Representations of N = 1 ADE quivers via reflection functors

Sheaves on Fibered Threefolds and Quiver Sheaves

“…This establishes a direct link to [14,21], according to which (in the generic case) irreducible representations of the finite N = 1 quiver with the given relations parametrize exceptional components of the contraction f s : X s →X s .…”

“…If C ∼ = A 1 , Theorem 3.1 can in some cases be re-written in terms of classical quiver representations: representations of a quiver in vector spaces. This will give an interpretation of an assertion of [6,14,21].…”

“…In this paper we generalize the work of [6,14,21] in two directions: we consider holomorphic D-branes, objects in the derived category of coherent sheaves, instead of exceptional components, and we study the semi-local case: the neighbourhood of a deformed ADE fibration X s → C over a general curve C. The main result is Theorem 3.1, which shows that certain holomorphic D-branes on the fibered threefold X s are classified by representations with relations of a Kronheimer-Nakajima-type quiver in the category Coh(C) of coherent sheaves on the curve C. In particular, moduli spaces of such holomorphic D-branes are quiver bundle varieties over C. If C ∼ = A 1 , a further dimensional reduction leads to Theorem 3.4, relating sheaves on the threefold to the zero-dimensional problem of ordinary matrix representations of the N = 1 ADE quiver of [6,14,21]. The loops in the N = 1 ADE quiver arise as the action by multiplication of a parameter t ∈ H 0 (O A 1 ) on spaces of sections of sheaves on the base A 1 .…”

“…Fibered local Calabi-Yau threefolds X → A 1 of this type, as well as their deformations X s → A 1 and extremal transitions, were thoroughly analized in [6,7] from the point of view of supersymmetric gauge theory. The paper [6] contains an assertion, made explicit in [14] and studied in [21], that exceptional components of a natural threefold contraction X s →X s are classified by irreducible representations of a certain quiver with loop edges, the N = 1 ADE quiver (see Figure 3.2 for an example), satisfying a specific set of relations. This statement is in the spirit of Gabriel's theorem classifying exceptional (not necessarily irreducible) rational curves in resolved ADE surfaces in terms of irreducible representations of the corresponding Dynkin quiver.…”

Noncommutative Resolutions of ADE Fibered Calabi-Yau Threefolds

“…2.1 below for details. The relevant N = 1 quiver gauge theory was written down in [11] (see also [10,23,44]). We will explicitly carry out the previous construction in the subsequent sections.…”