Moduli spaces for Bondal quivers

Given a sufficiently nice collection of sheaves on an algebraic variety V , Bondal explained how to build a quiver Q along with an ideal of relations in the path algebra of Q such that the derived category of representations of Q subject to these relations is equivalent to the derived category of coherent sheaves on V . We consider the case in which these sheaves are all locally free and study the moduli spaces of semistable representations of our quiver with relations for various stability conditions. We show that V can often be recovered as a connected component of such a moduli space and we describe the line bundle induced by a GIT construction of the moduli space in terms of the input data. In certain special cases, we interpret our results in the language of topological string theory. An algebraic variety V is completely determined by the abelian category Coh(V ) of coherent sheaves on V [Ga], and it is therefore a natural problem to find a way to describe this category in concrete terms. If V is affine, then Coh(V ) is nothing more than the category of finitely generated modules over the algebra of global functions on V . If we have a presentation of this algebra, this may be interpreted as a ‘presentation’ of the category Coh(V ). In the projective case, it is unreasonable to expect Coh(V ) to be equivalent to the category of modules over any ring. It is sometimes the case, however, that such an equivalence can be constructed after passing to the derived category DCoh(V ). The derived category carries less information than the abelian category Coh(V ), but it is enough to reconstruct such invariants as cohomology, K-theory, and higher Chow groups, as well as a great deal of information about the birational geometry of V . If V is Calabi-Yau, then an object of DCoh(V ) may be thought of as a D-brane in type IIB topological string theory on V [AD, Do, Sh]. This category is therefore of significant physical interest, and is a fundamental ingredient in the formulation of homological mirror symmetry [Ko]. Let us describe more concretely how one might attempt to construct such an equivalence. Given an object E of Coh(V ), there is a natural functor F from Coh(V ) to the category of finitely generated right modules over the endomorphism algebra End(E), or left modules over the opposite algebra End(E)op, taking a sheaf F to the module Hom(E,F). This functor will almost never be either faithful or essentially surjective, but if E satisfies certain technical conditions, then Rickard shows that the right derived functor RF from DCoh(V ) to the derived category of left modules over End(E)op will be an equivalence. (See Definition 1.2 and Theorem 1.3 for more details.) If E decomposes as a direct sum of smaller objects E = ⊕i=1Ei, then End(E) op may be expressed as the path algebra of a quiver with n nodes, modulo certain relations (which may not be admissible). One should think of the description of such a quiver along with its relations as an analogue of a presentation of the coordinate ring of an affine variety. Supported by the National Science Foundation under Grant Nos. PHY-0071512 and PHY-0455649, and the US Navy, Office of Naval Research, Grant Nos. N00014-03-1-0639 and N00014-04-1-0336, Quantum Optics Initiative. Preprint number UTTG–16–05. Supported by a National Science Foundation Postdoctoral Research Fellowship.