does not factor through K(A−fgmod). To see this, we will use the generalized Ringel resolution of [26,27] and discussed in [6]. This states that for an arbitrary representation of our CY quiver, Q, the following is a projective resolution:…”

“…For a more complete presentation of my view on the subject, please see [6]. The original references are given in the introduction.…”

“…The use of the topological B-model to explore the gauge theories living on D-branes at singularities has been extremely fruitful in recent years [1,2,3,4,5,6,7,8]. The fundamental feature of this technique is an equivalence of categories between the derived category of coherent sheaves on the resolved singularity and the derived category of representations of the algebra described by the quiver in the dual gauge theory.…”